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Journal of Applied Mathematics
Volume 2010 (2010), Article ID 561395, 14 pages
http://dx.doi.org/10.1155/2010/561395
Research Article

On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements

1Département de Mathématiques, CNRS AGM UMR 8088, Université de Cergy-Pontoise, 95302 Cergy-Pontoise Cedex, France
2Département de Mathématiques, Université des Sciences de Carthage, 7021 Bizerte, Tunisia

Received 29 March 2010; Revised 31 May 2010; Accepted 1 June 2010

Academic Editor: Christo I. Christov

Copyright © 2010 Christian Daveau 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

This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation in a bounded smooth domain in from partial (on part of the boundary) dynamic boundary measurements. In this paper, we prove that the knowledge of the partial Cauchy data for this class of hyperbolic PDE on any open subset of the boundary determines explicitly the coefficient provided that is known outside a bounded domain. Then, through construction of appropriate test functions by a geometrical control method, we derive a formula for calculating the coefficient from the knowledge of the difference between the local Dirichlet-to-Neumann maps.

1. Introduction

In this paper, we present a new method for multidimensional Coefficient Inverse Problems (CIPs) for a class of hyperbolic Partial Differential Equations (PDEs). In the literature, the reader can find many key investigations of this kind of inverse problems; see, for example, [111] and references cited there. Beilina and Klibanov have deeply studied this important problem in various recent works [2, 12]. In [2], the authors have introduced a new globally convergent numerical method to solve a coefficient inverse problem associated to a hyperbolic PDE. The development of globally convergent numerical methods for multidimensional CIPs has started, as a first generation, from the developments found in [1315]. Else, Ramm and Rakesh have developed a general method for proving uniqueness theorems for multidimensional inverse problems. For the two dimensional case, Nachman [7] proved a uniqueness result for CIPs for some elliptic equation. Moreover, we find the works of Pivrinta and Serov [16, 17] about the same issue, but for elliptic equations. In other manner, the author Chen has treated in [18] the Fourier transform of the hyperbolic equation similar to ours with the unknown coefficient . Unlike this, we derive, using as weights particular background solutions constructed by a geometrical control method, asymptotic formulas in terms of the partial dynamic boundary measurements (Dirichlet-to-Neumann map) that are caused by the small perturbations. These asymptotic formulae yield the inverse Fourier transform of unknown coefficient.

The ultimate objective of the work described in this paper is to determine, effectively, the unknown smooth coefficient entering a class of hyperbolic equations in a bounded smooth domain in from partial (on part of the boundary) dynamic boundary measurements. The main difficulty which appears in boundary measurements is that the formulation of our boundary value problem involves unknown boundary values. This problem is well known in the study of the classical elliptic equations, where the characterization of the unknown Neumann boundary value in terms of the given Dirichlet datum is known as the Dirichlet-to-Neumann map. But, the problem of determining the unknown boundary values also occurs in the study of hyperbolic equations formulated in a bounded domain.

As our main result, we develop, using as weights particular background solutions constructed by a geometrical control method, asymptotic formulas for appropriate averaging of the partial dynamic boundary measurements that are caused by the small perturbations of coefficient according to a parameter .

The final formula (3.44) represents a promising approach to the dynamical identification and reconstruction of the coefficient . Moreover, it improves the given asymptotic formula (2.3) of the coefficient . Assume that the coefficient is known outside a bounded domain , and suppose that we know explicitly the value of for . Then, the developed asymptotic formulae yield the inverse Fourier transform of the unknown part of this coefficient.

In the subject of small volume perturbations from a known background material associated to the full time-dependent Maxwell's equations, we have derived asymptotic formulas to identify their locations and certain properties of their shapes from dynamic boundary measurements [19]. The present paper represents a different investigation of this line of work.

As closely related stationary identification problems, we refer the reader to [7, 2022] and references cited there.

2. Problem Formulation

Let be a bounded domain with a smooth boundary and let (our assumption is necessary in order to obtain the appropriate regularity for the solution using classical Sobolev embedding; see Brezis [23]). For simplicity, we take to be , but this condition could be considerably weakened. Let denote the outward unit normal vector to at a point on . Let , , and let be a smooth subdomain of . We denote by a measurable smooth open part of the boundary .

Throughout this paper, we will use quite standard -based Sobolev spaces to measure regularity.

As the forward problem, we consider the initial boundary value problem for a hyperbolic PDE in the domain Here and are subject to the compatibility conditions which give that (2.6) has a unique solution in ; see [24]. It is also well known that (2.1) has a unique weak solution ; see [24, 25]. Indeed, from [25] we have that belongs to .

Equation (2.1) governs a wide range of applications, including, for example, propagation of acoustic and electromagnetic waves.

We assume that the coefficient of (2.1) is such that where for with where is a smooth subdomain of and is a positive constant. We also assume that , the order of magnitude of the small perturbations of coefficient, is sufficiently small that where is a positive constant.

Define to be the solution of the hyperbolic equation in the homogeneous situation (). Thus, satisfies

Now, we define , and we introduce the trace space

It is well known that the dual of is .

Then, one can write where is the Dirichlet-to-Neumann map (D-t-N) operator, and is the solution of (2.1).

Let be the Dirichlet-to-Neumann map (D-t-N) operator defined as in (2.8) for the case . Then, our problem can be stated as follows.

