First and Second Order Necessary Optimality Conditions for Controlled Stochastic Evolution Equations with Control and State Constraints

TL;DR

This paper develops first and second order necessary optimality conditions for stochastic evolution control problems with constraints, incorporating advanced set-valued analysis and backward stochastic equations.

Contribution

It introduces novel second order optimality conditions for controlled stochastic evolution equations with control and state constraints, utilizing new solution concepts for backward stochastic equations.

Findings

Established first order necessary optimality conditions.

Derived second order necessary optimality conditions involving correction terms.

Applied set-valued analysis and backward stochastic equations techniques.

Abstract

The purpose of this paper is to establish first and second order necessary optimality conditions for optimal control problems of stochastic evolution equations with control and state constraints. The control acts both in the drift and diffusion terms and the control region is a nonempty closed subset of a separable Hilbert space. We employ some classical set-valued analysis tools and theories of the transposition solution of vector-valued backward stochastic evolution equations and the relaxed-transposition solution of operator-valued backward stochastic evolution equations to derive these optimality conditions. The correction part of the second order adjoint equation, which does not appear in the first order optimality condition, plays a fundamental role in the second order optimality condition.