Controllability of a One-Dimensional Dynamic Debonding Model

TL;DR

This paper studies the controllability of a one-dimensional dynamic debonding model, combining wave equations with fracture criteria, and characterizes the states reachable through boundary controls.

Contribution

It provides a detailed analysis of the controllability of the model and constructs exact boundary controls for specific target states.

Findings

Characterization of reachable target states in different regularity settings

Construction of exact boundary controls for these states

Precise mathematical description of controllability conditions

Abstract

We investigate a one-dimensional dynamic debonding model, introduced by Freund (1990), in which the wave equation is coupled with a Griffith criterion governing the propagation of the fracture. In particular, we study the boundary controllability of the system to a prescribed target state. Our main results provide precise characterizations of the reachable target states, in both \( C^{ 0, 1 } \) and \( C^1 \) regularity settings, and construct exact controls toward these target states.