Recent zbMATH articles in MSC 49Khttps://www.zbmath.org/atom/cc/49K2021-07-26T21:45:41.944397ZWerkzeugAn optimization problem for some nonlinear elliptic equationhttps://www.zbmath.org/1463.352622021-07-26T21:45:41.944397Z"Zivari-Rezapour, Mohsen"https://www.zbmath.org/authors/?q=ai:zivari-rezapour.mohsenSummary: In this paper we prove existence and uniqueness of the optimal solution for an optimization problem related to a nonlinear elliptic equation. We use the concept \textit{tangent cones} to derive the optimality condition satisfied by optimal solution.Constraints optimal control governing by triple nonlinear hyperbolic boundary value problemhttps://www.zbmath.org/1463.490022021-07-26T21:45:41.944397Z"Al-Hawasy, Jamil A. Ali"https://www.zbmath.org/authors/?q=ai:al-hawasy.jamil-a-ali"Ali, Lamyaa H."https://www.zbmath.org/authors/?q=ai:ali.lamyaa-hSummary: The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.Efficient solution and value function for non-convex variational problemshttps://www.zbmath.org/1463.490032021-07-26T21:45:41.944397Z"Darabi, M."https://www.zbmath.org/authors/?q=ai:darabi.marzie"Zafarani, J."https://www.zbmath.org/authors/?q=ai:zafarani.jafarSummary: In this paper, we want to investigate a wide range of non-convex variational problems and obtain some sufficient and necessary conditions for existence of a feasible solution for these problems. Hence, we define optimal value function corresponding to these problems and obtain a relationship between subdifferential of the optimal value function and the set of Lagrange multipliers.Optimal control problem and maximum principle for fractional order cooperative systems.https://www.zbmath.org/1463.490052021-07-26T21:45:41.944397Z"Bahaa, G. M."https://www.zbmath.org/authors/?q=ai:bahaa.gaber-mohamedSummary: In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.Optimal control of nonlinear periodic population dynamic systems with individual scale in a polluted environmenthttps://www.zbmath.org/1463.490072021-07-26T21:45:41.944397Z"Gong, Wei"https://www.zbmath.org/authors/?q=ai:gong.wei.1|gong.wei"Wang, Zhanping"https://www.zbmath.org/authors/?q=ai:wang.zhanpingSummary: In this paper, we study an optimal problem of nonlinear periodic population with individual scale in a polluted environment. Firstly, we obtain the existence and uniqueness of nonnegative solution of the model by integral equation and operator theory. Then, the existence of optimal strategy is determined by using Mazur theorem, and the optimal condition of control problem is derived by means of tangent and normal cones in nonlinear analysis.On Tikhonov regularization of optimal Neumann boundary control problem for an ill-posed strongly nonlinear elliptic equation with an exponential type of non-linearity.https://www.zbmath.org/1463.490122021-07-26T21:45:41.944397Z"Manzo, Rosanna"https://www.zbmath.org/authors/?q=ai:manzo.rosannaSummary: We discuss the existence of solutions to an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for strongly non-linear elliptic equations with an exponential type of nonlinearity. A density of surface traction \(u\) acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution \(y_d\in L^2(\Omega)\) and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this, we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of ``defect''. We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.External and internal stability in set optimization using gamma convergencehttps://www.zbmath.org/1463.490262021-07-26T21:45:41.944397Z"Karuna"https://www.zbmath.org/authors/?q=ai:karuna.o-l"Lalitha, C. S."https://www.zbmath.org/authors/?q=ai:lalitha.c-sSummary: The main objective of this paper is to investigate the stability of solution sets of perturbed set optimization problems in the decision space as well as in the image space, by perturbing the objective maps. For a sequence of set-valued maps, a notion of gamma convergence is introduced to establish the external and internal stability in terms of Painlevé-Kuratowski convergence of sequence of solution sets of perturbed problems under certain compactness assumptions and domination properties.