Recent zbMATH articles in MSC 49https://www.zbmath.org/atom/cc/492022-05-16T20:40:13.078697ZUnknown authorWerkzeugPrefacehttps://www.zbmath.org/1483.000382022-05-16T20:40:13.078697ZFrom the text: The present volume is a Special Issue on Optimization and Differential Equations. The idea for this publication emerged from discussions at the conference IMAME 2019, International Meeting on Applied Mathematics \& Evolution, held in La Rochelle in April 2019.Mathematical analysis in interdisciplinary researchhttps://www.zbmath.org/1483.000422022-05-16T20:40:13.078697ZPublisher's description: This contributed volume provides an extensive account of research and expository papers in a broad domain of mathematical analysis and its various applications to a multitude of fields. Presenting the state-of-the-art knowledge in a wide range of topics, the book will be useful to graduate students and researchers in theoretical and applicable interdisciplinary research. The focus is on several subjects including: optimal control problems, optimal maintenance of communication networks, optimal emergency evacuation with uncertainty, cooperative and noncooperative partial differential systems, variational inequalities and general equilibrium models, anisotropic elasticity and harmonic functions, nonlinear stochastic differential equations, operator equations, max-product operators of Kantorovich type, perturbations of operators, integral operators, dynamical systems involving maximal monotone operators, the three-body problem, deceptive systems, hyperbolic equations, strongly generalized preinvex functions, Dirichlet characters, probability distribution functions, applied statistics, integral inequalities, generalized convexity, global hyperbolicity of spacetimes, Douglas-Rachford methods, fixed point problems, the general Rodrigues problem, Banach algebras, affine group, Gibbs semigroup, relator spaces, sparse data representation, Meier-Keeler sequential contractions, hybrid contractions, and polynomial equations. Some of the works published within this volume provide as well guidelines for further research and proposals for new directions and open problems.
The articles of mathematical interest will be reviewed individually.Semiconcavity and sensitivity analysis in mean-field optimal control and applicationshttps://www.zbmath.org/1483.301062022-05-16T20:40:13.078697Z"Bonnet, Benoît"https://www.zbmath.org/authors/?q=ai:bonnet.benoit"Frankowska, Hélène"https://www.zbmath.org/authors/?q=ai:frankowska.heleneSummary: In this article, we investigate some of the fine properties of the value function associated with an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems.Sharp bounds for the anisotropic \(p\)-capacity of Euclidean compact setshttps://www.zbmath.org/1483.310242022-05-16T20:40:13.078697Z"Li, Ruixuan"https://www.zbmath.org/authors/?q=ai:li.ruixuan"Xiong, Changwei"https://www.zbmath.org/authors/?q=ai:xiong.changwei.1|xiong.changweiSummary: We prove various sharp bounds for the anisotropic \(p\)-capacity \(\mathrm{Cap}_{F , p}(K)\) (\(1 < p < n\)) of compact sets \(K\) in the Euclidean space \(\mathbb{R}^n\) (\(n \geq 2\)). Our results are mainly the anisotropic generalizations of some isotropic ones in [\textit{M. Ludwig} et al., Math. Ann. 350, No. 1, 169--197 (2011; Zbl 1220.26020); \textit{J. Xiao}, Ann. Henri Poincaré 17, No. 8, 2265--2283 (2016; Zbl 1345.83014); Adv. Math. 308, 1318--1336 (2017; Zbl 1361.31008); Adv. Geom. 17, No. 4, 483--496 (2017; Zbl 1387.53024)]. Key ingredients in the proofs include the inverse anisotropic mean curvature flow (IAMCF), the anisotropic Hawking mass and its monotonicity property along IAMCF for certain surfaces, and the anisotropic isocapacitary inequality.Collision avoidance of multiagent systems on Riemannian manifoldshttps://www.zbmath.org/1483.310312022-05-16T20:40:13.078697Z"Goodman, Jacob R."https://www.zbmath.org/authors/?q=ai:goodman.jacob-r"Colombo, Leonardo J."https://www.zbmath.org/authors/?q=ai:colombo.leonardo-jesusExistence and relaxation for subdifferential inclusions with unbounded perturbationhttps://www.zbmath.org/1483.340352022-05-16T20:40:13.078697Z"Timoshin, Sergey A."https://www.zbmath.org/authors/?q=ai:timoshin.sergey-aSummary: We consider a differential inclusion of subdifferential type with a nonconvex and unbounded valued perturbation. Existence and relaxation results are obtained for this inclusion. By relaxation we mean approximation of a solution of the differential inclusion with convexified perturbation by solutions of the given inclusion. The traditional condition of Lipschitz continuity for such kind of problems is weakened and a somehow more appropriate in the context of unbounded valued multifunctions ``truncated'' version of it is considered instead.A control-based mathematical study on psoriasis dynamics with special emphasis on \(\text{IL}-21\) and \(\text{IFN} - \gamma\) interaction networkhttps://www.zbmath.org/1483.340652022-05-16T20:40:13.078697Z"Roy, Amit Kumar"https://www.zbmath.org/authors/?q=ai:roy.amit-kumar"Nelson, Mark"https://www.zbmath.org/authors/?q=ai:nelson.mark-p|nelson.mark-e|nelson.mark-ian"Roy, Priti Kumar"https://www.zbmath.org/authors/?q=ai:kumar-roy.pritiSummary: Psoriasis is characterized by the excessive growth of keratinocytes (skin cells), which is initiated by chaotic signaling within the immune system and irregular release of cytokines. Pro-inflammatory cytokines: Interleukin \(21 (\text{IL} - 21)\) and Interferon gamma \(( \text{IFN} - \gamma )\), released by \(\text{Th}_1\) cell and activated natural killer cells (NK cells) respectively, play central role in the disease pathogenesis. In this work, we have constructed two sets of nonlinear differential equations. One is representing the growth of three vital immune cells (T helper cells (type I and II) and activated NK cells) along with keratinocyte and the other set is for cytokines' \((\text{IL} - 21\) and \(\text{IFN} - \gamma )\) dynamics. The hazardous effects of these cytokines, preconditions for disease persistence and validation of the stability criteria of endemic equilibrium have been studied analytically. We have also observed the effect of the combined biologic therapy (anti \(\text{IFN} - \gamma\) and \(\text{IL} - 21\) inhibitor) by considering an optimal control problem. Analytical and numerical results reveal that the impact of activated NK cells on excessive formation of keratinocytes is mostly regulated by the effects of \(\text{IL} - 21\) and \(\text{IFN} - \gamma \).Optimal control problems for a neutral integro-differential system with infinite delayhttps://www.zbmath.org/1483.341022022-05-16T20:40:13.078697Z"Huang, Hai"https://www.zbmath.org/authors/?q=ai:huang.hai"Fu, Xianlong"https://www.zbmath.org/authors/?q=ai:fu.xianlongSummary: This work devotes to the study on problems of optimal control and time optimal control for a neutral integro-differential evolution system with infinite delay. The main technique is the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature. We first establish the existence and uniqueness of mild solutions and discuss the compactness of the solution operator for the considered control system. Then, we investigate the existence of optimal controls for the both cases of bounded and unbounded admissible control sets under some assumptions. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. An example is given at last to illustrate the applications of the obtained results.Second order nonlinear evolution equations with time dependent pseudomonotone and quasimonotone operators: an equilibrium problem approachhttps://www.zbmath.org/1483.350052022-05-16T20:40:13.078697Z"Saidi, Asma"https://www.zbmath.org/authors/?q=ai:saidi.asma"Chadli, Ouayl"https://www.zbmath.org/authors/?q=ai:chadli.ouayl"Yao, Jen-Chih"https://www.zbmath.org/authors/?q=ai:yao.jen-chihSummary: We study the existence of solutions for nonlinear second order evolution equations associated to time dependent pseudomonotone (respectively, quasimonotone) operators in the sense of Brézis by a new approach based on the theory of equilibrium problems. This new method leads us to improve and unify some results in the literature related to the problem considered.On the monotonicity of the best constant of Morrey's inequality in convex domainshttps://www.zbmath.org/1483.350072022-05-16T20:40:13.078697Z"Fărcăşeanu, Maria"https://www.zbmath.org/authors/?q=ai:farcaseanu.maria"Mihăilescu, Mihai"https://www.zbmath.org/authors/?q=ai:mihailescu.mihaiIn this interesting paper, the authors obtain some monotonicity properties of the best constant from Morrey's inequality in convex and bounded domains from the Euclidean space \(\mathbb{R}^{D}\) (\(D\ge1\)). Using these monotonicity properties, they give a new variational characterization of the best constant from Morrey's inequality on convex and bounded domains for which the maximum of the distance function to the boundary is small. The authors also show that this variational characterization does not hold true on convex and bounded domains for which the maximum of the distance function to the boundary is larger than one.
Reviewer: Meng Qu (Wuhu)Large time behavior for a Hamilton-Jacobi equation in a critical coagulation-fragmentation modelhttps://www.zbmath.org/1483.350372022-05-16T20:40:13.078697Z"Mitake, Hiroyoshi"https://www.zbmath.org/authors/?q=ai:mitake.hiroyoshi"Tran, Hung V."https://www.zbmath.org/authors/?q=ai:tran.hung-vinh"Van, Truong-Son"https://www.zbmath.org/authors/?q=ai:van.truong-sonThe large time behavior of viscosity solutions to
\begin{align*}
& \partial_t F(x,t) + \frac{1}{2} (\partial_x F(x,t)-1) (\partial_x F(x,t)-2) + \frac{F(x,t)}{x} - 1 = 0 \;\;\text{ in }\;\; (0,\infty)^2\,, \\
& 0 \le F(x,t) \le x \;\;\text{ on }\;\; [0,\infty)^2\,, \\
& F(x,0) = F_0(x) \;\;\text{ in }\;\; [0,\infty)\,,
\end{align*}
is investigated, when the initial condition \(F_0\) is sublinear with \(0\le F_0(x)\le x\) and \(0 \le \partial_x F_0\le 1\), and shown to depend upon the behavior of \(F_0(x)x^{-2/3}\) as \(x\to\infty\). More precisely, assuming that \(F_0(x)x^{-2/3}\to L\) as \(x\to\infty\) for some \(L\in [0,\infty]\), the family \(\{F(t) : t\ge 0\}\) converges as \(t\to\infty\) to a limit \(F_\infty\) uniformly on compact subsets of \((0,\infty)\) with \(F_\infty\equiv 0\) when \(L=0\), \(F_\infty(x)=x\) when \(L=\infty\), and \(F_\infty\) is a uniquely determined sublinear stationary solution (depending on \(L\)) when \(L\in (0,\infty)\). It is also proved that the family \(\{F(t) : t\ge 0\}\) need not have a limit as \(t\to\infty\) when \(F_0(x)x^{-2/3}\) does not have a limit as \(x\to\infty\). The proofs rely on the representation of viscosity solutions provided by optimal control theory and characteristics.
This problem is connected with the analysis of the large time behavior of weak solutions to the coagulation-fragmentation equation with multiplicative coagulation kernel, constant fragmentation kernel, and total mass equal to one, see [\textit{H. V. Tran} and \textit{T.-S. Van}, ``Coagulation-fragmentation equations with multiplicative coagulation kernel and constant fragmentation kernel'', Comm. Pure Appl. Math. (to appear)].
Reviewer: Philippe Laurençot (Toulouse)On some nonlocal problems in the calculus of variationshttps://www.zbmath.org/1483.350602022-05-16T20:40:13.078697Z"Chipot, Michel"https://www.zbmath.org/authors/?q=ai:chipot.michel"Mikayelyan, Hayk"https://www.zbmath.org/authors/?q=ai:mikayelyan.haykSummary: The goal of this paper is to investigate some nonlocal problems of the calculus of variations where pointwise comparisons principles fail but where some kind of nonlocal maximum principle or monotonicity property remains true for some quantities attached to the problem. A special attention will be given to a nonlocal energy recently introduced in [\textit{H. Mikayelyan}, ESAIM, Control Optim. Calc. Var. 24, No. 2, 859--872 (2018; Zbl 1402.49007)].On a family of torsional creep problems in Finsler metricshttps://www.zbmath.org/1483.350692022-05-16T20:40:13.078697Z"Fărcăşeanu, Maria"https://www.zbmath.org/authors/?q=ai:farcaseanu.maria"Mihăilescu, Mihai"https://www.zbmath.org/authors/?q=ai:mihailescu.mihai"Stancu-Dumitru, Denisa"https://www.zbmath.org/authors/?q=ai:stancu-dumitru.denisaSummary: The asymptotic behavior of solutions to a family of Dirichlet boundary value problems, involving differential operators in divergence form, on a domain equipped with a Finsler metric is investigated. Solutions are shown to converge uniformly to the distance function to the boundary of the domain, which takes into account the Finsler norm involved in the equation. This implies that a well-known result in the analysis of problems modeling torsional creep continues to hold in this more general setting.One-dimensional optimal control problems for time-discrete constrained quasilinear diffusion equations of Allen-Cahn typeshttps://www.zbmath.org/1483.351202022-05-16T20:40:13.078697Z"Kubota, Shodai"https://www.zbmath.org/authors/?q=ai:kubota.shodaiSummary: In this paper, we consider a class of optimal control problems for a one-dimensional time-discrete constrained quasilinear diffusion state-systems of singular Allen-Cahn types and its regularized approximating problems. We note that the control parameter for each system is given by physical temperature. The principal part of this paper is started with the verification of a Key-Theorem dealing with the decompositions of the subdifferentials of the governing convex energies of the state-systems. On this basis, we will prove five Main Theorems, concerned with: the solvability and precise regularity results of state-systems; the continuous-dependence of the solutions to state-systems including convergences in spatially \(C^1\)-topologies; the existence and parameter-dependence of optimal controls; the necessary optimality conditions for approximate optimal controls; precise characterizations of the approximating limit of the optimality conditions.Modeling, approximation, and time optimal temperature control for binder removal from ceramicshttps://www.zbmath.org/1483.351612022-05-16T20:40:13.078697Z"Chicone, Carmen"https://www.zbmath.org/authors/?q=ai:chicone.carmen-c"Lombardo, Stephen J."https://www.zbmath.org/authors/?q=ai:lombardo.stephen-j"Retzloff, David G."https://www.zbmath.org/authors/?q=ai:retzloff.david-gSummary: The process of binder removal from green ceramic components -- a reaction-gas transport problem in porous media -- has been analyzed with a number of mathematical techniques: 1) non-dimensionalization of the governing decomposition-reaction ordinary differential equation (ODE) and of the reaction gas-permeability partial differential equation (PDE); 2) development of a pseudo steady state approximation (PSSA) for the PDE, including error analysis via \(L^2\) norm and singular perturbation methods; 3) derivation and analysis of a discrete model approximation; and 4) development of a time optimal control strategy to minimize processing time with temperature and pressure constraints. Theoretical analyses indicate the conditions under which the PSSA and discrete models are viable approximations. Numerical results indicate that under a range of conditions corresponding to practical binder burnout conditions, utilization of the optimal temperature protocol leads to shorter cycle times as compared to typical industrial practice.First-order reduction and emergent behavior of the one-dimensional kinetic Cucker-Smale equationhttps://www.zbmath.org/1483.351732022-05-16T20:40:13.078697Z"Kim, Jeongho"https://www.zbmath.org/authors/?q=ai:kim.jeonghoSummary: In this paper, we introduce the kinetic description of the first-order Cucker-Smale (CS) flocking model on the real line. We reveal the equivalent relation between the measure-valued solution to the first- and second-order kinetic CS equations. The emergent behavior of the first-order kinetic CS equation and the characterization of the asymptotic solution are studied. We also provide the equivalent relation between classical/measure-valued solutions to the first-order kinetic CS equation and a classical solution to the second-order hydrodynamic CS equations, and present the corresponding analysis on the large-time behavior. The numerical experiments support our analysis and provide an efficient algorithm to obtain the asymptotic solution without simulating the model for a long time.Instability of ground states for the NLS equation with potential on the star graphhttps://www.zbmath.org/1483.351952022-05-16T20:40:13.078697Z"Ardila, Alex H."https://www.zbmath.org/authors/?q=ai:ardila.alex-hernandez"Cely, Liliana"https://www.zbmath.org/authors/?q=ai:cely.liliana"Goloshchapova, Nataliia"https://www.zbmath.org/authors/?q=ai:goloshchapova.nataliiaSummary: We study the nonlinear Schrödinger equation with an arbitrary real potential \(V(x)\in (L^1 +L^{\infty})(\Gamma)\) on a star graph \(\Gamma\). At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength \(-\gamma <0\). We show the existence of ground states \(\varphi_{\omega} (x)\) as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on \(V(x)\). Moreover, for \(V(x)=-\dfrac{\beta}{x^{\alpha}}\), in the supercritical case, we prove that the standing waves \(e^{i\omega t}\varphi_{\omega} (x)\) are orbitally unstable in \(H^1 (\Gamma)\) when \(\omega\) is large enough. Analogous result holds for an arbitrary \(\gamma \in \mathbb{R}\) when the standing waves have symmetric profile.Analysis and control of stationary inclusions in contact mechanicshttps://www.zbmath.org/1483.352592022-05-16T20:40:13.078697Z"Sofonea, Mircea"https://www.zbmath.org/authors/?q=ai:sofonea.mircea|sofonea.mircea-tSummary: We start with a mathematical model which describes the frictionless contact of an elastic body with an obstacle and prove that it leads to a stationary inclusion for the strain field. Then, inspired by this contact model, we consider a general stationary inclusion in a real Hilbert space, governed by three parameters. We prove the unique solvability of the inclusion as well as the continuous dependence of its solution with respect to the parameters. We use these results in the study of an associated optimal control problem for which we prove existence and convergence results. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact and provide the corresponding mechanical interpretations.Local well-posedness and sensitivity analysis for the self-organized kinetic modelhttps://www.zbmath.org/1483.352762022-05-16T20:40:13.078697Z"Jiang, Ning"https://www.zbmath.org/authors/?q=ai:jiang.ning"Zhang, Zeng"https://www.zbmath.org/authors/?q=ai:zhang.zengSummary: We consider the self-organized kinetic model (SOK), which was derived as the mean field limit of the Couzin-Vicsek algorithm. This model yields a singularity when the particle flux vanishes. By showing that the singularity does not happen in finite time, we obtain local existence and uniqueness of smooth solutions to SOK. Furthermore, considering uncertainties in the initial data and in the interaction kernel, we analyze the random SOK model (RSOK). We provide local sensitivity analysis to justify the regularity with respect to the random parameter and the stability of solutions to RSOK.Optimizing noisy complex systems liable to failurehttps://www.zbmath.org/1483.352782022-05-16T20:40:13.078697Z"Lunz, Davin"https://www.zbmath.org/authors/?q=ai:lunz.davinOptimal control problems governed by 1-D Kobayashi-Warren-Carter type systemshttps://www.zbmath.org/1483.352972022-05-16T20:40:13.078697Z"Antil, Harbir"https://www.zbmath.org/authors/?q=ai:antil.harbir"Kubota, Shodai"https://www.zbmath.org/authors/?q=ai:kubota.shodai"Shirakawa, Ken"https://www.zbmath.org/authors/?q=ai:shirakawa.ken"Yamazaki, Noriaki"https://www.zbmath.org/authors/?q=ai:yamazaki.noriakiSummary: This paper is devoted to the study of a class of optimal control problems governed by 1-D Kobayashi-Warren-Carter type systems, which are based on a phase-field model of grain boundary motion, proposed by [Kobayashi et al, Physica D, 140, 141-150, 2000]. The class consists of an optimal control problem for a physically realistic state-system of Kobayashi-Warren-Carter type, and its regularized approximating problems. The results of this paper are stated in three Main Theorems 1--3. The first Main Theorem 1 is concerned with the solvability and continuous dependence for the state-systems. Meanwhile, the second Main Theorem 2 is concerned with the solvability of optimal control problems, and some semi-continuous association in the class of our optimal control problems. Finally, in the third Main Theorem 3, we derive the first order necessary optimality conditions for optimal controls of the regularized approximating problems. By taking the approximating limit, we also derive the optimality conditions for the optimal controls for the physically realistic problem.Mixed-integer nonlinear PDE-constrained optimization for multi-modal chromatographyhttps://www.zbmath.org/1483.352982022-05-16T20:40:13.078697Z"Cebulla, Dominik H."https://www.zbmath.org/authors/?q=ai:cebulla.dominik-h"Kirches, Christian"https://www.zbmath.org/authors/?q=ai:kirches.christian"Potschka, Andreas"https://www.zbmath.org/authors/?q=ai:potschka.andreasSummary: Multi-modal chromatography emerged as a powerful tool for the separation of proteins in the production of biopharmaceuticals. In order to maximally benefit from this technology it is necessary to set up an optimal process control strategy. To this end, we present a mechanistic model with a recent kinetic adsorption isotherm that takes process controls such as pH and buffer salt concentration into account. Maximizing the yield of a target component subject to purity requirements leads to a mixed-integer nonlinear optimal control problem constrained by a partial differential equation. Computational experiments indicate that a good separation in a two-component system can be achieved.
