SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1152-1164, April 2024. Abstract. We consider the infinite horizon optimal control problems of the controlled Markov process. We verify the relationship between the controlled Markov process and its fluid limit by the viscosity solution approach. More precisely, we show that the value function of the controlled Markov process converges to one of its limit processes which is the viscosity solution of the associated Hamilton–Jacobi–Bellman equation.
{"title":"Markov Chain Approximation for Hamilton–Jacobi–Bellman Equation with Absorbing Boundary","authors":"Itsuki Watanabe","doi":"10.1137/23m1565723","DOIUrl":"https://doi.org/10.1137/23m1565723","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1152-1164, April 2024. <br/> Abstract. We consider the infinite horizon optimal control problems of the controlled Markov process. We verify the relationship between the controlled Markov process and its fluid limit by the viscosity solution approach. More precisely, we show that the value function of the controlled Markov process converges to one of its limit processes which is the viscosity solution of the associated Hamilton–Jacobi–Bellman equation.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140324395","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 2, Page 1122-1151, April 2024. Abstract. Currently, output-feedback control still necessitates severe constraints on systems, e.g., system nonlinearities cannot exceed certain degree and uncertainties should belong to specific types. In this paper, by exploiting dynamic-compensation mechanisms, we essentially extend system nonlinearities and uncertainties. Specifically, the nonlinearities heavily rely on unmeasured states and particularly have unknown arbitrary function-of-output growth rates. Unknown control coefficients whether with known or unknown bounds are admitted, which have been excluded before in the context of such inclusive nonlinearities. The key to our novel solution lies in realizing the potential of filter-based observers, dynamic high gains, design/analysis parameter designation, and composite Lyapunov functions. In detail, two dynamic-high-gain filters are worked out to provide available states for controller design. The filter states, after weighted by the unknown control coefficient, also make up the estimated states which lead to control-free and tractable error dynamics. Two dynamic high gains with new dynamics are put forward to counteract the nonlinearities and uncertainties and, meanwhile, to enable the adaptive controller to own a concise structure. During the controller design, crucial design parameters can no longer be expressed explicitly due to unknown control coefficients, but rather need to be pursued through a recursive algorithm. With a set of analysis parameters, important (dynamic-high-gain) input-to-state stable properties of some vital variables are uncovered, and exhaustive Lyapunov analysis is performed for the closed-loop boundedness and convergence.
{"title":"Global Output-Feedback Control by Exploiting High-Gain Dynamic-Compensation Mechanisms","authors":"Yuan Wang, Yungang Liu","doi":"10.1137/22m1536303","DOIUrl":"https://doi.org/10.1137/22m1536303","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1122-1151, April 2024. <br/> Abstract. Currently, output-feedback control still necessitates severe constraints on systems, e.g., system nonlinearities cannot exceed certain degree and uncertainties should belong to specific types. In this paper, by exploiting dynamic-compensation mechanisms, we essentially extend system nonlinearities and uncertainties. Specifically, the nonlinearities heavily rely on unmeasured states and particularly have unknown arbitrary function-of-output growth rates. Unknown control coefficients whether with known or unknown bounds are admitted, which have been excluded before in the context of such inclusive nonlinearities. The key to our novel solution lies in realizing the potential of filter-based observers, dynamic high gains, design/analysis parameter designation, and composite Lyapunov functions. In detail, two dynamic-high-gain filters are worked out to provide available states for controller design. The filter states, after weighted by the unknown control coefficient, also make up the estimated states which lead to control-free and tractable error dynamics. Two dynamic high gains with new dynamics are put forward to counteract the nonlinearities and uncertainties and, meanwhile, to enable the adaptive controller to own a concise structure. During the controller design, crucial design parameters can no longer be expressed explicitly due to unknown control coefficients, but rather need to be pursued through a recursive algorithm. With a set of analysis parameters, important (dynamic-high-gain) input-to-state stable properties of some vital variables are uncovered, and exhaustive Lyapunov analysis is performed for the closed-loop boundedness and convergence.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314814","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 2, Page 1093-1121, April 2024. Abstract. In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs two groups of interacting particles, one which performs a minimization over one variable while the other performs a maximization over the other variable. The two groups constantly exchange information through a suitably weighted average. This paradigm permits a passage to the mean-field limit, which makes the method amenable to theoretical analysis, and it allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, which most notably include nonconvex-nonconcave objectives. We further provide numerical evidence for the success of the algorithm.
