Parameter identification in a semilinear hyperbolic system

TL;DR

This paper addresses the inverse problem of identifying a nonlinear damping law in a damped wave equation from boundary measurements, establishing well-posedness, ill-posedness, and proposing a regularization method with numerical validation.

Contribution

It introduces a variational regularization approach for recovering nonlinear damping laws in hyperbolic systems, with theoretical analysis and numerical demonstrations.

Findings

Well-posedness established via semigroup theory

Inverse problem shown to be ill-posed

Numerical results illustrate the regularization method

Abstract

We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and contraction arguments. We then investigte the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a variational regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized solutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings.