OPTIMIZATION OF NON-CYLINDRICAL DOMAINS FOR THE EXACT NULL CONTROLLABILITY OF THE 1D WAVE EQUATION

This work is concerned with the null controllability of the one-dimensional wave equation over non-cylindrical distributed domains. The controllability in that case has been obtained by Castro et al. [SIAM J. Control Optim. 52 (2014)] for domains satisfying the usual geometric optic condition. We analyze the problem of optimizing the non-cylindrical support q of the control of minimal L(q)norm. In this respect, we prove a uniform observability inequality for a class of domains q satisfying the geometric optic condition. The proof based on the d’Alembert formula relies on arguments from graph theory. Numerical experiments are discussed and highlight the influence of the initial condition on the optimal domains. Mathematics Subject Classification. 49Q10, 93C20. Received July 16, 2020. Accepted January 19, 2021.


Introduction and main results
In the last decades, an important literature was devoted to the study of vibrating structures controlled or stabilized using piezoelectric actuators [1,12,27,28]. In most of the situations, the actuators occupy a fixed position in space. Nevertheless, some studies consider the controllability of elastic structures (plates or beams) by the action of mobile piezoelectric actuators [33] which might be available in the future. Actually, a mobile actuator can be emulated by an array of fixed actuators distributed along the structure under the constraint that only one actuator can be activated at a given time [20]. The present work aims to find the optimal trajectory of a piezoelectric actuator in order to control, with a minimal cost, the vibrations of a simplified one-dimensional structure corresponding to a given initial datum. Therefore, observing the state of the considered system, we can provide in real time a trajectory for the controller in order to minimize the cost of control. More precisely, we consider the null controllability of a one-dimensional wave equation by means of distributed controls acting in non-cylindrical domains. The main contribution of this work is the proof of a uniform observability inequality with respect to control domains in a precise class. A direct consequence of this uniform observability inequality is the existence of an optimal trajectory for a piezoelectric actuator of a given size in order to control to zero the vibrations of the considered structure with a control of minimal L 2 -norm. Keywords  Let T be a positive real, Ω = (0, 1), Q T the domain Ω × (0, T ), q a non-empty subset of Q T and Σ T = ∂Ω × (0, T ). We are concerned with the null distributed controllability of the 1D wave equation: in Q T , y = 0 on Σ T , (y, y t )(·, 0) = (y 0 , y 1 ) in Ω. (1.1) where y represents the transverse deflection of a simplified elastic structure of length 1 and v is the voltage applied to an actuator occupying the position q t = q ∩ {(x, t); x ∈ Ω} at time t.
We assume that (y 0 , y 1 ) ∈ V := H 1 0 (Ω) × L 2 (Ω); v is the control (a function in L 2 (q)) and y = y(x, t) is the associated state. 1 q from Q T to {0, 1} denotes the indicator function of q. We also use the notation Ly := y tt − y xx .
In the cylindrical case for which the control region q takes the form q := ω × (0, T ), where ω denotes a subset of Ω, the null controllability of (1.1) at any large T > T is well-known (for instance, see [2,18]). The critical time T is related to the measure of the set Ω \ ω. The controllability is also true in the multi-dimensional case assuming the celebrated geometric optic condition on the triplet (Ω, ω, T ) introduced by Bardos, Lebeau and Rauch in [2]. The optimal problem (1.4) has been studied by Periago in [29] with Q ad = Q ad,τ := {q = ω × (0, T ); ω ⊂ Ω, |ω| = τ |Ω|, T ≥ 2} parameterized by τ in (0, 1). In particular, the analysis is based on the following uniform property proved by using Fourier series: there exists a constant C > 0 which depends only on Q T and τ such that sup q∈Q ad,τ C obs (q) < C. (1.5) We also mention that the problem of the optimal shape and position of the support has been numerically investigated in [23,24] for the one and two dimensional wave equation. This problem is also considered by Privat, Trélat and Zuazua in [30] using spectral arguments. We also mention [31] which analyze the dependence of the observability constant with respect to the control region.
