A Concise Introduction to Control Theory for Stochastic Partial Differential Equations

TL;DR

This paper provides a concise overview of control theory applied to stochastic partial differential equations, focusing on controllability and optimal control, and introduces new analytical tools like the stochastic transposition method.

Contribution

It offers new results on controllability and optimal control for stochastic PDEs and introduces the stochastic transposition method as a key analytical tool.

Findings

Exact controllability of stochastic transport equations

Null and approximate controllability of stochastic parabolic equations

Lack of exact controllability of stochastic hyperbolic equations

Abstract

The aim of this notes is to give a concise introduction to control theory for systems governed by stochastic partial differential equations. We shall mainly focus on controllability and optimal control problems for these systems. For the first one, we present results for the exact controllability of stochastic transport equations, null and approximate controllability of stochastic parabolic equations and lack of exact controllability of stochastic hyperbolic equations. For the second one, we first introduce the stochastic linear quadratic optimal control problems and then the Pontryagin type maximum principle for general optimal control problems. It deserves mentioning that, in order to solve some difficult problems in this field, one has to develop new tools, say, the stochastic transposition method introduced in our previous works.