Stabilization Control for Linear Continuous-Time Mean-Field Systems

This paper investigates the stabilization and control problems for linear continuous-time mean-field systems. Under standard assumptions, the necessary and sufficient conditions to stabilize the mean-field systems in the mean-square sense are explored for the first time. It is shown that, under the assumption of exact detectability (exact observability), the mean-field system is stabilizable if and only if a coupled algebraic Riccati equation admits a unique positive-semidefinite solution (positive-definite solution), which coincides with the classical stabilization results for standard deterministic systems and stochastic systems.

[1]  W. Marsden.  I and J , 2012 .

[2]  Boualem Djehiche,et al.  A General Stochastic Maximum Principle for SDEs of Mean-field Type , 2011 .

[3]  D. Hinrichsen,et al.  Stochastic H∞ , 1998 .

[4]  Jiongmin Yong,et al.  Mean-field stochastic linear quadratic optimal control problems: closed-loop solvability , 2016, Mathematical Control & Related Fields.

[5]  P. Olver.  Nonlinear Systems , 2013 .

[7]  Juan Li,et al.  Reflected Mean-Field Backward Stochastic Differential Equations. Approximation and Associated Nonlinear PDEs , 2012, 1210.0628.

[8]  P. Caines,et al.  Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[9]  R. Carmona,et al.  Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov Dynamics , 2013, 1303.5835.

[10]  Xi Chen,et al.  Solvability and asymptotic behavior of generalized Riccati equations arising in indefinite stochastic LQ controls , 2001, IEEE Trans. Autom. Control..

[11]  H. McKean,et al.  A CLASS OF MARKOV PROCESSES ASSOCIATED WITH NONLINEAR PARABOLIC EQUATIONS , 1966, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Huanshui Zhang,et al.  Necessary and Sufficient Solution to Optimal Control for Linear Continuous Time Mean-field System , 2017 .

[13]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[14]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[15]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[16]  Minyue Fu,et al.  Optimal Stabilization Control for Discrete-Time Mean-Field Stochastic Systems , 2019, IEEE Transactions on Automatic Control.

[17]  Mokhtar Hafayed,et al.  A mean-field maximum principle for optimal control of forward–backward stochastic differential equations with Poisson jump processes , 2013 .

[18]  Xun Li,et al.  Discrete-time mean-field Stochastic linear-quadratic optimal control problems, II: Infinite horizon case , 2015, Autom..

[19]  Bor-Sen Chen,et al.  Generalized Lyapunov Equation Approach to State-Dependent Stochastic Stabilization/Detectability Criterion , 2008, IEEE Transactions on Automatic Control.

[20]  H. Pham,et al.  Bellman equation and viscosity solutions for mean-field stochastic control problem , 2015, 1512.07866.

[21]  T. Chan,et al.  Dynamics of the McKean-Vlasov equation , 1994 .

[22]  Boualem Djehiche,et al.  Mean-Field Backward Stochastic Differential Equations . A Limit Approach ∗ , 2007 .

[23]  R. Carmona,et al.  Control of McKean–Vlasov dynamics versus mean field games , 2012, 1210.5771.

[24]  J. Yong,et al.  A Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations in Infinite Horizon , 2012, 1208.5308.

[25]  M. Hp.  A class of markov processes associated with nonlinear parabolic equations. , 1966 .

[26]  Juan Li,et al.  Stochastic maximum principle in the mean-field controls , 2012, Autom..

[27]  Victor M. Becerra,et al.  Optimal control , 2008, Scholarpedia.

[28]  Rene Carmona,et al.  Mean field forward-backward stochastic differential equations , 2012, 1211.4186.

[29]  J. Yong.  Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations --- Time-Consistent Solutions , 2013, 1304.3964.

[30]  R. Carmona.  Lectures on Bsdes, Stochastic Control, and Stochastic Differential Games with Financial Applications , 2016 .

[31]  Jiongmin Yong,et al.  Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations , 2013, SIAM J. Control. Optim..

[32]  Huanshui Zhang,et al.  Infinite horizon linear quadratic optimal control for discrete‐time stochastic systems , 2008 .

[33]  D. Crisan,et al.  Approximate McKean–Vlasov representations for a class of SPDEs , 2005, math/0510668.

[34]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[35]  Bor-Sen Chen,et al.  On stabilizability and exact observability of stochastic systems with their applications , 2023, Autom..

[36]  Jingrui Sun.  Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Open-Loop Solvabilities , 2015, 1602.07825.

[37]  Xun Li,et al.  Discrete time mean-field stochastic linear-quadratic optimal control problems , 2013, Autom..

[38]  B. Anderson,et al.  Linear Optimal Control , 1971 .

[39]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[40]  Alain Bensoussan,et al.  Linear-Quadratic Mean Field Games , 2014, Journal of Optimization Theory and Applications.

[41]  J. Gärtner.  On the McKean‐Vlasov Limit for Interacting Diffusions , 1988 .