The non-cylindrical case is more involved on a mathematical viewpoint. It concerns however an increasing number of studies as it allows in many situations to obtain positive controllability results, in contrast to the cylindrical case. Focusing on the wave equation, one of the first contributions is due to Khapalov [16] providing observability results for a moving point sensor in the interior of the domain. This time-dependent observation allows to avoid the usual difficulties related to strategic or non-strategic points. In particular, in the 1D setting, for any T > 0, the existence of controls continuous almost everywhere in (0, T ) and supported over curves continuous almost everywhere is obtained for data in H 2 (Ω) ∩ H 1 0 (Ω) × H 1 0 (Ω). More recently, for initial data in W, Castro analyzes in [5] the controllability from a moving interior point. By the way of the d'Alembert formula, a uniform observability inequality is proved for a precise set of curves {(γ(t), t)} t∈[0,T ] leading to moving controls in H −1 (∪ t∈(0,T ) γ(t) × t). Still in the 1D setting, Ciu, Liu and Gao [11] and Haak and Hoang [13] analyze the boundary and moving interior point observability for wave equations posed on a sufficiently regular time-dependent domain. In the N -dimensional case, Liu and Yong [19] employ the multiplier method to prove that the wave equation is controllable under the hypothesis that the distributed control region q covers the whole space domain Ω before the time T . Under a similar hypothesis, we also mention the work [21] where the controllability of the damped wave equation y tt − y xx − εy txx = 0 defined on the 1D torus is established. Because of the presence of an essential spectrum, such property does not hold true in the cylindrical case.
In the 1D setting, the geometric assumption has been relaxed in [6]. Precisely, the observability inequality (1.2) is obtained assuming that the distributed control domain q ⊂ Q T is a finite union of connected open sets and satisfies the following hypothesis: any characteristic line starting from a point of Ω × {0} and following the laws of geometric optics when reflected on the boundary Σ T must meet the domain q (we refer to [6], Prop. 2.1). This geometric condition is the natural extension to non-cylindrical domains of the condition introduced in [2]. Following [5], the proof of this result is obtained by the way of the d'Alembert formula. It has been extended in [17] to the multi-dimensional case using microlocal analysis. We also mention the obtention of Carleman-type inequalities for general hyperbolic equations in [32].
With the aim to extend [29] to a non-cylindrical setting, we prove in a first part that the observability constant C obs (q) in (1.2) is uniformly bounded from above for a class of domains satisfying the geometric optic condition. More precisely, for any parameter ε > 0 small enough, we introduce the admissible set of control domains by Q ε ad = q ⊂ Q T ; q open and q ε verifies the geometric optic condition . (1.6) Here, q ε = q ε (q) denotes the ε-interior of any q ⊂ Q T and is defined by We obtain that, for any ε > 0, there exists a constant C > 0 which depends only on Q T and ε such that As in [29], this uniform property allows, in a second part, to analyze the problem (1.4) of the optimal distribution of the control domain q. Preliminarily, for numerical purposes, we restrict the analysis to control domains of the form for a given δ 0 > 0 and curves γ : (0, T ) → Ω in the following set consisting of uniformly Lipschitz functions of fixed constant M > 0. For T ≥ 2 and ε > 0 small enough, the class {q γ ; γ ∈ G ad } is a subset of Q ε ad . In this setting, the optimal problem (1.4) reads as follows: for δ 0 > 0, M > 0 and a given initial datum (y 0 , where v is the control of minimal L 2 (q γ )-norm distributed over q γ ⊂ Q T . This paper is organized as follows. In Section 2, we prove the uniform observability property (1.8) on Q ε ad and its variant on the subset {q γ ; γ ∈ G ad } (see Thm. 2.1). This is achieved by defining an appropriate decomposition of the observation domains in Q ε ad , and by using the d'Alembert formula. The proof also relies on arguments from graph theory. Then, in Section 3, following arguments from [15,29], we analyze a variant of the extremal problem (1.4). Introducing a C 1 -regularization of the support q γ , we prove that the underlying cost is continuous over G ad for the L ∞ (0, T )-norm, and admits at least one local minimum (see Prop. 2). Section 4 illustrates the result with numerical experiments for the regularized minimization problem introduced in Section 3.3. Minimization sequences for the regularized cost are constructed using a gradient method: each iteration requires the computation of a null control, performed using the space-time formulation developed in [9] and used in [6], well-suited to the description of the non-cylindrical domains where the control acts. As expected, the optimal domain we obtain are closely related to the travelling waves generated by the initial condition. Section 5 concludes the work with some perspectives.

