Observability And Optimal Control Of Stochastic Switched Systems Based On Non-Cooperative Dynamic Game

In this paper,we study the observability and optimal control problems of stochastic switched systems,which based on non-cooperative dynamic games.This kind of problem is a hierarchical structure:it includes a leader and numbers of followers.The leader formulate his own action strategies,and a non-cooperative dynamic game with Nash equilibrium may exist among followers.At the same time,the stochastic switched differential game system is composed of multiple subsystems and specific switching rules.This is a new direction of dynamic games from the view of control theory.It is also a new framework that combines traditional control problems with game theory.First,we study the exact observability of such stochastic switched differential game system.The algebraic Riccati equation is introduced,and the system model is transformed into a general form which is easy to analyze and process by constructing Hamilton equation.Then,the observer of system is constructed by using the linear operator theory and the knowledge of forward-backward stochastic differential equations.The observer here is not composed of Gram matrix in general sense.The purpose of constructed it is to obtain the dual system of stochastic system model.Then,the observability criterion of such dual system is proved.Based on the observability criterion of stochastic system and the definition of observability of system,the application of is given,we prove the equivalence relation among observability,stability and solutions of stochastic Lyapunov equation.Secondly,the classical linear quadratic optimal control and follower optimal control of the system are briefly introduced.In the meanwhile,two kinds of optimal control problems are introduced.One is the linear quadratic regular problem with special form of norm.The solvability and uniqueness of the optimal control are proved by the knowledge of forward-backward stochastic differential equations,and the minimum numerical solution of the optimal function is solved.The other is the operator optimization control problem which to study the characteristics of operators minimum norm.Then,the equivalent conditions of controllability,observability and optimal control of stochastic systems are given,the corresponding proofs are given through the backward stochastic differential equation,Ito formula and the open-loop operator theory.Finally,an example of stochastic differential game system in financial field is introduced and simulation of simple stochastic system is given.This section we discusses the optimal portfolio selection problem in the financial field.One investor has expected wealth goals,in addition to helping investors achieve wealth goals,two investment managers also want to maximize returns.They decide the portfolio according to the known trend.A simple stochastic system is analyzed and simulation by matlab.Through the above conclusions,the controller is designed and the feasibility is verified.