Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

In this paper, we are interested in the inverse problem of the determination of the unknown part zΩ, Γ0 of the boundary of a uniformly Lipschitzian domain Ω included in R from the measurement of the normal derivative znv on suitable part Γ0 of its boundary, where v is the solution of the wave equation zttv(x, t) − Δv(x, t) + p(x)v(x) � 0 in Ω × (0, T) and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of zΩ. From necessary conditions, we estimate a Lagrange multiplier k(Ω) which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. 'e Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.


Introduction and Main Result
e inverse problem in this paper means the problem of reconstructing object from observation data. We restrict ourselves to the case when the observation data are given as a boundary of the Cauchy data of a solution of a wave equation and the unknown object is a boundary. Let N ∈ N, T > 0 and let Ω ⊂ R N be a bounded domain with smooth boundary zΩ. Moreover, let us consider a partition of this boundary zΩ � Γ 0 ∪ Γ, Γ 0 ∩ Γ � ∅, where Γ 0 is the accessible regular part, for example, C 2 , and it satisfies the interior sphere condition (see [1]) and Γ � zΩ/Γ 0 is the unknown part of boundary. roughout this paper, let us take the functional v � v(x, t) with x ∈ Ω, t ∈ (0, T). We use the following notations: We consider the following wave equation: First of all, assume that p ∈ L ∞ (Ω), v 0 ∈ H 1 (Ω) , and v 1 ∈ L 2 (Ω) are given and verified the compatibility condition v 0 (x) � 0 for all x ∈ zΩ. e Cauchy problem (2) is known to be well posed and one can also prove the solution v ∈ C([0, T]; H 1 0 (Ω)) ∩ C 1 ([0, T]; L 2 (Ω)); this result can be found in [2].
Our inverse problem consists of determining Γ � zΩ, Γ 0 , the unknown part of boundary from Cauchy data (v 0 | zΩ , h) of a weak solution v of the following problem (3) with a given potential p(x).
where h is a given function, and the corresponding Neumann data measured on Γ 0 and n is outer normal vector unit. In this case, Ω and v are unknowns and we assume that the normal derivatives of function v can be measured by h.
In Section 1, we present the inverse problem which consists of finding a formula reconstructing the part of boundary Γ from the Cauchy data. e remainder of the paper is organized as follows.
In Section 2, we establish the shape optimization problem and prove the existence results. In Section 3, we study the derivation with respect to the domain and we prove the necessary conditions of optimality, that is, the existence of a Lagrange multiplier. Section 4 is devoted to auxiliary lemmas based on maximum principle theory for hyperbolic equations; see [3]. In Section 5, we give by a monotonicity result and under geometrical assumptions a uniqueness result of our inverse problem. e questions for the wave equation have all already received positive answers since the uniqueness result for the linear inverse problem has been proved by Klibanov (see [4]) and Lipschitz stability results (for both linear and nonlinear inverse problems) of Yamamoto (see [5]). Many results, to which we can refer concerning the wave equation, are related to the same type of inverse problem of determining a potential p(x). Some of them can be found in [6], for example. ese references are all based upon local Carleman estimates for the wave operator (see [5]) or global Carleman estimates for Schrödinger equation (see [7]) to prove uniqueness and stability estimate solution. Nevertheless, in our approach, for a given potential, the reconstruction of Γ from the Cauchy data is one of our aims and the estimation of the Lagrange multiplier which appears by derivation with respect to the domain of the energy of system in an admissible set of domains is another interesting one.
In [8], Isakov and Friedman studied the inverse spectral problems. is domain problem was formulated already by Sir A. Shuster who in 1882 introduced spectroscopy as a way to find a shape of a bell by means of the sounds which it is capable of sending out. More rigorously, it has been posed by Bochner in the 1950s and then in the well-known lecture of Kac (see [9]) "Can one hear the shape of a drum?" in 1966. He also studied inverse problem of gravimetry, inverse conductivity problem, tomography, and the inverse seismic problem and indicated their applications.
In [10], using conformal mapping technique, Kress studied mathematical modelling of electrostatic or thermal imaging methods in nondestructive testing and evaluation. In these applications, an unknown inclusion within a conducting host medium with constant conductivity is assessed from overdetermined Cauchy data on the accessible exterior boundary of the medium.

