Infinite Horizon Stochastic Maximum Principle for Stochastic Delay Evolution Equations in Hilbert Spaces

In the present work, we investigate infinite horizon optimal control problems driven by a class of stochastic delay evolution equations in Hilbert spaces and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation (ABSEE). We first establish a priori estimate for the solution to ABSEEs by imposing restriction on unbounded operator A∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*$$\end{document}, that is, the operator A∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^*$$\end{document} is maximal dissipative. In this way, the Itô inequality is applicable in our study, and we can also avoid the problem that neither Itô formula nor energy equation is available. Next, we obtain the existence and uniqueness results of solutions of linear backward stochastic evolution equations on infinite horizon by using approximating methods. Then, the existence and uniqueness results of ABSEEs on infinite horizon is obtained via the fixed-point theory. That is the highlight of innovation in this paper. Eventually, we establish necessary and sufficient conditions for optimality of the control problem on infinite horizon, in the form of Pontryagin’s maximum principle.