On Controllability of a Class of N -dimensional Hyperbolic Equations with Internal Single-point Degeneracy

TL;DR

This paper investigates the controllability of N-dimensional hyperbolic equations with a single interior degeneracy, establishing well-posedness and deriving exact controllability using Carleman estimates.

Contribution

It introduces a novel approach to construct weight functions in Carleman estimates that handle the degenerate region effectively.

Findings

Established well-posedness via Hardy inequality.

Derived observability inequality using a new Carleman estimate approach.

Proved exact controllability including control at the degenerate null point.

Abstract

This paper explores the controllability of a class of N-dimensional hyperbolic equations featuring a single interior degenerate point. Firstly, we establish the well-posedness of the equation through the application of the Hardy inequality. Following this, we primarily utilize the Carleman estimate method to derive the observability inequality. By leveraging the equivalence between observability and controllability, we deduce the exact controllability of the equation. It is worth noting that our selected control region includes the degenerate null point. In the Carleman estimate, we adopt a unique approach to construct the weight function, effectively negating the influence of the degenerate region.