Nash Equilibrium Precision Controllability And Linear Quadratic Optimal Control For Stochastic Switched Systems

In this paper,we study the non-cooperative dynamic game between leader and multiple followers.This is a new research direction for dynamic game,which is beyond the traditional control theory and game theory framework.Assuming that the Nash equilibrium point of the differential game problem exists under the symmetric information,leader is seen as third parties or other nonprofit organizations,then we can ignore the leader’s payoff function.The non-cooperative dynamic game of the followers is studied when the strategy is given by leader.In this system,we focus on the ability of the leader regulating and controlling.In a sense,it reflects the influence of the leader on the non-cooperative game system.First,this paper studies the maximum benefit of followers under the condition of system information equilibrium.This is actually an optimal control problem.In practice,the problems of optimization control are more and more concerned and in-depth study,such as financial markets,energy systems.Here we introduce the forward-backward stochastic differential equation(FBSDE)control theory.Considering the problem of linear quadratic non-zero-sum differential game under the symmetric information,the construction method of the followers’ maximum benefit solution is discussed,and the analytical expression of the optimal solution of the random switched system is given based on the stochastic maximum value principle.Secondly,this paper discusses the linear switched systems where the perturbation term coefficient contain the state and the controller.Under the condition that the random Nash equilibrium state exists,the stochastic control problem in the linear switched system are studied.We introduce the Riccati equation and the forward-backward stochastic differential equation(FBSDE)theory,to prove the existence and uniqueness of solutions for stochastic systems.Again,this paper studies the feasibility of the leader macro-control under the condition of Nash equilibrium.Using the theory of BSDE put forward by Peng,together with FBSDE theory,the necessary and sufficient conditions for the terminal exact controllability of the Nash equilibrium are obtained for the stochastic systems with switched parameters in the coefficient.Furthermore,we obtain the necessary and sufficient conditions for the exact controllability of Nash equilibrium.At the same time,the algebraic criteria for Nash equilibrium exact controllability is also given.Finally,in order to reflect the practical application value of the backward stochastic control system,this paper gives an example of the optimal portfolio in the market.Leader-decision is a control process,and the optimal income of the follower can be regulated.So the question we have studied is of practical significance.At the same time,the Matlab numerical simulation shows that the proposed controller can accurately control the system model.