SIAM Journal on Control and Optimization, Ahead of Print. Abstract. We consider the one-dimensional nonlinear Schrödinger equation with bilinear control. In the present paper, we study its local exact controllability near the ground state in the case of Dirichlet boundary conditions. To establish the controllability of the linearized equation, we use a bilinear control acting through four directions: three Fourier modes and one generic direction. The Fourier modes are appropriately chosen so that they satisfy a saturation property. These modes allow one to control approximately the linearzied Schrödinger equation. Then we show that the reachable set for the linearized equation is closed. This is achieved by representing the solution operator as a sum of two linear continuous mappings: one is surjective (here the control in generic direction is used) and the other is compact. A mapping with dense and closed image is surjective, so we conclude that the linearized Schrödinger equation is exactly controllable. The local exact controllability of the nonlinear equation then follows by the inverse mapping theorem.
{"title":"Local Exact Controllability of the One-Dimensional Nonlinear Schrödinger Equation in the Case of Dirichlet Boundary Conditions","authors":"Alessandro Duca, Vahagn Nersesyan","doi":"10.1137/23m1556034","DOIUrl":"https://doi.org/10.1137/23m1556034","url":null,"abstract":"SIAM Journal on Control and Optimization, Ahead of Print. <br/> Abstract. We consider the one-dimensional nonlinear Schrödinger equation with bilinear control. In the present paper, we study its local exact controllability near the ground state in the case of Dirichlet boundary conditions. To establish the controllability of the linearized equation, we use a bilinear control acting through four directions: three Fourier modes and one generic direction. The Fourier modes are appropriately chosen so that they satisfy a saturation property. These modes allow one to control approximately the linearzied Schrödinger equation. Then we show that the reachable set for the linearized equation is closed. This is achieved by representing the solution operator as a sum of two linear continuous mappings: one is surjective (here the control in generic direction is used) and the other is compact. A mapping with dense and closed image is surjective, so we conclude that the linearized Schrödinger equation is exactly controllable. The local exact controllability of the nonlinear equation then follows by the inverse mapping theorem.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2557-2589, October 2024. Abstract. We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as [math] by convention, but in terms of its conditional expectation [math] on a general subfiltration [math]. We thus term such a equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier and the recently developed BSDE with a mean reflection constraint as its two special and extreme cases: they exactly correspond to [math] being the full filtration to represent complete information and the degenerated filtration to deterministic scenario, respectively. For conditional RBSDE, we obtain its existence and uniqueness under mild conditions by combining the Snell envelope method with the Skorokhod lemma. We also discuss its connection, in the case of a linear driver, to a class of optimal stopping problems in the presence of partial information. As a by-product, a new version of the comparison theorem is obtained. With the help of this connection, we study weak formulations of a class of optimal control problems with reflected recursive functionals by characterizing the related optimal solution and value. Moreover, in the special case of recursive functionals being RBSDE with pointwise reflections, we study the strong formulations of related stochastic backward recursive control and zero-sum games, both in a non-Markovian framework, that are of their own interests and have not been fully explored by existing literature yet.
{"title":"Backward Stochastic Differential Equations with Conditional Reflection and Related Recursive Optimal Control Problems","authors":"Ying Hu, Jianhui Huang, Wenqiang Li","doi":"10.1137/22m1534985","DOIUrl":"https://doi.org/10.1137/22m1534985","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2557-2589, October 2024. <br/> Abstract. We introduce a new type of reflected backward stochastic differential equations (BSDEs) for which the reflection constraint is imposed on its main solution component, denoted as [math] by convention, but in terms of its conditional expectation [math] on a general subfiltration [math]. We thus term such a equation as conditionally reflected BSDE (for short, conditional RBSDE). Conditional RBSDE subsumes classical RBSDE with a pointwise reflection barrier and the recently developed BSDE with a mean reflection constraint as its two special and extreme cases: they exactly correspond to [math] being the full filtration to represent complete information and the degenerated filtration to deterministic scenario, respectively. For conditional RBSDE, we obtain its existence and uniqueness under mild conditions by combining the Snell envelope method with the Skorokhod lemma. We also discuss its connection, in the case of a linear driver, to a class of optimal stopping problems in the presence of partial information. As a by-product, a new version of the comparison theorem is obtained. With the help of this connection, we study weak formulations of a class of optimal control problems with reflected recursive functionals by characterizing the related optimal solution and value. Moreover, in the special case of recursive functionals being RBSDE with pointwise reflections, we study the strong formulations of related stochastic backward recursive control and zero-sum games, both in a non-Markovian framework, that are of their own interests and have not been fully explored by existing literature yet.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2590-2620, October 2024. Abstract. This paper is concerned with a long-standing optimal dividend payout problem subject to the so-called ratcheting constraint, that is, the dividend payout rate shall be nondecreasing over time and is thus self-path-dependent. The surplus process is modeled by a drifted Brownian motion process and the aim is to find the optimal dividend ratcheting strategy to maximize the expectation of the total discounted dividend payouts until the ruin time. Due to the self-path-dependent control constraint, the standard control theory cannot be directly applied to tackle the problem. The related Hamilton–Jacobi–Bellman (HJB) equation is a new type of variational inequality. In the literature, it is only shown to have a viscosity solution, which is not strong enough to guarantee the existence of an optimal dividend ratcheting strategy. This paper proposes a novel partial differential equation method to study the HJB equation. We not only prove the existence and uniqueness of the solution in some stronger functional space, but also prove the strict monotonicity, boundedness, and [math]-smoothness of the dividend ratcheting free boundary. Based on these results, we eventually derive an optimal dividend ratcheting strategy, and thus solve the open problem completely. Economically speaking, we find that if the surplus volatility is above an explicit threshold, then one should pay dividends at the maximum rate, regardless of the surplus level. Otherwise, by contrast, the optimal dividend ratcheting strategy relies on the surplus level and one should only ratchet up the dividend payout rate when the surplus level touches the dividend ratcheting free boundary. Moreover, our numerical results suggest that one should invest in those companies with stable dividend payout strategies since their income rates should be higher and volatility rates smaller.
