On a Hyperbolic Coefficient Inverse Problem via Partial Dynamic Boundary Measurements

Abstract

This paper is devoted to the identification of the unknown smooth coefficient c entering the hyperbolic equation 𝑐(𝑥)𝜕²_𝑡𝑢−Δ𝑢=0 in a bounded smooth domain in ℝ^𝑑 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 Γ of the boundary determines explicitly the coefficient 𝑐 provided that 𝑐 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 𝑐 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, [1–11] 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 [13–15]. 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 Päivärinta 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 𝑐(𝑥). 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 𝑐 entering a class of hyperbolic equations in a bounded smooth domain in ℝ^𝑑 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 𝛼.

The final formula (3.44) represents a promising approach to the dynamical identification and reconstruction of the coefficient 𝑐(𝑥). Moreover, it improves the given asymptotic formula (2.3) of the coefficient 𝑐(𝑥). Assume that the coefficient is known outside a bounded domain Ω, and suppose that we know explicitly the value of lim_𝛼→0⁺𝑐(𝑥) for 𝑥∈Ω. 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, 20–22] and references cited there.

2. Problem Formulation

Let Ω⊂ℝ^𝑑 be a bounded domain with a smooth boundary and let 𝑑=2,3 (our assumption 𝑑≤3 is necessary in order to obtain the appropriate regularity for the solution using classical Sobolev embedding; see Brezis [23]). For simplicity, we take 𝜕Ω to be 𝒞^∞, but this condition could be considerably weakened. Let 𝑛=𝑛(𝑥) denote the outward unit normal vector to Ω at a point on 𝜕Ω. Let 𝑇>0, 𝑥₀∈ℝ^𝑑⧵Ω, and let Ω^′ be a smooth subdomain of Ω. We denote by Γ⊂⊂𝜕Ω a measurable smooth open part of the boundary 𝜕Ω.

Throughout this paper, we will use quite standard 𝐿²-based Sobolev spaces to measure regularity.

As the forward problem, we consider the initial boundary value problem for a hyperbolic PDE in the domain Ω×(0,𝑇)(𝑐_𝛼𝜕²_𝑡−Δ)𝑢_𝛼𝑢=0,inΩ×(0,𝑇),_𝛼|_𝑡=0=𝜑,𝜕_𝑡𝑢_𝛼|_𝑡=0𝑢=𝜓inΩ,_𝛼|_{𝜕Ω×(0,𝑇)}=𝑓. (2.1) Here 𝜑,𝜓∈𝒞^∞(Ω) and 𝑓∈𝒞^∞(0,𝑇;𝒞^∞(𝜕Ω)) are subject to the compatibility conditions 𝜕_𝑡^2𝑙𝑓|_𝑡=0=(Δ^𝑙𝜑)|_𝜕Ω,𝜕_𝑡^2𝑙+1𝑓|_𝑡=0=(Δ^𝑙𝜓)|_𝜕Ω,𝑙=1,2,… (2.2) which give that (2.6) has a unique solution in 𝒞^∞([0,𝑇]×Ω); see [24]. It is also well known that (2.1) has a unique weak solution 𝑢_𝛼∈𝒞⁰(0,𝑇;𝐻¹(Ω))∩𝒞¹(0,𝑇;𝐿²(Ω)); see [24, 25]. Indeed, from [25] we have that (𝜕𝑢_𝛼/𝜕𝑛)|_𝜕Ω belongs to 𝐿²(0,𝑇;𝐿²(𝜕Ω)).

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

We assume that the coefficient 𝑐(𝑥) of (2.1) is such that𝑐𝑐(𝑥)=^𝛼(𝑥)=𝑐₀(𝑥)+𝛼𝑐₁𝑐(𝑥),for𝑥∈Ω,₂(𝑥)=const.>0,for𝑥∈ℝ^𝑑⧵Ω, (2.3) where 𝑐_𝑖(𝑥)∈𝒞²(Ω) for 𝑖=0,1 with 𝑐₁≡0inΩ⧵Ω^′,𝑀:=sup{𝑐₁(𝑥);𝑥∈Ω^′}, (2.4) where Ω^′ is a smooth subdomain of Ω and 𝑀 is a positive constant. We also assume that 𝛼>0, the order of magnitude of the small perturbations of coefficient, is sufficiently small that|𝑐_𝛼(𝑥)|≥𝑐_∗>0,𝑥∈Ω, (2.5) where 𝑐_∗ is a positive constant.