{"title":"Consensus-Based Optimization for Saddle Point Problems","authors":"Hui Huang, Jinniao Qiu, Konstantin Riedl","doi":"10.1137/22m1543367","DOIUrl":"https://doi.org/10.1137/22m1543367","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1093-1121, April 2024. <br/> Abstract. In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs two groups of interacting particles, one which performs a minimization over one variable while the other performs a maximization over the other variable. The two groups constantly exchange information through a suitably weighted average. This paradigm permits a passage to the mean-field limit, which makes the method amenable to theoretical analysis, and it allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, which most notably include nonconvex-nonconcave objectives. We further provide numerical evidence for the success of the algorithm.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140298737","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}
Michael Giegrich, Christoph Reisinger, Yufei Zhang
SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1060-1092, April 2024. Abstract. We study the global linear convergence of policy gradient (PG) methods for finite-horizon continuous-time exploratory linear-quadratic control (LQC) problems. The setting includes stochastic LQC problems with indefinite costs and allows additional entropy regularizers in the objective. We consider a continuous-time Gaussian policy whose mean is linear in the state variable and whose covariance is state-independent. Contrary to discrete-time problems, the cost is noncoercive in the policy and not all descent directions lead to bounded iterates. We propose geometry-aware gradient descents for the mean and covariance of the policy using the Fisher geometry and the Bures–Wasserstein geometry, respectively. The policy iterates are shown to satisfy an a priori bound, and converge globally to the optimal policy with a linear rate. We further propose a novel PG method with discrete-time policies. The algorithm leverages the continuous-time analysis, and achieves a robust linear convergence across different action frequencies. A numerical experiment confirms the convergence and robustness of the proposed algorithm.
{"title":"Convergence of Policy Gradient Methods for Finite-Horizon Exploratory Linear-Quadratic Control Problems","authors":"Michael Giegrich, Christoph Reisinger, Yufei Zhang","doi":"10.1137/22m1533517","DOIUrl":"https://doi.org/10.1137/22m1533517","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1060-1092, April 2024. <br/> Abstract. We study the global linear convergence of policy gradient (PG) methods for finite-horizon continuous-time exploratory linear-quadratic control (LQC) problems. The setting includes stochastic LQC problems with indefinite costs and allows additional entropy regularizers in the objective. We consider a continuous-time Gaussian policy whose mean is linear in the state variable and whose covariance is state-independent. Contrary to discrete-time problems, the cost is noncoercive in the policy and not all descent directions lead to bounded iterates. We propose geometry-aware gradient descents for the mean and covariance of the policy using the Fisher geometry and the Bures–Wasserstein geometry, respectively. The policy iterates are shown to satisfy an a priori bound, and converge globally to the optimal policy with a linear rate. We further propose a novel PG method with discrete-time policies. The algorithm leverages the continuous-time analysis, and achieves a robust linear convergence across different action frequencies. A numerical experiment confirms the convergence and robustness of the proposed algorithm.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140205599","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 2, Page 1034-1059, April 2024. Abstract. This paper studies the linearization of the viscous tank–liquid system. The linearization of the tank–liquid system gives a high-order partial differential equation, which is a combination of a wave equation with Kelvin–Voigt damping and a Euler–Bernoulli beam equation. The single input appears in two of the boundary conditions (boundary input). The paper provides results both for the open-loop system (existence/uniqueness of solutions and stability properties of the open-loop system) as well as results for the construction of feedback stabilizers. More specifically, the feedback design methodology is based on control Lyapunov functionals (CLFs). The proposed CLFs are modifications and augmentations of the total energy functionals for the tank–liquid system so that the dissipative effects of viscosity, friction, and surface tension are captured. By focusing on the linearized water–tank system, we are able to provide results that are not provided in the nonlinear case: (1) existence and uniqueness of solutions, (2) simultaneous presence of friction and surface tension, and (3) stabilization in a stronger norm, using a different CLF.
