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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 796539, 8 pages
http://dx.doi.org/10.1155/2015/796539
Research Article

An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation

Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science & Technology, P.O. Box 4700, Thuwal 23955-6900, Saudi Arabia

Received 12 February 2015; Revised 20 August 2015; Accepted 31 August 2015

Academic Editor: Herb Kunze

Copyright © 2015 Sharefa Asiri et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.