Optimality Conditions for Control Systems Governed by Monotone Stochastic Evolution Equations

TL;DR

This paper establishes necessary optimality conditions for control systems governed by nonlinear stochastic monotone equations, including a stochastic Pontryagin principle in specific cases, with applications to stochastic porous media equations.

Contribution

It provides the first order necessary conditions for a class of stochastic monotone control problems, extending the Pontryagin principle to these systems.

Findings

Derived first order necessary optimality conditions.

Established a stochastic Pontryagin principle for control systems.

Applied results to stochastic porous media equations.

Abstract

We study a class of optimal control problems governed by nonlinear stochastic equations of monotone type under certain coercivity and linear growth conditions. We give first order necessary conditions of optimality. A stochastic Pontryagin principle can be recovered in the case that the diffusion doesn't depend on the control. We give several applications, most notably for stochastic porous media equations in the Lipschitz case.