A Mini-Course on Stochastic Control

Abstract

This course is addressed to giving a short introduction to control theory of stochastic systems, governed by stochastic differential equations in both finite and infinite dimensions. We will mainly explain the new phenomenon and difficulties in the study of controllability and optimal control problems for these sort of equations. In particular, we will show by some examples that both the formulation of stochastic control problems and the tools to solve them may differ considerably from their deterministic counterpart.3 Controllability of stochastic (ordinary) differential equations 13 4 Pontryagin-type maximum principle for controlled stochastic (ordinary) differential equations 21 * This is a lecture notes of a short introduction to stochastic control. It was written for the LIASFMA

Exact Controllability for a Refined Stochastic Wave Equation

“…Compared with the deterministic situation, there are many new difficulties and phenomena appeared in the study of controllability problems for stochastic control systems, even for the systems governed by stochastic (ordinary) differential equations (e.g. [21,24]). For example, it was shown in [21] that there exist no Kalman-type rank condition for the null controllability/approximate controllability for controlled stochastic differential equations.…”

“…[21,24]). For example, it was shown in [21] that there exist no Kalman-type rank condition for the null controllability/approximate controllability for controlled stochastic differential equations. People will meet more obstacles and substantially extra difficulties in the study of controllability problems for stochastic PDEs.…”