We investigate local optimality conditions of first and second order for integer optimal control problems with total variation regularization via a finite-dimensional switching-point problem. We show the equivalence of local optimality for both problems, which will be used to derive conditions concerning the switching points of the control function. A non-local optimality condition treating back-and-forth switches will be formulated. For the numerical solution, we propose a proximal-gradient method. The emerging discretized subproblems will be solved by employing Bellman’s optimality principle, leading to an algorithm which is polynomial in the mesh size and in the admissible control levels. An adaption of this algorithm can be used to handle subproblems of the trust-region method proposed in Leyffer and Manns, ESAIM: Control Optim. Calc. Var. 28 (2022) 66. Finally, we demonstrate computational results.
通过有限维切换点问题，研究了具有全变分正则化的整数最优控制问题的一阶和二阶局部最优性条件。我们证明了这两个问题的局部最优性的等价性，这将用于导出关于控制函数的开关点的条件。一个处理来回切换的非局部最优性条件将被提出。对于数值解，我们提出了一种近似梯度法。采用Bellman最优性原理对出现的离散子问题进行求解，得到一个在网格大小和允许控制水平上都是多项式的算法。该算法的改进可用于处理Leyffer和Manns在ESAIM: Control Optim中提出的信任域方法的子问题。中国生物医学工程学报，28 (2022)最后，给出了计算结果。
{"title":"Integer optimal control problems with total variation regularization: Optimality conditions and fast solution of subproblems","authors":"Jonas Marko, Gerd Wachsmuth","doi":"10.1051/cocv/2023065","DOIUrl":"https://doi.org/10.1051/cocv/2023065","url":null,"abstract":"We investigate local optimality conditions of first and second order for integer optimal control problems with total variation regularization via a finite-dimensional switching-point problem. We show the equivalence of local optimality for both problems, which will be used to derive conditions concerning the switching points of the control function. A non-local optimality condition treating back-and-forth switches will be formulated. For the numerical solution, we propose a proximal-gradient method. The emerging discretized subproblems will be solved by employing Bellman’s optimality principle, leading to an algorithm which is polynomial in the mesh size and in the admissible control levels. An adaption of this algorithm can be used to handle subproblems of the trust-region method proposed in Leyffer and Manns, ESAIM: Control Optim. Calc. Var. 28 (2022) 66. Finally, we demonstrate computational results.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"95 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This manuscript aims to study finite horizon, first order Hamilton Jacobi Bellman (HJB) equations on stratified domains. This problem is related to optimal control problems with discontinuous dynamics. We use nonsmooth analysis techniques to derive a strong comparison principle as in the classical theory and deduce that the value function is the unique viscosity solution. Furthermore, we prove some stability results of the Hamilton Jacobi Bellman equation. Finally, we establish a general convergence result for monotone numerical schemes in the stratified case.
{"title":"A general comparison principle for Hamilton Jacobi Bellman equations on stratified domains 1\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 ","authors":"H. Zidani, O. Jerhaoui","doi":"10.1051/cocv/2022089","DOIUrl":"https://doi.org/10.1051/cocv/2022089","url":null,"abstract":"This manuscript aims to study finite horizon, first order Hamilton Jacobi Bellman (HJB) equations on stratified domains. This problem is related to optimal control problems with discontinuous dynamics. We use nonsmooth analysis techniques to derive a strong comparison principle as in the classical theory and deduce that the value function is the unique viscosity solution. Furthermore, we prove some stability results of the Hamilton Jacobi Bellman equation. Finally, we establish a general convergence result for monotone numerical schemes in the stratified case.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"31 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85857844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic.
{"title":"Homogeneous geodesics in sub-Riemannian geometry","authors":"A. Podobryaev","doi":"10.1051/cocv/2022086","DOIUrl":"https://doi.org/10.1051/cocv/2022086","url":null,"abstract":"We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"45 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88499097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate a class of mean field games containing a large number of major and minor players. Each player minimizes a quadratic-tracking type risk-sensitive cost functional, where the reference signal is a function of the state average term of the major and minor players. To reduce the complexity for solving the problem, we design a sequence of decentralized strategies by the Nash certainty equivalence principle. Firstly, for the optimal control problems with quadratic type risksensitive cost functionals, we propose a new verification theorem. Secondly, we apply the two-layer state aggregation method to construct the fixed-point equations for the estimations of the state average terms and give the conditions for the existence and uniqueness of the fixed points. Then, we design a sequence of decentralized strategies by the estimations of the state average terms based on local information. It is shown that the estimations of the state average terms are consistent with the true values for the closed-loop systems, and the sequence of strategies designed is a decentralized asymptotic Nash equilibrium. Finally, the effectiveness of the theoretical analysis is demonstrated by a numerical example.