Uniform observability with respect to the domain of observation
We prove in this section the uniform observability property (1.8) with respect to the domain of observation. Precisely, we prove the following equivalent result for regular data in V. Theorem 2.1. Let T > 0 and let ε > 0 be a small enough fixed parameter such that the set Q ε ad defined by (1.6) is non-empty. There exists a constant C ε obs > 0 such that for every q ∈ Q ε ad , the following inequality holds where ϕ is the solution of the wave equation (1.3) associated with the initial datum (ϕ 0 , ϕ 1 ).
We emphasize that the constant C ε obs does not depend on q ∈ Q ε ad . We also emphasize that the parameter ε is introduced here to quantify to which extent a domain q satisfies the geometric optic condition used in [6]. As ε goes to zero, the observability constant C ε obs blows up (see Rem. 2.18). The same is true when τ goes to zero in the study [29], see (1.5). Eventually, we emphasize that any domain q ∈ Q ε ad for some ε satisfies the geometric optic condition. Conversely, for any domain q satisfying the geometric optic condition, there exists ε > 0 such that q ∈ Q ε ad . In the remaining part of this section, we assume that the hypotheses of Theorem 2.1 are satisfied.

Proof of Theorem 2.1
The idea of the proof is to decompose the domain Q T as the sum of "elementary squares" on which, in view of the d'Alembert formula, the terms in the inequality (2.1) can be simply computed and compared. The proof based on arguments from graph theory allows notably to relate the value of the observability constant to the spectrum of a Laplacian matrix, defined in terms of the graph associated to any domain q ∈ Q ε ad .
The proof requires some notations and technical lemmas. We decompose it into seven steps.
Step 1 -Solution of the adjoint problem (1.3) associated with piecewise affine initial conditions.
Let N ∈ N * and κ N = 1/N . We denote by S N = (x N i ) 0≤i≤N the regular subdivision of Ω in N intervals such that x N i = i/N . For any (ϕ 0 , ϕ 1 ) ∈ V, we then associate the functions ϕ N 0 , ϕ N 1 as follows: for all x ∈ Ω, we set which are respectively affine and constant on each interval [x N i−1 , x N i ]. We also denote by (ϕ N 0 ) ∈ L 2 (Ω) the "derivative" of ϕ N 0 : Using that ϕ 0 ∈ H 1 0 (Ω), we easily check that In order to use the d'Alembert formula for the solution of (1.3) associated with the initial datum (ϕ N 0 , ϕ N 1 ), we extend these functions as odd functions to [−1, 1] and then by 2-periodicity to R. In this respect, we first extend the definition of x N i to i ∈ Z by putting x N i = i/N for every i ∈ Z, and then denote by I N i for every i ∈ Z * the following interval: (2.4) Definition 2.2. For any N ∈ N * and i ∈ Z * , we define the integer j N (i) as follows: Similarly, we extend α N i and β N i for every i ∈ Z * as follows: We extend the functions ϕ N 0 and ϕ N 1 as odd functions to [−1, 1] and by 2-periodicity to R.
