Null-controllability for a fourth order parabolic equation under general boundary conditions

TL;DR

This paper establishes null-controllability for a fourth order parabolic equation with general boundary conditions by deriving a spectral inequality through Carleman estimates, applicable to bounded domains and manifolds.

Contribution

It introduces a spectral inequality for fourth order parabolic equations with general boundary conditions, leading to null-controllability results under broad boundary operator assumptions.

Findings

Spectral inequality for bi-Laplace operator under general boundary conditions.

Null-controllability established for fourth order parabolic equations.

Carleman inequality used to derive key interpolation inequalities.

Abstract

In this paper, we consider a fourth order inner-controlled parabolic equation on an open bounded subset of $R^d$, or a smooth compact manifold with boundary, along with general boundary operators fulfilling the Lopatinskii-Sapiro condition. We derive a spectral inequality for the solution of the parabolic system that yields a null-controllability result. The spectral inequality is a consequence of an interpolation inequality obtained via a Carleman inequality for the bi-Laplace operator under the considered boundary conditions.