半线性双曲系统中参数识别
Parameter identification in a semilinear hyperbolic system
AI总结：
本文研究了一维阻尼波动方程中非线性摩擦定律的识别问题，通过半群理论和收缩论证建立系统well-posedness，并采用变分正则化方法解决逆问题，证明Tikhonov泛函的极小值存在性及正则化解的收敛性。
摘要：
我们考虑从额外的边界测量中识别一维阻尼波动方程中非线性摩擦定律的问题。通过半群理论和收缩论证建立了 governing 半线性双曲系统的 well-posedness。随后研究了从管道中压力降的额外边界测量中恢复未知非线性阻尼定律的逆问题。该系数逆问题被证明是病态的，并考虑了变分正则化方法以稳定求解。我们证明了 Tikhonov 泛函的极小值存在性，并讨论了在近似源条件下正则化解的收敛性。详细讨论了该条件的意义和其有效性的一些论证，并通过数值结果来说明理论发现。
英文摘要：
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.
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