A General Maximum Principle for Stochastic Systems with Delay

TL;DR

This paper develops a general maximum principle for stochastic control systems with delay, non-convex control domains, and control-dependent diffusion, enabling the solution of complex delayed control problems.

Contribution

It introduces a novel maximum principle for stochastic delayed systems with non-convex controls and control-dependent diffusion coefficients, expanding existing theoretical frameworks.

Findings

Derived a general maximum principle using spike variation and duality methods.

Applied the principle to a delayed linear-quadratic control problem.

Obtained an explicit optimal solution for the example problem.

Abstract

In this paper, we consider optimal control problems derived by stochastic systems with delay, where control domains are non-convex and the diffusion coefficients depend on control variables. By an estimate of the integral of x_{1}(t)x_{1}(t-\delta) term, we obtain a general maximum principle for the optimal control problems with a standard spike variational technique and duality method. The maximum principle is applied to study a delayed linear-quadratic optimal control problem with a non-convex control domain; an optimal solution is obtained.