For the entire collection see [Zbl 1468.90005].A novel \(W^{1,\infty}\) approach to shape optimisation with Lipschitz domainshttps://www.zbmath.org/1483.352992022-05-16T20:40:13.078697Z"Deckelnick, Klaus"https://www.zbmath.org/authors/?q=ai:deckelnick.klaus"Herbert, Philip J."https://www.zbmath.org/authors/?q=ai:herbert.philip-j"Hinze, Michael"https://www.zbmath.org/authors/?q=ai:hinze.michaelSummary: This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent steepest descent directions of the shape functional in the \(W^{1,\infty}\)-topology. The idea of our approach is demonstrated for shape optimisation of \(n\)-dimensional star-shaped domains, which we represent as functions defined on the unit \((n-1)\)-sphere. In this setting we provide the specific form of the shape derivative and prove the existence of solutions to the underlying shape optimisation problem. Moreover, we show the existence of a direction of steepest descent in the \(W^{1,\infty}-\) topology. We also note that shape optimisation in this context is closely related to the \(\infty\)-Laplacian, and to optimal transport, where we highlight the latter in the numerics section. We present several numerical experiments in two dimensions illustrating that our approach seems to be superior over a widely used Hilbert space method in the considered examples, in particular in developing optimised shapes with corners.Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domainshttps://www.zbmath.org/1483.353002022-05-16T20:40:13.078697Z"Egidi, Michela"https://www.zbmath.org/authors/?q=ai:egidi.michela"Nakić, Ivica"https://www.zbmath.org/authors/?q=ai:nakic.ivica"Seelmann, Albrecht"https://www.zbmath.org/authors/?q=ai:seelmann.albrecht"Täufer, Matthias"https://www.zbmath.org/authors/?q=ai:taufer.matthias"Tautenhahn, Martin"https://www.zbmath.org/authors/?q=ai:tautenhahn.martin"Veselić, Ivan"https://www.zbmath.org/authors/?q=ai:veselic.ivanSummary: We survey recent results on the control problem for the heat equation on unbounded and large bounded domains. First we formulate new uncertainty relations, respectively spectral inequalities. Then we present an abstract control cost estimate which improves upon earlier results. The latter is particularly interesting when combined with the earlier mentioned spectral inequalities since it yields sharp control cost bounds in several asymptotic regimes. We also show that control problems on unbounded domains can be approximated by corresponding problems on a sequence of bounded domains forming an exhaustion. Our results apply also for the generalized heat equation associated with a Schrödinger semigroup.
For the entire collection see [Zbl 1467.93008].Optimal control of a viscous generalized \(\theta\)-type dispersive equation with weak dissipationhttps://www.zbmath.org/1483.353012022-05-16T20:40:13.078697Z"Fan, Guobing"https://www.zbmath.org/authors/?q=ai:fan.guobing"Yang, Zhifeng"https://www.zbmath.org/authors/?q=ai:yang.zhifengSummary: In this paper, we investigate the problem for optimal control of a viscous generalized \(\theta\)-type dispersive equation (VG \(\theta\)-type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG \(\theta\)-type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.Optimal control of nonlinear renewal equationhttps://www.zbmath.org/1483.353022022-05-16T20:40:13.078697Z"Kakumani, Bhargav Kumar"https://www.zbmath.org/authors/?q=ai:kakumani.bhargav-kumarSummary: In this article, we study the optimal control for a nonlinear renewal equation popularly known as McKendrick-von Foerster (MV) equation. We establish necessary optimality conditions using the concept of normal cone. Finally, we prove the existence and uniqueness of optimal control using the Ekeland variational principle.Optimisation of the total population size with respect to the initial condition for semilinear parabolic equations: two-scale expansions and symmetrisationshttps://www.zbmath.org/1483.353042022-05-16T20:40:13.078697Z"Mazari, Idriss"https://www.zbmath.org/authors/?q=ai:mazari.idriss"Nadin, Grégoire"https://www.zbmath.org/authors/?q=ai:nadin.gregoire"Toledo Marrero, Ana Isis"https://www.zbmath.org/authors/?q=ai:toledo-marrero.ana-isisExtreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamishttps://www.zbmath.org/1483.353062022-05-16T20:40:13.078697Z"Tong, Shanyin"https://www.zbmath.org/authors/?q=ai:tong.shanyin"Vanden-Eijnden, Eric"https://www.zbmath.org/authors/?q=ai:vanden-eijnden.eric"Stadler, Georg"https://www.zbmath.org/authors/?q=ai:stadler.georgSummary: We propose and compare methods for the analysis of extreme events in complex systems governed by PDEs that involve random parameters, in situations where we are interested in quantifying the probability that a scalar function of the system's solution is above a threshold. If the threshold is large, this probability is small and its accurate estimation is challenging. To tackle this difficulty, we blend theoretical results from large deviation theory (LDT) with numerical tools from PDE-constrained optimization. Our methods first compute parameters that minimize the LDT-rate function over the set of parameters leading to extreme events, using adjoint methods to compute the gradient of this rate function. The minimizers give information about the mechanism of the extreme events as well as estimates of their probability. We then propose a series of methods to refine these estimates, either via importance sampling or geometric approximation of the extreme event sets. Results are formulated for general parameter distributions and detailed expressions are provided for Gaussian distributions. We give theoretical and numerical arguments showing that the performance of our methods is insensitive to the extremeness of the events we are interested in. We illustrate the application of our approach to quantify the probability of extreme tsunami events on shore. Tsunamis are typically caused by a sudden, unpredictable change of the ocean floor elevation during an earthquake. We model this change as a random process, which takes into account the underlying physics. We use the one-dimensional shallow water equation to model tsunamis numerically. In the context of this example, we present a comparison of our methods for extreme event probability estimation, and find which type of ocean floor elevation change leads to the largest tsunamis on shore.Ground states for the NLS equation with combined nonlinearities on noncompact metric graphshttps://www.zbmath.org/1483.353082022-05-16T20:40:13.078697Z"Pierotti, Dario"https://www.zbmath.org/authors/?q=ai:pierotti.dario"Soave, Nicola"https://www.zbmath.org/authors/?q=ai:soave.nicolaOptimal control of Clarke subdifferential type fractional differential inclusion with non-instantaneous impulses driven by Poisson jumps and its topological propertieshttps://www.zbmath.org/1483.370942022-05-16T20:40:13.078697Z"Durga, N."https://www.zbmath.org/authors/?q=ai:durga.nagarajan"Muthukumar, P."https://www.zbmath.org/authors/?q=ai:muthukumar.palanisamySummary: This article is devoted to studying the topological structure of a solution set for Clarke subdifferential type fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps. Initially, for proving the solvability result, we use a nonlinear alternative of Leray-Schauder fixed point theorem, Gronwall inequality, stochastic analysis, a measure of noncompactness, and the multivalued analysis. Furthermore, the mild solution set for the proposed problem is demonstrated with nonemptyness, compactness, and, moreover, \(R_\delta\)-set. By employing Balder's theorem, the existence of optimal control is derived. At last, an application is provided to validate the developed theoretical results.Instances of computational optimal recovery: dealing with observation errorshttps://www.zbmath.org/1483.410122022-05-16T20:40:13.078697Z"Ettehad, Mahmood"https://www.zbmath.org/authors/?q=ai:ettehad.mahmood"Foucart, Simon"https://www.zbmath.org/authors/?q=ai:foucart.simonThe paper presents new contributions to optimizing functions recovery from observational data. The proposed approach relies on considering inaccurate data through some boundedness models and the emphasis is set on computational realization. The efficient construction of optimal recovery maps is discussed in several instances: local optimality under linearly or semidefinitely describable models, global optimality for the estimation of linear functionals under approximability models, and global near-optimality under approximability models in the space of continuous functions.
Reviewer: Sorin-Mihai Grad (Paris)The measures with an associated square function operator bounded in \(L^2\)https://www.zbmath.org/1483.420122022-05-16T20:40:13.078697Z"Jaye, Benjamin"https://www.zbmath.org/authors/?q=ai:jaye.benjamin-j"Nazarov, Fedor"https://www.zbmath.org/authors/?q=ai:nazarov.fedor-l"Tolsa, Xavier"https://www.zbmath.org/authors/?q=ai:tolsa.xavierSummary: In this paper we provide an extension of a theorem of \textit{G. David} and \textit{S. Semmes} [Singular integrals and rectifiable sets in \(\mathbb{R}^n\). Astérisque 193 (1991; Zbl 0743.49018)] to general non-atomic measures. The result provides a geometric characterization of the non-atomic measures for which a certain class of square function operators, or singular integral operators, are bounded in \(L^2\). The description is given in terms of a modification of Jones' \(\beta\)-coefficients.Poincaré inequalities and uniform rectifiabilityhttps://www.zbmath.org/1483.460282022-05-16T20:40:13.078697Z"Azzam, Jonas"https://www.zbmath.org/authors/?q=ai:azzam.jonasSummary: We show that any \(d\)-Ahlfors regular subset of \(\mathbb{R}^n\) supporting a weak \((1, d)\)-Poincaré inequality with respect to surface measure is uniformly rectifiable.Fractional perimeters on the spherehttps://www.zbmath.org/1483.460312022-05-16T20:40:13.078697Z"Kreuml, Andreas"https://www.zbmath.org/authors/?q=ai:kreuml.andreas"Mordhorst, Olaf"https://www.zbmath.org/authors/?q=ai:mordhorst.olafSummary: This note treats several problems for the fractional perimeter or \(s\)-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as \(s \nearrow 1\) is proven. It is shown that their limit as \(s \searrow -\infty\) can be expressed in terms of the volume.Variational inclusion governed by \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappinghttps://www.zbmath.org/1483.470812022-05-16T20:40:13.078697Z"Gupta, Sanjeev"https://www.zbmath.org/authors/?q=ai:gupta.sanjeev-k"Husain, Shamshad"https://www.zbmath.org/authors/?q=ai:husain.shamshad"Mishra, Vishnu Narayan"https://www.zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this paper, we look into a new concept of accretive mappings called \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappings in Banach spaces. We extend the concept of proximal-point mappings connected with generalized \(m\)-accretive mappings to the \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappings and discuss its characteristics like single-valuable [sic] and Lipschitz continuity. Some illustration are given in support of \(\alpha \beta\)-\(H((.,.),(.,.))\)-mixed accretive mappings. Since proximal point mapping is a powerful tool for solving variational inclusion. Therefore, as an application of introduced mapping, we construct an iterative algorithm to solve variational inclusions and show its convergence with acceptable assumptions.Forward-backward approximation of nonlinear semigroups in finite and infinite horizonhttps://www.zbmath.org/1483.470882022-05-16T20:40:13.078697Z"Contreras, Andrés"https://www.zbmath.org/authors/?q=ai:contreras.andres-a"Peypouquet, Juan"https://www.zbmath.org/authors/?q=ai:peypouquet.juanThe authors consider the problem
\[
\begin{aligned}
-&\dot{u}(t)\in\left( A+B\right) u(t) \text{ for a.e. }t>0,\\
&u(0)=u_{0}\in D(A),
\end{aligned}
\]
in a class of Banach spaces, where \(A\) is \(m\)-accretive and \(B\) is coercive. First, the approximation of solutions is investigated. Solutions are approximated by trajectories constructed by interpolation of sequences generated using forward-backward iteration and these are shown to converge uniformly on a finite time interval, proving existence and uniqueness of solutions. Second, asymptotic equivalence results are given that connect the behaviour of forward-backward iterations as the number of iterations goes to infinity with the behaviour of the solution as time goes to infinity, for step sizes that are sufficiently small. These results are based on a certain inequality which the authors trace back to \textit{E. Hille} [Fysiogr. Sällsk. Lund Förh. 21, No. 14, 130--142 (1951; Zbl 0044.32902)].
Reviewer: Daniel C. Biles (Nashville)Proximal point algorithm for a countable family of weighted resolvent averageshttps://www.zbmath.org/1483.470952022-05-16T20:40:13.078697Z"Bagheri, Malihe"https://www.zbmath.org/authors/?q=ai:bagheri.malihe"Roohi, Mehdi"https://www.zbmath.org/authors/?q=ai:roohi.mehdiSummary: In this paper, we introduce a composite iterative algorithm for finding a common zero point of a countable family of weighted resolvent average of finite family of monotone operators in Hilbert spaces. We prove that the sequence generated by the iterative algorithm converges strongly to a common zero point. Finally, we apply our results to split common zero point problem.The iterative method for solving the proximal split feasibility problem with an application to LASSO problemhttps://www.zbmath.org/1483.470992022-05-16T20:40:13.078697Z"Ma, Xiaojun"https://www.zbmath.org/authors/?q=ai:ma.xiaojun"Liu, Hongwei"https://www.zbmath.org/authors/?q=ai:liu.hongwei|liu.hongwei.1"Li, Xiaoyin"https://www.zbmath.org/authors/?q=ai:li.xiaoyinSummary: In this paper, we investigate strong convergence of the iterative algorithm for solving the proximal split feasibility problem in real Hilbert spaces. The algorithm is motivated by the inertial method, the viscosity-type method and the split proximal algorithm with a self-adaptive stepsize. A~strong convergence theorem for the proposed algorithm is established without requiring firm nonexpansiveness of the involved operators. An application of our obtained results is offered. Finally, some numerical experiments are provided for illustration and comparison.A method of approximation for a zero of the sum of maximally monotone mappings In Hilbert spaceshttps://www.zbmath.org/1483.471052022-05-16T20:40:13.078697Z"Wega, Getahun Bekele"https://www.zbmath.org/authors/?q=ai:wega.getahun-bekele"Zegeye, Habtu"https://www.zbmath.org/authors/?q=ai:zegeye.habtuSummary: Our purpose of this study is to construct an algorithm for finding a zero of the sum of two maximally monotone mappings in Hilbert spaces and discuss its convergence. The assumption that one of the mappings is \(\alpha\)-inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.Approximation for the hierarchical constrained variational inequalities over the fixed points of nonexpansive semigroupshttps://www.zbmath.org/1483.471092022-05-16T20:40:13.078697Z"Zhu, Li-Jun"https://www.zbmath.org/authors/?q=ai:zhu.lijunSummary: The purpose of the present paper is to study the hierarchical constrained variational inequalities of finding a point \(x^\ast\) such that \(x^\ast \in \Omega\), \(\langle (A - \gamma f)x^\ast - (I - B)Sx^\ast, x - x^\ast \rangle \geq 0\), \(\forall x \in \Omega \), where \(\Omega\) is the set of the solutions of the following variational inequality: \(x^\ast \in \mathcal{F}\), \(\langle (A - S)x^\ast, x - x^\ast \rangle \geq 0\), \(\forall x \in \mathcal{F}\), where \(A, B\) are two strongly positive bounded linear operators, \(f\) is a \(\rho\)-contraction, \(S\) is a nonexpansive mapping, and \(\mathcal{F}\) is the fixed points set of a nonexpansive semigroup \(\{T(s)\}_{s \geq 0}\). We present a double-net convergence hierarchical to some elements in \(\mathcal{F}\) which solves the above hierarchical constrained variational inequalities.Optimal control of partial differential equations. Analysis, approximation, and applicationshttps://www.zbmath.org/1483.490012022-05-16T20:40:13.078697Z"Manzoni, Andrea"https://www.zbmath.org/authors/?q=ai:manzoni.andrea"Quarteroni, Alfio"https://www.zbmath.org/authors/?q=ai:quarteroni.alfio-m"Salsa, Sandro"https://www.zbmath.org/authors/?q=ai:salsa.sandroThis book is essentially devoted to optimal control problems (OCPs) governed by partial differential equations (PDEs). The authors' aim (in their words) ``is to cover the whole range going from the set up and the theoretical analysis of the OCP, to the derivation of the system of optimality conditions, the proposition of suitable numerical methods, their analysis, and the application to a wide class of problems of practical relevance.'' This ``broad span'' makes the book unique within the large set of existing textbooks and monographs in the field of OCPs.
The book comprises an introduction (Chapter 1), three parts (Chapters 2--11), two appendices, references, and an index.
Chapter 1 (entitled ``Introduction: Representative Examples, Mathematical Structure'') presents a general framework for OCPs governed by PDEs, as well as several examples from physics, fluid mechanics, engineering, etc. which motivate the theoretical treatment. The authors prepare the general setting and notations that are used throughout the book for the analysis of OCPs.
Part I (entitled ``A Preview on Optimization and Control in Finite Dimensions'') comprises Chapters 2--4. Chapter 2 (Prelude of Optimization: The Finite Dimensional Case) deals with optimization of functions in finite dimensional spaces. First-order optimality conditions are derived for both free (unconstrained) and constrained optimization problems. In particular, the authors introduce the Lagrange multipliers and the Karush-Kuhn-Tucker methods for treating the case of equality constraints and inequality constraints, respectively. Chapter 3 (Algorithms for Numerical Optimization) presents some of the well-known iterative algorithms for the approximation of local optimizers (minimizers or maximizers), such as \textit{descent methods} and \textit{trust region methods} for solving free optimization problems, and \textit{projection methods}, \textit{penalty} and \textit{augmented Lagrangian methods}, \textit{sequential quadratic programming method} for solving constrained optimization problems. Chapter 4 (Prelude on Control: The Case of Algebraic and ODE Systems) contains a description of two classes of finite dimensional OCPs, which are relevant in themselves and can serve in treating numerical discretization of OCPs governed by PDEs. In Section 4.1 the authors consider the minimization problem for a quadratic cost functional $J(y,u)=(1/2)|y-z_d|^2 + (\alpha /2)|u|^2$ governed by an algebraic linear equation in $\mathbb{R}^N$ of the form $Ay=f+Bu$, where $A\in \mathbb{R}^{N\times N}$ is a nonsingular matrix and $B\in \mathbb{R}^{N\times q}$ is a matrix of rank $q$. The authors present results on the existence and uniqueness of the solution to this OCP, optimality conditions, sensitivity analysis. In Section 4.2 the above OCP is regarded as a constrained optimization problem, thus the Lagrange multiplier rule comes into play and leads to a new approach. A similar approach, based on Karush-Kuhn-Tucker multipliers, can be used in the case of box control constraints. Section 4.3 is an introduction to (finite dimensional) control problems governed by ODEs in a time interval $[0,T]$, with initial and final conditions on the state variable $y$: $y(0)=\xi, \ y(T)\in Y_T$, where $Y_T\subset \mathbb{R}^N$ is the \textit{final target set}. The final time $T$ could be fixed or unknown. The problem proposed to be solved is to select an admissible (optimal) control $u^*$ which minimizes a cost functional $J(u)=\psi (y(T))+$$\int_0^Tf(y(s),u(s))\, ds$, where $y$ is the state corresponding to $u$. Section 4.4 is devoted to the particular case where the end point is free, the cost functional is of the form $J(u) = \eta \cdot y(T)$, were $\eta$ is a given nonzero vector, and the state $y$ satisfies a linear differential equation in $\mathbb{R}^N$ of the form $y'(t)=Ay(t)+Bu(t)$, $0<t<T, \ y(0)=\xi$. Admissible controls are piecewise continuous functions $u:[0,T]\to \mathbb{R}^m$ satisfying $|u(t)|\le 1, \ 0\le t\le T$. Section 4.5 is focused on the \textit{minimum time problem}, i.e. the problem of finding the minimum time $T$ such that $y(t)$ is driven from a given initial state $\xi$ to a given target state $y_{tar}$, under the dynamics $y'(t)=Ay(t)+Bu(t),$ $ \ 0<t<T, \ y(0)=\xi, \ y(T)=y_{tar}$. In Section 4.6 the minimization of a quadratic cost functional is considered, under the dynamics $y'(t)=Ay(t)+Bu(t), \ 0<t<T, \ y(0)=\xi$. Existence and uniqueness of an optimal control follow by standard arguments (see also Chapter 5). The Riccati equation gives the optimal control in feedback form. In Section 4.7 a possible strategy to approximate the solution of an OCP from the class of the OCPs discussed in Section 4.6 is presented.
Part II (Linear-quadratic Optimal Control Problems) comprises Chapters 5-8. Chapter 5 (Quadratic control problems governed by linear elliptic PDEs) is focused on OCPs involving a quadratic cost functional and a state problem associated with a linear elliptic equation. In Section 5.1 an unconstrained linear-quadratic OCP governed by an advection-diffusion equation is analyzed. Optimality conditions are derived by an adjoint-based approach, and also by using Lagrange multipliers. The constrained case is addressed in Section 5.2. Then, a general framework for linear-quadratic OCPs governed by variational problems in presence of control constraints is provided in Section 5.3. In Section 5.4 the weak and variational formulation of boundary value problems for second order elliptic equations in divergence form is recalled and their well-posedness is examined. Then, in Sections 5.5-5.11 the above analysis is applied to OCPs governed by such problems, with distributed or boundary controls and various quadratic cost functionals. A simple case of an OCP involving state constraints is considered in Section 5.12. Some OCPs governed by a Stokes system are considered in Section 5.13. Chapter 6 (Numerical Approximation of Linear-Quadratic OCPs) addresses the numerical solution of OCPs with quadratic functional and linear state equation. Two examples of state equations are considered: the advection-diffusion equation and the Stokes equations. Two general approaches (called \textit{optimize-then-discretize} and \textit{discretize-then-optimize}) are described in Section 6.2. In Section 6.3 iterative methods are addressed for both unconstrained OCPs and OCPs with control constraints. Then, the so called \textit{all-at-once methods} are described, that treat both control and state variables simultaneously as independent optimization variables, coupled through the PDE constraint, and therefore solve the state problem, the adjoint problem and the optimality condition as a unique system. Different numerical examples are solved and error estimates are provided. Chapter 7 (Control Problems Governed by Linear Evolution PDEs) is devoted to OCPs involving a quadratic cost functional and a state system described by a linear initial-boundary value problem (IBVP). Both unconstrained and constrained problems are considered. The whole analysis is similar to that of Chapter 5. First, an OCP governed by the heat equation with a distributed heat source as control function is studied in detail. This is followed by the case of box control constraints and the case in which the control function is the initial state. Then, a general framework for OCPs governed by parabolic PDEs with different boundary conditions is presented, including details on side controls, side observations, etc. Finally, an OCP related to time dependent Stokes equations is discussed, and the case of the wave equation is briefly mentioned. Chapter 8 (Numerical Approximation of Quadratic OCPs Governed by Linear Evolution PDEs) addresses the numerical solution of the OCPs discussed in the previous chapter, by closely following the road map of Chapter 6.
Part III (More general PDE-constrained optimization problems) comprises Chapters 9--11. Chapter 9 (A Mathematical Framework for Nonlinear OCPs) addresses rather general nonlinear OCPs and the corresponding set of optimality conditions. The state equation may be a nonlinear PDE, the control function may belong to a Banach space, and/or the cost functional is no longer quadratic. The results reported in this chapter extend those in Chapters 5 and 7 for the linear-quadratic case. Some model problems are analyzed and numerical results are obtained. Chapter 10 (Advanced Selected Applications) is devoted to OCPs involving mathematical models from fluid dynamics or cardiac electrophysiology. The authors handle nonlinear state equations, more involved cost functionals, and constraints required to ensure control feasibility. Both the theoretical results and numerical approaches are adaptations of what we have seen so far. Chapter 11 (Shape Optimization Problems) provides a short review of the most relevant analytical and numerical tools required to investigate shape optimization problems (SOPs). A SOP amounts to the minimization, over a set of admissible shapes, of a cost functional which depends on the unknown spatial domain $\Omega$ and on the solution of a state problem. Optimality conditions are derived in Section 11.4 for a simple SOP. Some techniques for the numerical approximation of SOPs are then described in Section 11.5. Finally, two applications are discussed in detail.
Appendix A (Toolbox of Functional Analysis) and Appendix B (Toolbox of Numerical Analysis) provide the readers with necessary auxiliary material.
This excellent book is suitable to people interested in mathematical and applied sciences.
Reviewer: Gheorghe Moroşanu (Cluj-Napoca)Generalized derivatives for solution operators of variational inequalities of obstacle typehttps://www.zbmath.org/1483.490022022-05-16T20:40:13.078697Z"Rauls, Anne-Therese"https://www.zbmath.org/authors/?q=ai:rauls.anne-theresePublisher's description: This thesis characterizes generalized derivatives for solution operators of obstacle problems. We consider the mathematical formulation of these problems as variational inequalities. It is well known that for every given force term these variational inequalities possess unique solutions and that the corresponding solutions depend Lipschitz continuously on the force terms. Due to the obstacle constraint, the solution operators are, in general, nonsmooth. However, a generalization of Rademacher's theorem to sufficiently regular infinite dimensional spaces states that the Lipschitz continuous solution operators of the variational inequalities are Gâteaux differentiable on a dense subset of their domains. For such operators, generalized derivatives can serve as a substitute for the classical Gâteaux derivatives in all points of the domain. These sets of generalized derivatives at a fixed point of the domain are defined as limits of Gâteaux derivatives at approximating sequences of the domain. For the convergence of the sequence in the domain and also for the Gâteaux derivatives, different combinations of topologies can be used in infinite dimensional spaces.
The base for our analysis is the characterization of the directional derivative as the solution operator of a variational inequality established by Mignot. This enables us to obtain a representation of the Gâteaux derivatives as solution operators of variational equations or Dirichlet problems on certain quasi-open domains. These quasi-open domains depend on the contact set and the contact forces acting between the obstacle and the solution. Subsequently, the convergence of these Gâteaux derivative operators is examined for different types of obstacle problems. We also study the numerical computation of subgradients for optimal control problems with respect to obstacle problems. An error estimate is presented that can be used for the implementation of an inexact Bundle method.Numerical approximation of optimal control problems for hyperbolic conservation lawshttps://www.zbmath.org/1483.490032022-05-16T20:40:13.078697Z"Schäfer Aguilar, Paloma"https://www.zbmath.org/authors/?q=ai:schafer-aguilar.palomaPublisher's description: Hyperbolic conservation laws are often used to model physical processes such as gas transport in networks or traffic flow. The crucial issue of nonlinear hyperbolic conservation laws is that even for smooth input data solutions develop moving discontinuities after finite time, so-called shocks. For this reason, the numerical approximation of hyperbolic conservation laws is quite involved, since several challenges in the analytical study as well as in the numerical analysis of the solutions appear.
The first part of this thesis provides a rigorous sensitivity and adjoint calculus for conservation laws with source terms. Due to the shock formations, the control-to-state mapping is at best differentiable in the weak topology of measures. Pfaff and S. Ulbrich developed an adjoint-based gradient formulation for tracking type functionals by using a suitable adjoint state, which is a solution to the associated adjoint equation. In the case of shocks, the correct definition of appropriate solutions to the adjoint and the associated sensitivity equation requires care. We analyze solutions to these equations and introduce convenient characterizations, which are suitable to show that the limit of discrete adjoints and discrete sensitivities are in fact the desired solution of the adjoint and sensitivity equation, respectively.
In the second part of this thesis, we derive consistent numerical discretization schemes for optimal boundary control problems of hyperbolic conservation laws with source terms. We use monotone difference schemes for the state equation and derive the associated sensitivity and adjoint scheme. We derive convergence results for these classes of schemes, wherefore we use suitable numerical flux functions to guarantee that the limit functions of the adjoint and sensitivity schemes attain the corresponding boundary data in the correct sense.Derivative-orthogonal wavelets for discretizing constrained optimal control problemshttps://www.zbmath.org/1483.490042022-05-16T20:40:13.078697Z"Ashpazzadeh, E."https://www.zbmath.org/authors/?q=ai:ashpazzadeh.elmira"Han, B."https://www.zbmath.org/authors/?q=ai:han.bin.1|khan.basar|han.baoyan|khan.baseem|han.baozhi|khan.burhan|han.banghe|khan.bivas|han.baiping|khan.bilal.1|khan.babar|han.bowen|han.bingkang|han.bingbing|han.biao|han.ben|khan.bilal|han.bing|han.brian|han.bohyung|han.bingyan|han.baichang|han.bingliang|khan.behram|han.byungrin|han.byoungcheon|han.baijing|han.bao|han.baoming|han.bangxian|han.benjamin|han.bochang|han.bingtao|han.bin|han.bo|han.baoling|khan.babu|han.brent|han.bensan|han.binling|han.boshun|han.bongtae|han.buhm|han.botang"Lakestani, M."https://www.zbmath.org/authors/?q=ai:lakestani.mehrdad"Razzaghi, M."https://www.zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: In this article, a pair of wavelets for Hermite cubic spline bases are presented. These wavelets are in \(C^1\) and supported on \([-1,1]\). These spline wavelets are then adapted to the interval \([0,1]\) and we prove that they form a Riesz wavelet in \(L_2([0,1])\). The wavelet bases are used to solve the linear optimal control problems. The operational matrices of integration and product are then utilised to reduce the given optimisation problems to the system of algebraic equations. Because of the sparsity nature of these matrices, this method is computationally very attractive and reduces CPU time and computer memory. In order to save the memory requirement and computation time, a threshold procedure is applied to obtain algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.Field theory for integrands with low regularityhttps://www.zbmath.org/1483.490052022-05-16T20:40:13.078697Z"Gratwick, Richard"https://www.zbmath.org/authors/?q=ai:gratwick.richard"Sychev, Mikhail A."https://www.zbmath.org/authors/?q=ai:sychev.mikhail-aIn this paper, the authors investigated a class of one-dimensional variational problems governed by integrands with low regularity and singular ellipticity.
Reviewer: Savin Treanta (Bucureşti)Fractional-order Bessel wavelet functions for solving variable order fractional optimal control problems with estimation errorhttps://www.zbmath.org/1483.490062022-05-16T20:40:13.078697Z"Dehestani, Haniye"https://www.zbmath.org/authors/?q=ai:dehestani.haniye"Ordokhani, Yadollah"https://www.zbmath.org/authors/?q=ai:ordokhani.yadollah"Razzaghi, Mohsen"https://www.zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: In the present paper, we apply the fractional-order Bessel wavelets (FBWs) for solving optimal control problems with variable-order (VO) fractional dynamical system. The VO fractional derivative operator is proposed in the sense of Caputo type. To solve the considered problem, the collocation method based on FBWFs, pseudo-operational matrix of VO fractional derivative and the dual operational matrix is proposed. In fact, we convert the problem with unknown coefficients in the constraint equations, performance index and conditions to an optimisation problem, by utilising the proposed method. Also, the convergence of the method with details is discussed. At last, to demonstrate the high precision of the numerical approach, we examine several examples.Unconstrained direct optimization of spacecraft trajectories using many embedded Lambert problemshttps://www.zbmath.org/1483.490072022-05-16T20:40:13.078697Z"Ottesen, David"https://www.zbmath.org/authors/?q=ai:ottesen.david"Russell, Ryan P."https://www.zbmath.org/authors/?q=ai:russell.ryan-pSummary: Direct optimization of many-revolution spacecraft trajectories is performed using an unconstrained formulation with many short-arc, embedded Lambert problems. Each Lambert problem shares its terminal positions with neighboring segments to implicitly enforce position continuity. Use of embedded boundary value problems (EBVPs) is not new to spacecraft trajectory optimization, including extensive use in primer vector theory, flyby tour design, and direct impulsive maneuver optimization. Several obstacles have prevented their use on problems with more than a few dozen segments, including computationally expensive solvers, lack of fast and accurate partial derivatives, unguaranteed convergence, and a non-smooth solution space. Here, these problems are overcome through the use of short-arc segments and a recently developed Lambert solver, complete with the necessary fast and accurate partial derivatives. These short arcs guarantee existence and uniqueness for the Lambert solutions when transfer angles are limited to less than a half revolution. Furthermore, the use of many short segments simultaneously approximates low-thrust and eliminates the need to specify impulsive maneuver quantity or location. For preliminary trajectory optimization, the EBVP technique is simple to implement, benefiting from an unconstrained formulation, the well-known Broyden-Fletcher-Goldfarb-Shanno line search direction, and Cartesian coordinates. Moreover, the technique is naturally parallelizable via the independence of each segment's EBVP. This new, many-rev EBVP technique is scalable and reliable for trajectories with thousands of segments. Several minimum fuel and energy examples are demonstrated, including a problem with 6143 segments for 256 revolutions, found within 5.5 h on a single processor. Smaller problems with only hundreds of segments take minutes.Optimisation of controller reconfiguration instant for spacecraft control systems with additive actuator faultshttps://www.zbmath.org/1483.490082022-05-16T20:40:13.078697Z"Tu, Yuanyuan"https://www.zbmath.org/authors/?q=ai:tu.yuanyuan"Wang, Dayi"https://www.zbmath.org/authors/?q=ai:wang.dayi"Li, Wenbo"https://www.zbmath.org/authors/?q=ai:li.wenbo"Li, Maodeng"https://www.zbmath.org/authors/?q=ai:li.maodengSummary: This paper aims to propose a method of optimising the controller reconfiguration (CR) instant for spacecraft control systems with additive actuator faults. Due to severe resource constraints, it is `expensive' for spacecraft to handle the faults in orbit. To overcome the constraints, we try to reduce the CR cost from a perspective of time management. The basic idea of this method is to propose a reconfigurability evaluation index to quantify the capability of the fault system in maintaining admissible performance by CR measures, and then to derive the mathematical relationship between the reconfigurability index and four crucial instants during the CR process. Based on this, the CR instant is optimised by maximising the reconfigurability index. Finally, the effectiveness of the proposed method is illustrated through an example. The results show that by enhancing the time management efficiency during the CR process, the limited resources can be economised to some extent.Necessary optimality conditions for a class of control problems with state constrainthttps://www.zbmath.org/1483.490092022-05-16T20:40:13.078697Z"Korytowski, Adam"https://www.zbmath.org/authors/?q=ai:korytowski.adam"Szymkat, Maciej"https://www.zbmath.org/authors/?q=ai:szymkat.maciejIn this work, the authors study necessary conditions of optimal control problems with pathwse state constraints involving ordinary differential equations. The motivation to study such a problem arises out of the fact that existing classifications of necessary conditions using gradient-based approaches are quite abstract and are difficult to verify. The authors use the idea of the needle variations in Pontryagin's maximum principle classification of optimality condition, to develop verifiability conditions that distinguish the controls to which the needle technique of optimality verification can be effectively applied. The authors also provide a geometric interpretations of such conditions. Finally, they demonstrate the validity of their theoretical results with several numerical examples.
The paper is well-written and easy to follow, and is certainly quite a good contribution to the field of optimal control. It will be interesting for researchers working on optimal control to use and validate the theoretical results of this paper in different applications. Moreoever, it will also be interesting to see if such an idea can be extended to optimal control problems involving partial differential equations.
Reviewer: Souvik Roy (Arlington)A robust pseudospectral method for numerical solution of nonlinear optimal control problemshttps://www.zbmath.org/1483.490102022-05-16T20:40:13.078697Z"Mehrpouya, Mohammad Ali"https://www.zbmath.org/authors/?q=ai:mehrpouya.mohammad-ali"Peng, Haijun"https://www.zbmath.org/authors/?q=ai:peng.haijunSummary: In the present paper, a robust pseudospectral method for efficient numerical solution of nonlinear optimal control problems is presented. In the proposed method, at first, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality which are led to the Hamiltonian boundary value problem are derived. Then, utilizing a pseudospectral method for discretization, the nonlinear optimal control problem is converted to a system of nonlinear algebraic equations. However, the need to have a good initial guess may lead to a challenging problem for solving the obtained system of nonlinear equations. So, an optimization approach is introduced to simplify the need of a good initial guess. Numerical findings of some benchmark examples are presented at the end and computational features of the proposed method are reported.Numerical solution of two-point BVPs in infinite-horizon optimal control theory: a combined quasilinearization method with exponential Bernstein functionshttps://www.zbmath.org/1483.490112022-05-16T20:40:13.078697Z"Nikooeinejad, Z."https://www.zbmath.org/authors/?q=ai:nikooeinejad.z"Heydari, M."https://www.zbmath.org/authors/?q=ai:heydari.mohammad-hossien"Loghmani, G. B."https://www.zbmath.org/authors/?q=ai:loghmani.ghasem-barid|loghmani.g-baridSummary: This study is aimed to relate nonlinear infinite-horizon optimal control problems (NLIHOCPs) with open-loop information. The difficulties of solving the two-point boundary value problems (TPBVPs) arising from NLIHOCPs can be assigned to the nonlinearity of differential equations, the combination of split boundary conditions, and how the transversality conditions in infinite-horizon are treated. In this paper, we propose a combined quasilinearization method (QLM) with the exponential Bernstein functions (EBFs) for solving nonlinear TPBVPs on the semi-infinite domain. First, the QLM is used to reduce the nonlinear TPBVP to a sequence of linear differential equations. Then, a collocation method based on the EBFs is utilized to find the approximate solution of the resulting linear differential equations. By applying the EBFs, the transversality conditions for TPBVP on the semi-infinite domain are satisfied. The convergence of the QLM + EBFs is proved. Some numerical experiments are performed to confirm the validity of the proposed computational scheme.Optimal control for systems of differential equations on the infinite interval of time scalehttps://www.zbmath.org/1483.490122022-05-16T20:40:13.078697Z"Stanzhytskyi, O."https://www.zbmath.org/authors/?q=ai:stanzhytskyi.oleksandr-m"Mogylova, V."https://www.zbmath.org/authors/?q=ai:mogylova.viktoriya-v"Lavrova, O."https://www.zbmath.org/authors/?q=ai:lavrova.o-ye|lavrova.olgaIn this work, the authors study an optimal control problem, involving ordinary differential equations, on a semi-infinite time domain. Traditional optimal control problems have been studied on finite time intervals and it is a major challenge to extend these results for the semi-infinite domain. The authors study the connection between solutions to optimal control problems on time scales and to the corresponding problem on semi-infinite real axis. They also propose a new method to construct a minimizing sequence for an optimal control problem on the semi-axis.
The paper is well-written and easy to follow, and is certainly quite a good contribution to the field of optimal control. A next step will be to validate the theoretical results using numerical examples.
For the entire collection see [Zbl 1470.53006].
Reviewer: Souvik Roy (Arlington)Non-convex functionals penalizing simultaneous oscillations along independent directions: rigidity estimateshttps://www.zbmath.org/1483.490132022-05-16T20:40:13.078697Z"Goldman, Michael"https://www.zbmath.org/authors/?q=ai:goldman.michael"Merlet, Benoit"https://www.zbmath.org/authors/?q=ai:merlet.benoitAuthors' abstract: We study a family of non-convex functionals \(\{\mathcal E\}\) on the space of measurable functions \(u: \Omega_1 \times \Omega_2 \subset \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \rightarrow \mathbb{R}\). These functionals vanish on the non-convex subset \(S(\Omega_1 \times \Omega_2)\) formed by functions of the form \(u(x_1,x_2)=u_1(x_1)\) or \(u(x_1,x_2)=u_2(x_2)\). We investigate under which conditions the converse implication ``\(\mathcal E(u) = 0 \Rightarrow u \in S(\Omega_1 \times \Omega_2)\)'' holds. In particular, we show that the answer depends strongly on the smoothness of \(u\). We also obtain quantitative versions of this implication by proving that (at least for some parameters) \(\mathcal E(u)\) controls in a strong sense the distance of \(u\) to \(S(\Omega_1 \times \Omega_2)\).
Reviewer: Costică Moroşanu (Iaşi)\(S\)-iterative algorithm for solving variational inequalitieshttps://www.zbmath.org/1483.490142022-05-16T20:40:13.078697Z"Ertürk, Müzeyyen"https://www.zbmath.org/authors/?q=ai:erturk.muzeyyen"Gürsoy, Faik"https://www.zbmath.org/authors/?q=ai:gursoy.faik"Şimşek, Necip"https://www.zbmath.org/authors/?q=ai:simsek.necipSummary: In this paper we propose an \(S\)-iterative algorithm for finding the common element of the set of solutions of the variational inequalities and the set of fixed points of nonexpansive mappings. We study the convergence criteria of the mentioned algorithm under some mild conditions. As an application, a modified algorithm is suggested to solve convex minimization problems. Numerical examples are given to validate the theoretical findings obtained herein. Our results may be considered as an improvement, refinement and complement of the previously known results.Probabilistic feasibility guarantees for solution sets to uncertain variational inequalitieshttps://www.zbmath.org/1483.490152022-05-16T20:40:13.078697Z"Fabiani, Filippo"https://www.zbmath.org/authors/?q=ai:fabiani.filippo"Margellos, Kostas"https://www.zbmath.org/authors/?q=ai:margellos.kostas"Goulart, Paul J."https://www.zbmath.org/authors/?q=ai:goulart.paul-jSummary: We develop a data-driven approach to the computation of a-posteriori feasibility certificates for sets of solutions of variational inequalities affected by uncertainty. Specifically, we focus on variational inequalities with a deterministic mapping and an uncertain feasible set, and represent uncertainty by means of scenarios. Building upon recent advances in the scenario approach literature, we quantify the robustness properties of the entire set of solutions of a variational inequality, with feasibility set constructed using the scenario approach, against a new unseen realization of the uncertainty. Our results extend existing ones that typically impose that the solution set is a singleton and require certain non-degeneracy properties: hence, we thereby offer probabilistic feasibility guarantees for any feasible solution of the underlying variational inequality. We show that assessing the violation probability of an entire set of solutions requires enumeration of the support constraints that ``shape'' this set, and also propose a procedure to enumerate the support constraints that does not require a description of the solution set. We illustrate our results through numerical simulations on a robust game involving an electric vehicle charging coordination problem.T-minima and application to the convergence of some integral functionals with infinite energy minimahttps://www.zbmath.org/1483.490162022-05-16T20:40:13.078697Z"Boccardo, Lucio"https://www.zbmath.org/authors/?q=ai:boccardo.lucioSummary: We consider integral functionals \(J\) as in [\textit{D. Arcoya} and \textit{L. Boccardo}, J. Funct. Anal. 268, No. 5, 1153--1166 (2015; Zbl 1317.35082)]. We study the Calderon-Zygmund theory for infinite energy minima \(u\) (T-minima) of \(J\) and the stability of the T-minima with respect to the convergence in [\textit{E. De Giorgi} and \textit{S. Spagnolo}, Boll. Unione Mat. Ital., IV. Ser. 8, 391--411 (1973; Zbl 0274.35002)], with \(V = W_0^{1, p}(\varOmega)\).Variational time discretization of Riemannian splineshttps://www.zbmath.org/1483.490172022-05-16T20:40:13.078697Z"Heeren, Behrend"https://www.zbmath.org/authors/?q=ai:heeren.behrend"Rumpf, Martin"https://www.zbmath.org/authors/?q=ai:rumpf.martin"Wirth, Benedikt"https://www.zbmath.org/authors/?q=ai:wirth.benediktSummary: We investigate a generalization of cubic splines to Riemannian manifolds. Spline curves are defined as minimizers of the spline energy -- a combination of the Riemannian path energy and the time integral of the squared covariant derivative of the path velocity -- under suitable interpolation conditions. A variational time discretization for the spline energy leads to a constrained optimization problem over discrete paths on the manifold. Existence of continuous and discrete spline curves is established using the direct method in the calculus of variations. Furthermore, the convergence of discrete spline paths to a continuous spline curve follows from the \(\Gamma \)-convergence of the discrete to the continuous spline energy. Finally, selected example settings are discussed, including splines on embedded finite-dimensional manifolds, on a high-dimensional manifold of discrete shells with applications in surface processing and on the infinite-dimensional shape manifold of viscous rods.Data-driven optimal control with a relaxed linear programhttps://www.zbmath.org/1483.490182022-05-16T20:40:13.078697Z"Martinelli, Andrea"https://www.zbmath.org/authors/?q=ai:martinelli.andrea"Gargiani, Matilde"https://www.zbmath.org/authors/?q=ai:gargiani.matilde"Lygeros, John"https://www.zbmath.org/authors/?q=ai:lygeros.johnSummary: The linear programming (LP) approach has a long history in the theory of approximate dynamic programming. When it comes to computation, however, the LP approach often suffers from poor scalability. In this work, we introduce a relaxed version of the Bellman operator for \(q\)-functions and prove that it is still a monotone contraction mapping with a unique fixed point. In the spirit of the LP approach, we exploit the new operator to build a relaxed linear program (RLP). Compared to the standard LP formulation, our RLP has only one family of constraints and half the decision variables, making it more scalable and computationally efficient. For deterministic systems, the RLP trivially returns the correct \(q\)-function. For stochastic linear systems in continuous spaces, the solution to the RLP preserves the minimizer of the optimal \(q\)-function, hence retrieves the optimal policy. Theoretical results are backed up in simulation where we solve sampled versions of the LPs with data collected by interacting with the environment. For general nonlinear systems, we observe that the RLP again tends to preserve the minimizers of the solution to the LP, though the relative performance is influenced by the specific geometry of the problem.On the convergence of solutions of variational problems with degeneration and unilateral functional constraints in variable domainshttps://www.zbmath.org/1483.490192022-05-16T20:40:13.078697Z"Rudakova, O. A."https://www.zbmath.org/authors/?q=ai:rudakova.olga-aA sequence of convex integral functionals defined on weighted Sobolev spaces is considered in this short paper. Several sufficient conditions are formulated for the convergence of minimizers and minimum values of functionals to the minimizer and the minimum of a given functional. The main result is Theorem 2, formulated in Section 2. It is given without proof.
Reviewer: Stepan Agop Tersian (Rousse)Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order Lagrangianhttps://www.zbmath.org/1483.490202022-05-16T20:40:13.078697Z"Zagatti, Sandro"https://www.zbmath.org/authors/?q=ai:zagatti.sandroSummary: We study the minimum problem for functionals of the form
\[
\mathcal{F}(u) = \int_I f(x, u(x), u^\prime(x), u^{\prime\prime}(x))\,dx,
\]
where the integrand \(f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} \) is not convex in the last variable. We provide an existence result assuming that the lower convex envelope \(\overline{f} = \overline{f}(x,p,q,\xi) \) of \(f \) with respect to \(\xi \) is regular and enjoys a special dependence with respect to the i-th single components \(p_i, q_i, \xi_i \) of the vector variables \(p,q,\xi \). More precisely, we assume that it is monotone in \(p_i \) and that it satisfies suitable affinity properties with respect to \(\xi_i \) on the set \(\{f> \overline{f}\} \) and with respect to \(q_i \) on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.Nonsmooth exact penalization second-order methods for incompressible bi-viscous fluidshttps://www.zbmath.org/1483.490212022-05-16T20:40:13.078697Z"González-Andrade, Sergio"https://www.zbmath.org/authors/?q=ai:gonzalez-andrade.sergio"López-Ordóñez, Sofía"https://www.zbmath.org/authors/?q=ai:lopez-ordonez.sofia"Merino, Pedro"https://www.zbmath.org/authors/?q=ai:merino.pedroThe authots consider the exact penalization of the incompressibility condition \(\operatorname{div}(\mathbf{u}) = 0\) for the velocity field of a bi-viscous fluid in terms of the \(L^1\)-norm. This results in a nonsmooth optimization problem for which a solution method relying on a steepest descent direction and extra generalized second-order information associated to the nonsmooth term is provided. The divergence-free property is enforced by the descent direction proposed by the method without the need of build-in divergence-free approximation schemes. An inexact penalization approach relying on the \(L^2\)-norm is also considered.
Reviewer: Radu Ioan Bot (Wien)Extended multi-interval Legendre-Gauss-Radau pseudospectral method for mixed-integer optimal control problem in engineeringhttps://www.zbmath.org/1483.490222022-05-16T20:40:13.078697Z"Liu, Zhe"https://www.zbmath.org/authors/?q=ai:liu.zhe"Li, Shurong"https://www.zbmath.org/authors/?q=ai:li.shurong"Zhao, Kai"https://www.zbmath.org/authors/?q=ai:zhao.kaiSummary: Many engineering optimisation problems can be summarised as mixed-integer optimal control problems (MIOCPs) owing to the needs for mixed-integer dynamic control decisions. However, the convergence theory of Legendre-Gauss-Radau (LGR) approximation fails to apply to such non-smooth and discontinuous optimal control problems. Therefore, this paper develops an extended multi-interval LGR pseudospectral method (EMLGR), which has the following features: (i) the mixed-integer controls at the end of each interval and the interval intersections are added as two new controls to avoid the unrestrained control and shorten the switching time of integer control, and (ii) a smart adaptive collocation monitor (SACM) is provided to optimise the polynomial order and interval structure for further reducing computational complexity and improving approximation precision. The detailed solution procedure of EMLGR is given in this study, and experimental studies including five challenging practical engineering MIOCPs are taken to verify the superiorities of the proposed EMLGR in efficiency, accuracy and stability.Structure of approximate solutions of autonomous variational problemshttps://www.zbmath.org/1483.490232022-05-16T20:40:13.078697Z"Zaslavski, Alexander J."https://www.zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study the structure of approximate solutions of autonomous variational problems on large finite intervals. Our goal is to show that approximate solutions are determined mainly by the integrand, and are esssentially independent of the choice of time interval and data. In the first part of the paper we discuss our recent results on Lagrange problems, the second part contains new results on the structure of approximate solutions of Bolza problems.Calculus of variations and optimal control for generalized functionshttps://www.zbmath.org/1483.490242022-05-16T20:40:13.078697Z"Frederico, Gastão S. F."https://www.zbmath.org/authors/?q=ai:frederico.gastao-s-f"Giordano, Paolo"https://www.zbmath.org/authors/?q=ai:giordano.paolo-robuffo"Bryzgalov, Alexandr A."https://www.zbmath.org/authors/?q=ai:bryzgalov.aleksandr-anatolevich"Lazo, Matheus J."https://www.zbmath.org/authors/?q=ai:lazo.matheus-jatkoskeThe authors introduce a framework for the calculus of variations and the theory of optimal control for a class of generalized functions, called generalized smooth functions (GSF, for short), which extend Sobolev-Schwartz distributions and Colombeau generalized functions. In a certain sense the article is a follow up of \textit{A. Lecke} et al. [Adv. Nonlinear Anal. 8, 779--808 (2019; Zbl 1448.49008)]. Firstly, after introducing some basic concepts of nonstandard analysis, the authors make use of fundamental results on GSF; it is presented, in a detailed way, the calculus of variations approach, which follows the traditional path, that is, initially proving, in this general context, the Fundamental Lemma, Euler-Lagrange equations, D' Alembert principle, du Bois-Reymond optimality condition and Noether's theorem in Lagrangian formalism among other results. Secondly, it is handled the theory of optimal control, that includes a version of Pontryagin maximum principle, and Noether's theorem in Hamiltonian formalism in this GSF setting. Some examples and applications of the theory afore presented close the article providing a study of a singularly variable length pendulum, oscillations damped by two media and Pais-Uhlenbeck oscillator with singular frequencies.
Reviewer: Antonio Roberto da Silva (Rio de Janeiro)Dimensional lower bounds for contact surfaces of Cheeger setshttps://www.zbmath.org/1483.490252022-05-16T20:40:13.078697Z"Caroccia, M."https://www.zbmath.org/authors/?q=ai:caroccia.marco"Ciani, S."https://www.zbmath.org/authors/?q=ai:ciani.simoneThe Cheeger problem for a bounded open set \(\Omega \subset \mathbb R^n\) consists in determining the minimum of the functional
\[
\mathcal{F}(E)=\frac{P(E)}{\mathcal{L}^n(E)},
\]
among all sets \(E\subset \Omega\) of finite perimeter, where \(P(E)\) denotes the distributional perimeter of \(E\). \\
The contact surface of a Cheeger set \(E\subset\Omega\) is the set of points \(\partial E \cap\partial \Omega\), where the two boundaries intersect.
In this paper it is proved a lower bound on the Hausdorff dimension of the contact surface \(\partial E \cap\partial \Omega\).
The main result of the paper gives sufficient conditions to infer that the contact surface has positive \((n-1)\) dimensional Hausdorff measure. Finally, by explicit examples it is proved that such bounds are optimal in dimension two.
Reviewer: Luca Esposito (Fisciano)A note on a bilevel problem for parameter learning for inverse problems with the wave equationhttps://www.zbmath.org/1483.490262022-05-16T20:40:13.078697Z"Günther, Wiebke"https://www.zbmath.org/authors/?q=ai:gunther.wiebke"Kröner, Axel"https://www.zbmath.org/authors/?q=ai:kroner.axelSummary: In this paper we consider a bilevel problem for determining the optimal regularization parameter in an inverse problem with the linear wave equation transferring results from \textit{G. Holler} et al. [Inverse Probl. 34, No. 11, Article ID 115012, 28 p. (2018; Zbl 1400.49046)], where a general function space setting and applications to (bilinear) elliptic problems have been addressed. We analyze the well-posedness and derive the optimality conditions for the bilevel problem for the wave equation. Moreover, for given noisy data the numerical performance of the approach to find the regularization parameter is compared for different choices of priors in the Tikohonov regularization term of the lower level problem.Optimal control of quasilinear parabolic PDEs with state-constraintshttps://www.zbmath.org/1483.490272022-05-16T20:40:13.078697Z"Hoppe, Fabian"https://www.zbmath.org/authors/?q=ai:hoppe.fabian"Neitzel, Ira"https://www.zbmath.org/authors/?q=ai:neitzel.iraMultigrid preconditioners for optimal control problems with stochastic elliptic PDE constraintshttps://www.zbmath.org/1483.490282022-05-16T20:40:13.078697Z"Soane, Ana Maria"https://www.zbmath.org/authors/?q=ai:soane.ana-mariaSummary: In this work, we construct multigrid preconditioners to be used in the solution process of pathwise optimal control problems constrained by elliptic partial differential equations with random coefficients. We use a sparse-grid collocation method to discretize in the stochastic space and multigrid techniques in the physical space. Numerical results show that the proposed preconditioners lead to significant computational savings, with the number of preconditioned conjugate gradient iterations decreasing as the resolution in the physical space increases.Regularized classical optimality conditions in iterative form for convex optimization problems for distributed Volterra-type systemshttps://www.zbmath.org/1483.490292022-05-16T20:40:13.078697Z"Sumin, Vladimir Iosifovich"https://www.zbmath.org/authors/?q=ai:sumin.v-i"Sumin, Mikhail Iosifovich"https://www.zbmath.org/authors/?q=ai:sumin.mikhail-iosifovichSummary: We consider the regularization of the \textit{classical optimality conditions} (COCs) -- the Lagrange principle and the Pontryagin maximum principle -- in a convex optimal control problem with functional constraints of equality and inequality type. The system to be controlled is given by a general linear functional-operator equation of the second kind in the space \(L^m_2\), the main operator of the right-hand side of the equation is assumed to be quasinilpotent. The objective functional of the problem is strongly convex. Obtaining regularized COCs in iterative form is based on the use of the iterative dual regularization method. The main purpose of the regularized Lagrange principle and the Pontryagin maximum principle obtained in the work in iterative form is stable generation of minimizing approximate solutions in the sense of J. Warga. Regularized COCs in iterative form are formulated as existence theorems in the original problem of minimizing approximate solutions. They ``overcome'' the ill-posedness properties of the COCs and are regularizing algorithms for solving optimization problems. As an illustrative example, we consider an optimal control problem associated with a hyperbolic system of first-order differential equations.Time optimal control for linear systems on Lie groupshttps://www.zbmath.org/1483.490302022-05-16T20:40:13.078697Z"Ayala, Victor"https://www.zbmath.org/authors/?q=ai:ayala.victor"Jouan, Philippe"https://www.zbmath.org/authors/?q=ai:jouan.philippe"Torreblanca, Maria Luisa"https://www.zbmath.org/authors/?q=ai:torreblanca.maria-luisa"Zsigmond, Guilherme"https://www.zbmath.org/authors/?q=ai:zsigmond.guilhermeAuthors' abstract: This paper is devoted to the study of time optimal control of linear systems on Lie groups. General necessary conditions of the existence of time minimizers are stated when the controls are unbounded. The results are applied to systems on various Lie groups.
Reviewer: Costică Moroşanu (Iaşi)A new class of orthonormal basis functions: application for fractional optimal control problemshttps://www.zbmath.org/1483.490312022-05-16T20:40:13.078697Z"Heydari, M. H."https://www.zbmath.org/authors/?q=ai:heydari.mohammad-hossien"Razzaghi, M."https://www.zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: This study aims to generate a novel set of basis functions called the orthonormal piecewise Chelyshkov functions to solve a certain category of optimal control problems whose dynamical system is governed by a nonlinear fractional differential equation. A new fractional integral matrix associated with these basis functions is derived. This matrix significantly reduces the computations in solving such problems. The proposed approach transforms the original problem into a nonlinear programming one by expanding the control and state variables in terms of the orthonormal piecewise Chelyshkov functions and employing the derived fractional integral matrix. Some numerical problems are examined for verification of the proposed method.On some extension of optimal control theoryhttps://www.zbmath.org/1483.490322022-05-16T20:40:13.078697Z"Karamzin, Dmitry Yu."https://www.zbmath.org/authors/?q=ai:karamzin.dmitry-yu"de Oliveira, Valeriano A."https://www.zbmath.org/authors/?q=ai:de-oliveira.valeriano-antunes"Pereira, Fernando L."https://www.zbmath.org/authors/?q=ai:lobo-pereira.fernando"Silva, Geraldo N."https://www.zbmath.org/authors/?q=ai:silva.geraldo-nunesSummary: Some problems of Calculus of Variations do not have solutions in the class of classic continuous and smooth arcs. This suggests the need of a relaxation or extension of the problem ensuring the existence of a solution in some enlarged class of arcs. This work aims at the development of an extension for a more general optimal control problem with nonlinear control dynamics in which the control function takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control. The proposed extension links various approaches on the issue of extension found in the literature.Data-driven optimal control of switched linear autonomous systemshttps://www.zbmath.org/1483.490332022-05-16T20:40:13.078697Z"Zhang, Chi"https://www.zbmath.org/authors/?q=ai:zhang.chi"Gan, Minggang"https://www.zbmath.org/authors/?q=ai:gan.minggang"Zhao, Jingang"https://www.zbmath.org/authors/?q=ai:zhao.jingangSummary: In this paper, a novel data-driven optimal control approach of switching times is proposed for unknown continuous-time switched linear autonomous systems with a finite-horizon cost function and a prescribed switching sequence. No a priori knowledge on the system dynamics is required in this approach. First, some formulas based on the Taylor expansion are deduced to estimate the derivatives of a cost function with respect to the switching times using system state data. Then, a data-driven optimal control approach based on the gradient decent algorithm is designed, taking advantage of the derivatives to approximate the optimal switching times. Moreover, the estimation errors are analysed and proven to be bounded. Finally, simulation examples are illustrated to validate the effectiveness of the approach.A generalization of multiplier rules for infinite-dimensional optimization problemshttps://www.zbmath.org/1483.490342022-05-16T20:40:13.078697Z"Yilmaz, Hasan"https://www.zbmath.org/authors/?q=ai:yilmaz.hasanThe main theorems give Fritz John type and KKT type necessary optimality conditions for optimization problems on normed spaces with finitely many constraints. In the case of inequality constrained problems, the objective functions and the constraint functions are assumed to be Gâteaux differentiable at the optimal point \(\widehat{x}\); whereas in the case when there are both inequality and equality constraints, the objective function and the equality constraint functions (the inequality constraint functions) are assumed to be Hadamard differentiable (resp., Gâteaux differentiable) at \(\widehat{x}\). In both cases, some extra mild assumptions are imposed.
Reviewer: Juan-Enrique Martínez-Legaz (Barcelona)Sensitivity analysis of optimal control problems governed by nonlinear Hilfer fractional evolution inclusionshttps://www.zbmath.org/1483.490352022-05-16T20:40:13.078697Z"Jiang, Yirong"https://www.zbmath.org/authors/?q=ai:jiang.yirong"Zhang, Qiongfen"https://www.zbmath.org/authors/?q=ai:zhang.qiongfen"Chen, An"https://www.zbmath.org/authors/?q=ai:chen.an"Wei, Zhouchao"https://www.zbmath.org/authors/?q=ai:wei.zhouchaoThe authors consider a control problem for a nonlinear evolution inclusion with a Hilfer fractional derivative in the time variable. This includes the Riemann-Liouville and the Caputo fractional derivatives. The initial condition is given by a Riemann-Liouville integral. The main results in the paper refer to the stability of the problem with respect to the initial conditions and parameters. For this purpose it is also studied the existence and compactness properties of the mild solutions corresponding to the differential inclusion.
