Global Carleman estimates for a fourth order parabolic equation and application to null controllability

The main objective of this paper is to establish the null controllability for fourth-order semilinear parabolic equations with nonlinearities involving the state and its gradient up to the second order. First of all, based on the optimal control theory of partial differential equations and global Carleman estimates obtained from literature, a global Carleman estimate inequality is established for the $ L^2(Q) $-weak solutions of a fourth-order linear parabolic equation with a low regularity external term and some linear terms including the derivatives of the state up to the second order. Then, we prove the null controllability of fourth-order linear parabolic equations by such global Carleman estimates, and furthermore verify the null controllability of fourth-order semilinear parabolic equations by means of Leray-Schauder's fixed point theorem.