Global null-controllability for stochastic semilinear parabolic equations

• Published in: Annales de l'Institut Henri Poincaré. Analyse non linéaire Vol. 40; no. 6; pp. 1415 - 1455
• Main Authors: Hernández-Santamaría, Víctor, Le Balc’h, Kévin, Peralta, Liliana
• Format: Journal Article
• Language: English
• Published: EMS 13-01-2023
• Subjects: Mathematics; Forward and backward semilinear stochastic parabolic equations; Global null-controllability; Banach fixed point method; Carleman estimates
• ISSN: 0294-1449; 1873-1430
• DOI: 10.4171/aihpc/69

In this paper we prove the small-time global null-controllability of forward (respectively backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in the drift and the diffusion terms (respectively 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 us 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.