Define 𝑢 to be the solution of the hyperbolic equation in the homogeneous situation (𝛼=0). Thus, 𝑢 satisfies(𝑐₀𝜕²_𝑡−Δ)𝑢=0,inΩ×(0,𝑇),𝑢|_𝑡=0=𝜑,𝜕_𝑡𝑢|_𝑡=0=𝜓inΩ,𝑢|_{𝜕Ω×(0,𝑇)}=𝑓. (2.6)

Now, we define Γ_𝑐:=𝜕Ω⧵Γ, and we introduce the trace space 𝐻^1/2(Γ)={𝑣∈𝐻^1/2(𝜕Ω×(0,𝑇)),𝑣≡0onΓ_𝑐×(0,𝑇)}. (2.7)

It is well known that the dual of 𝐻^1/2(Γ) is 𝐻^−1/2(Γ).

Then, one can writeΛ_𝛼(𝑓|_Γ)=𝜕𝑢_𝛼|𝜕𝑛_Γ,for𝑓|_Γ∈𝐻^1/2(Γ), (2.8) where Λ_𝛼 is the Dirichlet-to-Neumann map (D-t-N) operator, and 𝑢_𝛼 is the solution of (2.1).

Let Λ₀ be the Dirichlet-to-Neumann map (D-t-N) operator defined as in (2.8) for the case 𝛼=0. Then, our problem can be stated as follows.

Inverse Problem
Suppose that the smooth coefficient 𝑐(𝑥) satisfies (2.4), (2.5), and (2.6), where the positive number 𝑐₂ is given. Assume that the function 𝑐(𝑥) is unknown in the domain Ω. Is it possible to determine the coefficient 𝑐_𝛼(𝑥) from the knowledge of the difference between the local Dirichlet-to-Neumann maps Λ_𝛼−Λ₀ on Γ, if we know explicitly the value of lim_𝛼→0⁺𝑐_𝛼(𝑥) for 𝑥∈Ω ?
To give a positive answer, we will develop an asymptotic expansions of an "appropriate averaging" of 𝜕𝑢_𝛼/𝜕𝑛 on Γ×(0,𝑇), 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, 26–29]). It has been known for some time that the full knowledge of the (hyperbolic) Dirichlet to Neumann map (𝑢_𝛼|_{𝜕Ω×(0,𝑇)}↦(𝜕𝑢_𝛼/𝜕𝑛)|_{𝜕Ω×(0,𝑇)}) 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 𝛽(𝑥)∈𝒞^∞₀(Ω) such that 𝛽≡1 on Ω^′ and let 𝜂∈ℝ^𝑑.

We will take in what follows 𝜑(𝑥)=𝑒^{𝑖𝜂⋅𝑥},𝜓(𝑥)=−𝑖|𝜂|𝑒^{𝑖𝜂⋅𝑥},and𝑓(𝑥,𝑡)=𝑒^{𝑖𝜂⋅𝑥−𝑖|𝜂|𝑡} and assume that we are in possession of the boundary measurements of 𝜕𝑢_𝛼𝜕𝑛onΓ×(0,𝑇). (3.1) This particular choice of data 𝜑,𝜓, and 𝑓 implies that the background solution 𝑢 of the wave equation (2.6) in the homogeneous background medium can be given explicitly.

