Null Controllability for Stochastic Parabolic Equations with convection terms

TL;DR

This paper establishes null controllability for stochastic parabolic equations with convection terms using novel Carleman estimates, addressing open questions and providing improved control cost estimates.

Contribution

It introduces new Carleman estimates for stochastic parabolic equations with convection, solving previously open problems and refining control cost bounds.

Findings

Proved null controllability for stochastic parabolic equations with convection.

Developed new Carleman estimates for both forward and backward equations.

Provided more precise estimates of null-control costs.

Abstract

This paper addresses null controllability for both forward and backward linear stochastic parabolic equations by introducing convection terms on the drift parts with bounded coefficients. Moreover, the forward stochastic parabolic equation includes a convection term on the diffusion part. The null controllability results rely on novel Carleman estimates for both backward and forward stochastic parabolic equations, encompassing a divergence source term interpreted in the weak sense. These Carleman estimates are established through the application of the duality technique. In doing so, we resolve some previously unanswered questions (see Remarks 2.1-2.2 in [S. Tang, and X. Zhang, SIAM J. Control Optim., 48 (2009), p.p 2191-2216]). Additionally, we present a more accurate estimation of the null-control costs.