Lemma 2.4. For any N ∈ N * , the solution ϕ N of (1.3) associated with the initial datum (ϕ N 0 , ϕ N 1 ) satisfies: Proof. Using the notations above, we obtain Moreover, from the d'Alembert formula, the solution ϕ N of (1.3) associated with the initial datum (ϕ N 0 , ϕ N 1 ) is given as follows: Taking the derivative with respect to t and replacing the expressions (2.9) in the above equation, we deduce that for all (x, t) ∈ Q T , we have Using the properties of the function j N defined in (2.5), we deduce that for i, j ∈ Z * , and the result.
Step 2 -Representation of q ∈ Q ε ad using "elementary squares". In view of the expression (2.8), we introduce the following definition.
Definition 2.5. For any N ∈ N * and i, j ∈ Z * , we define the elementary square of indices (i, j) associated with the subdivision S N as the following closed set of R 2 : where the interval I N i is given by (2.4). We also denote by C N = {C N (i,j) ; i, j ∈ Z * } the set of all the elementary squares associated with the subdivision S N . We easily check that R 2 = i,j∈Z * C N (i,j) . Figure 1 illustrates the way the elementary squares are indexed, with the example of the subdivision S 4 . Remark 2.6. For every i, j ∈ Z * , the coordinates of the center of the elementary square C N (i,j) associated with the subdivision S N are given by The area of every elementary square C N (i,j) ∈ C N is given by |C N (i,j) | = 1 2N 2 . Notice that for every i, j ∈ Z * with |i|, |j| > 1, the elementary squares having one side in common with the elementary square and C N (i,j±1) .
Definition 2.7. For every q ∈ Q ε ad , we denote by C N (q) and C N (Q T ) the sets of the elementary squares in C N with their interior included in q and Q T respectively: If N is large enough, the sets C N (q) and C N (Q T ) are non-empty. We also define R N (q) as the union of the elementary squares in C N (q): With these notations, we can prove the following lemma.
is a cover of the ε-interior q ε of q (see (1.7)). Moreover, the set R N (q) defined by (2.11) satisfies q ε ⊂ R N (q) ⊂ q.
Proof. Let X ∈ q ε . By definition of q ε , we have X ∈ q and d(X, ∂q) > ε. Since R 2 is covered by squares in C N , Step 3 -Weighted graph associated with q ∈ Q ε ad . In order to write several expressions in a simpler form, we use elements from graph theory. More precisely, the introduction of a weighted graph in Definition 2.9 allows to reformulate the observability inequality for the solution ϕ N in terms of a spectral inequality involving the Laplacian matrix of the graph (see Def. 2.11). In this way, the observability constant associated with ϕ N is linked to the first strictly positive eigenvalue of the Laplacian matrix. Definition 2.9. Let q ∈ Q ε ad an observation domain. We define the weighted graph G N (q) as follows: -I N given by (2.6) is the set of vertices; -for every i ∈ I N , the degree of the vertex i is given by: -for every i, j ∈ I N , the weight of the edge linking the vertices i and j is Definition 2.10. Let q ∈ Q ε ad and let i, j ∈ I N be two vertices of the graph G N (q). We say that there is a path from i to j in G N (q) and we denote i N ∼ j if the vertices i and j are in the same connected component of G N (q). In particular, if w N i,j = 0, then i N ∼ j.
We then recall the definition of the Laplacian matrix associated with a graph (see [4,8]). . (2.12) Remark also that the Laplacian matrix A N (q) of the graph G N (q) verifies the following property (see [4,8]): From now on, we consider that the assumption of Lemma 2.8 holds true, i.e. we take N > 1/ε. More precisely, we fix N the smallest integer strictly greater than 1/ε.
