Boundary null controllability of a class of 2-d degenerate parabolic PDEs

TL;DR

This paper investigates boundary null controllability for a class of 2D degenerate parabolic PDEs on square domains, introducing new analytical techniques and extending results to higher dimensions.

Contribution

It provides the first analysis of boundary controllability for multidimensional degenerate parabolic equations, using combined classical methods and characterizing controllability via Kalman rank condition.

Findings

Boundary controllability established for 2D degenerate parabolic PDEs.

Extension of results to N-dimensional domains.

Characterization of controllability through Kalman rank condition.

Abstract

This article deals with the boundary null controllability of some degenerate parabolic equations posed on a square domain, presenting the first study of boundary controllability for such equations in multidimensional settings. The proof combines two classical techniques: the method of moments and the Lebeau-Robbiano strategy. A key novelty of this work lies in the analysis of boundary control localized on a subset of the boundary where the degeneracy occurs. Furthermore, we establish the Kalman rank condition as a full characterization of boundary controllability for coupled degenerate systems. The results are extended to $N$-dimensional domains, and potential extensions and open problems are discussed to motivate further research in this area.