SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2367-2386, August 2024. Abstract. In deterministic and stochastic control theory, relaxed or randomized control policies allow for versatile mathematical analysis (on continuity, compactness, convexity, and approximations) to be applicable with no artificial restrictions on the classes of control policies considered, leading to very general existence results on optimal measurable policies under various setups and information structures. On relaxed controls, two studied topologies are the Young and Borkar (weak[math]) topologies on spaces of functions from a state/measurement space to the space of probability measures on control action spaces; the former via a weak convergence topology on probability measures on a product space with a fixed marginal on the input (state) space, and the latter via a weak[math] topology on randomized policies viewed as maps from states/measurements to the space of signed measures with bounded variation. We establish implication and equivalence conditions between the Young and Borkar topologies on control policies. We then show that, under some conditions, for a controlled Markov chain with standard Borel spaces the invariant measure is weakly continuous on the space of stationary control policies defined by either of these topologies. An implication is near-optimality of quantized stationary policies in state and actions or continuous stationary and deterministic policies for average cost control under two sets of continuity conditions (with either weak continuity in the state-action pair or strong continuity in the action for each state) on transition kernels.
{"title":"On Borkar and Young Relaxed Control Topologies and Continuous Dependence of Invariant Measures on Control Policy","authors":"Serdar Yüksel","doi":"10.1137/23m1571940","DOIUrl":"https://doi.org/10.1137/23m1571940","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2367-2386, August 2024. <br/> Abstract. In deterministic and stochastic control theory, relaxed or randomized control policies allow for versatile mathematical analysis (on continuity, compactness, convexity, and approximations) to be applicable with no artificial restrictions on the classes of control policies considered, leading to very general existence results on optimal measurable policies under various setups and information structures. On relaxed controls, two studied topologies are the Young and Borkar (weak[math]) topologies on spaces of functions from a state/measurement space to the space of probability measures on control action spaces; the former via a weak convergence topology on probability measures on a product space with a fixed marginal on the input (state) space, and the latter via a weak[math] topology on randomized policies viewed as maps from states/measurements to the space of signed measures with bounded variation. We establish implication and equivalence conditions between the Young and Borkar topologies on control policies. We then show that, under some conditions, for a controlled Markov chain with standard Borel spaces the invariant measure is weakly continuous on the space of stationary control policies defined by either of these topologies. An implication is near-optimality of quantized stationary policies in state and actions or continuous stationary and deterministic policies for average cost control under two sets of continuity conditions (with either weak continuity in the state-action pair or strong continuity in the action for each state) on transition kernels.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204102","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 2346-2366, August 2024. Abstract. This paper investigates new sufficient conditions for the stability, asymptotic stability, and global asymptotic stability of nonlinear autonomous systems, specifically in cases where the first derivative of the Lyapunov function candidate may have both positive and negative values on its domain. The main contribution of this approach is the introduction of a new auxiliary function that relaxes the stability conditions, allowing the first derivative of the Lyapunov function candidate to be less than or equal to a nonnegative function. The suggested auxiliary function should be integrable within our first theorem. Meanwhile, our first corollary presents a technique that simplifies the task by establishing specific conditions related to differential inequalities. This weaker condition in the proposed results enables the establishment of stability properties in cases where the Lyapunov function candidate is not well chosen or finding a Lyapunov function is not straightforward. Additionally, it is proven that the original Lyapunov method for autonomous systems is a special case of our first theorem. Furthermore, it is demonstrated that assumptions in previous studies, such as Matrosov’s theorem or results on higher-order derivatives of the Lyapunov function, guarantee the existence of our auxiliary function. Finally, lemmas are provided to construct these auxiliary functions, and examples are presented to demonstrate the effectiveness of this approach. This work will contribute to the development of stability analysis techniques for nonlinear autonomous systems.
