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Stable invariant manifolds with application to control problems 稳定不变流形及其在控制问题中的应用
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2021-01-01 DOI: 10.3934/mcrf.2021040
A. Gorshkov
{"title":"Stable invariant manifolds with application to control problems","authors":"A. Gorshkov","doi":"10.3934/mcrf.2021040","DOIUrl":"https://doi.org/10.3934/mcrf.2021040","url":null,"abstract":"In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"86 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76238448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Maximum principle for discrete-time stochastic optimal control problem and stochastic game 离散时间随机最优控制问题和随机对策的极大值原理
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2021-01-01 DOI: 10.3934/mcrf.2021031
Zhen Wu, Feng Zhang
{"title":"Maximum principle for discrete-time stochastic optimal control problem and stochastic game","authors":"Zhen Wu, Feng Zhang","doi":"10.3934/mcrf.2021031","DOIUrl":"https://doi.org/10.3934/mcrf.2021031","url":null,"abstract":"This paper is first concerned with one kind of discrete-time stochastic optimal control problem with convex control domains, for which necessary condition in the form of Pontryagin's maximum principle and sufficient condition of optimality are derived. The results are then extended to two kinds of discrete-time stochastic games. Two illustrative examples are studied, for which the explicit optimal strategies are given. This paper establishes a rigorous version of discrete-time stochastic maximum principle in a clear and concise way and paves a road for further related topics.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"25 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84434236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Fractional optimal control problems on a star graph: Optimality system and numerical solution 星图上的分数阶最优控制问题:最优系统与数值解
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2021-01-01 DOI: 10.3934/mcrf.2020033
Vaibhav Mehandiratta, M. Mehra, G. Leugering
{"title":"Fractional optimal control problems on a star graph: Optimality system and numerical solution","authors":"Vaibhav Mehandiratta, M. Mehra, G. Leugering","doi":"10.3934/mcrf.2020033","DOIUrl":"https://doi.org/10.3934/mcrf.2020033","url":null,"abstract":"In this paper, we study optimal control problems for nonlinear fractional order boundary value problems on a star graph, where the fractional derivative is described in the Caputo sense. The adjoint state and the optimality system are derived for fractional optimal control problem (FOCP) by using the Lagrange multiplier method. Then, the existence and uniqueness of solution of the adjoint equation is proved by means of the Banach contraction principle. We also present a numerical method to find the approximate solution of the resulting optimality system. In the proposed method, the begin{document}$ L2 $end{document} scheme and the Grunwald-Letnikov formula is used for the approximation of the Caputo fractional derivative and the right Riemann-Liouville fractional derivative, respectively, which converts the optimality system into a system of linear algebraic equations. Two examples are provided to demonstrate the feasibility of the numerical method.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"78 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81736391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 31
Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems 一类时间不一致随机最优控制问题的开环平衡点
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2021-01-01 DOI: 10.3934/mcrf.2021053
I. Alia
{"title":"Open-loop equilibriums for a general class of time-inconsistent stochastic optimal control problems","authors":"I. Alia","doi":"10.3934/mcrf.2021053","DOIUrl":"https://doi.org/10.3934/mcrf.2021053","url":null,"abstract":"This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As special cases, a mean-variance portfolio selection problem and a time-inconsistent problem under non-exponential discounting are discussed.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"59 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84840846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems 随机跳跃-扩散系统的线性二次平均场型stackelberg微分对策
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2021-01-01 DOI: 10.3934/MCRF.2021026
Jun Moon
{"title":"Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems","authors":"Jun Moon","doi":"10.3934/MCRF.2021026","DOIUrl":"https://doi.org/10.3934/MCRF.2021026","url":null,"abstract":"In this paper, we consider linear-quadratic (LQ) leader-follower Stackelberg differential games for mean-field type stochastic systems with jump diffusions, where the system includes mean-field variables, i.e., the expected value of state and control variables. We first solve the LQ mean-field type control problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the coupled integro-Riccati differential equations (CIRDEs) via the Four-Step Scheme. Next, we solve the problem of the leader, which is the LQ control problem subject to the mean-field type forward-backward stochastic system with jump diffusions, where the constraint characterizes the rational behavior of the follower. Using the variational approach, we obtain the (mean-field type) stochastic maximum principle. However, to obtain the state-feedback representation of the open-loop optimal solution of the leader, there is a technical challenge due to the jump process. We consider two different cases, in which the state-feedback type control in terms of the CIRDEs can be characterized by generalizing the Four-Step Scheme. We finally show that the state-feedback type controls of the open-loop optimal solutions for the leader and the follower constitute the Stackelberg equilibrium.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84913688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
