Finite-horizon closed-loop Nash game for stochastic large-scale systems with multiple decision makers

The purpose of this paper is to investigate a finite-time horizon closed-loop Nash game for a class of stochastic systems. First, the condition necessary to attain Nash equilibrium is derived by means of stochastic maximum principle. When this principle is used, the existence conditions consist of cross-coupled forward-backward stochastic differential equations (CFBSDEs). Second, these results are then used to investigate closed-loop Nash games for a class of stochastic large-scale linear systems. Finally, simple examples are solved to show the validity of the proposed method.

[1]  H. Mukaidani,et al.  Nonlinear Stochastic Dynamic Games * , 2014 .

[2]  W. Zhang,et al.  State Feedback H_∞Control for a Class of Nonlinear Stochastic Systems , 2009 .

[3]  Gang Feng,et al.  Nonlinear Stochastic $H_2/H_\infty$ Control with $(x,u,v)$-Dependent Noise: Infinite Horizon Case , 2008, IEEE Transactions on Automatic Control.

[4]  Z. Gajic,et al.  Parallel Algorithms for Optimal Control of Large Scale Linear Systems , 1993 .

[5]  Minyi Huang,et al.  Large-Population Cost-Coupled LQG Problems With Nonuniform Agents: Individual-Mass Behavior and Decentralized $\varepsilon$-Nash Equilibria , 2007, IEEE Transactions on Automatic Control.

[6]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[7]  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..

[8]  Bor-Sen Chen,et al.  Stochastic H2/H∞ control with (x, u, v)-dependent noise: Finite horizon case , 2006, 2006 9th International Conference on Control, Automation, Robotics and Vision.

[9]  Hiroaki Mukaidani,et al.  Nonlinear stochastic H2/H∞ control with multiple decision makers , 2013, 52nd IEEE Conference on Decision and Control.

[10]  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).

[11]  M. Ishikawa.  Optimal Vaccination Strategy under Saturated Treatment using the Stochastic SIR Model (Special Issue on the 44th ISCIE International Symposium on Stochastic Systems Theory and Its Applications(1)) , 2013 .

[12]  Weihai Zhang,et al.  State Feedback HINFINITY Control for a Class of Nonlinear Stochastic Systems , 2006, SIAM J. Control. Optim..

[13]  Bor-Sen Chen,et al.  Stochastic H2/Hinfinity control with (x, u, v)-dependent noise: Finite horizon case , 2006, Autom..

[14]  Hiroaki Mukaidani,et al.  Soft-constrained stochastic Nash games for weakly coupled large-scale systems , 2009, Autom..

[15]  J. Yong,et al.  Solving forward-backward stochastic differential equations explicitly — a four step scheme , 1994 .

[16]  Ishikawa Masaaki.  Optimal Strategies for Vaccination using the Stochastic SIRV Model , 2012 .

[17]  Bor-Sen Chen,et al.  Stochastic H2/H∞ control with state-dependent noise , 2004, IEEE Trans. Autom. Control..

[18]  T. Başar,et al.  Solutions to a class of linear-quadratic-Gaussian (LQG) stochastic team problems with nonclassical information , 1987 .