Insensitizing controls for a fourth order semi-linear parabolic equations

TL;DR

This paper investigates the existence of insensitizing controls for a fourth order semilinear parabolic equation, ensuring certain functionals remain unaffected by small initial data perturbations, using advanced controllability techniques.

Contribution

It establishes null controllability results for a nonlinear cascade system related to the equation via Carleman estimates and fixed point arguments, advancing control theory for complex PDEs.

Findings

Proved null controllability for linear fourth order parabolic equations.

Extended controllability results to semi-linear cases using fixed point methods.

Demonstrated insensitizing controls can be constructed for the system.

Abstract

This paper is concerned with the existence of insensitizing controls for a fourth order semilinear parabolic equation. Here, the initial data is partially unknown, we would like to find controls such that a specific functional is insensitive for small perturbations of the initial data. In general, this kind of problems can be recast as a null controllability problem for a nonlinear cascade system. We will first prove a null controllability result for a linear problem by global Carleman estimates and dual arguments. Then, by virtue of Leray-Schauder's fixed points theorem, we conclude the null controllability for the cascade system in the semi-linear case.