Carleman inequality for a class of super strong degenerate parabolic operators and applications

TL;DR

This paper develops a new Carleman inequality for a class of super strong degenerate parabolic operators and uses it to establish null controllability results for nonlinear parabolic systems.

Contribution

Introduces a novel Carleman estimate for degenerate parabolic equations and applies it to achieve null controllability for certain nonlinear systems.

Findings

Established a global null controllability for semilinear equations.

Achieved local null controllability for equations with nonlocal terms.

Provided a new analytical tool for degenerate parabolic control problems.

Abstract

In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in the general situations. Additionally, we also apply the aformentioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem.