Recent zbMATH articles in MSC 49Khttps://www.zbmath.org/atom/cc/49K2022-05-16T20:40:13.078697ZWerkzeugSemiconcavity 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.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.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 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)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.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 \(V = W_0^{1, p}(\varOmega)\).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.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.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).Sharp 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.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.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.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.