Reviewer: Juan Casado-Díaz (Sevilla)The solution of fuzzy variational problem and fuzzy optimal control problem under granular differentiability concepthttps://www.zbmath.org/1483.490362022-05-16T20:40:13.078697Z"Mustafa, Altyeb Mohammed"https://www.zbmath.org/authors/?q=ai:mustafa.altyeb-mohammed"Gong, Zengtai"https://www.zbmath.org/authors/?q=ai:gong.zengtai"Osman, Mawia"https://www.zbmath.org/authors/?q=ai:osman.mawiaSummary: In this paper, the fuzzy variational problem and fuzzy optimal control problem are considered. The granular Euler-Lagrange condition for the fuzzy variational problem and necessary conditions of Pontryagin-type for fixed and free final state fuzzy optimal control problem are derived based on the concepts of horizontal membership function (HMF) and granular differentiability with the calculus of variations. Further, based on the proposed solution method, the solutions of fuzzy optimal control problem, i.e., optimal fuzzy control, and corresponding optimal fuzzy state are always fuzzy functions. Finally, the proposed algorithm is used to summarize the main steps of solving the fuzzy variational problem and fuzzy optimal control problem numerically using He's variational iteration method (VIM).An improved iterative computational approach to the solution of the Hamilton-Jacobi equation in optimal control problems of affine nonlinear systems with applicationhttps://www.zbmath.org/1483.490372022-05-16T20:40:13.078697Z"Aliyu, M. D. S."https://www.zbmath.org/authors/?q=ai:aliyu.m-d-sSummary: In this paper, we improve an earlier iterative successive approximation method for solving the Hamilton-Jacobi equation (HJE) arising in deterministic optimal control of affine nonlinear systems. The new methods generate smooth approximate solutions for systems with polynomial nonlinearities, compared with the former method that generates rational functions with possible singularities in the domain. We prove quadratic convergence of the methods and demonstrate their effectiveness with some examples. Application to factorisation of nonlinear systems is also discussed.Finite-horizon optimal control for continuous-time uncertain nonlinear systems using reinforcement learninghttps://www.zbmath.org/1483.490382022-05-16T20:40:13.078697Z"Zhao, Jingang"https://www.zbmath.org/authors/?q=ai:zhao.jingang"Gan, Minggang"https://www.zbmath.org/authors/?q=ai:gan.minggangSummary: This paper investigates finite-horizon optimal control problem of continuous-time uncertain nonlinear systems. The uncertainty here refers to partially unknown system dynamics. Unlike the infinite-horizon, the difficulty of finite-horizon optimal control problem is that the Hamilton-Jacobi-Bellman (HJB) equation is time-varying and must meet certain terminal boundary constraints, which brings greater challenges. At the same time, the partially unknown system dynamics have also caused additional difficulties. The main innovation of this paper is the proposed cyclic fixed-finite-horizon-based reinforcement learning algorithm to approximately solve the time-varying HJB equation. The proposed algorithm mainly consists of two phases: the data collection phase over a fixed-finite-horizon and the parameters update phase. A least-squares method is used to correlate the two phases to obtain the optimal parameters by cyclic. Finally, simulation results are given to verify the effectiveness of the proposed cyclic fixed-finite-horizon-based reinforcement learning algorithm.A novel non-uniform optimal control approach for hypersonic cruise vehicle with waypoint and no-fly zone constraintshttps://www.zbmath.org/1483.490392022-05-16T20:40:13.078697Z"Lv, Lu"https://www.zbmath.org/authors/?q=ai:lv.lu"Liu, Xinggao"https://www.zbmath.org/authors/?q=ai:liu.xinggao"Xiao, Long"https://www.zbmath.org/authors/?q=ai:xiao.long"Ma, Weihua"https://www.zbmath.org/authors/?q=ai:ma.weihua"Qi, Zhenqiang"https://www.zbmath.org/authors/?q=ai:qi.zhenqiang"Ye, Song"https://www.zbmath.org/authors/?q=ai:ye.song"Xu, Guoqiang"https://www.zbmath.org/authors/?q=ai:xu.guoqiang"Zheng, Zongzhun"https://www.zbmath.org/authors/?q=ai:zheng.zongzhun"Wang, Sen"https://www.zbmath.org/authors/?q=ai:wang.sen"Zhang, Zeyin"https://www.zbmath.org/authors/?q=ai:zhang.zeyinSummary: Dynamic optimisation for hypersonic cruise vehicle (HCV) is a challenge for the complex nonlinear constraints with waypoint and no-fly zone during a global strike mission. Because the passing time of each waypoint is unknown, it is difficult to guarantee the precision of waypoint constraints on non-collocation points for traditional pseudospectral method. A novel optimal control approach is therefore proposed, where waypoints are firstly transformed into variable optimisation parameters of pseudospectral method, which can satisfy the waypoint constraints completely. Then, a novel non-uniform pseudospectral approach with mesh refinement is further proposed by introducing the adaptive gradient analysis to provide a time grid iteratively. With this refinement, the proposed approach needs fewer computational time to achieve better performance of accuracy compared with traditional uniform refinement pseudospectral method. The proposed approaches are finally applied to a classic HCV problem with waypoints and no-fly zone constraints and compared with other methods reported in literature in detail. The research results validate both the effectiveness and the time saving benefit of the proposed methods.Monotone systems involving variable-order nonlocal operatorshttps://www.zbmath.org/1483.490402022-05-16T20:40:13.078697Z"Yangari, Miguel"https://www.zbmath.org/authors/?q=ai:yangari.miguelIn this paper, the author investigates the existence (via Perron's method) and uniqueness of bounded viscosity solutions for some monotone systems of parabolic Hamilton-Jacobi type. Here, the diffusion term is determined by variable-order nonlocal operators whose kernels depend on the space-time variable.
Reviewer: Savin Treanta (Bucureşti)Optimal control of the Keilson-Storer master equation in a Monte Carlo frameworkhttps://www.zbmath.org/1483.490412022-05-16T20:40:13.078697Z"Bartsch, Jan"https://www.zbmath.org/authors/?q=ai:bartsch.jan"Nastasi, Giovanni"https://www.zbmath.org/authors/?q=ai:nastasi.giovanni"Borzì, Alfio"https://www.zbmath.org/authors/?q=ai:borzi.alfioThe paper considers the formulation of an optimal control problem governed by the Keilson-Storer master equation, and discusses the optimality system for the problem using the Lagrange framework. The implementation of the Monte Carlo scheme for the Keilson-Storer problem is illustrated. The Keilson-Storer optimal control framework is validated by some numerical experiments.
Reviewer: Hang Lau (Montréal)Total variation diminishing schemes in optimal control of scalar conservation lawshttps://www.zbmath.org/1483.490422022-05-16T20:40:13.078697Z"Hajian, Soheil"https://www.zbmath.org/authors/?q=ai:hajian.soheil"Hintermüller, Michael"https://www.zbmath.org/authors/?q=ai:hintermuller.michael"Ulbrich, Stefan"https://www.zbmath.org/authors/?q=ai:ulbrich.stefanSummary: Optimal control problems subject to a nonlinear scalar conservation law, the state system, are studied. Such problems are challenging at both the continuous level and the discrete level since the control-to-state operator poses difficulties such as nondifferentiability. Therefore, discretization of the control problem has to be designed with care to provide stability and convergence. Here, the discretize-then-optimize approach is pursued, and the state is removed by the solution of the underlying state system, thus providing a reduced control problem. An adjoint calculus is then applied for computing the reduced gradient in a gradient-related descent scheme for solving the optimization problem. The time discretization of the underlying state system relies on total variation diminishing Runge-Kutta (TVD-RK) schemes, which guarantee stability, best possible order and convergence of the discrete adjoint to its continuous counterpart. While interesting in its own right, it also influences the quality and accuracy of the numerical reduced gradient and, thus, the accuracy of the numerical solution. In view of these demands, it is proven that providing a state scheme that is a strongly stable TVD-RK method is enough to ensure stability of the discrete adjoint state. It is also shown that requiring strong stability for both the discrete state and adjoint is too strong, confining one to a first-order method, regardless of the number of stages employed in the TVD-RK scheme. Given such a discretization, we further study order conditions for the discrete adjoint such that the numerical approximation is of the best possible order. Also, convergence of the discrete adjoint state towards its continuous counterpart is studied. In particular, it is shown that such a convergence result hinges on a regularity assumption at final time for a tracking-type objective in the control problem. Finally, numerical experiments validate our theoretical findings.High-order fully actuated system approaches. VIII: Optimal control with application in spacecraft attitude stabilisationhttps://www.zbmath.org/1483.490432022-05-16T20:40:13.078697Z"Duan, Guangren"https://www.zbmath.org/authors/?q=ai:duan.guangrenSummary: In this paper, the optimal control problem for dynamical systems represented by general high-order fully actuated (HOFA) models is formulated. The problem aims to minimise an objective in the quadratic form of the states and their derivatives of certain orders. The designed controller is a combination of the linearising nonlinear controller and an optimal quadratic controller for a converted linear system. In the infinite-time output regulation case, the solution is in essence a nonlinear state feedback dependent on a well-known Riccati algebraic equation. In the sub-fully actuated system case, the feasibility of the controller is investigated and guaranteed by properly characterising a ball restriction area of the system initial values. Application of the optimal control technique for sub-fully actuated systems to a spacecraft attitude control provides very smooth and steady responses and well demonstrates the effect and simplicity of the proposed approach.
For Part I--VII, see
[the auhor, ibid. 52, No. 2, 422--435 (2021; Zbl 1480.93184);
ibid. 52, No. 3, 437--454 (2021; Zbl 1480.93138);
ibid. 52, No. 5, 952--971 (2021; Zbl 1480.93096);
ibid. 52, No. 5, 972--989 (2021; Zbl 1480.93223);
ibid. 52, No. 10, 2129--2143 (2021; Zbl 1480.93097);
ibid. 52, No. 10, 2161--2181 (2021; Zbl 1480.93282);
ibid. 52, No. 14, 3091--3114 (2021; Zbl 1480.93038)].Optimal control and duality-based observer design for a hyperbolic PDEs system with application to fixed-bed reactorhttps://www.zbmath.org/1483.490442022-05-16T20:40:13.078697Z"Aksikas, Ilyasse"https://www.zbmath.org/authors/?q=ai:aksikas.ilyasseSummary: This paper is devoted to the design of an optimal infinite-dimensional Luenberger observer combined with a linear-quadratic state feedback controller for a system of hyperbolic PDEs. The design is based on the duality fact between the control design and the observer design. Both the original linear-quadratic and dual control problems have been solved by using the associated Riccati equations. A general algorithm that combines the designed observer together with the (estimated) state-feedback controller has been developed. The theoretical development has been applied to a fixed-bed reactor to validate the performances of the designed observer-controller via numerical simulation. Estimation and control of the temperature and the reactant concentration in a fixed-bed reactor is investigated by using the developed algorithm, which lead to express the jacket temperature (manipulated variable) as a feedback of the estimated temperature and concentration in the reactor.The piecewise parametric optimal control of uncertain linear quadratic modelshttps://www.zbmath.org/1483.490452022-05-16T20:40:13.078697Z"Li, Bo"https://www.zbmath.org/authors/?q=ai:li.bo.1"Zhu, Yuanguo"https://www.zbmath.org/authors/?q=ai:zhu.yuanguoSummary: The optimal control of linear quadratic model is given in a feedback form and determined by the solution of a Riccati equation. However, the control-related Riccati equation usually cannot be solved analytically such that the form of optimal control will become more complex. In this paper, we consider a piecewise parametric optimal control problem of uncertain linear quadratic model for simplifying the form of optimal control. By introducing a piecewise control parameter, a piecewise parametric optimal control model is established. Then we present a parametric optimisation method for solving the optimal piecewise control parameter. Finally, an uncertain inventory-promotion optimal control problem is discussed and a comparison is made to show the effectiveness of proposed piecewise parametric optimal control model.On guarantee optimization in control problem with finite set of disturbanceshttps://www.zbmath.org/1483.490462022-05-16T20:40:13.078697Z"Gomoyunov, Mikhail Igorevich"https://www.zbmath.org/authors/?q=ai:gomoyunov.mikhail-igorevich"Serkov, Dmitriĭ Aleksandrovich"https://www.zbmath.org/authors/?q=ai:serkov.dmitrij-aSummary: In this paper, we deal with a control problem under conditions of disturbances, which is stated as a problem of optimization of the guaranteed result. Compared to the classical formulation of such problems, we assume that the set of admissible disturbances is finite and consists of piecewise continuous functions. In connection with this additional functional constraint on the disturbance, we introduce an appropriate class of non-anticipative control strategies and consider the corresponding value of the optimal guaranteed result. Under a technical assumption concerning a property of distinguishability of the admissible disturbances, we prove that this result can be achieved by using control strategies with full memory. As a consequence, we establish unimprovability of the class of full-memory strategies. A key element of the proof is a procedure of recovering the disturbance acting in the system, which allows us to associate every non-anticipative strategy with a full-memory strategy providing a close guaranteed result. The paper concludes with an illustrative example.Finite element approach to linear parabolic pointwise control problems of incomplete datahttps://www.zbmath.org/1483.490472022-05-16T20:40:13.078697Z"Mahoui, Sihem"https://www.zbmath.org/authors/?q=ai:mahoui.sihem"Moulay, Mohamed Said"https://www.zbmath.org/authors/?q=ai:moulay.mohamed-said"Omrane, Abdennebi"https://www.zbmath.org/authors/?q=ai:omrane.abdennebiSummary: In this paper we give a priori error estimates for finite element approximations of linear parabolic problems with pointwise control and incomplete data. We discretise the optimal control problem by using piecewise linear and continuous finite elements for the space discretisation of the state, and we use the backward Euler scheme for time discretisation. We prove a priori error estimates for the state, the adjoint-state as well as for the low-regret pointwise optimal control.Optimal damping stabilisation based on LQR synthesishttps://www.zbmath.org/1483.490482022-05-16T20:40:13.078697Z"Veremey, Evgeny I."https://www.zbmath.org/authors/?q=ai:veremey.evgeny-iSummary: Significant attention is currently being paid to the synthesis of stabilising controllers for nonlinear and non-autonomous plants. We aimed to present a new method for nonlinear time-dependent control law design based on the application of Zubov's optimal damping concept. This theory is used to reduce significant computational costs in solving optimal stabilisation problems. The main contribution is the proposition of a new methodology for selecting the functional to be damped. The central idea is to perform parameterisation of a set of admissible items for the mentioned functional. As a particular case, a new method of this parameterisation has been developed, which can be used for constructing an approximate solution of the classical optimisation problem. The emphasis is on the specific choice of the functional to be damped using LQR control to provide the desirable stability and performance features of the closed-loop connection. The applicability and effectiveness of the proposed approach are confirmed using a practical numerical example of the convey-crane control.Gradient estimates of \(\omega\)-minimizers to double phase variational problems with variable exponentshttps://www.zbmath.org/1483.490492022-05-16T20:40:13.078697Z"Byun, Sun-Sig"https://www.zbmath.org/authors/?q=ai:byun.sun-sig"Lee, Ho-Sik"https://www.zbmath.org/authors/?q=ai:lee.ho-sikThis paper concerns the integral functionals involving non-uniformly elliptic operators of the type
\[
\mathcal{F}(u,\Omega):=\int_{\Omega}\Bigl( f_1(x,Du) +a(x)f_2(x,Du)\Bigr)\ dx
\]
whose model case is \(f_1(x,z)=|z|^{p(x)}\), \(f_2(x,z)=|z|^{q(x)}\). Regarding the model case, the functions \(p(x),q(x),a(x)\) are assumed to be continuous and satisfying,
\[
0\leq a(x)\in C^{0,\alpha},\;\;\;1<\gamma_1\leq p(x)\leq q(x)\leq \gamma_2<\infty, \;\;\frac{q(x)}{p(x)}\leq 1+\frac{\alpha}{n}
\]
for some constant \(\alpha \in (0,1]\) and \(\gamma_1,\gamma_2 \in \mathbb R^n\).
\\
The authors establish a local Calderón-Zygmund theory for \(\omega\)-minimizers of functional \(\mathcal{F}\).
Reviewer: Luca Esposito (Fisciano)Lipschitz bounds and nonautonomous integralshttps://www.zbmath.org/1483.490502022-05-16T20:40:13.078697Z"De Filippis, Cristiana"https://www.zbmath.org/authors/?q=ai:de-filippis.cristiana"Mingione, Giuseppe"https://www.zbmath.org/authors/?q=ai:mingione.giuseppeAuthors' abstract: We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range from those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and also yield new, optimal regularity criteria in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.
Reviewer: Mohammed El Aïdi (Bogotá)On properties of one functional used in software constructions for solving differential gameshttps://www.zbmath.org/1483.490512022-05-16T20:40:13.078697Z"Chentsov, Aleksandr Georgievich"https://www.zbmath.org/authors/?q=ai:chentsov.alexander-gSummary: Nonlinear differential game (DG) is investigated; relaxations of the game problem of guidance are investigated also. The variant of the program iterations method realized in the space of position functions and delivering in limit the value function of the minimax-maximin DG for special functionals of a trajectory is considered. For every game position, this limit function realizes the least size of the target set neighborhood for which, under proportional weakening of phase constraints, the player interested in a guidance yet guarantees its realization. Properties of above-mentioned functionals and limit function are investigated. In particular, sufficient conditions for realization of values of given function under fulfilment of finite iteration number are obtained.Mean-field limit for a class of stochastic ergodic control problemshttps://www.zbmath.org/1483.490522022-05-16T20:40:13.078697Z"Albeverio, Sergio"https://www.zbmath.org/authors/?q=ai:albeverio.sergio-a"De Vecchi, Francesco C."https://www.zbmath.org/authors/?q=ai:de-vecchi.francesco-carlo"Romano, Andrea"https://www.zbmath.org/authors/?q=ai:romano.andrea"Ugolini, Stefania"https://www.zbmath.org/authors/?q=ai:ugolini.stefaniaSharp quantitative stability for isoperimetric inequalities with homogeneous weightshttps://www.zbmath.org/1483.490532022-05-16T20:40:13.078697Z"Cinti, E."https://www.zbmath.org/authors/?q=ai:cinti.eleonora"Glaudo, F."https://www.zbmath.org/authors/?q=ai:glaudo.federico"Pratelli, A."https://www.zbmath.org/authors/?q=ai:pratelli.aldo"Ros-Oton, X."https://www.zbmath.org/authors/?q=ai:ros-oton.xavier"Serra, J."https://www.zbmath.org/authors/?q=ai:serra.joaquimSummary: We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights.
Inspired by the proof of such isoperimetric inequalities through the ABP method (see [\textit{X. Cabré} et al., J. Eur. Math. Soc. (JEMS) 18, No. 12, 2971--2998 (2016; Zbl 1357.28007)]), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set \(E\) and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of \textit{A. Figalli} et al. [Invent. Math. 182, No. 1, 167--211 (2010; Zbl 1196.49033)] and prove that if \(E\) is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations.
As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.A sufficient criterion to determine planar self-Cheeger setshttps://www.zbmath.org/1483.490542022-05-16T20:40:13.078697Z"Saracco, Giorgio"https://www.zbmath.org/authors/?q=ai:saracco.giorgioLet \(\Omega\) be a plane Jordan set. The author is interested in the minimizers of the following functional \(\mathcal{F}_\kappa(E)=P(E)-\kappa |E|\) among subsets \(E\subset \Omega\) and, more particularly, in cases where \(\Omega\) is itself a minimizer. A possible motivation is to give simple conditions ensuring that \(\Omega\) is a self-Cheeger set, meaning that it minimizes the ratio \(P(E)/|E|\) among its subsets.
First of all, let us recall that a set \(\Omega\) satisfies the interior rolling disk property of radius \(R\) if its complement has a reach greater or equal to \(R\). Moreover, it satisfies the strict interior rolling disk property of radius \(R\) if any such ball of radius \(R\) does not contain two antipodal points of the boundary. of \(\Omega\).
The main result of the paper is the following: If \(\Omega\) satisfies the interior rolling disk property of radius \(R\) with \(R\leq P(\Omega)/|\Omega|\) then it is a minimizer of \(\mathcal{F}_\kappa\) with \(\kappa=R^{-1}\). Moreover, if it satisfies the strict interior rolling disk property, it is the unique minimizer.
A consequence of this result: If \(\Omega\) satisfies the interior rolling disk property of radius \(R\) with \(R = P(\Omega)/|\Omega|\), then \(\Omega\) is self-Cheeger (and the Cheeger constant of \(\Omega\) is \(R^{-1}\)). Moreover, if it satisfies the strict property it is the unique Cheeger set.
Reviewer: Antoine Henrot (Vandœuvre-lès-Nancy)The Orlicz-Minkowski problem for polytopeshttps://www.zbmath.org/1483.520052022-05-16T20:40:13.078697Z"Jiang, Meiyue"https://www.zbmath.org/authors/?q=ai:jiang.meiyue"Wang, Chu"https://www.zbmath.org/authors/?q=ai:wang.chuThe classical Minkowski problem and various modern versions of it, known as Minkowski-type problems, are crucial in the study of convex bodies throughout the 20th and 21st century and even more so in the last few decades. Some of the most influential Minkowski-type problems include the \(L_p\) Minkowski problem (which contains the well-known logarithmic Minkowski problem) and the recently posed dual Minkowski problems. These problems, with enough regularity assumed, reduce to the study of a family of Monge-Ampere type equations. The uniqueness of these solutions are often tied to isoperimetric inequalities and spetrum analysis of certain second order nonlinear elliptic operators.
The current paper establishes some existence result for the Orlicz Minkowski problem. It uses a variational approach and studies particularly the discrete case.
