A concise introduction to control theory for stochastic partial differential equations

Qi Lü; Xu Zhang

doi:10.3934/mcrf.2021020

Article Contents

2022, Volume 12, Issue 4: 847-954. Doi: 10.3934/mcrf.2021020

This issue Previous Article Preface special issue on recent advances in mathematical control theory Next Article Control of reaction-diffusion models in biology and social sciences

A concise introduction to control theory for stochastic partial differential equations

Qi Lü^, and
Xu Zhang

School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China

^* Corresponding author: Qi Lü
^* Corresponding author: Qi Lü

Received: June 2020

Early access: March 2021

Published: December 2022

The first author is supported by NSF of China under grants 12025105, 11971334 and 11931011, by the Chang Jiang Scholars Program from the Chinese Education Ministry, and by the Science Development Project of Sichuan University under grants 2020SCUNL201. The second author is supported by the NSF of China under grants 11931011 and 11821001, and by the Science Development Project of Sichuan University under grant 2020SCUNL201

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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.

Keywords:

stochastic partial differential equation,
controllability,
observability,
optimal control,
linear quadratic control problem,
Pontryagin type maximum principle.

Mathematics Subject Classification: Primary: 93E20, 60H15, 93B05, 93B07.

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