Null Controllability for Cascade systems of Coupled Backward Stochastic Parabolic Equations with One Distributed Control

TL;DR

This paper establishes the null controllability of a cascade system of coupled backward stochastic parabolic equations using a novel Carleman estimate and duality, with implications for control cost estimation.

Contribution

It introduces a new global Carleman estimate for stochastic parabolic systems and proves null controllability for coupled backward stochastic equations with a single distributed control.

Findings

Proved null controllability for the cascade system.

Developed a new Carleman estimate for stochastic systems.

Provided estimates for control costs relative to final time and potentials.

Abstract

We prove the null controllability of a cascade system of \(n\) coupled backward stochastic parabolic equations involving both reaction and convection terms, as well as general second-order parabolic operators, with \(n \geq 2\). 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 appropriate observability inequality, and by employing the classical duality argument, we establish our null controllability result. Additionally, we provide an estimate for the null control cost with respect to the final time \(T\) and the potentials.