Reviewer: Yiming Zhao (Syracuse)First explicit constrained Willmore minimizers of non-rectangular conformal classhttps://www.zbmath.org/1483.530122022-05-16T20:40:13.078697Z"Heller, Lynn"https://www.zbmath.org/authors/?q=ai:heller.lynn"Ndiaye, Cheikh Birahim"https://www.zbmath.org/authors/?q=ai:ndiaye.cheikh-birahimIn this interesting paper the authors study immersed tori in 3-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes \((0, b)\) with \(b \sim 1\) the homogeneous tori \(f^ b\) are known to be the unique constrained Willmore minimizers (up to invariance). The authors generalize this result and show that the candidates constructed in [\textit{L. Heller} and \textit{F. Pedit}, in: Willmore energy and Willmore conjecture. Boca Raton, FL: CRC Press. 119--138 (2018; Zbl 1384.53007)] are indeed constrained Willmore minimizers in certain non-rectangular conformal classes \((a, b)\). It is shown that the minimal Willmore energy \(\omega(a, b)\) is real analytic and concave in \(a \in (0, a^b)\) for some \(a^b > 0\) and fixed \(b \sim 1\), \(b \ne 1\).
Reviewer: Andreas Arvanitoyeorgos (Patras)The isoperimetric problem in Carnot-Carathéodory spaceshttps://www.zbmath.org/1483.530532022-05-16T20:40:13.078697Z"Franceschi, Valentina"https://www.zbmath.org/authors/?q=ai:franceschi.valentinaPansu's conjectured isoperimetric profile of the Heisenberg group \(({\mathbb H}^1,g_{\mathrm{cc}})\), where \(g_{\mathrm{cc}}\) denotes the standard Carnot-Caratheodory metric, remains one of the signature open problems in sub-Riemannian geometric analysis. In this paper, the author provides an overview of several related results obtained in two papers with R. Monti and F. Montefalcone.
In [the author and \textit{R. Monti}, Rev. Mat. Iberoam. 32, No. 4, 1227--1258 (2016; Zbl 1368.49051)], the authors solve the corresponding isoperimetric problem in a class of sub-Riemannian spaces of Grushin type, with applications to the classication of \(x\)-symmetric isoperimetric minimizers in \(H\)-type Carnot groups. Their result generalizes prior work by [\textit{R. Monti} and \textit{D. Morbidelli}, J. Geom. Anal. 14, No. 2, 355--368 (2004; Zbl 1076.53035)].
In [\textit{V. Franceschi} et al., Anal. Geom. Metr. Spaces 7, 109--129 (2019; Zbl 1428.53073)], the authors consider the isoperimetric problem for rotationally symmetric surfaces in a family of Riemannian Heisenberg groups. Building on prior work of \textit{P. Tomter} [Proc. Symp. Pure Math. 54, 485--495 (1993; Zbl 0799.53073)], they considered a two-parameter family of left-invariant Riemannian metrics, denoted \(g_{\epsilon,\sigma}\), on the first Heisenberg group. Here the parameter \(\sigma \ne 0\) governs the degree of non-commutativity of the horizontal vector fields, while \(\epsilon>0\) is a penalty parameter restricting non-horizontal motion. Fixing \(\sigma\), respectively \(\epsilon\), the resulting Riemannian 3-manifold structure converges to the Euclidean space \({\mathbb R}^3\), respectively, to the sub-Riemannian Heisenberg group \({\mathbb H}^1\). Moreover, the Riemannian volume and perimeter functionals, suitably normalized, converge to their respective analogues in these two spaces. It follows that isoperimetric solutions in \(({\mathbb R}^3,g_{\epsilon,\sigma})\) converge to their analogues in the two limit spaces. In particular, under the degenerating limiting operation \(({\mathbb R}^3,g_{\epsilon,\sigma}) \stackrel{\epsilon \to 0}{\longrightarrow} ({\mathbb H}^1,g_{\mathrm{cc}})\), the Riemannian isoperimetric solutions converge to sub-Riemannian ones. The authors consider the isoperimetric problem in \(({\mathbb R}^3,g_{\epsilon,\sigma})\) under a topological assumption on the extremals. They identify isoperimetric minimizers which are topological balls, and recover Pansu's conjectured sub-Riemannian extremals in the limit as \(\epsilon \to 0\).
For the entire collection see [Zbl 1411.35007].
Reviewer: Jeremy Tyson (Urbana)A differential perspective on gradient flows on \(\mathbf{\mathsf{CAT}} (\kappa)\)-spaces and applicationshttps://www.zbmath.org/1483.530612022-05-16T20:40:13.078697Z"Gigli, Nicola"https://www.zbmath.org/authors/?q=ai:gigli.nicola"Nobili, Francesco"https://www.zbmath.org/authors/?q=ai:nobili.francescoAuthors' abstract: We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on \(\mathsf{CAT} (\kappa)\)-spaces and prove that they can be characterized by the same differential inclusion \(y_t^{\prime}\in -\partial^- \mathsf{E} (y_t)\) one uses in the smooth setting and more precisely that \(y_t^{\prime}\) selects the element of minimal norm in \(-\partial^- \mathsf{E} (y_t)\). This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar-Schoen energy functional on the space of \(L^2\) and \(\mathsf{CAT}(0)\) valued maps: we define the Laplacian of such \(L^2\) map as the element of minimal norm in \(-\partial^- \mathsf{E}(u)\), provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is \(L^2\)-dense. Basic properties of this Laplacian are then studied.
Reviewer: Mohammed El Aïdi (Bogotá)Isoperimetric inequalities in metric measure spaces [after F. Cavalletti \& A. Mondino]https://www.zbmath.org/1483.530672022-05-16T20:40:13.078697Z"Villani, Cédric"https://www.zbmath.org/authors/?q=ai:villani.cedricSummary: The synthetic theory of Ricci curvature in metric measure spaces obtained its first successes a decade ago, and grew rapidly since then; but it was stumbling on questions which were as fundamental as they appeared tricky, such as the Lévy-Gromov inequality or other geometric inequalities where the effective dimension and the optimal constants are crucial. The recent works of Cavalletti and Mondino, adapting the localization techniques of Klartag, allow to bypass these obstacles, and in particular prove the first nonsmooth version of the Lévy-Gromov inequality.
For the entire collection see [Zbl 1416.00029].New developments on the p-Willmore energy of surfaceshttps://www.zbmath.org/1483.530792022-05-16T20:40:13.078697Z"Aulisa, Eugenio"https://www.zbmath.org/authors/?q=ai:aulisa.eugenio"Gruber, Anthony"https://www.zbmath.org/authors/?q=ai:gruber.anthony"Toda, Magdalena"https://www.zbmath.org/authors/?q=ai:toda.magdalena"Tran, Hung"https://www.zbmath.org/authors/?q=ai:tran.hung-tuan|tran-hung.|tran.hung-thanh|tran-cong-hung.|tran.hung-vinhThe Willmore energy is an important conformal invariant of immersed surfaces \(\Sigma\) in the Euclidean 3-space. The article surveys some recent results [the second author et al., Ann. Global Anal. Geom. 56, No. 1, 147--165 (2019; Zbl 1417.58009)] on the \(p\)-Willmore energy \(W^p(\Sigma) = \int_\Sigma H^p dS\) (i.e., different powers of the mean curvature \(H\) of \(\Sigma\) in the integrand) for nonnegative integer \(p\) of surfaces (with boundary) in 3-dimensional space forms. Section 2 gives the first and second variations of \(W^p\), and presented a flux formula, which reveals (in Section 3) a connection between its critical points and the minimal surfaces. Section 4 reformulates the \(p\)-Willmore flow problem and presents (and visualizes through computer implementation) a model for the \(p\)-Willmore flow of graphs.
For the entire collection see [Zbl 1445.53003].
Reviewer: Vladimir Yu. Rovenskij (Nesher)On the KKM theory on ordered spaceshttps://www.zbmath.org/1483.540322022-05-16T20:40:13.078697Z"Park, Sehie"https://www.zbmath.org/authors/?q=ai:park.sehieSummary: Since \textit{C. D. Horvath} and \textit{J. V. Llinares Cisar} [J. Math. Econ. 25, No. 3, 291--306 (1996; Zbl 0852.90006)] began to study maximal elements and fixed points for binary relations on topological ordered spaces, there have appeared many works related to the KKM theory on such spaces by several authors. Independently to these works, we began to study the KKM theory on abstract convex spaces [the author, Nonlinear Anal. Forum 11, No. 1, 67--77 (2006; Zbl 1120.47038)]. Our aim in the present paper is to extend the known results on topological ordered spaces to the corresponding ones on abstract convex spaces.Marginal and dependence uncertainty: bounds, optimal transport, and sharpnesshttps://www.zbmath.org/1483.600312022-05-16T20:40:13.078697Z"Bartl, Daniel"https://www.zbmath.org/authors/?q=ai:bartl.daniel"Kupper, Michael"https://www.zbmath.org/authors/?q=ai:kupper.michael"Lux, Thibaut"https://www.zbmath.org/authors/?q=ai:lux.thibaut"Papapantoleon, Antonis"https://www.zbmath.org/authors/?q=ai:papapantoleon.antonis"Eckstein, Stephan"https://www.zbmath.org/authors/?q=ai:eckstein.stephanLow correlation noise stability of symmetric setshttps://www.zbmath.org/1483.600332022-05-16T20:40:13.078697Z"Heilman, Steven"https://www.zbmath.org/authors/?q=ai:heilman.steven-mGaussian noise stability is a well-studied topic with connections to geometry of minimal surfaces [\textit{T. H. Colding} and \textit{W. P. Minicozzi II}, in: Surveys in geometric analysis and relativity. Dedicated to Richard Schoen in honor of his 60th birthday. Somerville, MA: International Press; Beijing: Higher Education Press. 73--143 (2011; Zbl 1261.53006)], hypercontractivity and invariance principles [\textit{E. Mossel} et al., Ann. Math. (2) 171, No. 1, 295--341 (2010; Zbl 1201.60031)], isoperimetric inequalities [\textit{D. M. Kane}, Comput. Complexity 23, No. 2, 151--175 (2014; Zbl 1314.68138)], sharp unique games hardness results in theoretical computer science [\textit{S. Khot} et al., SIAM J. Comput. 37, No. 1, 319--357 (2007; Zbl 1135.68019)], social choice theory, learning theory [\textit{A. R. Klivans} et al. ``Learning geometric concepts via Gaussian surface area'', in: IEEE 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS'08. Los Alamitos, CA: IEEE Computer Society. 541--550 (2008; \url{doi:10.1109/FOCS.2008.64})] and communication complexity [\textit{A. Chakrabarti} and \textit{O. Regev}, in: Proceedings of the 43rd annual ACM symposium on theory of computing, STOC '11. San Jose, CA, USA, June 6--8, 2011. New York, NY: Association for Computing Machinery (ACM). 51--60 (2011; Zbl 1288.90005)]. The author studies the Gaussian noise stability of subsets \(A\) of Euclidean space satisfying \(A =-A.\) It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. On the other hand, in dimension two and higher, the ball or its complement does not always maximize noise stability among symmetric sets of fixed Gaussian measure.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Least gradient functions in metric random walk spaceshttps://www.zbmath.org/1483.600682022-05-16T20:40:13.078697Z"Górny, Wojciech"https://www.zbmath.org/authors/?q=ai:gorny.wojciech"Mazón, José M."https://www.zbmath.org/authors/?q=ai:mazon-ruiz.jose-mThe authors study least gradient functions in the general setting of metric random walk spaces. A metric random walk space \([X,d,m]\) is a Polish metric space \((X,d)\) with a random walk \(m\), i.e., a family of probability measures \((m_x)_{x\in X}\) on the Borel \(\sigma\)-algebra of \(X\) describing the distribution of jumps from \(x\). Assuming the existence and uniqueness of an invariant and reversible measure \(\nu\) with \(\nu(X)<\infty\) and validity of the following \emph{\(p\)-Poincaré Inequality} in some \(\Omega\) with \(0<\nu(\Omega)<\nu(X)\) for all \(p\geq1\):
\begin{align*}
&\text{ there exists }\,\lambda>0\,:\quad \text{ for all }\,\, u\in L^p(\Omega,\nu),\quad\text{ for all }\,\,\psi\in L^p(\partial_m\Omega),\\
&\lambda\int_{\Omega}|u(x)|^pd\nu(x)\leq \int_\Omega\int_{\Omega_m}|u_\psi(y)-u(x)|^pdm_x(y)d\nu(x)+\int_{\partial_m\Omega}|\psi(x)|^pd\nu(y),
\end{align*}
where \(\partial_m\Omega:=\{x\in X\,:\,m_x(\Omega)>0 \}\) is \emph{\(m\)-boundary} of \(\Omega\), \(\Omega_m:=\Omega\cup\partial_m\Omega\), \(u_\psi(x):=u(x)\) for \(x\in\Omega\) and \(u_\psi(x):=\psi(x)\) for \(x\in\partial_m\Omega\), the authors prove the following theorem:
Consider the space of functions of \(m\)-bounded variation \[BV_m(X):=\left\{f\,:\,X\to\mathbb{R}\,\,\nu\text{-measurable}\,\,:\,\,\int_X\int_X|f(y)-f(x)|dm_x(y)d\nu(x)<\infty \right\}.\] Let \(\psi\in L^\infty(X\setminus\Omega)\) and \(u\in BV_m(X)\) such that \(u=\psi\) \(\,\nu\)-a.s. on \(X\setminus\Omega\). Then the following statements are equivalent:
\begin{itemize}
\item[(i)] \(u\big|_\Omega\) is a minimizer of the energy functional \(\mathcal{J}_\psi\), where
\begin{align*}
\mathcal{J}_\psi(u):=TV_m(u_\psi):=\frac12\int_{\Omega_m}\int_{\Omega_m}|u_\psi(y)-u_\psi(x)|dm_x(y)d\nu(x);
\end{align*}
\item[(ii)] \(u\big|_\Omega\) is a solution to the nonlocal \(1\)-Laplace problem with Dirichlet boundary condition \(\psi\)
\begin{align*}
\left\{ \begin{array}{ll} -\Delta_1^m u(x)=0, & x\in\Omega,\\
u(x)=\psi(x), & x\in\partial_m\Omega, \end{array} \right.
\end{align*}
where \(\Delta_1^m u(x):=\int_{\Omega_m}\frac{u_\psi(y)-u_\psi(x)}{|u_\psi(y)-u_\psi(x)|}dm_x(y)\);
\item[(iii)] \(u\) is a function of \(m\)-least gradient in \(\Omega\), i.e., \(TV_m(u)\leq TV_m(u+g)\) for every \(g\in BV_m(X)\) with \(g\equiv0\) \(\,\nu\)-a.e. on \(X\setminus\Omega\);
\item[(iv)] \(u\big|_{\Omega_m}\) is a solution of the \(m\)-least gradient problem
\begin{align*}
\min\left\{TV_m(u)\,\,:\,\, u\in BV_m(\Omega_m)\,\,\text{such that}\,\, u=\psi \,\,\nu\text{-a.s. on}\,\partial_m\Omega \right\}.
\end{align*}
\end{itemize}
Further, the authors present several examples of \((m,\nu)\) satisfying a \(p\)-Poincaré inequality as well as some counterexamples.
Reviewer: Yana Kinderknecht (Saarbrücken)Mean-field backward-forward stochastic differential equations and nonzero sum stochastic differential gameshttps://www.zbmath.org/1483.600782022-05-16T20:40:13.078697Z"Chen, Yinggu"https://www.zbmath.org/authors/?q=ai:chen.yinggu"Djehiche, Boualem"https://www.zbmath.org/authors/?q=ai:djehiche.boualem"Hamadène, Said"https://www.zbmath.org/authors/?q=ai:hamadene.saidEmpirical regularized optimal transport: statistical theory and applicationshttps://www.zbmath.org/1483.620552022-05-16T20:40:13.078697Z"Klatt, Marcel"https://www.zbmath.org/authors/?q=ai:klatt.marcel"Tameling, Carla"https://www.zbmath.org/authors/?q=ai:tameling.carla"Munk, Axel"https://www.zbmath.org/authors/?q=ai:munk.axelTransport analysis of infinitely deep neural networkhttps://www.zbmath.org/1483.620722022-05-16T20:40:13.078697Z"Sonoda, Sho"https://www.zbmath.org/authors/?q=ai:sonoda.sho"Murata, Noboru"https://www.zbmath.org/authors/?q=ai:murata.noboruSummary: We investigated the feature map inside deep neural networks (DNNs) by tracking the transport map. We are interested in the role of depth -- why do DNNs perform better than shallow models? -- and the interpretation of DNNs -- what do intermediate layers do? Despite the rapid development in their application, DNNs remain analytically unexplained because the hidden layers are nested and the parameters are not faithful. Inspired by the integral representation of shallow NNs, which is the continuum limit of the width, or the hidden unit number, we developed the flow representation and transport analysis of DNNs. The flow representation is the continuum limit of the depth, or the hidden layer number, and it is specified by an ordinary differential equation (ODE) with a vector field. We interpret an ordinary DNN as a transport map or an Euler broken line approximation of the flow. Technically speaking, a dynamical system is a natural model for the nested feature maps. In addition, it opens a new way
to the
coordinate-free treatment of DNNs by avoiding the redundant parametrization of DNNs. Following Wasserstein geometry, we analyze a flow in three aspects: dynamical system, continuity equation, and Wasserstein gradient flow. A key finding is that we specified a series of transport maps of the denoising autoencoder (DAE), which is a cornerstone for the development of deep learning. Starting from the shallow DAE, this paper develops three topics: the transport map of the deep DAE, the equivalence between the stacked DAE and the composition of DAEs, and the development of the double continuum limit or the integral representation of the flow representation. As partial answers to the research questions, we found that deeper DAEs converge faster and the extracted features are better; in addition, a deep Gaussian DAE transports mass to decrease the Shannon entropy of the data distribution. We expect that further investigations on these questions lead to the development of an interpretable and principled alternatives
to DNNs.Solvers for systems of linear algebraic equations with block-band matriceshttps://www.zbmath.org/1483.650532022-05-16T20:40:13.078697Z"Shteĭnberg, Boris Yakovlevich"https://www.zbmath.org/authors/?q=ai:shteinberg.boris-yakovlevich"Vasilenko, Aleksandr Aleksandrovich"https://www.zbmath.org/authors/?q=ai:vasilenko.aleksandr-aleksandrovich"Veselovskiĭ, Vadim Vladimirovich"https://www.zbmath.org/authors/?q=ai:veselovskii.vadim-vladimirovich"Zhivykh, Nikita Aleksandrovich"https://www.zbmath.org/authors/?q=ai:zhivykh.nikita-aleksandrovichSummary: The article proposes methods for constructing fast solvers for systems of linear algebraic equations with block-band matrices. A data structure for efficient storage of such matrices in RAM and a fast algorithm for solving systems of linear equations with such matrices based on this structure are proposed. The article is focused on the creation of solvers based on iterative algorithms for solving systems of linear equations with both symmetric matrices and matrices having a saddle-point singularity. It is proposed to develop and use a special precompiler to speed up the solver. The experimental solver is implemented in C, and the preliminary compilation is based on the Optimizing Parallelizing System in this paper. The results of numerical experiments that demonstrate high efficiency of the developed methods, including the efficiency of the precompiler, are presented.Linear convergence of accelerated conditional gradient algorithms in spaces of measureshttps://www.zbmath.org/1483.650892022-05-16T20:40:13.078697Z"Pieper, Konstantin"https://www.zbmath.org/authors/?q=ai:pieper.konstantin"Walter, Daniel"https://www.zbmath.org/authors/?q=ai:walter.danielSummary: A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear \(\mathcal{O}(1/k)\) rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear \(\mathcal{O}(\zeta/k)\) convergence rate is obtained locally.The back-and-forth method for Wasserstein gradient flowshttps://www.zbmath.org/1483.651042022-05-16T20:40:13.078697Z"Jacobs, Matt"https://www.zbmath.org/authors/?q=ai:jacobs.matthew"Lee, Wonjun"https://www.zbmath.org/authors/?q=ai:lee.wonjun"Léger, Flavien"https://www.zbmath.org/authors/?q=ai:leger.flavienSummary: We present a method to efficiently compute Wasserstein gradient flows. Our approach is based on a generalization of the back-and-forth method (BFM) introduced in [\textit{M. Jacobs} and \textit{F. Léger}, Numer. Math. 146, No. 3, 513--544 (2020; Zbl 1451.65078)] to solve optimal transport problems. We evolve the gradient flow by solving the dual problem to the JKO scheme. In general, the dual problem is much better behaved than the primal problem. This allows us to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies.Parameter estimation for a morphochemical reaction-diffusion model of electrochemical pattern formationhttps://www.zbmath.org/1483.651512022-05-16T20:40:13.078697Z"Sgura, Ivonne"https://www.zbmath.org/authors/?q=ai:sgura.ivonne"Lawless, Amos S."https://www.zbmath.org/authors/?q=ai:lawless.amos-s"Bozzini, Benedetto"https://www.zbmath.org/authors/?q=ai:bozzini.benedettoThis paper presents the first example of the fitting of an experimental electrochemical morphochemical distribution with a mathematical model of pattern formation, with corresponding parameter estimation. The process of electrodeposition can be modeled in terms of a two-variable nonlinear reaction-diffusion partial differential equations (PDE) system that simulates both the dynamics of the morphology profile and the chemical composition. Here we fit such a model to the different patterns present in a range of electrodeposited and electrochemically modified alloys employing PDE constrained optimization. Experiments with simulated data show how the parameter space of the model can be divided into zones that correspond to the different physical patterns by examining the structure of an appropriate cost function. We then use real data to demonstrate how numerical optimization of the cost function can allow the model to fit the rich variety of patterns generated by experiments. The computational technique developed provides a potential tool for tuning experimental parameters so as to exhibit desired patterns.