{"title":"Control of a Linearized Viscous Liquid–Tank System with Surface Tension","authors":"Iasson Karafyllis, Miroslav Krstic","doi":"10.1137/23m158749x","DOIUrl":"https://doi.org/10.1137/23m158749x","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1034-1059, April 2024. <br/> Abstract. This paper studies the linearization of the viscous tank–liquid system. The linearization of the tank–liquid system gives a high-order partial differential equation, which is a combination of a wave equation with Kelvin–Voigt damping and a Euler–Bernoulli beam equation. The single input appears in two of the boundary conditions (boundary input). The paper provides results both for the open-loop system (existence/uniqueness of solutions and stability properties of the open-loop system) as well as results for the construction of feedback stabilizers. More specifically, the feedback design methodology is based on control Lyapunov functionals (CLFs). The proposed CLFs are modifications and augmentations of the total energy functionals for the tank–liquid system so that the dissipative effects of viscosity, friction, and surface tension are captured. By focusing on the linearized water–tank system, we are able to provide results that are not provided in the nonlinear case: (1) existence and uniqueness of solutions, (2) simultaneous presence of friction and surface tension, and (3) stabilization in a stronger norm, using a different CLF.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140201735","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}
Lin Lin, James Lam, Min Meng, Xiaochen Xie, Panshuo Li, Daotong Zhang, Peng Shi
SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1006-1033, April 2024. Abstract.This paper investigates the sampled-data stabilization of continuous-time probabilistic logical control networks (CT-PLCNs). CT-PLCNs can provide quantitative and accurate descriptions for the transient kinetics in comparing discrete-time probabilistic logical control networks (DT-PLCNs). First, CT-PLCNs are transformed into switched continuous-time probabilistic logical networks by regarding the control input as a switching signal. In this setup, CT-PLCNs can be classified into two types: one with stable modes and the other with only unstable modes. Then the concept of average [math]-sample dwell time is proposed to describe the scenario, where the dwell time of each mode is an integral multiple of the sampling period [math]. Based on this, the stabilization conditions for CT-PLCNs are established by restricting the sampling dwell time of controller modes. Furthermore, a copositive Lyapunov function is constructed for the case with stable modes and is discretized for the case without stable modes, providing a new framework for studying the stabilization of CT-PLCNs. Finally, a chemical model generated by GINsim is provided to demonstrate the feasibility of the obtained theoretical results. Overall, this paper provides new insights into the stabilization of CT-PLCNs and presents practical applications for chemical models.
{"title":"Stabilization of Continuous-Time Probabilistic Logical Networks Under Sampling Dwell Time Constraints","authors":"Lin Lin, James Lam, Min Meng, Xiaochen Xie, Panshuo Li, Daotong Zhang, Peng Shi","doi":"10.1137/23m1566388","DOIUrl":"https://doi.org/10.1137/23m1566388","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1006-1033, April 2024. <br/>Abstract.This paper investigates the sampled-data stabilization of continuous-time probabilistic logical control networks (CT-PLCNs). CT-PLCNs can provide quantitative and accurate descriptions for the transient kinetics in comparing discrete-time probabilistic logical control networks (DT-PLCNs). First, CT-PLCNs are transformed into switched continuous-time probabilistic logical networks by regarding the control input as a switching signal. In this setup, CT-PLCNs can be classified into two types: one with stable modes and the other with only unstable modes. Then the concept of average [math]-sample dwell time is proposed to describe the scenario, where the dwell time of each mode is an integral multiple of the sampling period [math]. Based on this, the stabilization conditions for CT-PLCNs are established by restricting the sampling dwell time of controller modes. Furthermore, a copositive Lyapunov function is constructed for the case with stable modes and is discretized for the case without stable modes, providing a new framework for studying the stabilization of CT-PLCNs. Finally, a chemical model generated by GINsim is provided to demonstrate the feasibility of the obtained theoretical results. Overall, this paper provides new insights into the stabilization of CT-PLCNs and presents practical applications for chemical models.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140298816","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}
Sebastian Jaimungal, Silvana M. Pesenti, Leandro Sánchez-Betancourt
SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 982-1005, April 2024. Abstract. Given an [math]-dimensional stochastic process [math] driven by [math]-Brownian motions and Poisson random measures, we search for a probability measure [math], with minimal relative entropy to [math], such that the [math]-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterize the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting—including an algorithm to simulate under the optimal measure—and explore an example where an agent seeks to answer the question what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process?
{"title":"Minimal Kullback–Leibler Divergence for Constrained Lévy–Itô Processes","authors":"Sebastian Jaimungal, Silvana M. Pesenti, Leandro Sánchez-Betancourt","doi":"10.1137/23m1555697","DOIUrl":"https://doi.org/10.1137/23m1555697","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 982-1005, April 2024. <br/> Abstract. Given an [math]-dimensional stochastic process [math] driven by [math]-Brownian motions and Poisson random measures, we search for a probability measure [math], with minimal relative entropy to [math], such that the [math]-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterize the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary variance, and show that pinned measures arise as a limiting case of optimal measures. The results are illustrated in a risk management setting—including an algorithm to simulate under the optimal measure—and explore an example where an agent seeks to answer the question what dynamics are induced by a perturbation of the Value-at-Risk and the average time spent below a barrier on the reference process?","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140168841","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 2, Page 953-981, April 2024. Abstract. In this paper, we study optimal stochastic control problems for stochastic systems driven by non-Markov subdiffusion [math], which have mixed features of deterministic and stochastic controls. Here [math] is the standard Brownian motion on [math], and [math] is the inverse of a subordinator [math] with drift [math] that is independent of [math]. We obtain stochastic maximum principles (SMPs) for these systems using both convex and spiking variational methods, depending on whether or not the domain is convex. To derive SMPs, we first establish a martingale representation theorem for subdiffusions [math], and then use it to derive the existence and uniqueness result for the solutions of backward stochastic differential equations (BSDEs) driven by subdiffusions, which may be of independent interest. We also derive sufficient SMPs. Application to a linear quadratic system is given to illustrate the main results of this paper.