i the support of the characteristic line "x + t = x N i ", starting from x N i in the direction of decreasing x and following the laws of geometric optics for its reflection on Σ T . Since q ∈ Q ε ad , q ε satisfies the geometric optic condition; consequently, there exists (x * , t * ) ∈ q ε ∩ D + i . From Lemma 2.8, we have q ε ⊂ R N (q), so (x * , t * ) belongs to the common side of two elementary squares in C N (q): Therefore i N ∼ i + 1 and, so, the vertices {1, . . . , N } are in the same connected component. Similarly, we denote by D − i the support of the characteristic line "x − t = x N i ", starting from x N i in the direction of increasing x and following the laws of geometric optics for its reflection on Σ T . Since q ε satisfies the geometric optic condition, there exists ( , so (x * , t * ) belongs to the common side of two elementary squares in C N (q): Therefore −i N ∼ − i − 1 and, so, the vertices {−N, . . . , −1} are in the same connected component. It remains to show that the vertices N and −N belong to the same connected component. We denote by D + N the support of the characteristic line "x + t = x N N ", starting from x N N in the direction of decreasing x and following the laws of geometric optics for its reflection on Σ T . Since q ε satisfies the geometric optic condition, there exists (x * , t * ) ∈ q ε ∩ D + N . From Lemma 2.8, we have q ε ⊂ R N (q), so (x * , t * ) belongs to the common side of two elementary squares in C N (q): Remark 2.14. A well-known graph theory result (see, for instance, [4], Prop. 1.3.7) states that the graph G N (q) is connected if and only if dim(ker(A N (q))) = 1. Moreover, if G N (q) is connected, then ker(A N (q)) = Vect(1 2N ), where 1 2N is the vector in R 2N with all its component equal to 1. Let us denote λ N (q) > 0 the smallest non-zero eigenvalue of the matrix A N (q). This eigenvalue is known in graph theory as the algebraic connectivity of the graph. We also define λ N by (2.14) Note that since the set {G N (q); q ∈ Q ε ad } has a finite number of elements, λ N is well defined.
Step 4p-refinement of the graph and connection with the spectrum of the Laplacian matrix.
Definition 2.15. For every p ∈ N * , we denote by C p N (q) the set formed by the elementary squares associated with the subdivision S pN and having their interior in R N (q): We then define the graph G p N (q) following Definition 2.9, substituting N by pN and substituting C N (q) by C p N (q) in the definitions of the vertex degrees and the edge weights. Finally, we denote A p N (q) ∈ M 2pN (R) the Laplacian matrix associated with the graph G p N (q). This matrix has the following block form: where I p , J p ∈ M p (R) are respectively the identity matrix and the matrix with all its elements equal to 1.
Definition 2.16. For any i ∈ Z * , we define the set J p i by For any p ∈ N * , the following lemma makes the link between the spectrum of the Laplacian matrix A p N (q) (see Def. 2.15) and the spectrum of the Laplacian matrix A N (q) (see Def. 2.11).
Lemma 2.17. Let p ∈ N * . The spectrum of the Laplacian matrix 1 p A p N (q) (see (2.15)) is composed of the spectrum of the Laplacian matrix A N (q) (see (2.12)), and the diagonal elements of A N (q) repeated p − 1 times. Moreover, dim(ker(A p N (q))) = 1 and ker(A p N (q)) = Vect(1 2pN ).
Proof. Let η = (η −pN , . . . , η −1 , η 1 , . . . , η pN ) ∈ R 2pN . For every i ∈ I N , we denote by Γ i = (η i ) i ∈J p i ∈ R p and we group these vectors in the matrix In view of (2.15), it follows that Since J p is a real symmetric matrix, there exists an orthonormal basis Then, there exists a diagonal matrix D ∈ M p (R) and a unitary matrix Q ∈ M p (R) such that J p = QDQ T . These matrices have the following form: We also define the matrix U = Q T Γ ∈ M p,2N (R), and denote respectively the rows and the columns of U . Then, for i, j ∈ I N , we have The spectrum of the matrix 1 p A p N (q) can now be computed from Indeed, the expression above shows that A p N (q) is unitarily similar to the block diagonal matrix .