{"title":"Constituting an Extension of Lyapunov’s Direct Method","authors":"M. Akbarian, N. Pariz, A. Heydari","doi":"10.1137/23m1595242","DOIUrl":"https://doi.org/10.1137/23m1595242","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2346-2366, August 2024. <br/> Abstract. This paper investigates new sufficient conditions for the stability, asymptotic stability, and global asymptotic stability of nonlinear autonomous systems, specifically in cases where the first derivative of the Lyapunov function candidate may have both positive and negative values on its domain. The main contribution of this approach is the introduction of a new auxiliary function that relaxes the stability conditions, allowing the first derivative of the Lyapunov function candidate to be less than or equal to a nonnegative function. The suggested auxiliary function should be integrable within our first theorem. Meanwhile, our first corollary presents a technique that simplifies the task by establishing specific conditions related to differential inequalities. This weaker condition in the proposed results enables the establishment of stability properties in cases where the Lyapunov function candidate is not well chosen or finding a Lyapunov function is not straightforward. Additionally, it is proven that the original Lyapunov method for autonomous systems is a special case of our first theorem. Furthermore, it is demonstrated that assumptions in previous studies, such as Matrosov’s theorem or results on higher-order derivatives of the Lyapunov function, guarantee the existence of our auxiliary function. Finally, lemmas are provided to construct these auxiliary functions, and examples are presented to demonstrate the effectiveness of this approach. This work will contribute to the development of stability analysis techniques for nonlinear autonomous systems.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204103","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}
{"title":"The Unconditional Consensus Control through Leadership for the Delayed Hegselmann–Krause Model","authors":"Linglong Du, Jianwen Zhu, Feng Xie","doi":"10.1137/23m1588858","DOIUrl":"https://doi.org/10.1137/23m1588858","url":null,"abstract":"","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141928757","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 2319-2345, August 2024. Abstract. We investigate time-inconsistent mean-field stopping games under nonexponential discounting in discrete time. At the intrapersonal level, each player plays against her future selves as a result of the time inconsistency caused by nonexponential discounting. At the interpersonal level, she plays against other players due to players’ interaction via the proportion of players that have stopped. We look for sharp mean-field equilibria (MFEs), such that given other players’ stopping policies, the representative player’s strategy not only is an intrapersonal equilibrium, but also an optimal one among all such intrapersonal equilibria. We analyze two classes of examples. The first one is on time-inconsistent bank-run models, and we construct an (optimal) sharp MFE by a monotone iteration scheme. The second one has a Markovian setup and no common noise, and we show the existence of a sharp MFE based on the Tikhonov fixed-point theorem.
{"title":"Sharp Equilibria for Time-Inconsistent Mean-Field Stopping Games","authors":"Ziyuan Wang, Zhou Zhou","doi":"10.1137/23m1625512","DOIUrl":"https://doi.org/10.1137/23m1625512","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2319-2345, August 2024. <br/> Abstract. We investigate time-inconsistent mean-field stopping games under nonexponential discounting in discrete time. At the intrapersonal level, each player plays against her future selves as a result of the time inconsistency caused by nonexponential discounting. At the interpersonal level, she plays against other players due to players’ interaction via the proportion of players that have stopped. We look for sharp mean-field equilibria (MFEs), such that given other players’ stopping policies, the representative player’s strategy not only is an intrapersonal equilibrium, but also an optimal one among all such intrapersonal equilibria. We analyze two classes of examples. The first one is on time-inconsistent bank-run models, and we construct an (optimal) sharp MFE by a monotone iteration scheme. The second one has a Markovian setup and no common noise, and we show the existence of a sharp MFE based on the Tikhonov fixed-point theorem.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938545","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 2273-2296, August 2024. Abstract. We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the [math]-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights.
