Statistical null-controllability of stochastic nonlinear parabolic equations

Springer Science and Business Media LLC - Tập 10 - Trang 190-222 - 2021
Víctor Hernández-Santamaría1, Kévin Le Balc’h2, Liliana Peralta3
1Instituto de Matemáticas, Universidad Nacional Autónoma de México, CDMX, Mexico
2Institut de Mathématiques de Bordeaux, Bordeaux, France
3Centro de Investigación en Matemáticas, UAEH, Pachuca, Mexico

Tóm tắt

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.

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