Suppose now that 𝑇 and the part Γ of the boundary 𝜕Ω are such that they geometrically control Ω which roughly means that every geometrical optic ray, starting at any point 𝑥∈Ω at time 𝑡=0, hits Γ before time 𝑇 at a nondiffractive point; see [32]. It follows from [33] (see also [34]) that there exists (a unique) 𝑔_𝜂∈𝐻¹₀(0,𝑇;𝑇𝐿²(Γ)) (constructed by the Hilbert Uniqueness Method) such that the unique weak solution 𝑤_𝜂 to the wave equation(𝑐₀𝜕²_𝑡−Δ)𝑤_𝜂𝑤=0inΩ×(0,𝑇),_𝜂|_𝑡=0=𝛽(𝑥)𝑒^{𝑖𝜂⋅𝑥}∈𝐻¹₀𝜕(Ω),_𝑡𝑤_𝜂|_𝑡=0𝑤=0inΩ,_𝜂|_Γ×(0,𝑇)=𝑔_𝜂,𝑤_𝜂|_{𝜕Ω⧵Γ×(0,𝑇)}=0 (3.2) satisfies 𝑤_𝜂(𝑇)=𝜕_𝑡𝑤_𝜂(𝑇)=0.

Let 𝜃_𝜂∈𝐻¹(0,𝑇;𝐿²(Γ)) denote the unique solution of the Volterra equation of second kind𝜕_𝑡𝜃_𝜂(𝑥,𝑡)+^𝑇_𝑡𝑒^{−𝑖|𝜂|(𝑠−𝑡)}(𝜃_𝜂(𝑥,𝑠)−𝑖|𝜂|𝜕_𝑡𝜃_𝜂(𝑥,𝑠))𝑑𝑠=𝑔_𝜂𝜃(𝑥,𝑡),for𝑥∈Γ,𝑡∈(0,𝑇),_𝜂(𝑥,0)=0,for𝑥∈Γ. (3.3) 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 𝜃_𝜂 in 𝐻¹(0,𝑇;𝐿²(Γ)) for any 𝜂∈ℝ^𝑑 can be established using the resolvent kernel. However, observing from differentiation of (3.3) with respect to 𝑡 that 𝜃_𝜂 is the unique solution of the ODE:𝜕²_𝑡𝜃_𝜂−𝜃_𝜂=𝑒^{𝑖|𝜂|𝑡}𝜕_𝑡(𝑒^{−𝑖|𝜂|𝑡}𝑔_𝜂𝜃),for𝑥∈Γ,𝑡∈(0,𝑇),_𝜂(𝑥,0)=0,𝜕_𝑡𝜃_𝜂(𝑥,𝑇)=0,for𝑥∈Γ, (3.4) the function 𝜃_𝜂 may be found (in practice) explicitly with variation of parameters and it also immediately follows from this observation that 𝜃_𝜂 belongs to 𝐻²(0,𝑇;𝐿²(Γ)).

We introduce 𝑣_𝜂 as the unique weak solution (obtained by transposition) in 𝒞⁰(0,𝑇;𝐿²(Ω))∩𝒞¹(0,𝑇;𝐻⁻¹(Ω)) to the wave equation(𝑐₀𝜕²_𝑡−Δ)𝑣_𝜂𝑣=0inΩ×(0,𝑇),_𝜂|_𝑡=0𝜕=0inΩ,_𝑡𝑣_𝛼,𝜂|_𝑡=0=𝑖∇⋅(𝜂𝑐₁(𝑥)𝑒^{𝑖𝜂⋅𝑥})∈𝐿²𝑣(Ω),_𝜂|_{𝜕Ω×(0,𝑇)}=0. (3.5) Then, the following holds.

Proposition 3.1. Suppose that Γ and 𝑇 geometrically control Ω. For any 𝜂∈ℝ^𝑑, we have ^𝑇_0Γ𝑔_𝜂Λ₀(𝑣_𝜂)𝑑𝜎(𝑥)𝑑𝑡=|𝜂|²_Ω′𝑐₁(𝑥)𝑒^{2𝑖𝜂⋅𝑥}𝑑𝑥. (3.6) Here 𝑑𝜎(𝑥) means an elementary surface for 𝑥∈Γ.

Proof. Let 𝑣_𝜂 be the solution of (3.5). From [25,Theorem 4.1, page 44], it follows that Λ₀(𝑣_𝜂)=(𝜕𝑣_𝜂/𝜕𝑛)|_Γ∈𝐿²(0,𝑇;𝐿²(Γ)). Then, multiplying the equation (𝜕²_𝑡+Δ)𝑣_𝜂=0 by 𝑤_𝜂 and integrating by parts over (0,𝑇)×Ω, for any 𝜂∈ℝ^𝑑, we have ^𝑇_0Ω(𝜕²_𝑡−Δ)𝑣_𝜂𝑤_𝜂=𝑖_Ω∇⋅(𝜂𝑐₁(𝑥)𝑒^{𝑖𝜂⋅𝑥})𝛽(𝑥)𝑒^{𝑖𝜂⋅𝑥}𝑑𝑥−^𝑇_0Γ𝑔_𝜂𝜕𝑣_𝜂𝜕𝑛=0. (3.7) Therefore, |𝜂|²_Ω′𝑐₁(𝑥)𝑒^{2𝑖𝜂⋅𝑥}𝑑𝑥=^𝑇_0Γ𝑔_𝜂𝜕𝑣_𝜂,𝜕𝑛 (3.8) since 𝑐₁≡0 on Ω⧵Ω^′.

In terms of the function 𝑣_𝜂 as solution of (3.3), we introduce𝑢_𝛼(𝑥,𝑡)=𝑢(𝑥,𝑡)+𝛼^𝑑𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝑣_𝜂(𝑥,𝑡−𝑠)𝑑𝑠,𝑥∈Ω,𝑡∈(0,𝑇). (3.9) Moreover, for 𝑧(𝑡)∈𝒞^∞₀(]0,𝑇[) and for any 𝑣∈𝐿¹(0,𝑇;𝐿²(Ω)), we define𝑣(𝑥)=^𝑇₀𝑣(𝑥,𝑡)𝑧(𝑡)𝑑𝑡∈𝐿²(Ω). (3.10)

The following lemma is useful to prove our main result.

Lemma 3.2. Consider an arbitrary function 𝑐(𝑥) satisfying condition (2.3), and assume that conditions (2.4) and (2.5) hold. Let 𝑢 and 𝑢_𝛼 be solutions of (2.6) and (2.1), respectively. Then, using (3.9) the following estimates hold: ‖𝑢_𝛼−𝑢‖_𝐿^∞_(0,𝑇;𝐿²_(Ω))≤𝐶𝛼, (3.11) where 𝐶 is a positive constant. And ‖𝑢_𝛼−𝑢_𝛼‖_𝐿^∞_(0,𝑇;𝐿²_(Ω))≤𝐶^′𝛼^𝑑+1, (3.12) where 𝐶^′ is a positive constant.