{"title":"Risk-sensitive mean field games with major and minor players","authors":"Yan Chen, Taoying Li, Zhixian Xin","doi":"10.1051/cocv/2022082","DOIUrl":"https://doi.org/10.1051/cocv/2022082","url":null,"abstract":"We investigate a class of mean field games containing a large number of major and minor players. Each player minimizes a quadratic-tracking type risk-sensitive cost functional, where the reference signal is a function of the state average term of the major and minor players. To reduce the complexity for solving the problem, we design a sequence of decentralized strategies by the Nash certainty equivalence principle. Firstly, for the optimal control problems with quadratic type risksensitive cost functionals, we propose a new verification theorem. Secondly, we apply the two-layer state aggregation method to construct the fixed-point equations for the estimations of the state average terms and give the conditions for the existence and uniqueness of the fixed points. Then, we design a sequence of decentralized strategies by the estimations of the state average terms based on local information. It is shown that the estimations of the state average terms are consistent with the true values for the closed-loop systems, and the sequence of strategies designed is a decentralized asymptotic Nash equilibrium. Finally, the effectiveness of the theoretical analysis is demonstrated by a numerical example.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"17 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79105288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg–de Vries (KdV)–Benjamin–Bona–Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [L. Rosier and B.-Y. Zhang, J. Diff. Equ. 254 (2013) 141–178], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.
{"title":"Unique Continuation and Time Decay for a Higher-Order Water Wave Model","authors":"A. Pazoto, M. Soto","doi":"10.1051/cocv/2023040","DOIUrl":"https://doi.org/10.1051/cocv/2023040","url":null,"abstract":"This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg–de Vries (KdV)–Benjamin–Bona–Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in [L. Rosier and B.-Y. Zhang, J. Diff. Equ. 254 (2013) 141–178], which combines energy estimates, multipliers and compactness arguments, the problem is reduced to prove the Unique Continuation Property (UCP) for weak solutions of the model. Then, this is done by deriving Carleman estimates for a system of coupled elliptic-hyperbolic equations.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"8 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73311844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.
{"title":"Insensitizing control for linear and semi-linear heat equations with partially unknown domain","authors":"P. Lissy, Y. Privat, Y. Simporé","doi":"10.1051/COCV/2018035","DOIUrl":"https://doi.org/10.1051/COCV/2018035","url":null,"abstract":"We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83016116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization.
{"title":"Kernel representation of Kalman observer and associated H-matrix based discretization","authors":"M. Aussal, P. Moireau","doi":"10.1051/cocv/2022071","DOIUrl":"https://doi.org/10.1051/cocv/2022071","url":null,"abstract":"In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"216 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89095581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies an equivalence theorem for three different kinds of optimal control problems,which are optimal time control problems, optimal norm control problems, and optimal target control problems. The controlled systems in this paper are internally controlled linear heat equations with memory.
{"title":"Equivalence of three kinds of optimal control problems for linear heat equations with memory","authors":"Lijuan Wang, Xiuxiang Zhou","doi":"10.1051/cocv/2022072","DOIUrl":"https://doi.org/10.1051/cocv/2022072","url":null,"abstract":"This paper studies an equivalence theorem for three different kinds of optimal control problems,\u0000which are optimal time control problems, optimal norm control problems, and optimal target control problems. The controlled systems in this paper are internally controlled linear heat equations with memory.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"10 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77970400","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ensemble optimal control problems governed by a Fokker–Planck equation with space–time dependent controls are investigated. These problems require the minimisation of objective functionals of probability type and aim at determining robust control mechanisms for the ensemble of trajectories of the stochastic system defining the Fokker–Planck model. In this work, existence of optimal controls is proved and a detailed analysis of their characterization by first– and second–order optimality conditions is presented. For this purpose, the well–posedness of the Fokker–Planck equation, and new estimates concerning an inhomogeneous Fokker–Planck model are discussed, which are essential to prove the necessary regularity and compactness of the control–to–state map appearing in the first–and second–order analysis.
{"title":"Second–order analysis of Fokker–Planck ensemble optimal control problems","authors":"Jacob Körner, A. Borzì","doi":"10.1051/cocv/2022066","DOIUrl":"https://doi.org/10.1051/cocv/2022066","url":null,"abstract":"Ensemble optimal control problems governed by a Fokker–Planck equation with space–time dependent controls are investigated. These problems require the minimisation of objective functionals of probability type and aim at determining robust control mechanisms for the ensemble of trajectories of the stochastic system defining the Fokker–Planck model. In this work, existence of optimal controls is proved and a detailed analysis of their characterization by first– and second–order optimality conditions is presented. For this purpose, the well–posedness of the Fokker–Planck equation, and new estimates concerning an inhomogeneous Fokker–Planck model are discussed, which are essential to prove the necessary regularity and compactness of the control–to–state map appearing in the first–and second–order analysis.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"33 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89910880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.
{"title":"Probabilistic representation of viscosity solutions to quasi-variational inequalities with non-local drivers","authors":"M. Perninge","doi":"10.1051/cocv/2023015","DOIUrl":"https://doi.org/10.1051/cocv/2023015","url":null,"abstract":"We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"88 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80920044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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