Study of the Shape Optimization Problem
2.1. Auxiliary Results. We describe some fundamental properties which will be useful in the following. We consider a fixed and bounded domain D in R N which contains all open subsets we used. Definition 1. Let K 1 and K 2 be two compact subsets of D. Let and we call Hausdorff distance of K 1 and K 2 , the following positive number, denoted by d H (K 1 , K 2 ). Let (Ω n ) be a sequence of open subsets of D and let Ω be an open subset of D. We say that the sequence (Ω n ) converges on Ω in the Hausdorff sense and we denote it by sequence of open sets of R N and let Ω be an open set of R N . We say that the sequence (Ω n ) converges on Ω in the sense of L p , 1 ≤ p < ∞ if χ Ω n converges on χ Ω in L p loc (R N ), χ Ω being the characteristic functions of Ω. Remark 1. Let K n be a sequence of compact sets included in a fixed and bounded set D of IR N ; then there are a compact set K and n k such that K n k converges on K in the sense of Hausdorff.
We have the following lemmas.
For detailed proof, see [11]. If v Ω n is the solution of problem 2 in Ω n for all n ∈ N and v Ω is the solution of this problem in Ω, For detailed proof, see [11].
. For detailed proof, see [11]. For all v(x, t) solution of at time t, the energy of v is defined by and it verifies Let J be the functional defined by where v w (x, t) is the solution: We study the existence of the result of the following op- where D is a bounded domain of R N containing all Ω and V 0 is positive real.

Existence of Solution of Shape Optimization Problem.
We study the existence result of the following shape optimization problem.
For the proof, we take the following: e fact that sequence (Ω n ) n∈N ∈ O is bounded ensures the existence of a subsequence (Ω n k ) n k ∈N ∈ O and a domain Ω ∈ O such that (Ω n k ) n k converges to Ω in the sense of Hausdorff according to Lemma 1.
International Journal of Mathematics and Mathematical Sciences erefore, ). If not, J(Ω n ) converges to +∞, which is a contradiction.
) and always according to Lemma 3.

Derivation with respect to the Domain
ese results would allow us to assume regularity C 2 on Ω solution of the shape optimization problem to proceed with the derivation with respect to the domain and to show the result of monotony.
Let E: O ⟶ X, with O being the set of domains having the ϵ− cône property and X being a normal vector space.
Let us consider θ ⟶ ξ(θ) � E(I + θ)(Ω), where θ varies around 0 in a normalized vector space Θ of applications from R N to R N . We can introduce the differentiability in the classic sense of Frechet for the application θ ∈ Θ ⟶ ξ(θ) ∈ X. is is efficient in proving the regularity properties of the shape functional, in using the derivation calculations, and in clearly identifying the derivatives structures so-called "of shape".
We will use a numerical variable to be comfortable in the calculations.
Let us choose function ϕ: y↦I + yθ with θ being a regular vector field from R N to R N ; y ⟶ ϕ(y) ∈ Θ. We analyse the derivative of ϕ(y) and the expression of Let us consider v y � v Ω y . e question is, how can we derive the function y↦v y ∈ H 1 (Ω y ) where Ω y is a variable domain?
(1) We know that v y is extended by 0 because v y ∈ H 1 0 (Ω y ) (2) We know that function u y � v y°ϕ (y) is always defined on the fixed domain Ω; it belongs to space To derive function v y , it suffices to "transport" it by ϕ y , because y↦u y has more regularity than y↦v y . erefore, it is more strategic to study this problem.
We fix Ω ⊂ R N as measurable. It is easy to verify that Ω y is measurable and that if Ω is open, then Ω y is also measurable.

Notations. Let
where R N is provided with the Euclidean norm ‖. Let I denote the identity of R N . We recall that this space is identified with the subspace of L ∞ (R N ), whose partial derivatives in the sense of distributions are functions of L ∞ (R N ). In addition, the functions of W 1,∞ are a.e. differentiable, and we have where the norms of differential are understood as linear operators of R N . However, the consideration of W 1,∞ is interesting for deformations of the Lipschitzian domain. If ‖θ‖ 1,∞ < 1, by the theorem of fixed point, 1 + θ is inversible such that (1 + θ) − 1 ∈ W 1,∞ and we have (see [11]) and its differentiability is the opposite of identity Let us consider ϕ: Because ϕ(y) is close to the identity in W 1,∞ (R N ) for y close to 0, it is inversible and even if it decreases Y according to 7.
We write independently ϕ(y)(x) or ϕ(y, x) (same for all other functions). Let J(y, x) � det(D x ϕ(y)(x)) be the 4 International Journal of Mathematics and Mathematical Sciences Jacobian of ϕ(y) in x (which is, therefore, a.e. defined in Subsequently, we note Ω y f(y) � Ω f(y, ϕ(y))J(y).

Derivation Formula.
To calculate derivation, we use the following theorem.

Theorem 3.1. Let us consider ϕ verifying (19). We suppose that
en, t ⟶ I(y) � Ω y f(y) is derivable in 0, and we have If, in addition, Ω is a Lipschitzian domain, then For detailed proof, see [11].

