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

TL;DR

This paper develops new Carleman estimates for stochastic fourth order parabolic equations and applies them to establish null controllability for a stochastic Cahn-Hilliard type equation with complex derivatives.

Contribution

It introduces improved global Carleman estimates for stochastic fourth order equations and uses them to prove null controllability for a stochastic Cahn-Hilliard type model.

Findings

Established new Carleman estimates for stochastic fourth order equations.

Proved null controllability for the stochastic Cahn-Hilliard type equation.

Developed a fundamental identity for stochastic fourth order operators.

Abstract

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