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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Stochastics and Partial Differential Equations: Analysis and Computations
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
