Null Controllability for a Multi-Dimensional Degenerate Parabolic Equation with Degenerated Interior Point

TL;DR

This paper establishes null controllability for a multi-dimensional degenerate parabolic PDE with an interior degeneracy point, using a novel approximation approach and Carleman estimates.

Contribution

It introduces a new approximation method for degenerate PDEs and proves null controllability without control over the degenerate point.

Findings

Derived Carleman estimates for approximate equations

Established observability inequality for the system

Proved null controllability despite interior degeneracy

Abstract

In this study, we study the null controllability of a multi-dimensional degenerate parabolic equation characterized by a degenerate interior point. The control domain, which is an arbitrary inner region, does not encompass the degenerate point. To tackle this problem, we adopt a new approximation methodology. Specifically, we approximate the degenerate partial differential equations (PDEs) with a series of uniformly elliptic PDEs, notwithstanding their limited regularity. We then derive the Carleman estimate for these approximate uniformly parabolic equations and establish the observability inequality, which ultimately paves the way for demonstrating the null controllability of the system.