Proof. Let 𝑦_𝛼 be defined by 𝑦_𝛼∈𝐻¹₀(Ω),Δ𝑦_𝛼=𝑐_𝛼𝜕_𝑡(𝑢_𝛼−𝑢)inΩ. (3.13) We have _Ω𝑐_𝛼𝜕²_𝑡(𝑢_𝛼−𝑢)𝑦_𝛼+_Ω∇(𝑢_𝛼−𝑢)⋅∇𝑦_𝛼=𝛼_Ω𝑐₁𝑐₀∇𝑢⋅∇𝑦_𝛼. (3.14) Since _Ω∇(𝑢_𝛼−𝑢)⋅∇𝑦_𝛼=−_Ω𝑐_𝛼𝜕_𝑡(𝑢_𝛼−𝑢)(𝑢_𝛼1−𝑢)=−2𝜕_𝑡Ω𝑐_𝛼(𝑢_𝛼−𝑢)²,_Ω𝑐_𝛼𝜕²_𝑡(𝑢_𝛼−𝑢)𝑦_𝛼1=−2𝜕_𝑡Ω|∇𝑦_𝛼|², (3.15) we obtain 𝜕_𝑡Ω|∇𝑦_𝛼|²+𝜕_𝑡Ω𝑐_𝛼(𝑢_𝛼−𝑢)²=−2𝛼_Ω𝑐₁𝑐₀∇𝑢⋅∇𝑦_𝛼≤𝐶𝛼‖∇𝑦_𝛼‖_{𝐿∞(0,𝑇;𝐿2(Ω))}. (3.16) From the Gronwall Lemma, it follows that ‖𝑢_𝛼−𝑢‖_{𝐿∞(0,𝑇;𝐿2(Ω))}≤𝐶𝛼. (3.17) As a consequence, by using (3.10), one can see that the function 𝑢_𝛼−𝑢 solves the following boundary value problem: Δ(𝑢_𝛼−(𝑢𝑢)=𝑂(𝛼)inΩ,_𝛼−𝑢)|_𝜕Ω=0. (3.18) Integration by parts immediately gives ‖grad(𝑢_𝛼−𝑢)‖_𝐿2(Ω)=𝑂(𝛼). (3.19) Taking into account that grad(𝑢_𝛼−𝑢)∈𝐿^∞(0,𝑇;𝐿²(Ω)), we find by using the above estimate that ‖grad(𝑢_𝛼−𝑢)‖_𝐿2(Ω)=𝑂(𝛼)a.e.𝑡∈(0,𝑇). (3.20) Under relation (3.9), one can define the function 𝑦_𝛼 as a solution of 𝑦_𝛼∈𝐻¹₀Δ𝑦(Ω),_𝛼=𝑐_𝛼𝜕_𝑡(𝑢_𝛼−𝑢_𝛼)inΩ. (3.21) Integrating by parts immediately yields _Ω𝑐_𝛼𝜕²_𝑡(𝑢_𝛼−𝑢_𝛼)𝑦_𝛼1=−2𝜕_𝑡Ω|∇𝑦_𝛼|²,_Ω∇(𝑢_𝛼−𝑢_𝛼)∇𝑦_𝛼1=−2𝜕_𝑡Ω𝑐_𝛼(𝑢_𝛼−𝑢_𝛼)². (3.22) To proceed with the proof of estimate (3.12), we firstly remark that the function 𝑢_𝛼 given by (3.9) is a solution of (𝑐₀𝜕²_𝑡−Δ)𝑢_𝛼=𝑖𝛼^𝑑∇⋅(𝜂𝑐₁(𝑥)𝑒^{𝑖𝜂⋅𝑥})𝑒^{−𝑖|𝜂|𝑡}∈𝐿²𝑢(Ω)inΩ×(0,𝑇),_𝛼|_𝑡=0𝜕=𝜑(𝑥)inΩ,_𝑡𝑢_𝛼|_𝑡=0𝑢=𝜓(𝑥)inΩ,_𝛼|_{𝜕Ω×(0,𝑇)}=𝑒^{𝑖𝜂⋅𝑥−𝑖|𝜂|𝑡}. (3.23) Then, we deduce that 𝑢_𝛼−𝑢_𝛼 solves the following initial boundary value problem: (𝑐_𝛼𝜕²_𝑡−∇⋅Δ)(𝑢_𝛼−𝑢_𝛼)=𝛼^𝑑∇⋅(𝑐₁(𝑥)grad(^𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝑣_𝜂(𝑥,𝑡−𝑠)𝑑𝑠))inΩ×(0,𝑇),(𝑢_𝛼−𝑢_𝛼)|_𝑡=0𝜕=0inΩ,_𝑡(𝑢_𝛼−𝑢_𝛼)|_𝑡=0=0inΩ,(𝑢_𝛼−𝑢_𝛼)|_{𝜕Ω×(0,𝑇)}=0. (3.24) Finally, we can use (3.24) to find by integrating by parts that 𝜕_𝑡Ω|∇𝑦_𝛼|²+𝜕_𝑡Ω𝑐_𝛼(𝑢_𝛼−𝑢_𝛼)²=2𝛼^𝑑_Ω𝑐₁grad(𝑢−𝑢_𝛼)⋅grad𝑦_𝛼 (3.25) which, from the Gronwall Lemma and by using (3.20), yields ‖𝑢_𝛼−𝑢_𝛼‖_{𝐿∞(0,𝑇;𝐿2(Ω))}≤𝐶^′𝛼^𝑑+1. (3.26) This achieves the proof.