{"title":"Stochastic Maximum Principle for Subdiffusions and Its Applications","authors":"Shuaiqi Zhang, Zhen-Qing Chen","doi":"10.1137/23m157168x","DOIUrl":"https://doi.org/10.1137/23m157168x","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 953-981, April 2024. <br/> Abstract. In this paper, we study optimal stochastic control problems for stochastic systems driven by non-Markov subdiffusion [math], which have mixed features of deterministic and stochastic controls. Here [math] is the standard Brownian motion on [math], and [math] is the inverse of a subordinator [math] with drift [math] that is independent of [math]. We obtain stochastic maximum principles (SMPs) for these systems using both convex and spiking variational methods, depending on whether or not the domain is convex. To derive SMPs, we first establish a martingale representation theorem for subdiffusions [math], and then use it to derive the existence and uniqueness result for the solutions of backward stochastic differential equations (BSDEs) driven by subdiffusions, which may be of independent interest. We also derive sufficient SMPs. Application to a linear quadratic system is given to illustrate the main results of this paper.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153966","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 2, Page 924-952, April 2024. Abstract. This paper provides some sufficient conditions for the existence and uniqueness and the stochastic stability of the global solution for nonlinear neutral stochastic functional differential equations. When the drift term and the diffusion term satisfy a locally Lipschitz condition, and the Lyapunov monotonicity condition has a sign-changed time-varying coefficient, the existence and uniqueness of the global solution for such equations will be studied by using the Lyapunov–Krasovskii function approach and the theory of stochastic analysis. The stability in [math]th-moment, the asymptotical stability in [math]th-moment, and the exponential stability in [math]th-moment will be investigated. Different characterizations for these three kinds of stochastic stability in moment will be established, which are presented with respect to integration conditions. These results have seldom been reported in the existing literature. The almost surely exponential stability for the global solution of such equations is also discussed. Some discussions and comparisons are provided. Two examples are given to check the effectiveness of the theoretical results obtained.
{"title":"Stability Analysis for Nonlinear Neutral Stochastic Functional Differential Equations","authors":"Huabin Chen, Chenggui Yuan","doi":"10.1137/22m1523066","DOIUrl":"https://doi.org/10.1137/22m1523066","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 924-952, April 2024. <br/> Abstract. This paper provides some sufficient conditions for the existence and uniqueness and the stochastic stability of the global solution for nonlinear neutral stochastic functional differential equations. When the drift term and the diffusion term satisfy a locally Lipschitz condition, and the Lyapunov monotonicity condition has a sign-changed time-varying coefficient, the existence and uniqueness of the global solution for such equations will be studied by using the Lyapunov–Krasovskii function approach and the theory of stochastic analysis. The stability in [math]th-moment, the asymptotical stability in [math]th-moment, and the exponential stability in [math]th-moment will be investigated. Different characterizations for these three kinds of stochastic stability in moment will be established, which are presented with respect to integration conditions. These results have seldom been reported in the existing literature. The almost surely exponential stability for the global solution of such equations is also discussed. Some discussions and comparisons are provided. Two examples are given to check the effectiveness of the theoretical results obtained.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140057298","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 2, Page 903-923, April 2024. Abstract. An optimal control problem in the space of probability measures and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a smooth Fourier–Wasserstein metric. A comparison result between the Lipschitz viscosity sub- and supersolutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution.
{"title":"Viscosity Solutions for McKean–Vlasov Control on a Torus","authors":"H. Mete Soner, Qinxin Yan","doi":"10.1137/22m1543732","DOIUrl":"https://doi.org/10.1137/22m1543732","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 903-923, April 2024. <br/> Abstract. An optimal control problem in the space of probability measures and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a smooth Fourier–Wasserstein metric. A comparison result between the Lipschitz viscosity sub- and supersolutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140046769","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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