Stochastic Maximum Principle For Delayed Doubly Stochastic Control Systems And Their Applications

Optimal control theory is the main content of modern control theory which devel-oped rapidly in 1960s.Dynamic programming and maximum principle are the basic content and common methods of optimal control theory.The maximum principle was formulated and derived by Pontryagin and his group.It states that the optimal control along with the optimal state trajectory should satisfy the Hamiltonian systems,which is a two point boundary value problem plus a maximum condition of Hamiltonian func-tion.As a milestone of optimal control theory,the maximum principle is an extension of the classical variational method.Due to maximum principle relaxed the precondi-tion solving the problem,many engineering problems have been solved which can not be dealed with by classical variational method or dynamic programming method.Along with the intensives research,people come to realize many problems in real world have the relation not only with the present time but also with their past history,which can be identified as time delay problems.The equation describing this kind of problem is called delay equation.The problem of time delay is widely used in many fields,such as life science,engineering,physics,control theory and finance.Many scholars study and discuss this problem and extend their work to more widely fields.Based on the abundant literature we give the stochastic maximum principle for delayed doubly stochastic control systems.The stochastic maximum principle is a set of necessary conditions that must be satisfied by any optimal solution which the state equation of the control system is described by delayed doubly stochastic differential equations.In chapter 3,we considered the followed delayed stochastic control system.with the cost functionalThe optimal control problem can be stated as maximizing the costing functional over U/[0,T].For optimal control u*(·)satisfyingFirstly,we investigate a class of doubly stochastic differential equations with time delay.The existence and uniqueness of solution for the delayed doubly stochastic differential equation can be deduced by means of martingale representation theorem and contraction mapping principle under some assumptions.We assume that the following conditions hold:When the control domain is convex,we deduce a stochastic maximum principle as a necessary condition of the optimal control by using classical variational technique.Theorem 2 Let(y*(·),z*(·),u*(·))be an optimal triple of the delayed doubly stochastic control system(1)-(3).If the time delay ? is sufficiently small,there is a unique Ft-adapted solution satisfying the associated adjoint equations such thatAt the same time,under certain assumptions,a sufficient condition of optimality is obtained by using the duality method.Then u*(·)is an optimal control for delayed stochastic optimal problems(1)-(3).At the end of chapter 3,we apply our stochastic maximum principle to a class of linear quadratic optimal control problem and obtain the explicit expression of the optimal control.In chapter 4,we study the linear forward-backward doubly stochastic Hamilto-nian systems.We give the uniqueness and existence of solution for these equations by defining the matrix valued Riccati equation.Theorem 4 We assume that in some interval[t1,t2](?)[0,T],Riccati equation has a solution(K(t),M(t),N(t)).Then the followed forward-backward doubly stochas-tic Hamiltonian system has a solution where p(t)is the solution of If we assume that and H13 is nonsingular,then the solution is unique.In chapter 5,we give the existence condition of a class of stochastic Riccati equa-tion.Because of its complex structure,the solvability of the stochastic Riccati equation is very difficult,but is the key to solve the stochastic linear quadratic optimal control problem.The main idea is to replace the stochastic Riccati equation by a backward stochastic differential equation(without including algebraic constraint),and the exis-tence of solution for backward stochastic differential equation enforces the algebraic constraint to be satisfied automatically.We also give the application example of the stochastic Riccati equation in the linear quadratic optimal control problem.admits a unique bounded positive definite solution.admits a unique bounded positive definite solution.