Simultaneous Identification of Damping Coefficient and Initial Value in PDEs from boundary measurement

TL;DR

This paper presents a novel algorithm for simultaneously identifying damping coefficients and initial conditions in PDEs using boundary measurements, with proven convergence and demonstrated effectiveness through numerical examples.

Contribution

It introduces a new identification method leveraging system output decomposition into exponential and periodic functions, enabling simultaneous parameter and initial value estimation in PDEs.

Findings

Algorithm accurately estimates damping coefficients and initial values.

Numerical simulations confirm the method's effectiveness across different PDEs.

Convergence and error bounds are rigorously analyzed.

Abstract

In this paper, the simultaneous identification of damping or anti-damping coefficient and initial value for some PDEs is considered. An identification algorithm is proposed based on the fact that the output of system happens to be decomposed into a product of an exponential function and a periodic function. The former contains information of the damping coefficient, while the latter does not. The convergence and error analysis are also developed. Three examples, namely an anti-stable wave equation with boundary anti-damping, the Schr\"odinger equation with internal anti-damping, and two connected strings with middle joint anti-damping, are investigated and demonstrated by numerical simulations to show the effectiveness of the proposed algorithm.