Necessary Conditions for Stochastic Optimal Control Problems in Infinite Dimensions

TL;DR

This paper develops first and second order necessary conditions for stochastic optimal control problems in infinite-dimensional spaces, addressing challenges with adjoint equations and control set nonconvexity.

Contribution

It introduces a novel approach to derive second order necessary conditions using only first and second order adjoint equations in infinite dimensions.

Findings

Established first order necessary conditions for stochastic controls.

Derived second order necessary conditions avoiding higher order adjoint equations.

Applicable to nonconvex control sets with stochastic evolution equations.

Abstract

The purpose of this paper is to establish the first and second order necessary conditions for stochastic optimal controls in infinite dimensions. The control system is governed by a stochastic evolution equation, in which both drift and diffusion terms may contain the control variable and the set of controls is allowed to be nonconvex. Only one adjoint equation is introduced to derive the first order optimality necessary condition either by means of the classical variational analysis approach or under some assumption which is quite natural in the deterministic setting to guarantee the existence of optimal controls. More importantly, in order to avoid the essential difficulty with the well-posedness of higher order adjoint equations, using again the classical variational analysis approach, only the first and the second order adjoint equations are needed to formulate the second order necessary optimality condition, in which the solutions to the second order adjoint equation are understood in the sense of relaxed transposition.