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

Theorem 3.3. Let 𝜂∈ℝ^𝑑, 𝑑=2,3. Suppose that the smooth coefficient 𝑐(𝑥) satisfies (2.3), (2.4), and (2.5). Let 𝑢_𝛼 be the unique solution in 𝒞⁰(0,𝑇;𝐻¹(Ω))∩𝒞¹(0,𝑇;𝐿²(Ω)) to the wave equation (2.1) with 𝜑(𝑥)=𝑒^{𝑖𝜂⋅𝑥},𝜓(𝑥)=−𝑖|𝜂|𝑒^{𝑖𝜂⋅𝑥}, and 𝑓(𝑥,𝑡)=𝑒^{𝑖𝜂⋅𝑥−𝑖|𝜂|𝑡}. Let 𝑓=𝑓|_Γ∈𝐻^1/2(Γ). Suppose that Γ and 𝑇 geometrically control Ω; then we have ^𝑇_0Γ(𝜃_𝜂+𝜕_𝑡𝜃_𝜂𝜕_𝑡⋅)(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡=𝛼^𝑑−1|𝜂|²_Ω′(𝑐_𝛼−𝑐₀)(𝑥)𝑒^{2𝑖𝜂⋅𝑥}𝑑𝑥+𝑂(𝛼^𝑑+1)=𝛼^𝑑|𝜂|²_Ω′𝑐₁(𝑥)𝑒^{2𝑖𝜂⋅𝑥}𝑑𝑥+𝑂(𝛼^𝑑+1), (3.27)(3.28) where 𝜃_𝜂 is the unique solution to the ODE (3.4) with 𝑔_𝜂 defined as the boundary control in (3.2). The term 𝑂(𝛼^𝑑+1) is independent of the function 𝑐₁. It depends only on the bound 𝑀.