{"title":"Optimal Ratcheting of Dividend Payout Under Brownian Motion Surplus","authors":"Chonghu Guan, Zuo Quan Xu","doi":"10.1137/23m159250x","DOIUrl":"https://doi.org/10.1137/23m159250x","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2590-2620, October 2024. <br/> Abstract. This paper is concerned with a long-standing optimal dividend payout problem subject to the so-called ratcheting constraint, that is, the dividend payout rate shall be nondecreasing over time and is thus self-path-dependent. The surplus process is modeled by a drifted Brownian motion process and the aim is to find the optimal dividend ratcheting strategy to maximize the expectation of the total discounted dividend payouts until the ruin time. Due to the self-path-dependent control constraint, the standard control theory cannot be directly applied to tackle the problem. The related Hamilton–Jacobi–Bellman (HJB) equation is a new type of variational inequality. In the literature, it is only shown to have a viscosity solution, which is not strong enough to guarantee the existence of an optimal dividend ratcheting strategy. This paper proposes a novel partial differential equation method to study the HJB equation. We not only prove the existence and uniqueness of the solution in some stronger functional space, but also prove the strict monotonicity, boundedness, and [math]-smoothness of the dividend ratcheting free boundary. Based on these results, we eventually derive an optimal dividend ratcheting strategy, and thus solve the open problem completely. Economically speaking, we find that if the surplus volatility is above an explicit threshold, then one should pay dividends at the maximum rate, regardless of the surplus level. Otherwise, by contrast, the optimal dividend ratcheting strategy relies on the surplus level and one should only ratchet up the dividend payout rate when the surplus level touches the dividend ratcheting free boundary. Moreover, our numerical results suggest that one should invest in those companies with stable dividend payout strategies since their income rates should be higher and volatility rates smaller.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2529-2556, October 2024. Abstract. We consider reinforcement learning for continuous-time Markov decision processes (MDPs) in the infinite-horizon, average-reward setting. In contrast to discrete-time MDPs, a continuous-time process moves to a state and stays there for a random holding time after an action is taken. With unknown transition probabilities and rates of exponential holding times, we derive instance-dependent regret lower bounds that are logarithmic in the time horizon. Moreover, we design a learning algorithm and establish a finite-time regret bound that achieves the logarithmic growth rate. Our analysis builds upon upper confidence reinforcement learning, a delicate estimation of the mean holding times, and stochastic comparison of point processes.
{"title":"Logarithmic Regret Bounds for Continuous-Time Average-Reward Markov Decision Processes","authors":"Xuefeng Gao, Xun Yu Zhou","doi":"10.1137/23m1584101","DOIUrl":"https://doi.org/10.1137/23m1584101","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2529-2556, October 2024. <br/> Abstract. We consider reinforcement learning for continuous-time Markov decision processes (MDPs) in the infinite-horizon, average-reward setting. In contrast to discrete-time MDPs, a continuous-time process moves to a state and stays there for a random holding time after an action is taken. With unknown transition probabilities and rates of exponential holding times, we derive instance-dependent regret lower bounds that are logarithmic in the time horizon. Moreover, we design a learning algorithm and establish a finite-time regret bound that achieves the logarithmic growth rate. Our analysis builds upon upper confidence reinforcement learning, a delicate estimation of the mean holding times, and stochastic comparison of point processes.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2506-2528, October 2024. Abstract. In this paper we establish a Lebeau–Robbiano spectral inequality for a degenerate one-dimensional elliptic operator, with an optimal dependency with the frequency parameter. The proof relies on a combination of uniform local Carleman estimates away from the degeneracy and a moment method adapted for a degenerate elliptic operator. We also provide an application to the null-controllability on a measurable set in time for the associated degenerate heat equation.
