New global Carleman estimates and null controllability for a stochastic Cahn-Hilliard type equation

Abstract

In this paper, we study the null controllability for a stochastic semilinear Cahn-Hilliard type equation, whose semilinear term contains first and second order derivatives of solutions. To start with, an improved global Carleman estimate for linear backward fourth order stochastic parabolic equations with $L^2$-valued source terms is derived, which is based on a new fundamental identity for a fourth order stochastic parabolic operator. Based on it, we establish a new global Carleman estimate for linear backward fourth order stochastic parabolic equations with $H^{-2}$-valued source terms, which, together with a fixed point argument, derive the desired null controllability for the stochastic Cahn-Hilliard type equation.