Global Carleman estimate on a network for the wave equation and application to an inverse problem

Lucie Baudouin,Emmanuelle Crépeau,Julie Valein

doi:10.3934/mcrf.2011.1.307

Abstract

We are interested in an inverse problem for the wave equation with potential on a starshaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.

Highlights

Introduction and main resultIn this paper we consider a star-shaped network R of n + 1 edges ej, of length lj > 0, j ∈ {0, .., n}, connected at one vertex that we assume to be the origin 0 of all the edges
More precisely we consider on this plane 1-D network a wave equation with a different potential on each string, given by the following system uj,tt(x, t) − uj,xx(x, t) + pj (x)uj (x, t) = gj (x, t), uj = 0
One should quote for instance [29] for a case of two unknowns to recover and [3] that concerns the determination of potential but for an hyperbolic equation where the principal part of the operator has a discontinuity on an interface. These references are all based upon the use of local or global Carleman estimates

Summary

Introduction and main result

In this paper we consider a star-shaped network R of n + 1 edges ej, of length lj > 0, j ∈ {0, .., n}, connected at one vertex that we assume to be the origin 0 of all the edges (see Figure 1). One should quote for instance [29] for a case of two unknowns to recover and [3] that concerns the determination of potential but for an hyperbolic equation where the principal part of the operator has a discontinuity on an interface These references are all based upon the use of local or global Carleman estimates. The control, observation and stabilization problems of networks have been the object of intensive research (see [11, 18, 31] and the references therein) These works use results from several domains: non-harmonic Fourier series, Diophantine approximations, graph theory, wave propagation techniques.

A global Carleman estimate on a network

Stability of the inverse problem