Inverse Problem
Suppose that the smooth coefficient satisfies (2.4), (2.5), and (2.6), where the positive number is given. Assume that the function is unknown in the domain . Is it possible to determine the coefficient from the knowledge of the difference between the local Dirichlet-to-Neumann maps on , if we know explicitly the value of for ?
To give a positive answer, we will develop an asymptotic expansions of an "appropriate averaging" of on , using particular background solutions as weights. These particular solutions are constructed by a control method as it has been done in the original work [10] (see also [11, 2629]). It has been known for some time that the full knowledge of the (hyperbolic) Dirichlet to Neumann map uniquely determines conductivity; see [30, 31]. Our identification procedure can be regarded as an important attempt to generalize the results of [30, 31] in the case of partial knowledge (i.e., on only part of the boundary) of the Dirichlet-to-Neumann map to determine the coefficient of the hyperbolic equation considered above. The question of uniqueness of this inverse problem can be addressed positively via the method of Carleman estimates; see, for example, [6, 14].

3. The Identification Procedure

Before describing our identification procedure, let us introduce the following cutoff function such that on and let .

We will take in what follows and assume that we are in possession of the boundary measurements of This particular choice of data and implies that the background solution of the wave equation (2.6) in the homogeneous background medium can be given explicitly.

Suppose now that and the part of the boundary are such that they geometrically control which roughly means that every geometrical optic ray, starting at any point at time , hits before time at a nondiffractive point; see [32]. It follows from [33] (see also [34]) that there exists (a unique) (constructed by the Hilbert Uniqueness Method) such that the unique weak solution to the wave equation satisfies .

Let denote the unique solution of the Volterra equation of second kind We can refer to the work of Yamamoto in [11] who conceived the idea of using such Volterra equation to apply the geometrical control for solving inverse source problems.

The existence and uniqueness of this in for any can be established using the resolvent kernel. However, observing from differentiation of (3.3) with respect to that is the unique solution of the ODE: the function may be found (in practice) explicitly with variation of parameters and it also immediately follows from this observation that belongs to .

We introduce as the unique weak solution (obtained by transposition) in to the wave equation Then, the following holds.

Proposition 3.1. Suppose that and geometrically control . For any , we have Here means an elementary surface for .

Proof. Let be the solution of (3.5). From [25,Theorem , page 44], it follows that . Then, multiplying the equation by and integrating by parts over , for any , we have Therefore, since on .

In terms of the function as solution of (3.3), we introduce Moreover, for and for any , we define

The following lemma is useful to prove our main result.

Lemma 3.2. Consider an arbitrary function satisfying condition (2.3), and assume that conditions (2.4) and (2.5) hold. Let and be solutions of (2.6) and (2.1), respectively. Then, using (3.9) the following estimates hold: where is a positive constant. And where is a positive constant.

Proof. Let be defined by We have Since we obtain From the Gronwall Lemma, it follows that As a consequence, by using (3.10), one can see that the function solves the following boundary value problem: Integration by parts immediately gives Taking into account that , we find by using the above estimate that Under relation (3.9), one can define the function as a solution of Integrating by parts immediately yields To proceed with the proof of estimate (3.12), we firstly remark that the function given by (3.9) is a solution of Then, we deduce that solves the following initial boundary value problem: Finally, we can use (3.24) to find by integrating by parts that which, from the Gronwall Lemma and by using (3.20), yields This achieves the proof.

Now, we identify the function by using the difference between local Dirichlet to Neumann maps and the function as a solution to the Volterra equation (3.3) or equivalently the ODE (3.4), as a function of . Then, the following main result holds.

Theorem 3.3. Let , . Suppose that the smooth coefficient satisfies (2.3), (2.4), and (2.5). Let be the unique solution in to the wave equation (2.1) with and Let . Suppose that and geometrically control ; then we have where is the unique solution to the ODE (3.4) with defined as the boundary control in (3.2). The term is independent of the function . It depends only on the bound .

Proof. Since the extension of to is , then by conditions and we have Therefore, the term may be simplified as follows:
On the other hand, we have where .
Given that, satisfies the Volterra equation (3.4) and we obtain by integrating by parts over that and so, from Proposition 3.1, we obtain Thus, to prove Theorem 3.3, it suffices then to show that
From definition (3.10), we have which gives by system (3.24) that Thus, by (3.9) and (3.24) again, we see that the function is the solution of Taking into account estimate (3.12) given by Lemma 3.2, then by using standard elliptic regularity (see, e.g., [24]) for the boundary value problem (3.38), we find that By the fact that,we deduce, as done in the proof of Lemma 3.2, that which implies that This completes the proof of our Theorem.

We are now in position to describe our identification procedure which is based on Theorem 3.3. Let us neglect the asymptotically small remainder in the asymptotic formula (3.27). Then, it follows that The method of reconstruction we propose here consists in sampling values of at some discrete set of points and then calculating the corresponding inverse Fourier transform.

In the following, a better approximation than (2.3) is derived. It is not hard to prove the more convenient approximation in terms of the values of local Dirichlet-to-Neumann maps and at .

Corollary 3.4. Let and let . Suppose that and geometrically control ; then we have the following better approximation: where the boundary control is defined by (3.2).

Proof. The term , given in Theorem 3.3, has to be interpreted as follows: because and . In fact, in view of the ODE (3.4), the term may be simplified after integration by parts over and using of the fact that is the solution to the ODE (3.4) to become Then, the desired approximation is established.

4. Conclusion

The use of approximate formula (3.27), including the difference between the local Dirichlet to Neumann maps, represents a promising approach to the dynamical identification and reconstruction of a coefficient which is unknown in a bounded domain (but it is known outside of this domain) for a class of hyperbolic PDE. We believe that this method will yield a suitable approximation to the dynamical identification of small conductivity ball (of the form ) in a homogeneous medium in from the boundary measurements. We will present convenable numerical implementations for this investigation. This issue will be considered in a forthcoming work.

Acknowledgments

The authors are grateful to the editor and the anonymous referees for their valuable comments and helpful suggestions which have much improved the presentation of the article.

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