{"title":"An Optimal Spectral Inequality for Degenerate Operators","authors":"Rémi Buffe, Kim Dang Phung, Amine Slimani","doi":"10.1137/23m1605211","DOIUrl":"https://doi.org/10.1137/23m1605211","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2506-2528, October 2024. <br/> Abstract. In this paper we establish a Lebeau–Robbiano spectral inequality for a degenerate one-dimensional elliptic operator, with an optimal dependency with the frequency parameter. The proof relies on a combination of uniform local Carleman estimates away from the degeneracy and a moment method adapted for a degenerate elliptic operator. We also provide an application to the null-controllability on a measurable set in time for the associated degenerate heat equation.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2475-2505, October 2024. Abstract. We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean–Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the “fast” variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose “fast” component converges to the optimal control process, while the “slow” component converges to the optimal state process. The interest in such a procedure is that it allows one to approximate the solution of the control problem, avoiding the usual step of the minimization of the Hamiltonian.
{"title":"A Tikhonov Theorem for McKean–Vlasov Two-Scale Systems and a New Application to Mean Field Optimal Control Problems","authors":"Matteo Burzoni, Alekos Cecchin, Andrea Cosso","doi":"10.1137/22m1543070","DOIUrl":"https://doi.org/10.1137/22m1543070","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2475-2505, October 2024. <br/> Abstract. We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean–Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the “fast” variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose “fast” component converges to the optimal control process, while the “slow” component converges to the optimal state process. The interest in such a procedure is that it allows one to approximate the solution of the control problem, avoiding the usual step of the minimization of the Hamiltonian.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Umberto De Maio, Antonio Gaudiello, Cătălin-George Lefter
SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2456-2474, October 2024. Abstract. This paper is devoted to studying the null internal controllability of a Kirchoff–Love thin plate with a middle surface having a comb-like shaped structure with a large number of thin fingers described by a small positive parameter [math]. It is often impossible to directly approach such a problem numerically, due to the large number of thin fingers. So an asymptotic analysis is needed. In this paper, we first prove that the problem is null controllable at each level [math]. We then prove that the sequence of the respective controls with minimal [math] norm converges, as [math] vanishes, to a limit control function ensuring the optimal null controllability of a degenerate limit problem set in a domain without fingers.
{"title":"Null Internal Controllability for a Kirchhoff–Love Plate with a Comb-Like Shaped Structure","authors":"Umberto De Maio, Antonio Gaudiello, Cătălin-George Lefter","doi":"10.1137/24m1647825","DOIUrl":"https://doi.org/10.1137/24m1647825","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2456-2474, October 2024. <br/> Abstract. This paper is devoted to studying the null internal controllability of a Kirchoff–Love thin plate with a middle surface having a comb-like shaped structure with a large number of thin fingers described by a small positive parameter [math]. It is often impossible to directly approach such a problem numerically, due to the large number of thin fingers. So an asymptotic analysis is needed. In this paper, we first prove that the problem is null controllable at each level [math]. We then prove that the sequence of the respective controls with minimal [math] norm converges, as [math] vanishes, to a limit control function ensuring the optimal null controllability of a degenerate limit problem set in a domain without fingers.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2433-2455, October 2024. Abstract. We study optimal stochastic control problems for fully coupled forward-backward stochastic differential equations driven by anomalous subdiffusion, which have nontrivial mixed features of deterministic and stochastic controls. Both the stochastic maximum principle (SMP) and sufficient SMP are obtained by using a convex variational method. The paper ends with an application of the main results of this paper to a linear quadratic problem in the subdiffusive setting, which is solved explicitly.