\(B\)-well-posedness for set optimization problems involving set order relationshttps://www.zbmath.org/1463.490282021-07-26T21:45:41.944397Z"Vui, Pham Thi"https://www.zbmath.org/authors/?q=ai:vui.pham-thi"Anh, Lam Quoc"https://www.zbmath.org/authors/?q=ai:anh.lam-quoc"Wangkeeree, Rabian"https://www.zbmath.org/authors/?q=ai:wangkeeree.rabianSummary: In this paper, both pointwise and global \(B\)-well-posedness for set optimization problems involving three kinds of set order relations are investigated. Characterizations and sufficient and/or necessary conditions of these types of well-posedness are given. Moreover, pointwise \(L\)-well-posedness and relationships between these kinds of pointwise well-posedness are studied.The optimal control of the parabolic system related to a class of non-Newtonian fluidshttps://www.zbmath.org/1463.490302021-07-26T21:45:41.944397Z"Hassan, H. M."https://www.zbmath.org/authors/?q=ai:hassan.h-m"el Badawy, M. A."https://www.zbmath.org/authors/?q=ai:el-badawy.m-aSummary: In this paper, we consider an optimal control problem whose states are governed by a type of a parabolic system related to a class of non-Newtonian fluids. We define the necessary and sufficient conditions of optimal control for the system related to a class of non-Newtonian fluids.A novel successive approximation method for solving a class of optimal control problemshttps://www.zbmath.org/1463.490312021-07-26T21:45:41.944397Z"Shirazian, Mohammad"https://www.zbmath.org/authors/?q=ai:shirazian.mohammad"Effati, Sohrab"https://www.zbmath.org/authors/?q=ai:effati.sohrabSummary: This paper presents a successive approximation method (SAM) for solving a large class of optimal control problems. The proposed analytical-approximate method, successively solves the Two-Point Boundary Value Problem (TPBVP), obtained from the Pontryagin's Maximum Principle (PMP). The convergence of this method is proved and a control design algorithm with low computational complexity is presented. Through the finite number of algorithm iterations, a suboptimal control law is obtained for the optimal control problem. An illustrative example is given to demonstrate the efficiency of the proposed method.Pontryagin's maximum principle for optimal control problems with nonlocal boundary conditionshttps://www.zbmath.org/1463.490322021-07-26T21:45:41.944397Z"Zeĭnally, F. M."https://www.zbmath.org/authors/?q=ai:zeinally.f-m(no abstract)On a Bolza problemhttps://www.zbmath.org/1463.490332021-07-26T21:45:41.944397Z"Krastanov, Mikhail I."https://www.zbmath.org/authors/?q=ai:krastanov.mikhail-ivanov"Ribarska, Nadezhda K."https://www.zbmath.org/authors/?q=ai:ribarska.nadezhda-kIn this paper classical problem of the calculus of variations is investigated assuming that the integrand is a continuous function. The authors apply the main ideas of non-smooth analysis in order to prove a non-smooth version of the classical Euler equation. The usual assumption of existence of a common \( L^1 \)-upper bound of a family of summable functions is replaced by uniform integrability of the same family. The proposed in the paper technique do not use variational principles. A necessary optimality condition for the basic problem of calculus of variations is obtained.Optimal control problems for some classes of functional-differential equations on the semi-axishttps://www.zbmath.org/1463.490342021-07-26T21:45:41.944397Z"Kichmarenko, O."https://www.zbmath.org/authors/?q=ai:kichmarenko.olga-d"Stanzhytskyi, O."https://www.zbmath.org/authors/?q=ai:stanzhytsky.o-m|stanzhytskyi.oleksandr-mSummary: In this paper we study functional-differential equations on the semi-axis, which are non-linear with respect to the phase variables and linear with respect to the control. Sufficient conditions for existence of optimal control in terms of the right-hand side and the quality criterion are obtained. Relation between the solutions of the problems on infinite and finite intervals is studied and results that about these connections are proven.Continuity of the solution mappings to primal and dual vector equilibrium problemshttps://www.zbmath.org/1463.490352021-07-26T21:45:41.944397Z"Bantaojai, Thanatporn"https://www.zbmath.org/authors/?q=ai:bantaojai.thanatporn"Duy, Tran Quoc"https://www.zbmath.org/authors/?q=ai:duy.tran-quocSummary: In this paper, we consider strong forms of the primal and dual vector equilibrium problems in Hausdorff topological vector spaces. Under suitable assumptions, some continuity properties of solution mappings to such problems are established. The mains results improve recent existing ones in the literature. Some examples are provided to illustrate our results. Applications to vector optimization problem and vector variational inequalities are also discussed.Monotonicity and stability of optimal solutions of a minimization problemhttps://www.zbmath.org/1463.490362021-07-26T21:45:41.944397Z"Liu, Yichen"https://www.zbmath.org/authors/?q=ai:liu.yichen"Emamizadeh, Behrouz"https://www.zbmath.org/authors/?q=ai:emamizadeh.behrouzSummary: This paper is concerned with a minimization problem modeling the minimum displacement of an isotropic elastic membrane subject to a vertical force such as a load distribution. In addition to proving existence and uniqueness of optimal solutions, we show that these solutions are monotone and stable, in a certain sense. The main mathematical tool used in the analysis is the tangent cones from convex analysis, which helps to derive the optimality condition. Our results are compatible