Parameter identification in a semilinear hyperbolic system
This work provides a theoretical and numerical framework for recovering nonlinear damping laws in hyperbolic systems, relevant for applications like pipe flow modeling, but the results are incremental and domain-specific.
The paper addresses the identification of a nonlinear friction law in a damped wave equation from boundary measurements, proving well-posedness and ill-posedness of the inverse problem, and proposing a variational regularization method with convergence analysis and numerical results.
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.