Global Carleman estimates for the fourth order parabolic equations and application to null controllability

TL;DR

This paper develops global Carleman estimates for fourth order parabolic equations and uses them to prove null controllability for certain semilinear systems involving second order derivatives.

Contribution

It introduces new global Carleman estimates for low regularity solutions of fourth order parabolic equations and applies them to establish null controllability.

Findings

Established global Carleman estimates for weak solutions with low regularity external forces.

Proved null controllability of fourth order semilinear parabolic equations.

Extended controllability results to equations involving second order derivatives.

Abstract

The main objective of this paper is to establish the null controllability for the fourth order semilinear parabolic equations with the nonlinearities involving the state and its gradient up to second order. First of all, based on optimal control theory of partial differential equations and global Carleman estimates obtained in \cite{gs} for fourth order parabolic equation with $L^2(Q)$-external force, we establish the global Carleman estimates for the $L^2(Q)$ weak solutions of the same system with a low regularity external term and some linear terms including the derivatives of the state up to second order. Then we prove the null controllability of the fourth order semilinear parabolic equations by such global Carleman estimates and the Leray-Schauder's fixed points theorem.