with physical expectations.On the generic stability of the constrained largest element of a set family with vector parameterhttps://www.zbmath.org/1463.490372021-07-26T21:45:41.944397Z"Lu, Meihua"https://www.zbmath.org/authors/?q=ai:lu.meihua"Xiao, Li'na"https://www.zbmath.org/authors/?q=ai:xiao.lina"Zuo, Yonghua"https://www.zbmath.org/authors/?q=ai:zuo.yonghuaSummary: In this paper, the correspondence method is used to define the constrained generalized largest element. We discuss the generic stability of the constrained generalized largest element, when the maximum meta-mapping perturbation and vector parameter are disturbed, especially when the constraint conditions are disturbed. Under the most extensive perturbation, we obtain the generic stability theorem.Sufficient condition for near-optimal control of a stochastic SIRS epidemic modelhttps://www.zbmath.org/1463.490382021-07-26T21:45:41.944397Z"Mu, Xiaojie"https://www.zbmath.org/authors/?q=ai:mu.xiaojie"Zhang, Qimin"https://www.zbmath.org/authors/?q=ai:zhang.qimin"Wang, Zong"https://www.zbmath.org/authors/?q=ai:wang.zongSummary: The parameters of the usual epidemic model are determined, but the parameters of the model are difficult to be accurately obtained because of the influence of various uncertain factors. This paper discusses the near-optimal control of a stochastic SIRS model with imprecise parameters. The objective function for the cost of the treatment of disease is as small as possible. The error bounds of the near-optimal control are given. The sufficient conditions for the near-optimal control are established by using the Hamiltonian function, and the effects of the control variables on the disease are verified by a numerical example.Finite-time stochastic linear quadratic optimal control based on \(Q\)-learninghttps://www.zbmath.org/1463.490392021-07-26T21:45:41.944397Z"Wang, Tao"https://www.zbmath.org/authors/?q=ai:wang.tao.7|wang.tao.3|wang.tao.9|wang.tao.6|wang.tao.5|wang.tao.8|wang.tao.2"Luo, Minna"https://www.zbmath.org/authors/?q=ai:luo.minna"Wang, Na"https://www.zbmath.org/authors/?q=ai:wang.na"Cui, Lili"https://www.zbmath.org/authors/?q=ai:cui.liliSummary: A \(Q\)-learning iteration algorithm based on value iteration is adopted to obtain the finite-time stochastic linear quadratic (SLQ) optimal control for stochastic linear discrete-time systems of partially unknown parameters with state and control dependent on noises. First, the condition of the attainability and well-posedness for the SLQ optimal control problem is given. In the meantime, the optimal control gain matrix sequence and the corresponding stochastic algebra Riccati equations (SAREs) are obtained by the matrix Lagrange multiplier algorithm. Secondly, a \(Q\) function is defined by the value iteration algorithm, and a \(Q\)-learning iteration algorithm is introduced to get the iteration control gain matrix sequence and \(H\) matrix sequence for every optimal control gain matrix. The algorithm relies on the system state information, which partially gets rid of the restriction of the system unknown parameters. Then, the convergence analysis of the iteration algorithm is presented to prove that the control gain matrix sequences converge to the respective optimal control gain matrix and the \(H\) matrix sequences converge to the respective optimal \(H\) matrix. Lastly, one simulation example is provided to verify the effectiveness of the theoretical discussions.Minimizing almost smooth control variation in nonlinear optimal control problemshttps://www.zbmath.org/1463.490402021-07-26T21:45:41.944397Z"Zhang, Ying"https://www.zbmath.org/authors/?q=ai:zhang.ying.5|zhang.ying.4"Yu, Changjun"https://www.zbmath.org/authors/?q=ai:yu.changjun"Xu, Yingtao"https://www.zbmath.org/authors/?q=ai:xu.yingtao"Bai, Yanqin"https://www.zbmath.org/authors/?q=ai:bai.yanqinSummary: In this paper, we consider an optimal control problem in which the control is almost smooth and the state and control are subject to terminal state constraints and continuous state and control inequality constraints. By introducing an extra set of differential equations for this almost smooth control, we transform this constrained optimal control problem into an equivalent problem involving both control function and system parameter vector as decision variables. Then, by the control parametrization technique and a time scaling transformation, the equivalent problem is approximated by a sequence of constrained optimal parameter selection problems, each of which is a finite dimensional optimization problem. For each of these constrained optimal parameter selection problems, a novel exact penalty function method is constructed by appending penalized constraint violations to the cost function. This gives rise to a sequence of unconstrained optimal parameter selection problems; and