Proof. Since the extension of (Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡) to 𝜕Ω×(0,𝑇) is (𝜕𝑢_𝛼/𝜕𝑛−𝜕𝑢/𝜕𝑛), then by conditions 𝜕_𝑡𝜃_𝜂(𝑇)=0 and (𝜕𝑢_𝛼/𝜕𝑛−𝜕𝑢/𝜕𝑛)|_𝑡=0=0, we have (Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)|_𝑡=0=0. Therefore, the term ^𝑇_0Γ𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡 (3.29) may be simplified as follows: ^𝑇_0Γ𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡=−^𝑇_0Γ𝜕²_𝑡𝜃_𝜂(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡. (3.30)
On the other hand, we have ^T_0Γ[𝜃_𝜂(Λ_𝛼−Λ₀)(𝑓)+𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼−Λ₀=)(𝑓)](𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡_𝑇0Γ[𝜃_𝜂(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)))+𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)+))](𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡_𝑇0Γ[𝜃_𝜂𝛼^𝑑𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡−𝑠)𝑑𝑠+𝛼^𝑑𝜕_𝑡𝜃_𝜂𝜕_𝑡^𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡−𝑠)𝑑𝑠]𝑑𝜎(𝑥)𝑑𝑡, (3.31) where Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇))=Λ₀(𝑓)+𝛼^𝑑∫^𝑡₀𝑒^{−𝑖|𝜂|𝑠}Λ₀(𝑣_𝜂|_Γ)(𝑥,𝑡−𝑠)𝑑𝑠.
Given that, 𝜃_𝜂 satisfies the Volterra equation (3.4) and 𝜕_𝑡(^𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡−𝑠)𝑑𝑠)=𝜕_𝑡(−𝑒^{−𝑖|𝜂|𝑡𝑡}₀𝑒^{𝑖|𝜂|𝑠}𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑠)𝑑𝑠)=𝑖|𝜂|𝑒^{−𝑖|𝜂|𝑡𝑡}₀𝑒^{𝑖|𝜂|𝑠}𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑠)𝑑𝑠+𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡), (3.32) we obtain by integrating by parts over (0,𝑇) that ^𝑇_0Γ[𝜃_𝜂^𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡−𝑠)𝑑𝑠+𝜕_𝑡𝜃_𝜂𝜕_𝑡^𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡−𝑠)𝑑𝑠]𝑑𝜎(𝑥)𝑑𝑡=^𝑇_0Γ(𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡)(𝜕_𝑡𝜃_𝜂+^𝑇_𝑡𝜃_𝜂(𝑠)𝑒^{𝑖|𝜂|(𝑡−𝑠)}𝑑𝑠)−𝑖|𝜂|(𝑒^{−𝑖|𝜂|𝑡}𝜕_𝑡𝜃_𝜂(𝑡))^𝑡₀𝑒^{𝑖|𝜂|𝑠}𝜕𝑣_𝜂=𝜕𝑛(𝑥,𝑠)𝑑𝑠)𝑑𝜎(𝑥)𝑑𝑡_𝑇0Γ𝜕𝑣_𝜂𝜕𝑛(𝑥,𝑡)(𝜕_𝑡𝜃_𝜂+^𝑇_𝑡(𝜃_𝜂(𝑠)−𝑖|𝜂|𝜕_𝑡𝜃_𝜂(𝑠))𝑒^{𝑖|𝜂|(𝑡−𝑠)}=𝑑𝑠)𝑑𝜎(𝑥)𝑑𝑡_𝑇0Γ𝑔_𝜂(𝑥,𝑡)Λ₀(𝑣_𝜂|_Γ)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡, (3.33) and so, from Proposition 3.1, we obtain ^𝑇_0Γ[𝜃_𝜂(Λ_𝛼−Λ₀)(𝑓)+𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼−Λ₀)(𝑓)](𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡=𝛼^𝑑|𝜂|²_Ω′𝑐₁(𝑥)𝑒^{2𝑖𝜂⋅𝑥}𝑑𝑥+^𝑇_0Γ[𝜃_𝜂(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)))+𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)))]𝑑𝜎(𝑥)𝑑𝑡+𝑂(𝛼^𝑑+1). (3.34) Thus, to prove Theorem 3.3, it suffices then to show that ^𝑇_0Γ[𝜃_𝜂(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)))+𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)))]𝑑𝜎(𝑥)𝑑𝑡=𝑂(𝛼^𝑑+1). (3.35)
From definition (3.10), we have 𝑢_𝛼−𝑢_𝛼=^𝑇₀(𝑢_𝛼−𝑢_𝛼)𝑧(𝑡)𝑑𝑡, (3.36) which gives by system (3.24) that Δ(𝑢_𝛼−𝑢_𝛼)=^𝑇₀𝑐_𝛼𝜕²_𝑡(𝑢_𝛼−𝑢_𝛼)𝑧(𝑡)𝑑𝑡+𝛼^𝑑𝑇₀∇⋅(𝑐₁(𝑥)grad(^𝑡₀𝑒^{−𝑖|𝜂|𝑠}𝑣_𝜂(𝑥,𝑡−𝑠)𝑑𝑠))𝑧(𝑡)𝑑𝑡. (3.37) Thus, by (3.9) and (3.24) again, we see that the function 𝑢_𝛼−𝑢_𝛼 is the solution of −Δ(𝑢_𝛼−𝑢_𝛼)=−^𝑇₀𝑐_𝛼(𝑢_𝛼−𝑢_𝛼)𝑧^′′(𝑡)𝑑𝑡+∇⋅(𝑐₁(𝑥)grad(𝑢_𝛼−(𝑢𝑢))inΩ,_𝛼−𝑢_𝛼)|_𝜕Ω=0. (3.38) 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 ‖𝜕(𝑢𝜕𝑛^𝛼−𝑢_𝛼)‖_𝐿2(Γ)=𝑂(𝛼^𝑑+1). (3.39)By the fact thatΛ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)):=(𝜕/𝜕𝑛)(𝑢_𝛼−𝑢_𝛼)∈𝐿^∞(0,𝑇;𝐿²(Γ)),we deduce, as done in the proof of Lemma 3.2, that ‖Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇))‖_𝐿2(Γ)=𝑂(𝛼^𝑑+1), (3.40) which implies that ^𝑇_0Γ[𝜃_𝜂(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)))+𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼(𝑓)−Λ_𝛼(𝑢_𝛼|_Γ×(0,𝑇)))]𝑑𝜎(𝑥)𝑑𝑡=𝑂(𝛼^𝑑+1). (3.41) 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 that𝑐_𝛼(𝑥)−𝑐₀2(𝑥)≈𝛼^𝑑−1_ℝ^𝑑𝑒^{−2𝑖𝜂⋅𝑥}|𝜂|^2𝑇0Γ(𝜃_𝜂+𝜕_𝑡𝜃_𝜂𝜕_𝑡⋅)(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑦)𝑑𝑡𝑑𝜂,𝑥∈Ω. (3.42) The method of reconstruction we propose here consists in sampling values of1|𝜂|^2𝑇0Γ(𝜃_𝜂+𝜕_𝑡𝜃_𝜂𝜕_𝑡⋅)(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡 (3.43) at some discrete set of points 𝜂 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 Λ_𝛼 and Λ₀ at 𝑓.