Optimality Conditions of the Problem. Again, y↦ϕ(y)
verifying (19) , Ω ⊂ R N is open bounded Lipschitzian domain, and Ω y � ϕ(y)(Ω). If y is close to 0, v y is a solution of the problem defined by its variational formulation (3). As in (11), we are interested in the following functional: We set , a vector field with compact support and y sufficiently small such that Id + yθ defines a diffeomorphism (ii) I(y) � J(Ω y ) � J(v Ω y , Ω y ) We look for the derivative of J with respect to domain Ω in direction θ, that is, I ′ (0).
As regards derivative of J, for a choice of θ, as mentioned above, we deform only the boundary part zΩ\Γ 0 that we will denote by Γ.
To calculate the derivative y↦I(y), it is useful to derive y↦v y in the appropriate direction v y ∈ H 1 0 (Ω y ). For the functional derivative, we have is gives, according to Hadamard and the boundary conditions, International Journal of Mathematics and Mathematical Sciences erefore, We suppose that it is possible to estimate the normal derivative of v Ω (x, t) on Γ 0 ; that is, there exists h: where n is the exterior normal unit vector defined on Γ 0 . We have the following necessary conditions of optimality.

Proposition 2.
If Ω is the solution of the shape optimization problem min using the derivative with respect to domain, in the direction of vector field, we show that there exists k(Ω) Lagrange multiplier such that where is gives us the following relationship according to (27): For more details on the expression in (29), see [14]. Let us take v Ω � v. To estimate k(Ω), it suffices to recognize Ω and if we suppose that zΩ\Γ 0 � Γ is of class C 2 , since v � 0 on Γ, then we have where n is the outer normal vector. Note that |∇v| � (− k(Ω)) 1/2 is an optimality condition and if we situate Γ, we will be able to estimate |∇v| on Γ. erefore, we deduce an approximation of Lagrange multiplier k(Ω).

Auxiliary Lemmas
In this section, we sum up some fundamental lemmas for the algorithm which we will present in the next sections. ese lemmas are based only on maximum principle theory in wave equation in high dimension. We assume also that u i ∈ C 2 (Ω) ∩ C(Ω\Γ i ), i � 1, 2.
In sequel, we need some hypothesis for the operator and the initial value problem in order to apply maximum principle for hyperbolic problem to obtain additional information about functions which satisfy (H) If the hypothesis (H) holds, then the solution v satisfies v ≤ 0, for all t > 0; see [3], page 234.

Numerical Simulations
In this part, we solve our inverse problem numerically using polar coordinates. Let h be a function as defined in the Introduction; we seek to determine Γ of class C 2 under the constraints given. We choose function h defined by cos(nr sin(θ))ch(nr cos(θ)) + cos(θ)sin(nr sin(θ))sh(nr cos(θ))), with N ∈ N, λ n ∈ R + , ch the hyperbolic cosine function and sh the hyperbolic sine function. e integer n takes values from 0 to N and λ n is a multiple for r depending on n. We consider that h is the normal derivative of a function u unique solution of problem 1 in a domain Ω of which a part Γ of the boundary is unknown. We seek to determine its optimal shape of Γ. With Matlab, by varying the parameters N, λ n , θ and t, we determine Γ of class C 2 which we describe as a trajectory.
us, we choose arbitrary values for r and λ n and a maximum value for time t. By varying the angle θ and N, we obtain the geometrical optimal shapes of Γ in the below figures. For each given unit of time, we vary N and θ. us, for each fixed time, we obtain an optimal shape when N � 300 and θ is equal to the value indicated in the legend.
In Figure 1 where the observation time is 5 seconds for a maximum angle equal to 300 × π/28, the length of Γ is equal to 5.43 kilometers.
In Figure 2 where the observation time is 8 seconds for a maximum angle equal to 300 × π/18, the length of Γ is equal to 23.79 kilometers.
In Figure 3 where the observation time is 10 seconds for a maximum angle equal to 10π, the length of Γ is equal to 195.15 kilometers.
We notice that if the observation time is small, then the optimal shape of Γ looks like an arc of circle. But if it is larger, the optimal shape of Γ is a part of a hyperbola.

Conclusion
In this paper, we prove the existence result of the solution for our inverse problem by determining the optimal shape in Section 2 and we prove the existence of a Lagrange multiplier, which appears in the optimality condition of the problem. By maximum principle for hyperbolic equations, we prove the uniqueness and the Lipschitz stability of the solution for our inverse problem. We make some numerical simulations with Matlab to illustrate the theoretical results and then identify the optimal shape of Γ . It would be interesting to study the problem with nonsmooth boundaries; topological optimization is the technique to use in future researches.

Data Availability
e data used to support the findings of this study are included within the article.