{"title":"An Observer for Pipeline Flow with Hydrogen Blending in Gas Networks: Exponential Synchronization","authors":"Martin Gugat, Jan Giesselmann","doi":"10.1137/23m1563840","DOIUrl":"https://doi.org/10.1137/23m1563840","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2273-2296, August 2024. <br/> Abstract. We consider a state estimation problem for gas flows in pipeline networks where hydrogen is blended into the natural gas. The flow is modeled by the quasi-linear isothermal Euler equations coupled to an advection equation on a graph. The flow through the vertices where the pipes are connected is governed by algebraic node conditions. The state is approximated by an observer system that uses nodal measurements. We prove that the state of the observer system converges to the original system state exponentially fast in the [math]-norm if the measurements are exact. If measurement errors are present we show that the observer state approximates the original system state up to an error that is proportional to the maximal measurement error. The proof of the synchronization result uses Lyapunov functions with exponential weights.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938547","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}
Pengju Ning, Sergey N. Dashkovskiy, Changchun Hua, Kuo Li
SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2254-2272, August 2024. Abstract. This paper investigates the prescribed-time output feedback stabilization problem for a class of nonlinear time-delay systems. First, a novel dual-gain function is put forward by exploiting the dynamic gain and the time-varying gain function to design the reduced-order observer for reconstructing unavailable states. Then, by utilizing the Lyapunov–Krasovskii functional and state variables of the reduced-order observer, a new prescribed-time controller is presented based on the nonscaling design framework. Since no state scaling is required in controller design process under this framework, our control strategy is simpler and can greatly reduce the computational burden. Further, compared with the previous prescribed-time stabilization results, our designed controller acts on the entire time domain, not just a limited time interval. Based on our proposed stability criterion, it is proved that the controller can render that all system state variables converge to the origin within the prescribed time. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control strategy.
{"title":"Dual-Gain Function Based Prescribed-Time Output Feedback Control Nonlinear Time-Delay Systems","authors":"Pengju Ning, Sergey N. Dashkovskiy, Changchun Hua, Kuo Li","doi":"10.1137/23m1556496","DOIUrl":"https://doi.org/10.1137/23m1556496","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2254-2272, August 2024. <br/> Abstract. This paper investigates the prescribed-time output feedback stabilization problem for a class of nonlinear time-delay systems. First, a novel dual-gain function is put forward by exploiting the dynamic gain and the time-varying gain function to design the reduced-order observer for reconstructing unavailable states. Then, by utilizing the Lyapunov–Krasovskii functional and state variables of the reduced-order observer, a new prescribed-time controller is presented based on the nonscaling design framework. Since no state scaling is required in controller design process under this framework, our control strategy is simpler and can greatly reduce the computational burden. Further, compared with the previous prescribed-time stabilization results, our designed controller acts on the entire time domain, not just a limited time interval. Based on our proposed stability criterion, it is proved that the controller can render that all system state variables converge to the origin within the prescribed time. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control strategy.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938548","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 2229-2253, August 2024. Abstract. We consider a singularly perturbed system of stochastic differential equations proposed by Chaudhari et al. (Res. Math. Sci. 2018) to approximate the entropic gradient descent in the optimization of deep neural networks via homogenization. We embed it in a much larger class of two-scale stochastic control problems and rely on convergence results for Hamilton–Jacobi–Bellman equations with unbounded data proved recently by ourselves (ESAIM Control Optim. Calc. Var. 2023). We show that the limit of the value functions is itself the value function of an effective control problem with extended controls and that the trajectories of the perturbed system converge in a suitable sense to the trajectories of the limiting effective control system. These rigorous results improve the understanding of the convergence of the algorithms used by Chaudhari et al., as well as of their possible extensions where some tuning parameters are modeled as dynamic controls.
