Boundary null controllability of degenerate heat equation as the limit of internal controllability

TL;DR

This paper establishes boundary null controllability for a degenerate heat equation by analyzing the limit of internal controllability solutions using Carleman estimates and weak convergence techniques.

Contribution

It introduces a novel approach to derive boundary controllability results for degenerate heat equations via asymptotic analysis of internal control problems.

Findings

Boundary null controllability is achieved for the degenerate heat equation.

A new set of trace operator inequalities are developed for the degenerate case.

The approach relies on asymptotic analysis and Carleman estimates.

Abstract

In this paper, we recover the boundary null controllability for the degenerate heat equation by analyzing the asymptotic behavior of an eligible family of state-control pairs $((u_{\varepsilon}, h_{\varepsilon}))_{\varepsilon >0}$ solving corresponding singularly perturbed internal null controllability problems. As in other situations studied in the literature, our approach relies on Carleman estimates and meticulous weak convergence results. However, for the degenerate parabolic case, some specific trace operator inequalities must be obtained, in order to justify correctly the passage to the limit argument.