2020/09/24 by Víctor Hernández-Santamaría, Hernandez-Santamaria, Victor, Kévin Le Balc’h +3
Computer Science · Economics, Econometrics and Finance · Engineering · #93B05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2009.11914
openalex publication_date 2020/09/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In this paper, we consider forward stochastic nonlinear parabolic equations,\nwith a control localized in the drift term. Under suitable assumptions, we\nprove the small-time global null-controllability, with a truncated\nnonlinearity. We also prove the statistical local null-controllability of the\ntrue system. The proof relies on a precise estimation of the cost of\nnull-controllability of the stochastic heat equation and on an adaptation of\nthe source term method to the stochastic setting. The main difficulty comes\nfrom the estimation of the nonlinearity in the fixed point argument due to the\nlack of regularity (in probability) of the functional spaces where stochastic\nparabolic equations are well-posed. This main issue is tackled through a\ntruncation procedure. As relevant examples that are covered by our results, let\nus mention the stochastic Burgers equation in the one dimensional case and the\nAllen-Cahn equation up to the three-dimensional setting.\n