Null controllability for some systems of two backward stochastic heat equations with one control force

Abstract

In this paper, we establish the null controllability for system coupled by two backward stochastic parabolic equations. The desired controllability result is obtained by means of proving a suitable observability estimate for the dual system of the controlled system.

Controllability and observability of some coupled stochastic parabolic systems

“…Compared to the known results in [10], the coupling appears not only in drift terms, but also in diffusion terms in our paper. Also, the requirement for regularity on coefficients of principal parts may be relaxed to W 1,∞ (G), while the coefficients in diffusion terms are required to be in L ∞ (G), rather than W 1,∞ (G) in [10].…”

“…By a duality technique, this is indeed a special case of Theorem 1.2 for n = 2, since the controllability result in [10] is equivalent to the observability for a coupled system of two forward stochastic parabolic equations through only one observer. Compared to the known results in [10], the coupling appears not only in drift terms, but also in diffusion terms in our paper. Also, the requirement for regularity on coefficients of principal parts may be relaxed to W 1,∞ (G), while the coefficients in diffusion terms are required to be in L ∞ (G), rather than W 1,∞ (G) in [10].…”

“…Also, further explanations on the necessity of (H 2 ) are given in section 3 for n = 3. On the other hand, in [10], a null controllability result for some coupled systems by two backward stochastic parabolic equations by one control was obtained. By a duality technique, this is indeed a special case of Theorem 1.2 for n = 2, since the controllability result in [10] is equivalent to the observability for a coupled system of two forward stochastic parabolic equations through only one observer.…”

“…On the other hand, in [10], a null controllability result for some coupled systems by two backward stochastic parabolic equations by one control was obtained. By a duality technique, this is indeed a special case of Theorem 1.2 for n = 2, since the controllability result in [10] is equivalent to the observability for a coupled system of two forward stochastic parabolic equations through only one observer. Compared to the known results in [10], the coupling appears not only in drift terms, but also in diffusion terms in our paper.…”

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.…”