Controllability results for stochastic coupled systems of fourth- and second-order parabolic equations

Abstract

In this paper, we study some controllability and observability problems for stochastic systems coupling fourth-and second-order parabolic equations. The main goal is to control both equations with only one controller localized on the drift of the fourth-order equation. We analyze two cases: on one hand, we study the controllability of a backward system where the couplings are made through first-order terms. The key point is to use suitable Carleman estimates for the heat equation and the fourth-order operator with the same weight to deduce an observability inequality for the adjoint system. On the other hand, we study the controllability of a coupled model of forward equations. This case, which is well-known to be harder to solve, requires to prove an observability inequality for an adjoint backward system with initial datum in a finite dimensional space and employ the classical iterative method introduced in the seminal work by G. Lebeau and L. Robbiano.