| This thesis mainly study H2/H∞control problem of forward and backward stochas-tic systems. This thesis consists of four chapters.One of the most important robust control approaches is the so-called H∞control, which demands that one designs a controller to eliminate the external disturbance below a given disturbance attenuation levelγ>0. Obviously, there may be more than one controller to H∞control problem, among which, one selects u* not only to restrain the exogenous disturbance, but also to minimize a cost function when the worst case disturbance d* is implemented, this is the so-called H2/H∞control problem. Therefore, in chapter 1, we first give a brief review of H2/H∞theory. We also introduce the main results of this thesis.The objective of Chapter 2 is to deal with the H2/H∞control problem of linear forward stochastic systems with (x,u)-dependent noise in both Finite and infinite time horizons. For finite horizon case, by virtue of the solution of the forward-backward stochastic differential equation (FBSDE), we obtain an explicit form of the solution for the stochastic H2/H∞control problem. We also give the linear feedback form of the solution using the solution of a generalized matrix-valued Riccati-type equation, which is uniquely solvable. Moreover, we show that the linear state feedback solution to the stochastic H2/H∞control is unique. For infinite horizon case, the control u* is demanded to stabilize system internally. By virtue of a algebraic Riccati equation, We obtain a sufficient condition for (u*,d*) to be a solution for the infinite horizon stochastic H2/H∞control.In Chapter 3, we discuss the properties of the storage functions for a class of nonlinear stochastic systems. We find that a function which is smooth enough, then it is a storage function for the nonlinear stochastic systems if and only if it is a supersolution of a second-order partial differential equation. However, in practice, a storage function is not necessarily smooth enough, then we have that for a continuous function, it is a storage function for the nonlinear stochastic systems if and only if it is a viscosity supersolution of the second-order partial differential equation. Specially, the available storage function is a solution of the second-order partial differential equation when it is smooth enough and a viscosity solution when it is continuous. As applications, the finite and infinite horizon nonlinear stochastic H∞controls for systems with state, control, and external disturbance dependent noise are investigated, which generalize the previous results.In the last chapter, mixed H2/H∞control problems of the backward stochastic systems with exogenous disturbance signal in both finite and infinite horizon cases and the backward doubly stochastic systems in finite horizon case arc discussed. For backward stochastic systems in finite horizon case, the solutions of this problem are obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique. Two equivalent expressions for the H2/H∞control are presented. Contrary to forward deterministic and stochastic cases, the solutions of the backward stochastic H2/H∞control are no longer feedbacks of the current state; rather, they arc feedbacks of the entire history of the state. For backward stochastic systems in infinite horizon case, by virtue of the solution of an infinite horizon forward-backward stochastic differential equation (FBSDE), we obtain the explicit form of the solution for this control problem. The solutions arc still feedbacks of the entire history of the state. We also discuss the solvability of the infinite horizon Riccati-type equation and give the solution for the control problem using the solution of this kind of Riccati equation. For mixed H2/H∞control problem of linear backward doubly stochastic differential equations (BDSDEs), We give a sufficient condition to get the explicit form of the H2/H∞control for linear backward doubly stochastic systems.
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