Reviewer: Carlos A. de Moura (Rio de Janeiro)Truncated nonsmooth Newton multigrid methods for block-separable minimization problemshttps://www.zbmath.org/1483.651982022-05-16T20:40:13.078697Z"Gräser, Carsten"https://www.zbmath.org/authors/?q=ai:graser.carsten"Sander, Oliver"https://www.zbmath.org/authors/?q=ai:sander.oliverSummary: The Truncated Nonsmooth Newton Multigrid method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere.Analysis of driving styles of a GP2 car via minimum lap-time direct trajectory optimizationhttps://www.zbmath.org/1483.700442022-05-16T20:40:13.078697Z"Gabiccini, M."https://www.zbmath.org/authors/?q=ai:gabiccini.marco"Bartali, L."https://www.zbmath.org/authors/?q=ai:bartali.l"Guiggiani, M."https://www.zbmath.org/authors/?q=ai:guiggiani.massimoSummary: This paper addresses the problem of the link between the driving style of an ideal driver, modelled as an optimal controller, and fundamental set-up parameters of a vehicle in the GP2 motorsport class. The aim is to evaluate quantitatively how set-up parameters, like distribution of aerodynamic loads, weight and roll stiffness between front and rear axles, affect the driving style, encoded in the shape of the optimal trajectory and in the acceleration, brake and steer inputs.
To this aim, we develop an optimization code that includes a double-track vehicle model capable of solving the minimum lap-time problem (MLTP) on a given track. The track is represented via NURBS curves and the MLTP is framed and solved as an optimal control problem by transcription into a nonlinear program using direct collocation. To assess the accuracy of the vehicle model and the optimization pipeline, we also validate our results against real telemetry data.
The developed software framework lends itself to easily perform both sensitivity analysis and concurrent trajectory planning and set-up parameter optimization: this is obtained by simple promotion of static parameters of interests to variables in the optimal control problem. Some results along these lines are also included.Poroelastic medium with non-penetrating crack driven by hydraulic fracture: variational inequality and its semidiscretizationhttps://www.zbmath.org/1483.740282022-05-16T20:40:13.078697Z"Kovtunenko, Victor A."https://www.zbmath.org/authors/?q=ai:kovtunenko.viktor-anatolievichSummary: A new class of unilateral variational models appearing in the theory of poroelasticity is introduced and studied. A poroelastic medium consists of solid phase and pores saturated with a Newtonian fluid. The medium contains a fluid-driven crack, which is subjected to non-penetration between the opposite crack faces. The fully coupled poroelastic system includes elliptic-parabolic governing equations under the unilateral constraint. Well-posedness of the corresponding variational inequality is established based on the Rothe semi-discretization in time, after subsequent passing time step to zero. The NLCP-formulation of non-penetration conditions is given which is useful for a semi-smooth Newton solution strategy.Dynamic frictional thermoviscoelastic contact problem with normal compliance and damagehttps://www.zbmath.org/1483.740722022-05-16T20:40:13.078697Z"Chadi, Khelifa"https://www.zbmath.org/authors/?q=ai:chadi.khelifa"Selmani, Mohamed"https://www.zbmath.org/authors/?q=ai:selmani.mohamedSummary: We study a dynamic problem describing the frictional contact between a thermoviscoelastic body and a foundation. The thermoviscoelastic constitutive law includes a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition and a temperature effect described by the first order evolution equation. The contact is modeled with normal compliance condition with friction. We present a variational formulation of the problem and establish an existence and uniqueness of the weak solution. The proof is based on parabolic variational inequalities of first and second kind, first order evolutionary variational equations and fixed point arguments.Linear optimal control of transient growth in turbulent channel flowshttps://www.zbmath.org/1483.760292022-05-16T20:40:13.078697Z"Song, Yang"https://www.zbmath.org/authors/?q=ai:song.yang"Xu, Chunxiao"https://www.zbmath.org/authors/?q=ai:xu.chunxiao"Huang, Weixi"https://www.zbmath.org/authors/?q=ai:huang.weixiSummary: This work investigates the suppression of linear transient growth in turbulent channel flows via linear optimal control. Two control algorithms are employed, i.e. the linear quadratic regulator (LQR) control based on full information of flow fields, and the linear quadratic Gaussian (LQG) control based on the information measured at walls. The influence of these controls on the development of both small-scale and large-scale perturbations is considered. The results show that the energy amplification of large-scale perturbations is significantly suppressed by both LQR and LQG controls, while small-scale perturbations are affected only by LQR control. The effects of the weighting parameters and control price on control performance are also analyzed for both controls, which reveals that different weighting parameters in the cost function do not qualitatively change the evaluation of control performance. As the control price increases, the effectiveness of both controls decreases markedly. For small-scale perturbations, the upper limit of the effective range of control price is lower than that for large-scale perturbations. When the Reynolds number is increased, it indicates that both LQR and LQG control become more effective in suppressing the energy amplification of large-scale perturbations.A quantum computing based numerical method for solving mixed-integer optimal control problemshttps://www.zbmath.org/1483.810422022-05-16T20:40:13.078697Z"Liu, Zhe"https://www.zbmath.org/authors/?q=ai:liu.zhe"Li, Shurong"https://www.zbmath.org/authors/?q=ai:li.shurongSummary: Mixed-integer optimal control problems (MIOCPs) usually play important roles in many real-world engineering applications. However, the MIOCP is a typical NP-hard problem with considerable computational complexity, resulting in slow convergence or premature convergence by most current heuristic optimization algorithms. Accordingly, this study proposes a new and effective hybrid algorithm based on quantum computing theory to solve the MIOCP. The algorithm consists of two parts: (i) Quantum Annealing (QA) specializes in solving integer optimization with high efficiency owing to the unique annealing process based on quantum tunneling, and (ii) Double-Elite Quantum Ant Colony Algorithm (DEQACA) which adopts double-elite coevolutionary mechanism to enhance global searching is developed for the optimization of continuous decisions. The hybrid QA/DEQACA algorithm integrates the strengths of such algorithms to better balance the exploration and exploitation abilities. The overall evolution performs to seek out the optimal mixed-integer decisions by interactive parallel computing of the QA and the DEQACA. Simulation results on benchmark functions and practical engineering optimization problems verify that the proposed numerical method is more excel at achieving promising results than other two state-of-the-art heuristics.Schrödinger dynamics and optimal transport of measureshttps://www.zbmath.org/1483.810712022-05-16T20:40:13.078697Z"Zanelli, Lorenzo"https://www.zbmath.org/authors/?q=ai:zanelli.lorenzoWilson loops in SYM \(\mathcal{N}=4\) do not parametrize an orientable spacehttps://www.zbmath.org/1483.811302022-05-16T20:40:13.078697Z"Agarwala, Susama"https://www.zbmath.org/authors/?q=ai:agarwala.susama"Marcott, Cameron"https://www.zbmath.org/authors/?q=ai:marcott.cameronSummary: We explore the geometric space parametrized by (tree level) Wilson loops in SYM \(\mathcal{N}=4\). We show that this space can be seen as a vector bundle over a totally non-negative subspace of the Grassmannian, \(\mathcal{W}_{k,n}\). Furthermore, we explicitly show that this bundle is non-orientable in the majority of the cases, and conjecture that it is non-orientable in the remaining situation. Using the combinatorics of the Deodhar decomposition of the Grassmannian, we identify subspaces \(\Sigma(W)\subset\mathcal{W}_{k,n}\) for which the restricted bundle lies outside the positive Grassmannian. Finally, while probing the combinatorics of the Deodhar decomposition, we give a diagrammatic algorithm for reading equations determining each Deodhar component as a semialgebraic set.Toward emulating nuclear reactions using eigenvector continuationhttps://www.zbmath.org/1483.811592022-05-16T20:40:13.078697Z"Drischler, C."https://www.zbmath.org/authors/?q=ai:drischler.c"Quinonez, M."https://www.zbmath.org/authors/?q=ai:quinonez.m"Giuliani, P. G."https://www.zbmath.org/authors/?q=ai:giuliani.p-g"Lovell, A. E."https://www.zbmath.org/authors/?q=ai:lovell.a-e"Nunes, F. M."https://www.zbmath.org/authors/?q=ai:nunes.f-mSummary: We construct an efficient emulator for two-body scattering observables using the general (complex) Kohn variational principle and trial wave functions derived from eigenvector continuation. The emulator simultaneously evaluates an array of Kohn variational principles associated with different boundary conditions, which allows for the detection and removal of spurious singularities known as Kohn anomalies. When applied to the \(K\)-matrix only, our emulator resembles the one constructed by \textit{R. J. Furnstahl} et al. [Phys. Lett., B 809, Article ID 135719, 6 p. (2020; Zbl 1473.81174)] although with reduced numerical noise. After a few applications to real potentials, we emulate differential cross sections for \(^{40}\mathrm{Ca}(n, n)\) scattering based on a realistic optical potential and quantify the model uncertainties using Bayesian methods. These calculations serve as a proof of principle for future studies aimed at improving optical models.Double-inertial proximal gradient algorithm for difference-of-convex programminghttps://www.zbmath.org/1483.901242022-05-16T20:40:13.078697Z"Wang, Tanxing"https://www.zbmath.org/authors/?q=ai:wang.tanxing"Cai, Xingju"https://www.zbmath.org/authors/?q=ai:cai.xingju"Song, Yongzhong"https://www.zbmath.org/authors/?q=ai:song.yongzhong"Gao, Xue"https://www.zbmath.org/authors/?q=ai:gao.xueSummary: We study a class of difference-of-convex programming whose objective function is the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a proper closed concave function composited with a linear operator. First, we consider the primal-dual reformulation of difference-of-convex programming. Then, adopting the framework of the double-proximal gradient algorithm (DPGA) and the inertial technique for accelerating the first-order algorithms, we propose a double-inertial proximal gradient algorithm (DiPGA) which includes some classical algorithms as its special cases. Under the assumption that the underlying function satisfies the Kurdyka-Łojasiewicz (KL) property and some suitable conditions on the parameters, we prove that each bounded sequence generated by DiPGA globally converges to a critical point of the objective function. Finally, we apply the algorithm to image processing model and compare it with DPGA to show its efficiency.A new approach to strong duality for composite vector optimization problemshttps://www.zbmath.org/1483.901482022-05-16T20:40:13.078697Z"Cánovas, María J."https://www.zbmath.org/authors/?q=ai:canovas.maria-josefa"Dinh, Nguyen"https://www.zbmath.org/authors/?q=ai:dinh.nguyen-van|dinh.nguyen-pham|dinh.nguyen-n"Long, Dang H."https://www.zbmath.org/authors/?q=ai:long.dang-hai"Parra, Juan"https://www.zbmath.org/authors/?q=ai:parra.juanThe paper concerns with duality results for the class of vector optimization problems involving composite mappings, called composite vector optimization problems. Different recent results in the literature are generalized and extended. Moreover, a new methodology is proposed regarding such vector optimization problems involving composite mappings in locally convex Hausdorff topological vector spaces. An application to convex semi-vector bilevel optimization problem is also provided.
Reviewer: Silvija Vlah Jerić (Zagreb)A proximal bundle-based algorithm for nonsmooth constrained multiobjective optimization problems with inexact datahttps://www.zbmath.org/1483.901522022-05-16T20:40:13.078697Z"Hoseini Monjezi, N."https://www.zbmath.org/authors/?q=ai:hoseini-monjezi.najmeh|monjezi.n-hoseini"Nobakhtian, S."https://www.zbmath.org/authors/?q=ai:nobakhtian.soghraSummary: In this paper, a proximal bundle-based method for solving nonsmooth nonconvex constrained multiobjective optimization problems with inexact information is proposed and analyzed. In this method, each objective function is treated individually without employing any scalarization. Using the improvement function, we transform the problem into an unconstrained one. At each iteration, by the proximal bundle method, a piecewise linear model is built and by solving a convex subproblem, a new candidate iterate is obtained. For locally Lipschitz objective and constraint functions, we study the problem of computing an approximate substationary point (a substationary point), when only inexact (exact) information about the functions and subgradient values are accessible. At the end, some numerical experiments are provided to illustrate the effectiveness of the method and certify the theoretical results.A unified concept of approximate and quasi efficient solutions and associated subdifferentials in multiobjective optimizationhttps://www.zbmath.org/1483.901532022-05-16T20:40:13.078697Z"Huerga, L."https://www.zbmath.org/authors/?q=ai:huerga.lidia"Jiménez, B."https://www.zbmath.org/authors/?q=ai:jimenez.bienvenido"Luc, D. T."https://www.zbmath.org/authors/?q=ai:dinh-the-luc."Novo, V."https://www.zbmath.org/authors/?q=ai:novo-sanjurjo.vicenteIn this article are introduced some new notions of quasi efficiency and quasi proper efficiency for multi-objective optimization problems to which the most important concepts of approximate and quasi efficient solutions given up to now are reduced. The authors establish some important properties and provide characterizations for these solutions by linear and nonlinear scalarizations. With the help of quasi efficient solutions, they introduce a generalized subdifferential of a vector mapping, which generates an approximate number of subdifferentials frequently used in optimization in a unifying way. The generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization. In the present paper, the authors use the error depending on the decision variable and extend, in the framework of multi-objective optimization, the notion of quasi efficiency given in and the notion of weak subdifferential to a more general setting. The paper is structured as follows. In Section 2 are given the necessaries preliminaries. In Section 3 are presented some generalizations of the corresponding notions of proper, efficient, and weak efficient solutions for the classical constrained multi-objective optimization problem that unify the most known concepts of exact, approximate, and quasi efficiency given in the literature.
In Section 4 are provided characterizations for generalized solutions by linear and nonlinear scalarizations. Section 5 presents an application of the new notion of quasi efficient solutions, used for the introduction of the efficient and proper subdifferentials for vector mappings. The result related to the existence result for proper efficient subdifferential and optimality conditions for quasi solutions is in terms of these generalized subdifferentials. Some conclusions are presented in Section 6.
Reviewer: Doina Carp (Bucureşti)Optimality conditions for vector variational inequalities via image space analysishttps://www.zbmath.org/1483.901562022-05-16T20:40:13.078697Z"Mastroeni, G."https://www.zbmath.org/authors/?q=ai:mastroeni.giandomenico"Pappalardo, M."https://www.zbmath.org/authors/?q=ai:pappalardo.massimoSaddle point and Karush-Kuh-Tucker type optimality conditions for vector variational inequalities in Hausdorff locally convex topological spaces are provided by means of a separation scheme in the framework of image space analysis.
Reviewer: Sorin-Mihai Grad (Paris)A bundle-type quasi-Newton method for nonconvex nonsmooth optimizationhttps://www.zbmath.org/1483.901602022-05-16T20:40:13.078697Z"Tang, Chunming"https://www.zbmath.org/authors/?q=ai:tang.chunming"Chen, Huangyue"https://www.zbmath.org/authors/?q=ai:chen.huangyue"Jian, Jinbao"https://www.zbmath.org/authors/?q=ai:jian.jinbao"Liu, Shuai"https://www.zbmath.org/authors/?q=ai:liu.shuaiSummary: We propose a bundle-type quasi-Newton method for minimizing a nonconvex nonsmooth function. The method is based on the redistributed bundle method with an on-the-fly convexification technique. At each iteration, the convexification parameter and the prox-parameter are suitably modified to guarantee that the proximal point of a piecewise affine model of a local convexification function approximates well-enough the proximal point of the objective function \(f\) at \(x^k\). A quasi-Newton procedure is added at the end of each serious step. Specifically, we construct a suitable search direction \(d^k\) via the BFGS update and monitor the reduction in the norm of the approximate subgradient to recognize whether an Armijo-type line search on \(f\) should be executed. Global convergence of the algorithm is established in the sense that there exists an accumulation point of the serious iterations such that it is a stationary point of \(f\). Superlinear convergence is proved under suitable assumptions. Preliminary numerical results are reported to illustrate that the method is efficient and has advantages over the redistributed bundle method.Two descent Dai-Yuan conjugate gradient methods for systems of monotone nonlinear equationshttps://www.zbmath.org/1483.901612022-05-16T20:40:13.078697Z"Waziri, Mohammed Yusuf"https://www.zbmath.org/authors/?q=ai:waziri.mohammed-yusuf"Ahmed, Kabiru"https://www.zbmath.org/authors/?q=ai:ahmed.kabiruSummary: In this paper, we present two Dai-Yuan type iterative methods for solving large-scale systems of nonlinear monotone equations. The methods can be considered as extensions of the classical Dai-Yuan conjugate gradient method for unconstrained optimization. By employing two different approaches, the Dai-Yuan method is modified to develop two different search directions, which are combined with the hyperplane projection technique of Solodov and Svaiter. The first search direction was obtained by carrying out eigenvalue study of the search direction matrix of an adaptive DY scheme, while the second is obtained by minimizing the distance between two adaptive versions of the DY method. Global convergence of the methods are established under mild conditions and preliminary numerical results show that the proposed methods are promising and more effective compared to some existing methods in the literature.Hölder continuity results for parametric set optimization problems via improvement setshttps://www.zbmath.org/1483.901632022-05-16T20:40:13.078697Z"Xu, Yingrang"https://www.zbmath.org/authors/?q=ai:xu.yingrang"Li, Shengjie"https://www.zbmath.org/authors/?q=ai:li.shengjieSummary: We consider a class of parametric set optimization problems, where both objective functions and constraint functions are perturbed by different parameters. Firstly, upper and lower set orderings with respect to improvement sets are introduced and used to define solution mappings. Then, some assumptions including strong domination properties are proposed to study the Hölder continuity of solution mappings and corresponding optimal value mappings. Our results generalize the upper Hölder continuity of efficient solution mappings for parametric vector optimization problems.Strongly stable C-stationary points for mathematical programs with complementarity constraintshttps://www.zbmath.org/1483.901662022-05-16T20:40:13.078697Z"Hernández Escobar, Daniel"https://www.zbmath.org/authors/?q=ai:hernandez-escobar.daniel"Rückmann, Jan-J."https://www.zbmath.org/authors/?q=ai:ruckmann.jan-joachimThe authors prove a topological and an equivalent algebraic characterization of the strong stability of a C-stationary point of mathematical programs with complementarity constraints. They adapt the concept of strong stability, starting from the standard nonlinear optimization, which includes the local uniqueness existence and continuous dependence of a solution for each sufficiently small perturbed problem where perturbations up to second order are allowed. For example, the set of Lagrange vectors is frequently not convex in this framework, which arises the need of replacing this concept by a more useful one in characterizing strongly stable C-stationary points. Then, the authors introduce the set of basic Lagrange vectors, which plays a similar role to that of the set of extreme points. A necessary second order condition, called Condition \(C^{\ast}\), for the strong stability of a C-stationary point is introduced. The case of convexity of the set of Lagrange vectors is studied, proving that it is necessary for the strong stability of C-stationary points under two additional assumptions, denoted by A1 and A2. An overview on some existing concepts of stationarity for mathematical programs with complementarity constraints are finally mentioned, which opens the need of further research on characterization of strong stability in each case.
