On Boundary Exact Controllability of One-Dimensional Wave Equations with Weak and Strong Interior Degeneration

TL;DR

This paper investigates the boundary controllability of one-dimensional wave equations with interior degeneracy, establishing conditions for controllability and well-posedness using the HUM method.

Contribution

It provides a rigorous analysis of controllability conditions for wave equations with interior degeneracy, including well-posedness and the impact of degeneracy rates.

Findings

Derived conditions for exact boundary controllability

Established well-posedness of degenerate wave systems

Identified degeneracy rates that hinder controllability

Abstract

In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.