Study Of Optimal Control Problems For Stochastic Evolution Equations

Stochastic evolution equations are the popular topics of dynamics research. Based on the existed literatures, this thesis studies the nonlinear Kolmogorov equations associ-ated with stochastic delay evolution equations and applications to stochastic optimal control problem. Moreover, this thesis considers the optimal control problem for stochastic evolu-tion equations and stochastic delay evolution equations in Hilbert spaces.This Ph.D. thesis is divided into four chapters.In Chapter 1, we introduce the historical background, the current situation and the main results of this thesis.In Chapter 2, nonlinear Kolmogorov equations and its applications to optimal control for stochastic delay evolution equations are studied. Firstly, two basic Lemmas are proved to ensure that the nonlinear Kolmogorov equations be well defined. Then forward-backward system is considered, regular dependence on parameters and Malliavin differentiability are showed, and we define a determined function v:[0, T]×C→R by the solution of backward stochastic differential equations. We proved that the function v is a classical solution of the nonlinear Kolmogorov equations if v is sufficiently regular. Moreover, under the assump-tions of this chapter, we prove that there exists a unique mild solution v of the nonlinear Kolmogorov equation and the function v has the property of associated Theorem. Finally, as an application, an optimal control of stochastic delay evolution equations is given to illus-trate our results. By constructing an optimal feedback, we obtain optimal control and show that v is the value function. The results is new.In Chapter 3, we consider an optimal control problem in which the controlled state dy-namics is governed by a stochastic evolution equation in Hilbert spaces, the cost functional has a quadratic growth and the control process takes values in a closed set (not necessar-ily compact). The backward stochastic differential equation corresponding to our control problem has a quadratic growth in the Z variable. The monotone stability for this class of backward stochastic differential equations in finitedimensional has been obtained in the case of bounded terminal value. By the similar approach, we can show that the results also hold true if the BSDE is driven by a cylindrical Wiener process. We study the backward stochas-tic differential equations with quadratic growth and unbounded terminal terminal value and show that the maximal solution (Y, Z) of the BSDE has the property of associated Theo-rem, then we can construct an optimal feedback and the optimal control is obtained. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results. The results is new.Chapter 4 is concerned with an optimal control problem in which the controlled state dynamics is governed by a general system of stochastic delay evolution equations in Hilbert spaces. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations. As an application, an optimal control of stochastic delay partial differential equations is also given to illustrate our results. Some known results are generalized and improved.