Statistical null-controllability of stochastic nonlinear parabolic equations

TL;DR

This paper establishes small-time global null-controllability for stochastic nonlinear parabolic equations with localized control, and explores statistical local null-controllability of the true system, using cost estimates and truncation techniques.

Contribution

It introduces a novel approach to control stochastic nonlinear parabolic equations, including the stochastic Burgers and Allen-Cahn equations, under minimal regularity assumptions.

Findings

Proved small-time global null-controllability for stochastic nonlinear parabolic equations.

Established statistical local null-controllability of the true stochastic system.

Developed a truncation method to handle nonlinearity and regularity issues.

Abstract

In this paper, we consider forward stochastic nonlinear parabolic equations, with a control localized in the drift term. Under suitable assumptions, we prove the small-time global null-controllability, with a truncated nonlinearity. We also prove the statistical local null-controllability of the true system. The proof relies on a precise estimation of the cost of null-controllability of the stochastic heat equation and on an adaptation of the source term method to the stochastic setting. The main difficulty comes from the estimation of the nonlinearity in the fixed point argument due to the lack of regularity (in probability) of the functional spaces where stochastic parabolic equations are well-posed. This main issue is tackled through a truncation procedure. As relevant examples that are covered by our results, let us mention the stochastic Burgers equation in the one dimensional case and the Allen-Cahn equation up to the three-dimensional setting.