A linear quadratic stochastic Stackelberg differential game with time delay 具有时滞的线性二次随机Stackelberg微分对策
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2020-12-28 DOI: 10.3934/mcrf.2021035
Weijun Meng, Jingtao Shi
{"title":"A linear quadratic stochastic Stackelberg differential game with time delay","authors":"Weijun Meng, Jingtao Shi","doi":"10.3934/mcrf.2021035","DOIUrl":"https://doi.org/10.3934/mcrf.2021035","url":null,"abstract":". This paper is concerned with a linear quadratic stochastic Stackelberg differential game with time delay. The model is general, in which the state delay and the control delay both appear in the state equation, moreover, they both enter into the diffusion term. By introducing two Pseudo-Riccati equations and a special matrix equation, the state feedback representation of the open-loop Stackelberg strategy is derived, under some assumptions. Fi- nally, two examples are given to illustrate the applications of the theoretical results.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"15 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82478987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Exact noise cancellation for 1d-acoustic propagation systems 一维声传播系统的精确噪声消除
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2020-10-19 DOI: 10.3934/MCRF.2020055
Jérôme Lohéac, C. Boultifat, P. Chevrel, M. Yagoubi
{"title":"Exact noise cancellation for 1d-acoustic propagation systems","authors":"Jérôme Lohéac, C. Boultifat, P. Chevrel, M. Yagoubi","doi":"10.3934/MCRF.2020055","DOIUrl":"https://doi.org/10.3934/MCRF.2020055","url":null,"abstract":"This paper deals with active noise control applied to a one-dimensional acoustic propagation system. The aim here is to keep over time a zero noise level at a given point. We aim to design this control using noise measurement at some point in the spatial domain. Based on symmetry property, we are able to design a feedback boundary control allowing this fact. Moreover, using D'Alembert formula, an explicit formula of the control can be computed. Even if the focus is made on the wave equation, this approach is easily extendable to more general operators.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"89 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75713500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Networks of geometrically exact beams: Well-posedness and stabilization 几何上精确的梁网:稳健性和稳定性
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2020-09-15 DOI: 10.3934/MCRF.2021002
C. Rodriguez
{"title":"Networks of geometrically exact beams: Well-posedness and stabilization","authors":"C. Rodriguez","doi":"10.3934/MCRF.2021002","DOIUrl":"https://doi.org/10.3934/MCRF.2021002","url":null,"abstract":"In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) -- in the sense that large motions (deflections, rotations) are accounted for in addition to shearing -- and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin & Coron in cite{BC2016} that the zero steady state of this network is exponentially stable for the $H^1$ and $H^2$ norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"73 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86379004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
State-constrained semilinear elliptic optimization problems with unrestricted sparse controls 无约束稀疏控制的状态约束半线性椭圆优化问题
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2020-09-01 DOI: 10.3934/mcrf.2020009
E. Casas, F. Tröltzsch
{"title":"State-constrained semilinear elliptic optimization problems with unrestricted sparse controls","authors":"E. Casas, F. Tröltzsch","doi":"10.3934/mcrf.2020009","DOIUrl":"https://doi.org/10.3934/mcrf.2020009","url":null,"abstract":"In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"81 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83943168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Sparse optimal control for the heat equation with mixed control-state constraints 具有混合控制状态约束的热方程稀疏最优控制
IF 1.2 4区 数学
Mathematical Control and Related Fields Pub Date : 2020-09-01 DOI: 10.3934/mcrf.2020007
E. Casas, F. Tröltzsch
{"title":"Sparse optimal control for the heat equation with mixed control-state constraints","authors":"E. Casas, F. Tröltzsch","doi":"10.3934/mcrf.2020007","DOIUrl":"https://doi.org/10.3934/mcrf.2020007","url":null,"abstract":"A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the begin{document}$ L^1 $end{document} -norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"78 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85527514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
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