{"title":"Stochastic Maximum Principle for Fully Coupled Forward-Backward Stochastic Differential Equations Driven by Subdiffusion","authors":"Shuaiqi Zhang, Zhen-Qing Chen","doi":"10.1137/23m1620168","DOIUrl":"https://doi.org/10.1137/23m1620168","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2433-2455, October 2024. <br/> Abstract. We study optimal stochastic control problems for fully coupled forward-backward stochastic differential equations driven by anomalous subdiffusion, which have nontrivial mixed features of deterministic and stochastic controls. Both the stochastic maximum principle (SMP) and sufficient SMP are obtained by using a convex variational method. The paper ends with an application of the main results of this paper to a linear quadratic problem in the subdiffusive setting, which is solved explicitly.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2412-2432, August 2024. Abstract. The title of the present work is a nod to the paper “The hybrid maximum principle is a consequence of Pontryagin maximum principle” by Dmitruk and Kaganovich [Systems Control Lett., 57 (2008), pp. 964–970]. We investigate a similar framework of hybrid optimal control problems that is also different from Dmitruk and Kaganovich’s. Precisely, we consider a general control system that is described by a differential equation involving a spatially heterogeneous dynamics. In that context, the sequence of dynamics followed by the trajectory and the corresponding switching times are fully constrained by the state position. We prove with an explicit counterexample that the augmentation technique used by Dmitruk and Kaganovich cannot be fully applied to our setting, but we show that it can be adapted by introducing a new notion of local solution to classical optimal control problems and by establishing a corresponding Pontryagin maximum principle. Thanks to this method, we derive a hybrid maximum principle adapted to our setting, with a simple proof that does not require any technical tools (such as implicit function arguments) to handle the dynamical discontinuities.
SIAM 控制与优化期刊》第 62 卷第 4 期第 2412-2432 页，2024 年 8 月。 摘要。本论文的标题是对 Dmitruk 和 Kaganovich 的论文 "The hybrid maximum principle is a consequence of Pontryagin maximum principle" [Systems Control Lett.我们研究了一个类似的混合最优控制问题框架，它也不同于 Dmitruk 和 Kaganovich 的研究。确切地说，我们考虑的是由涉及空间异质动力学的微分方程描述的一般控制系统。在这种情况下，轨迹所遵循的动力学序列和相应的切换时间完全受状态位置的约束。我们通过一个明确的反例证明，德米特鲁克和卡加诺维奇使用的增强技术不能完全适用于我们的环境，但我们通过引入经典最优控制问题的局部解这一新概念，并建立相应的庞特里亚金最大原则，证明了这一技术是可以调整的。由于采用了这种方法，我们推导出了适合我们的混合最大原理，其证明简单，不需要任何技术工具（如隐函数参数）来处理动态不连续性。
{"title":"The Hybrid Maximum Principle for Optimal Control Problems with Spatially Heterogeneous Dynamics is a Consequence of a Pontryagin Maximum Principle for [math]-Local Solutions","authors":"Térence Bayen, Anas Bouali, Loïc Bourdin","doi":"10.1137/23m155311x","DOIUrl":"https://doi.org/10.1137/23m155311x","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2412-2432, August 2024. <br/> Abstract. The title of the present work is a nod to the paper “The hybrid maximum principle is a consequence of Pontryagin maximum principle” by Dmitruk and Kaganovich [Systems Control Lett., 57 (2008), pp. 964–970]. We investigate a similar framework of hybrid optimal control problems that is also different from Dmitruk and Kaganovich’s. Precisely, we consider a general control system that is described by a differential equation involving a spatially heterogeneous dynamics. In that context, the sequence of dynamics followed by the trajectory and the corresponding switching times are fully constrained by the state position. We prove with an explicit counterexample that the augmentation technique used by Dmitruk and Kaganovich cannot be fully applied to our setting, but we show that it can be adapted by introducing a new notion of local solution to classical optimal control problems and by establishing a corresponding Pontryagin maximum principle. Thanks to this method, we derive a hybrid maximum principle adapted to our setting, with a simple proof that does not require any technical tools (such as implicit function arguments) to handle the dynamical discontinuities.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fritz Colonius, Alexandre J. Santana, Eduardo C. Viscovini
SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2387-2411, August 2024. Abstract. For linear control systems with bounded control range, chain controllability properties are analyzed. It is shown that there exists a unique chain control set and that it equals the sum of the control set around the origin and the center Lyapunov space of the homogeneous part. For the proof, the linear control system is extended to a bilinear control system on an augmented state space. This system induces a control system on projective space. For the associated control flow, attractor-repeller decompositions are used to show that the control system on projective space has a unique chain control set that is not contained in the equator. It is given by the image of the chain control set of the original linear control system.
{"title":"Chain Controllability of Linear Control Systems","authors":"Fritz Colonius, Alexandre J. Santana, Eduardo C. Viscovini","doi":"10.1137/23m1626347","DOIUrl":"https://doi.org/10.1137/23m1626347","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2387-2411, August 2024. <br/> Abstract. For linear control systems with bounded control range, chain controllability properties are analyzed. It is shown that there exists a unique chain control set and that it equals the sum of the control set around the origin and the center Lyapunov space of the homogeneous part. For the proof, the linear control system is extended to a bilinear control system on an augmented state space. This system induces a control system on projective space. For the associated control flow, attractor-repeller decompositions are used to show that the control system on projective space has a unique chain control set that is not contained in the equator. It is given by the image of the chain control set of the original linear control system.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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