Control theory of stochastic distributed parameter systems: recent progress and open problems

Abstract

In recent years, important progresses have been made in the control theory for stochastic distributed parameter control systems (SDPSs for short). However, the theory is far from being complete. The primary difficulty is that many effective tools and methods for deterministic distributed parameter control systems and stochastic finite-dimensional control systems do not work anymore for SDPSs. One has to develop new mathematical tools, such as stochastic transposition method and stochastic Carleman estimate, even for some very simple SDPSs. The objectives of this paper are to provide some new results, to show some new phenomena, to explain the new difficulties, and to present some new methods for the control theory of SDPSs. We mainly focus on our works for the controllability for stochastic hyperbolic equations, and the Pontryagin-type maximum principle for controlled stochastic evolution equations. At last, a number of open questions and future directions of research are given.