{"title":"Deep Relaxation of Controlled Stochastic Gradient Descent via Singular Perturbations","authors":"Martino Bardi, Hicham Kouhkouh","doi":"10.1137/23m1544878","DOIUrl":"https://doi.org/10.1137/23m1544878","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2229-2253, August 2024. <br/> Abstract. We consider a singularly perturbed system of stochastic differential equations proposed by Chaudhari et al. (Res. Math. Sci. 2018) to approximate the entropic gradient descent in the optimization of deep neural networks via homogenization. We embed it in a much larger class of two-scale stochastic control problems and rely on convergence results for Hamilton–Jacobi–Bellman equations with unbounded data proved recently by ourselves (ESAIM Control Optim. Calc. Var. 2023). We show that the limit of the value functions is itself the value function of an effective control problem with extended controls and that the trajectories of the perturbed system converge in a suitable sense to the trajectories of the limiting effective control system. These rigorous results improve the understanding of the convergence of the algorithms used by Chaudhari et al., as well as of their possible extensions where some tuning parameters are modeled as dynamic controls.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780273","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 2203-2228, August 2024. Abstract. We consider a class of zero-sum stopper versus singular-controller games in which the controller can only act on a subset [math] of the [math] coordinates of a controlled diffusion. Due to the constraint on the control directions these games fall outside the framework of recently studied variational methods. In this paper we develop an approximation procedure, based on [math]-stability estimates for the controlled diffusion process and almost sure convergence of suitable stopping times. That allows us to prove existence of the game’s value and to obtain an optimal strategy for the stopper under continuity and growth conditions on the payoff functions. This class of games is a natural extension of (single-agent) singular control problems, studied in the literature, with similar constraints on the admissible controls.
{"title":"Zero-Sum Stopper Versus Singular-Controller Games with Constrained Control Directions","authors":"Andrea Bovo, Tiziano De Angelis, Jan Palczewski","doi":"10.1137/23m1579558","DOIUrl":"https://doi.org/10.1137/23m1579558","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2203-2228, August 2024. <br/> Abstract. We consider a class of zero-sum stopper versus singular-controller games in which the controller can only act on a subset [math] of the [math] coordinates of a controlled diffusion. Due to the constraint on the control directions these games fall outside the framework of recently studied variational methods. In this paper we develop an approximation procedure, based on [math]-stability estimates for the controlled diffusion process and almost sure convergence of suitable stopping times. That allows us to prove existence of the game’s value and to obtain an optimal strategy for the stopper under continuity and growth conditions on the payoff functions. This class of games is a natural extension of (single-agent) singular control problems, studied in the literature, with similar constraints on the admissible controls.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740263","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}
{"title":"On Markov Perfect Equilibria in Discounted Stochastic ARAT Games","authors":"Anna Jaśkiewicz, Andrzej S. Nowak","doi":"10.1137/23m1592365","DOIUrl":"https://doi.org/10.1137/23m1592365","url":null,"abstract":"","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141641863","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}
Axel Ringh, Isabel Haasler, Yongxin Chen, Johan Karlsson
SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2176-2202, August 2024. Abstract. In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problem. This type of problem, which can be seen as a generalization of multimarginal optimal transport problems with graph-structured costs, appears in many applications. Examples are unbalanced optimal transport and multispecies potential mean field games, where the latter is a class of nonlinear density control problems. The method we develop is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures it is possible to derive efficient methods for computing these projections, and here we specifically consider the graph structure that occurs in multispecies potential mean field games. We also illustrate the methodology on a numerical example from this problem class.
{"title":"Graph-Structured Tensor Optimization for Nonlinear Density Control and Mean Field Games","authors":"Axel Ringh, Isabel Haasler, Yongxin Chen, Johan Karlsson","doi":"10.1137/23m1571587","DOIUrl":"https://doi.org/10.1137/23m1571587","url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 4, Page 2176-2202, August 2024. <br/> Abstract. In this work we develop a numerical method for solving a type of convex graph-structured tensor optimization problem. This type of problem, which can be seen as a generalization of multimarginal optimal transport problems with graph-structured costs, appears in many applications. Examples are unbalanced optimal transport and multispecies potential mean field games, where the latter is a class of nonlinear density control problems. The method we develop is based on coordinate ascent in a Lagrangian dual, and under mild assumptions we prove that the algorithm converges globally. Moreover, under a set of stricter assumptions, the algorithm converges R-linearly. To perform the coordinate ascent steps one has to compute projections of the tensor, and doing so by brute force is in general not computationally feasible. Nevertheless, for certain graph structures it is possible to derive efficient methods for computing these projections, and here we specifically consider the graph structure that occurs in multispecies potential mean field games. We also illustrate the methodology on a numerical example from this problem class.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717954","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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