Infinite Horizon Optimal Control of Stochastic Delay Evolution Equations in Hilbert Spaces

Xueping Zhu,Jianjun Zhou

doi:10.1155/2013/791786

Abstract

The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamics is governed by a stochastic delay evolution equation in Hilbert spaces. The existence and uniqueness of the optimal control are obtained by means of associated infinite horizon backward stochastic differential equations without assuming the Gâteaux differentiability of the drift coefficient and the diffusion coefficient. An optimal control problem of stochastic delay partial differential equations is also given as an example to illustrate our results.

Highlights

In this paper, we consider a controlled stochastic evolution equation of the following form: dXu (s) = AXu (s) ds + F (s, Xsu) ds + G (s, Xsu) R (s, Xsu, u (s)) ds + G (s, Xsu) dW (s), s ≥ t, Xtu = x, (1) whereXsu (l) = Xu (s + l), l ∈ [−τ, 0], x ∈ C ([−τ, 0], H) . (2)u is the control process in a measurable space (U, U),and W is a cylindrical Wiener process in a Hilbert space Ξ
We list some notations that are used in this paper
We use the symbol | ⋅ | to denote the norm in a Banach space F, with a subscript if necessary

Summary

Introduction

Fuhrman and Tessiture [24] dealt with an infinite horizon optimal control problem for the stochastic evolution equation in Hilbert space, and the optimal control is showed by means of infinite horizon backward stochastic differential equation in infinite dimensional spaces and Malliavin calculus. In Fuhrman [26], a class of optimal control problems governed by stochastic evolution equations in Hilbert spaces which includes state constraints is considered, and the optimal control is obtained by the Fleming logarithmic transformation. We study the infinite horizon optimal control problem for stochastic delay evolution equations in Hilbert spaces, and by using Theorem 10, the optimal control is obtained. Since we do not relate the optimal feedback law with the gradient of the value function and do not consider the associated Hamilton-Jacobi-Bellman equation, we can drop the Gateaux differentiability of the drift term and the diffusion term.

Preliminaries

The Backward-Forward System

The Fundamental Relation

Existence of Optimal Control

Applications