Controllability and observability of some coupled stochastic parabolic systems

Abstract

This paper is devoted to a study of controllability and observability problems for some stochastic coupled linear parabolic systems only by one control and through an observer, respectively. In order to get a null controllability result, the Lebeau-Robbiano technique is adopted. The key point is to prove an observability inequality for certain stochastic coupled backward parabolic system by an iteration, when terminal values belong to a finite dimensional space. Different from deterministic systems, Kalman-type rank conditions for the controllability of stochastic coupled parabolic systems do not hold any more. Meanwhile, based on the Carleman estimates method, an observability inequality and unique continuation property for general stochastic linear coupled parabolic systems through an observer are derived.

Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

“…In the following, we always suppose that B is a generator of GE-semigroup U(t) induced by A. Now, we consider the mild solution of stochastic singular linear system (58).…”

“…Definition 32. (a) Stochastic singular linear system ( 58) is called to be exactly controllable on [0, b], if for all x 0 ∈ L 2 (Ω, F 0 , P, D 1 ), x b ∈ L 2 (Ω, F b , P, D 1 ), there exists v(t) ∈ L 2 ([0, b], Ω, U), such that the mild solution x(t, x 0 ) to stochastic singular linear system (58) which is given by (61) satisfies x(T, x 0 ) = x b ; (b) Stochastic singular linear system ( 58) is called to be approximately controllable on [0, b], if for any state x b ∈ L 2 (Ω, F b , P, D 1 ), any initial state x 0 ∈ L 2 (Ω, F 0 , P, D 1 ), and any > 0, existence v ∈ L 2 ([0, b], Ω, U) makes that the mild solution x(t, x 0 ) which is given by (61) satisfies x(b, x 0 ) − x b L 2 (Ω,F b ,P,D 1 ) < .…”

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

“…As we have mentioned before, this situation is more complicated than for a single equation and in the stochastic setting even more difficulties appear. Indeed, there are only a handful of works studying controllability problems for coupled stochastic systems with less controls than equations, see, [LL12,Liu14a,LL18]. In particular, in [LL18], for controlling several parabolic equations with few controls, well-known facts such as Kalman-type conditions that are true in the deterministic setting (see e.g.…”

“…Indeed, there are only a handful of works studying controllability problems for coupled stochastic systems with less controls than equations, see, [LL12,Liu14a,LL18]. In particular, in [LL18], for controlling several parabolic equations with few controls, well-known facts such as Kalman-type conditions that are true in the deterministic setting (see e.g. [AKBGBdT11, Theorem 5.1]) are not longer valid for the stochastic setting.…”

Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem

“…One successful application of Carleman estimate in stochastic partial differential equations is to study related control problems for various mathematical models with stochastic effect [15,[20][21][22][23]. As for null controllability for the deterministic degenerate equations, we refer to [2,3] for degenerate parabolic equation, [1,4] for degenerate parabolic equation with the gradient terms, [24][25][26][27] for coupled degenerate systems and so on.…”