Global null-controllability for stochastic semilinear parabolic equations

In this paper, we prove the small-time global null-controllability of forward (resp. backward) semilinear stochastic parabolic equations with globally Lipschitz nonlinearities in drift and diffusion terms term). particular, solve open question posed by S. Tang X. Zhang, 2009. We propose a new twist on classical strategy for controlling linear systems. By employing refined Carleman estimate, obtain controllability result weighted space system source terms. The main novelty here is that parameters are made explicit then used Banach fixed point method. This allows to circumvent well-known problem lack compactness embeddings solutions spaces arising study problems PDEs.