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

2022/11/01 by Bo You, You, Bo, Fangjuan Li +1
Engineering · Mathematics · #34D06 #35Q30 #37C50 #76B75 #93C20 #Advanced Mathematical Physics Problems #FOS: Mathematics #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2211.00647

openalex publication_date 2022/11/01 · openalex created_date 2022/11/08 · openalex updated_date 2026/07/28

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 \citegs for fourth order parabolic equation with L²(Q)-external force, we establish the global Carleman estimates for the L²(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.

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