Controllability of Forward Stochastic Reaction--Convection--Diffusion Systems with Cascade Structure

Abstract

We investigate the null and approximate controllability of coupled linear forward stochastic reaction--convection--diffusion systems under suitable cascade coupling conditions. The model consists of two forward stochastic parabolic equations governed by general second-order differential operators with time-, space-, and random-dependent coefficients. We consider a localized control acting on the drift term of the first equation together with controls on the diffusion terms. By a duality argument, the controllability problem is reduced to an observability problem for the associated adjoint backward stochastic parabolic system. The main contribution of this paper is the establishment of a new global Carleman estimate for coupled backward stochastic parabolic systems whose drift terms belong to a negative Sobolev space. This estimate yields the required observability properties and, consequently, the null and approximate controllability of the original system.