each of which can be solved by existing optimization algorithms or software packages. Finally, a practical container crane operation problem is solved, showing the effectiveness and applicability of the proposed approach.Optimal control in a model of chemotherapy-induced radiosensilisationhttps://www.zbmath.org/1463.490532021-07-26T21:45:41.944397Z"Bajger, Piotr"https://www.zbmath.org/authors/?q=ai:bajger.piotr"Fujarewicz, Krzysztof"https://www.zbmath.org/authors/?q=ai:fujarewicz.krzysztof"Świerniak, Andrzej"https://www.zbmath.org/authors/?q=ai:swierniak.andrzej-pSummary: In this work, we consider a simple mathematical model of radiochemotherapy which includes a term responsible for radiosensitization. We focus on finding theoretically optimal controls which maximise tumour cure probability for a finite, fixed therapeutic horizon. We prove that the optimal controls for both therapies are of 0-bang type, a result which is not altered by the inclusion of the radiosensilization term. By means of numerical simulations, we show that optimal control offers a moderate increase in survival time over a sequential treatment. We then revisit in more detail a question of measuring the synergy between the therapies by means of isobolograms, a common experimental technique for measuring the additivity of two treatments.Highly accurate numerical schemes for stochastic optimal control via FBSDEshttps://www.zbmath.org/1463.650052021-07-26T21:45:41.944397Z"Fu, Yu"https://www.zbmath.org/authors/?q=ai:fu.yu"Zhao, Weidong"https://www.zbmath.org/authors/?q=ai:zhao.weidong"Zhou, Tao"https://www.zbmath.org/authors/?q=ai:zhou.taoSummary: This work is concerned with numerical schemes for stochastic optimal control problems (SOCPs) by means of forward backward stochastic differential equations (FBSDEs). We first convert the stochastic optimal control problem into an equivalent stochastic optimality system of FBSDEs. Then we design an efficient second order FBSDE solver and an quasi-Newton type optimization solver for the resulting system. It is noticed that our approach admits the second order rate of convergence even when the state equation is approximated by the Euler scheme. Several numerical examples are presented to illustrate the effectiveness and the accuracy of the proposed numerical schemes.A priori error estimates of finite element methods for an optimal control problem governed by a one-prey and one-predator modelhttps://www.zbmath.org/1463.653182021-07-26T21:45:41.944397Z"Zheng, Ruirui"https://www.zbmath.org/authors/?q=ai:zheng.ruirui"Sun, Tongjun"https://www.zbmath.org/authors/?q=ai:sun.tongjunSummary: An optimal control problem governed by a one-prey and one-predator model is considered. The co-state equations and optimality conditions are established using optimal control theory. In order to construct the fully discrete approximation, the state and co-state variables are approximated by piecewise linear continuous functions and the control variable is approximated by piecewise constant functions. A priori error estimates for the state variables, co-state variables and control variable are proved.Closedness of the optimal solution sets for general vector alpha optimization problemshttps://www.zbmath.org/1463.901962021-07-26T21:45:41.944397Z"Su, Tran Van"https://www.zbmath.org/authors/?q=ai:su.tran-van"Hang, Dinh Dieu"https://www.zbmath.org/authors/?q=ai:hang.dinh-dieuSummary: The aim of paper is to study the closedness of the optimal solution sets for general vector alpha optimization problems in Hausdorff locally convex topological vector spaces. Firstly, we present the relationships between the optimal solution sets of primal and dual general vector alpha optimization problems. Secondly, making use of the upper semicontinuity of a set-valued mapping, we discuss the results on closedness of the optimal solution sets for general vector alpha optimization problems in infinite dimensional spaces.A strongly convergent generalized gradient projection method for minimax optimization with general constraintshttps://www.zbmath.org/1463.902372021-07-26T21:45:41.944397Z"Ma, Guodong"https://www.zbmath.org/authors/?q=ai:ma.guodongSummary: In this paper, the minimax optimization problem with inequality and equality constraints is discussed. The original problem is transformed into an associated simple problem with a penalty term and only inequality constraints. Then a new generalized gradient projection algorithm is presented. The main characters of the proposed algorithm are as follows: the improved search direction is generated by only one generalized gradient projection explicit formula; the new optimal identification function is introduced; the algorithm is globally and strongly convergent under some mild assumptions. Finally, the numerical results show that the proposed algorithm is promising.