Global null-controllability for stochastic semilinear parabolic equations

TL;DR

This paper establishes the small-time global null-controllability of certain stochastic semilinear parabolic equations, solving an open problem and introducing a novel control strategy using refined Carleman estimates and fixed point methods.

Contribution

It introduces a new control approach for stochastic PDEs with explicit Carleman parameters, overcoming compactness issues in controllability analysis.

Findings

Proves small-time global null-controllability for stochastic semilinear parabolic equations.

Develops a refined Carleman estimate with explicit parameters for control purposes.

Employs a fixed point method to handle nonlinearities in stochastic control problems.

Abstract

In this paper, we prove the small-time global null-controllability of forward (resp. backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and diffusion terms (resp. in the drift term). In particular, we solve the open question posed by S. Tang and X. Zhang, in 2009. We propose a new twist on a classical strategy for controlling linear stochastic systems. By employing a new refined Carleman estimate, we obtain a controllability result in a weighted space for a linear system with source terms. The main novelty here is that the Carleman parameters are made explicit and are then used in a Banach fixed point method. This allows to circumvent the well-known problem of the lack of compactness embeddings for the solutions spaces arising in the study of controllability problems for stochastic PDEs.