Reviewer: Gabriela Cristescu (Arad)Performance guarantees for model-based approximate dynamic programming in continuous spaceshttps://www.zbmath.org/1483.901722022-05-16T20:40:13.078697Z"Beuchat, Paul Nathaniel"https://www.zbmath.org/authors/?q=ai:beuchat.paul-nathaniel"Georghiou, Angelos"https://www.zbmath.org/authors/?q=ai:georghiou.angelos"Lygeros, John"https://www.zbmath.org/authors/?q=ai:lygeros.johnEditorial remark: No review copy delivered.Approximation of value function of differential game with minimal costhttps://www.zbmath.org/1483.910432022-05-16T20:40:13.078697Z"Averboukh, Yurii Vladimirovich"https://www.zbmath.org/authors/?q=ai:averboukh.yurii-vladimirovichSummary: The paper is concerned with the approximation of the value function of the zero-sum differential game with the minimal cost, i. e., the differential game with the payoff functional determined by the minimization of some quantity along the trajectory by the solutions of continuous-time stochastic games with the stopping governed by one player. Notice that the value function of the auxiliary continuous-time stochastic game is described by the Isaacs-Bellman equation with additional inequality constraints. The Isaacs-Bellman equation is a parabolic PDE for the case of stochastic differential game and it takes a form of system of ODEs for the case of continuous-time Markov game. The approximation developed in the paper is based on the concept of the stochastic guide first proposed by \textit{N. N. Krasovskii} and \textit{A. N. Kotel'nikova} [Proc. Steklov Inst. Math. 269, 191--213 (2010; Zbl 1250.49039); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 15, No. 4, 146--166 (2009)].A dynamic theory of spatial externalitieshttps://www.zbmath.org/1483.910442022-05-16T20:40:13.078697Z"Boucekkine, Raouf"https://www.zbmath.org/authors/?q=ai:boucekkine.raouf"Fabbri, Giorgio"https://www.zbmath.org/authors/?q=ai:fabbri.giorgio"Federico, Salvatore"https://www.zbmath.org/authors/?q=ai:federico.salvatore"Gozzi, Fausto"https://www.zbmath.org/authors/?q=ai:gozzi.faustoSummary: We characterize the shape of spatial externalities in a continuous time and space differential game with transboundary pollution. We posit a realistic spatiotemporal law of motion for pollution (diffusion and advection), and tackle spatiotemporal non-cooperative (and cooperative) differential games. Precisely, we consider a circle partitioned into several states where a local authority decides autonomously about its investment, production and depollution strategies over time knowing that investment/production generates pollution, and pollution is transboundary. The time horizon is infinite. We allow for a rich set of geographic heterogeneities across states. We solve \textbf{analytically} the induced non-cooperative differential game and characterize its long-term spatial distributions. In particular, we prove that there exist a perfect Markov equilibrium, unique among the class of the affine feedbacks. We further provide with a full exploration of the free riding problem and the associated border effect.Mini-max incentive strategy for leader-follower games under uncertain dynamicshttps://www.zbmath.org/1483.910522022-05-16T20:40:13.078697Z"Rodríguez-Carreón, Celeste"https://www.zbmath.org/authors/?q=ai:rodriguez-carreon.celeste"Jiménez-Lizárraga, Manuel"https://www.zbmath.org/authors/?q=ai:jimenez-lizarraga.manuel"Villarreal, César Emilio"https://www.zbmath.org/authors/?q=ai:villarreal.cesar-emilio"Quiroz-Vázquez, Ignacio"https://www.zbmath.org/authors/?q=ai:quiroz-vazquez.ignacioSummary: This paper studies the problem of designing an incentive strategy for a leader-follower dynamic game affected by some sort of uncertainties. As is traditionally understood in the standard theory of incentives, the leader has complete knowledge of the game parameters, including the follower's performance index. So then he can compute the strategy that will lead the game to the global optimum that is favourable for him. Most of the current work is devoted to this situation. Nevertheless, such an assumption is unrealistic. This paper proposes an incentive scheme in which the game's dynamic depends on an unknown value that belongs to a finite set. The solution of the incentive strategy is computed in terms of the worst-case scenario, of the team's optimal solution. Based on the robust maximum principle, the new incentive is presented in the form of a mini-max feedback control. Two numerical examples illustrate the effectiveness of the approach.Optimal management of an insurer's exposure in a competitive general insurance markethttps://www.zbmath.org/1483.911922022-05-16T20:40:13.078697Z"Emms, Paul"https://www.zbmath.org/authors/?q=ai:emms.paul"Haberman, Steven"https://www.zbmath.org/authors/?q=ai:haberman.stevenSummary: The qualitative behavior of the optimal premium strategy is determined for an insurer in a finite and an infinite market using a deterministic general insurance model. The optimization problem leads to a system of forward-backward differential equations obtained from Pontryagin's maximum principle. The focus of the modelling is on how this optimization problem can be simplified by the choice of demand function and the insurer's objective. Phase diagrams are used to characterize the optimal control. When the demand is linear in the relative premium, the structure of the phase diagram can be determined analytically. Two types of premium strategy are identified for an insurer in an infinite market, and which is optimal depends on the existence of equilibrium points in the phase diagram. In a finite market there are four more types of premium strategy, and optimality depends on the initial exposure of the insurer and the position of a saddle point in the phase diagram. The effect of a nonlinear demand function is examined by perturbing the linear price function. An analytical optimal premium strategy is also found using inverse methods when the price function is nonlinear.A mean field game of optimal portfolio liquidationhttps://www.zbmath.org/1483.912152022-05-16T20:40:13.078697Z"Fu, Guanxing"https://www.zbmath.org/authors/?q=ai:fu.guanxing"Graewe, Paulwin"https://www.zbmath.org/authors/?q=ai:graewe.paulwin"Horst, Ulrich"https://www.zbmath.org/authors/?q=ai:horst.ulrich"Popier, Alexandre"https://www.zbmath.org/authors/?q=ai:popier.alexandreSummary: We consider a mean field game (MFG) of optimal portfolio liquidation under asymmetric information. We prove that the solution to the MFG can be characterized in terms of a forward-backward stochastic differential equation (FBSDE) with a possibly singular terminal condition on the backward component or, equivalently, in terms of an FBSDE with a finite terminal value yet a singular driver. Extending the method of continuation to linear-quadratic FBSDEs with a singular driver, we prove that the MFG has a unique solution. Our existence and uniqueness result allows proving that the MFG with a possibly singular terminal condition can be approximated by a sequence of MFGs with finite terminal values.Optimal control of pattern formations for an SIR reaction-diffusion epidemic modelhttps://www.zbmath.org/1483.921262022-05-16T20:40:13.078697Z"Chang, Lili"https://www.zbmath.org/authors/?q=ai:chang.lili"Gao, Shupeng"https://www.zbmath.org/authors/?q=ai:gao.shupeng"Wang, Zhen"https://www.zbmath.org/authors/?q=ai:wang.zhen.1Summary: Patterns arising from the reaction-diffusion epidemic model provide insightful aspects into the transmission of infectious diseases. For a classic SIR reaction-diffusion epidemic model, we review its Turing pattern formations with different transmission rates. A quantitative indicator, ``normal serious prevalent area (\textit{NSPA})'', is introduced to characterize the relationship between patterns and the extent of the epidemic. The extent of epidemic is positively correlated to \textit{NSPA}. To effectively reduce \textit{NSPA} of patterns under the large transmission rates, taken removed (recovery or isolation) rate as a control parameter, we consider the mathematical formulation and numerical solution of an optimal control problem for the SIR reaction-diffusion model. Numerical experiments demonstrate the effectiveness of our method in terms of control effect, control precision and control cost.Dynamic analysis and optimal control of a class of SISP respiratory diseaseshttps://www.zbmath.org/1483.921472022-05-16T20:40:13.078697Z"Shi, Lei"https://www.zbmath.org/authors/?q=ai:shi.lei.3|shi.lei|shi.lei.1|shi.lei.2|shi.lei.4"Qi, Longxing"https://www.zbmath.org/authors/?q=ai:qi.longxingSummary: In this paper, the actual background of the susceptible population being directly patients after inhaling a certain amount of \(\mathrm{PM_{2.5}}\) is taken into account. The concentration response function of \(\mathrm{PM_{2.5}}\) is introduced, and the SISP respiratory disease model is proposed. Qualitative theoretical analysis proves that the existence, local stability and global stability of the equilibria are all related to the daily emission \(P_0\) of \(\mathrm{PM_{2.5}}\) and \(\mathrm{PM_{2.5}}\) pathogenic threshold \(K\). Based on the sensitivity factor analysis and time-varying sensitivity analysis of parameters on the number of patients, it is found that the conversion rate \(\beta\) and the inhalation rate \(\eta\) has the largest positive correlation. The cure rate \(\gamma\) of infected persons has the greatest negative correlation on the number of patients. The control strategy formulated by the analysis results of optimal control theory is as follows: The first step is to improve the clearance rate of \(\mathrm{PM_{2.5}}\) by reducing the \(\mathrm{PM_{2.5}}\) emissions and increasing the intensity of dust removal. Moreover, such removal work must be maintained for a long time. The second step is to improve the cure rate of patients by being treated in time. After that, people should be reminded to wear masks and go out less so as to reduce the conversion rate of susceptible people becoming patients.Frequency-weighted \(\mathcal{H}_2\)-optimal model order reduction via oblique projectionhttps://www.zbmath.org/1483.930502022-05-16T20:40:13.078697Z"Zulfiqar, Umair"https://www.zbmath.org/authors/?q=ai:zulfiqar.umair"Sreeram, Victor"https://www.zbmath.org/authors/?q=ai:sreeram.victor"Ahmad, Mian Ilyas"https://www.zbmath.org/authors/?q=ai:ahmad.mian-ilyas"Du, Xin"https://www.zbmath.org/authors/?q=ai:du.xinSummary: In projection-based model order reduction, a reduced-order approximation of the original full-order system is obtained by projecting it onto a reduced subspace that contains its dominant characteristics. The problem of frequency-weighted \(\mathcal{H}_2\)-optimal model order reduction is to construct a local optimum in terms of the \(\mathcal{H}_2\)-norm of the weighted error transfer function. In this paper, a projection-based model order reduction algorithm is proposed that constructs a reduced-order model, which nearly satisfies the first-order optimality conditions for the frequency-weighted \(\mathcal{H}_2\)-optimal model order reduction problem. It is shown that as the order of the reduced model is increased, the deviation in the satisfaction of the optimality conditions reduces further. Numerical methods are also discussed that improve the computational efficiency of the proposed algorithm. Four numerical examples are presented to demonstrate the efficacy of the proposed algorithm.\(\mathcal{H}_2/\mathcal{H}_\infty\) formulation of LQR controls based on LMI for continuous-time uncertain systemshttps://www.zbmath.org/1483.931252022-05-16T20:40:13.078697Z"Caun, Rodrigo da Ponte"https://www.zbmath.org/authors/?q=ai:caun.rodrigo-da-ponte"Assunção, Edvaldo"https://www.zbmath.org/authors/?q=ai:assuncao.edvaldo"Teixeira, Marcelo Carvalho Minhoto"https://www.zbmath.org/authors/?q=ai:carvalho-minhoto-teixeira.marceloSummary: The classical theory of linear matrix inequality (LMI)-based linear quadratic regulator (LQR) control that is well established in literature does not provide, in the controller project phase, the occurrence of exogenous inputs acting on the plants. Therefore, this study proposes sufficient conditions for the robust synthesis of a mixed \(\mathcal{H}_2/\mathcal{H}_\infty\) control of the LQR problem. A new mixed polytopic and norm-bounded uncertainties representation for linear time-invariant uncertain systems allows to express different uncertainty models in one single LMI by using the S-procedure. As an additional objective, we propose the study of the robustness of controllers concerned with the sensibility of their coefficients, in such a way that the variations in controller gain matrix are described by norm-bounded uncertainties. A practical implementation in Quanser\(^\circledR\) active suspension evaluates the control projects through the rejection level of vibrations caused by track irregularities.LQG control on mixed \(H_2/H_\infty\) problem: the discrete-time casehttps://www.zbmath.org/1483.931332022-05-16T20:40:13.078697Z"Li, Xiaoqian"https://www.zbmath.org/authors/?q=ai:li.xiaoqian"Wang, Wei"https://www.zbmath.org/authors/?q=ai:wang.wei.23|wang.wei.30|wang.wei.19|wang.wei.17"Xu, Juanjuan"https://www.zbmath.org/authors/?q=ai:xu.juanjuan"Zhang, Huanshui"https://www.zbmath.org/authors/?q=ai:zhang.huanshuiSummary: In this paper, we are concerned with linear quadratic Gaussian (LQG) control on the mixed \(H_2/H_\infty\) problem. The mixed \(H_2/H_\infty\) problem can be formulated as a kind of constraint optimisation problems where the control input is to minimise the \(H_2\)-norm subject to the \(H_\infty\)-constraint dealt with by the disturbance. The main contributions are twofold. First, when the state variables can be obtained exactly, necessary and sufficient conditions are given to guarantee the existence of the strategies. Second, the optimal LQG controller in terms of three decoupled equations (two Riccati equations and one Lyapunov equation) is obtained. The key technique is to apply the Stackelberg game approach by treating the disturbance as the follower and the control input as the leader respectively. A practical example is given to verify the efficiency of the proposed approach.Less conservative conditions for robust LQR-state-derivative controller design: an LMI approachhttps://www.zbmath.org/1483.931882022-05-16T20:40:13.078697Z"Beteto, Marco Antonio Leite"https://www.zbmath.org/authors/?q=ai:beteto.marco-antonio-leite"Assunção, Edvaldo"https://www.zbmath.org/authors/?q=ai:assuncao.edvaldo"Teixeira, Marcelo Carvalho Minhoto"https://www.zbmath.org/authors/?q=ai:carvalho-minhoto-teixeira.marcelo"Silva, Emerson Ravazzi Pires da"https://www.zbmath.org/authors/?q=ai:silva.emerson-ravazzi-pires-da"Buzachero, Luiz Francisco Sanches"https://www.zbmath.org/authors/?q=ai:buzachero.luiz-francisco-sanches"da Ponte Caun, Rodrigo"https://www.zbmath.org/authors/?q=ai:caun.rodrigo-da-ponteSummary: This study proposes less conservative conditions for robust linear quadratic regulator controllers using state-derivative feedback (SDF). The algebraic Riccati equation was formulated using the SDF, and its solution was obtained by linear matrix inequalities. SDF was chosen owing to the presence of accelerometers as sensors. Since accelerometers are the main sensors in mechanical systems, the proposed technique may be used to control/attenuate their vibrations/oscillations. Moreover, to formulate the less conservative conditions, some methods in the specialised literature were used, such as, for example, slack variables by Finsler's lemma. The paper also offers necessary and sufficient conditions for an arbitrary convex combination of square real matrices \(A_1, A_2, \dots, A_r\) to be a nonsingular matrix, and thus an invertible one: \(A_1\) must be nonsingular and all the real eigenvalues of \(A_1^{-1}A_2, A_1^{-1}A_3, \dots, A_1^{-1}A_r\) must be positive. This result is important in the formulation of the proposed less conservative conditions since it was assumed that a given convex combination is nonsingular. A feasibility analysis demonstrates that the proposed conditions reduce the conservatism. Thereby, it is possible to stabilise a higher number of systems and to reduce the guaranteed cost. Furthermore, a practical implementation illustrated the application of the proposed conditions.On sufficient optimality conditions for a guaranteed control in the speed problem for a linear time-varying discrete-time system with bounded controlhttps://www.zbmath.org/1483.933612022-05-16T20:40:13.078697Z"Ibragimov, D. N."https://www.zbmath.org/authors/?q=ai:ibragimov.d-n"Novozhilin, N. M."https://www.zbmath.org/authors/?q=ai:novozhilin.n-m"Portseva, E. Yu."https://www.zbmath.org/authors/?q=ai:portseva.e-yuSummary: We consider the solution of the speed problem for linear time-varying discrete-time systems with convex control constraints. A method is proposed for reducing the general case of the speed problem to the case of linear control constraints using polyhedral approximation algorithms. Sufficient optimality conditions for the guaranteed solution are stated and proved. Examples are given. Based on the methods obtained, the speed-optimal damping problem for a high-rise structure located in a seismic activity zone is solved.A note on stability and control of sampled-data asynchronous switched systemshttps://www.zbmath.org/1483.934712022-05-16T20:40:13.078697Z"Şen, Büşra"https://www.zbmath.org/authors/?q=ai:sen.busra"Türker, Türker"https://www.zbmath.org/authors/?q=ai:turker.turker"Eren, Yavuz"https://www.zbmath.org/authors/?q=ai:eren.yavuzSummary: High-performance control of switched systems is needed in various applications involving sampled state variables and discrete control signals. However, asynchronous switching phenomenon existing in sampled-data switched systems may introduce undesired response characteristics and even instability. After demonstrating such unfavorable effects, the conditions to check the stability in these systems are provided in this study. Then the control design problem is considered, and an optimal controller design procedure for assorted performance measures is presented for a class of sampled-data switched systems. The performance of the controller is validated through numerical simulations. The proposed control structure is viable in many different applications containing switching phenomena with a discrete control structure.Synchronisation of uncertain chaotic systems via fuzzy-regulated adaptive optimal control approachhttps://www.zbmath.org/1483.935172022-05-16T20:40:13.078697Z"Zhang, Haiyun"https://www.zbmath.org/authors/?q=ai:zhang.haiyun"Meng, Deyuan"https://www.zbmath.org/authors/?q=ai:meng.deyuan"Wang, Jin"https://www.zbmath.org/authors/?q=ai:wang.jin"Lu, Guodong"https://www.zbmath.org/authors/?q=ai:lu.guodongSummary: This study investigates the adaptive synchronisation of uncertain chaotic systems with unmodelled nonlinearities, dynamic mismatch, parametric perturbations, and external disturbances. A fuzzy-regulated adaptive optimal control (FRAOC) scheme is derived to realise chaotic synchronisation under both structure and parameter uncertainties. In the proposed scheme, the uncertain chaotic dynamics is firstly captured and estimated using a self-organising learning algorithm in an online fuzzy rule database. Based on the complicated and uncertain chaotic information, control strategy is adaptively regulated and configured in the form of weighting matrix for the subsequent optimal controller by the use of fuzzy logic inference. An adaptive optimal controller is then developed, so that its control behaviour and performance are adaptively adjusted for the chaotic synchronisation and compound uncertainty compensation. A supervisory compensator with recursive adaptation law is also designed to attenuate the residual compensation error and guarantee the synchronisation stability. Chaotic synchronisation convergence using the proposed approach can be mathematically ensured and speeded up with satisfactory robustness. Simulation results also demonstrate the effectiveness of the proposed control method in comparison with robust quadratic optimal control based on linear matrix inequality approach.Optimal control for unknown mean-field discrete-time system based on Q-learninghttps://www.zbmath.org/1483.937042022-05-16T20:40:13.078697Z"Ge, Yingying"https://www.zbmath.org/authors/?q=ai:ge.yingying"Liu, Xikui"https://www.zbmath.org/authors/?q=ai:liu.xikui"Li, Yan"https://www.zbmath.org/authors/?q=ai:li.yan.6Summary: Solving the optimal mean-field control problem usually requires complete system information. In this paper, a Q-learning algorithm is discussed to solve the optimal control problem of the unknown mean-field discrete-time stochastic system. First, through the corresponding transformation, we turn the stochastic mean-field control problem into a deterministic problem. Second, the \(H\) matrix is obtained through Q-function, and the control strategy relies only on the \(H\) matrix. Therefore, solving \(H\) matrix is equivalent to solving the mean-field optimal control. The proposed Q-learning method iteratively solves \(H\) matrix and gain matrix according to input system state information, without the need for system parameter knowledge. Next, it is proved that the control matrix sequence obtained by Q-learning converge to the optimal control, which shows theoretical feasibility of the Q-learning. Finally, two simulation cases verify the effectiveness of Q-learning algorithm.A diffusion wavelets-based multiscale framework for inverse optimal control of stochastic systemshttps://www.zbmath.org/1483.937052022-05-16T20:40:13.078697Z"Ha, Jung-Su"https://www.zbmath.org/authors/?q=ai:ha.jung-su"Chae, Hyeok-Joo"https://www.zbmath.org/authors/?q=ai:chae.hyeok-joo"Choi, Han-Lim"https://www.zbmath.org/authors/?q=ai:choi.han-limSummary: This work presents a multiscale framework to solve a class of inverse optimal control (IOC) problems in the context of robot motion planning and control in a complex environment. In order to handle complications resulting from a large decision space and complex environmental geometry, two key concepts are adopted: (a) a diffusion wavelet representation of the Markov chain for hierarchical abstraction of the state space; and (b) a desirability function-based representation of the Markov decision process (MDP) to efficiently calculate the optimal policy. An IOC problem constructed on a `abstract state' is solved, which is much more tractable than using the original bases set; moreover, the solution can be obtained recursively in the `coarse to fine' direction by utilizing the hierarchical structure of basis functions. The resulting multiscale plan is utilized to finally compute a continuous-time optimal control policy within a receding horizon implementation. Illustrative numerical experiments on a robot path control in a complex environment and on a quadrotor ball-catching task are presented to verify the proposed method.