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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.
@article{AIHPC_2023__40_6_1415_0,
author = {Hern\'andez-Santamar{\'\i}a, V{\'\i}ctor and Le Balc{\textquoteright}h, K\'evin and Peralta, Liliana},
title = {Global null-controllability for stochastic semilinear parabolic equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1415--1455},
volume = {40},
number = {6},
year = {2023},
doi = {10.4171/aihpc/69},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpc/69/}
}
TY - JOUR AU - Hernández-Santamaría, Víctor AU - Le Balc’h, Kévin AU - Peralta, Liliana TI - Global null-controllability for stochastic semilinear parabolic equations JO - Annales de l'I.H.P. Analyse non linéaire PY - 2023 SP - 1415 EP - 1455 VL - 40 IS - 6 UR - http://geodesic.mathdoc.fr/articles/10.4171/aihpc/69/ DO - 10.4171/aihpc/69 LA - en ID - AIHPC_2023__40_6_1415_0 ER -
%0 Journal Article %A Hernández-Santamaría, Víctor %A Le Balc’h, Kévin %A Peralta, Liliana %T Global null-controllability for stochastic semilinear parabolic equations %J Annales de l'I.H.P. Analyse non linéaire %D 2023 %P 1415-1455 %V 40 %N 6 %U http://geodesic.mathdoc.fr/articles/10.4171/aihpc/69/ %R 10.4171/aihpc/69 %G en %F AIHPC_2023__40_6_1415_0
Hernández-Santamaría, Víctor; Le Balc’h, Kévin; Peralta, Liliana. Global null-controllability for stochastic semilinear parabolic equations. Annales de l'I.H.P. Analyse non linéaire, Tome 40 (2023) no. 6, pp. 1415-1455. doi: 10.4171/aihpc/69
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