Distributed source identification for wave equations: an observer-based approach (full paper)

TL;DR

This paper presents an observer-based method for identifying unknown source terms in 1D and N-dimensional wave equations using boundary measurements, achieving minimal observation time through iterative schemes.

Contribution

It introduces a novel back-and-forth iterative observer approach for source identification in wave equations, extending to higher dimensions.

Findings

Successful source reconstruction in 1D wave equations

Extension of method to N-dimensional wave equations

Achieves minimal observation time for source identification

Abstract

In this paper, we consider the 1D wave equation where the spatial domain is a bounded interval. Assuming the initial conditions to be known, we are here interested in identifying an unknown source term, while we take the Neumann derivative of the solution on one of the boundaries as the measurement output. Applying a back-and-forth iterative scheme and constructing well-chosen observers, we retrieve the source term from the measurement output in the minimal observation time. We further provide an extension of the method to the case of wave equations with N dimensional spatial domain.