Null controllability for cascade systems of coupled backward stochastic parabolic equations with one distributed control

Abstract

We prove the null controllability of a cascade system of n coupled backward stochastic parabolic equations, which involve both reaction and convection terms, as well as general second-order parabolic operators, with $n\ge 2$. To achieve this, we apply a single distributed control to the first equation, while the other equations are controlled through the coupling. To obtain our results, we develop a new global Carleman estimate for the forward stochastic parabolic adjoint system, with some terms in the $H^{-1}$-space. Subsequently, we derive the corresponding observability inequality, and using the classical duality argument, we establish our null controllability result. Additionally, we provide an accurate estimate for the null control cost in terms of the final time T and the potentials of the system.