Corollary 3.4. Let 𝜂∈ℝ^𝑑 and let 𝑓=𝑓|_Γ∈𝐻^1/2(Γ). Suppose that Γ and 𝑇 geometrically control Ω; then we have the following better approximation: 𝑐_𝛼(𝑥)≈𝑐₀−2(𝑥)𝛼^𝑑−1_ℝ^𝑑𝑒^{−2𝑖𝜂⋅𝑥}|𝜂|^2𝑇0Γ[𝑒^{𝑖|𝜂|𝑡}𝜕_𝑡(𝑒^{−𝑖|𝜂|𝑡}𝑔_𝜂(𝑦,𝑡))(Λ_𝛼−Λ₀)(𝑓)(𝑦,𝑡)]𝑑𝜎(𝑦)𝑑𝑡𝑑𝜂,𝑥∈Ω, (3.44) where the boundary control 𝑔_𝜂 is defined by (3.2).

Proof. The term ∫^𝑇₀∫_Γ𝜕_𝑡𝜃_𝜂𝜕_𝑡(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡, given in Theorem 3.3, has to be interpreted as follows: ^𝑇_0Γ𝜕_𝑡𝜃_𝜂⋅𝜕_𝑡(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡=−^𝑇_0Γ𝜕²_𝑡𝜃_𝜂⋅(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡, (3.45) because 𝜃_𝜂|_𝑡=𝑇=0 and 𝜕_𝑡(𝜕𝑢_𝛼/𝜕𝑛−𝜕𝑢/𝜕𝑛)|_𝑡=0=0. In fact, in view of the ODE (3.4), the term ∫^𝑇₀∫_Γ[𝜃_𝜂(Λ_𝛼−Λ₀)+𝜕_𝑡𝜃_𝜂⋅𝜕_𝑡(Λ_𝛼−Λ₀)]𝑓(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡 may be simplified after integration by parts over (0,𝑇) and using of the fact that 𝜃_𝜂 is the solution to the ODE (3.4) to become −^𝑇_0Γ𝑒^{𝑖|𝜂|𝑡}𝜕_𝑡(𝑒^{−𝑖|𝜂|𝑡}𝑔_𝜂)⋅(Λ_𝛼−Λ₀)(𝑓)(𝑥,𝑡)𝑑𝜎(𝑥)𝑑𝑡. (3.46) 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 𝑧+𝛼𝐷) in a homogeneous medium in ℝ^𝑑 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.