Optimal Distributed and Tangential Boundary Control for the Unsteady Stochastic Stokes Equations

TL;DR

This paper develops a framework for optimal control of unsteady stochastic Stokes equations with boundary and distributed controls, deriving explicit formulas via a stochastic maximum principle to improve control strategies.

Contribution

It introduces a novel approach combining boundary and distributed controls for stochastic Stokes equations and derives explicit optimal control formulas using a stochastic maximum principle.

Findings

Explicit formulas for optimal controls are derived.

Controls can be effectively applied both inside the domain and on the boundary.

The method enhances control precision for stochastic fluid dynamics problems.

Abstract

We consider a control problem constrained by the unsteady stochastic Stokes equations with nonhomogeneous boundary conditions in connected and bounded domains. In this paper, controls are defined inside the domain as well as on the boundary. Using a stochastic maximum principle, we derive necessary and sufficient optimality conditions such that explicit formulas for the optimal controls are derived. As a consequence, we are able to control the stochastic Stokes equations using distributed controls as well as boundary controls in a desired way.