Let N be the smallest integer strictly greater than ε −1 . We prove an observability inequality for the function ϕ N given by (2.10) and associated with the initial datum (ϕ N 0 , ϕ N 1 ) defined in (2.2)-(2.3). In view of Lemma 2.4, the function (ϕ N t ) 2 is constant on each elementary square C N (i,j) in C N : where γ N j N (i) is given by (2.7) and j N by (2.5). In view of the definition of the set R N (q) (see (2.11)), we can estimate from below the L 2 (q)-norm of ϕ N t as follows: In the previous equality, we used that the area of every elementary square in C N is equal to 1 2N 2 . Combining (2.17) with the relation (2.13), we obtain and γ N i given by (2.7). It is easy to see that γ N ∈ ker(A N (q)) ⊥ . Indeed, applying Lemma 2.13, the graph G N (q) is connected. Then, from Remark 2.14, we have ker(A N (q)) = Vect(1 2N ) and, since Then, using λ N defined in (2.14), it follows that where the constant 4N λ N is independent of the domain q and the initial datum (ϕ 0 , ϕ 1 ).
Step 6 -Uniform observability for the solution ϕ pN associated with (ϕ pN 0 , ϕ pN 1 ) w.r.t. q ∈ Q ε ad and p ∈ N * . The next step consists in obtaining a uniform observability inequality for an initial datum (ϕ pN 0 , ϕ pN 1 ) of the form (2.2)-(2.3) with respect to q ∈ Q ε ad and p ∈ N * . As in the previous step, we get that (ϕ pN t ) 2 is constant on every elementary square C pN (i ,j ) ∈ C pN : where ϕ pN is the solution of (1.3) associated with the initial datum (ϕ pN 0 , ϕ pN 1 ) given by (2.2)-(2.3), and γ pN is defined by (2.7). Then, remark that every elementary square C N (i,j) ∈ C N is the union of p 2 elementary squares in C pN , or more precisely that Using the above expression in the evaluation of the L 2 (q)-norm of ϕ pN t , we have Since G N (q) is a connected graph, the degree d N i of every vertex i ∈ I N verifies d N i ≥ 1. Applying Lemma 2.17, the smallest non-zero eigenvalue λ p N (q) of 1 p A p N (q) verifies The vector γ pN = (γ pN i ) i ∈I pN belongs to ker(A p N (q)) ⊥ . Indeed, Setting λ N = min(λ N , 1), it follows that Consequently, combining the above relation with (2.20), we obtain the following observability inequality: where the constant 4N λ N is independent of the domain q, the initial datum (ϕ 0 , ϕ 1 ) and the integer p.
Step 7 -Passing to the limit p → ∞ and conclusion.
In order to finish the proof, we pass to the limit when p → ∞ in the observability inequality (2.21). We easily check that as p tends to ∞, Moreover, since the solution ϕ of the wave equation (1.3) depends continuously on its initial condition (ϕ 0 , ϕ 1 ) ∈ V, we can write Eventually, passing to the limit in (2.21), we get which concludes the proof with C ε obs = max 4N, 4N λ N . We recall that N depends on ε by the condition N > 1/ε. Remark 2.18. Let q ⊂ Q T be a finite union of open sets. If q verifies the usual geometric optic condition, there exists ε > 0 small enough such that q ε still verifies the geometric optic condition. We then set N = 1/ε + 1. The associated graph G N (q) being connected, there exists a relation (see, for instance, [22]) between the algebraic connectivity λ N (q), the number of vertices N V and the diameter D G of the graph. More precisely, we have λ N (q) ≥ 4 N V D G . Since in our case N V = 2N and D G ≤ 2N , we deduce that λ N (q) ≥ 1 N 2 and therefore that C obs (q) ≤ 4N 3 . Thus, in the worst situation, we could have an observability constant of order 1/ε 3 . If we see ε as a measure of the "thickness" of the observation domain q, we get the same estimate of the observability constant as the one given in Proposition 2.1 of [29].

One explicit example
We illustrate the proof of Theorem 2.1 on a simple example for which the observation domain q depicted in Figure 3 (colored in red) is well adapted to the subdivision S 4 . The study of this example is also the opportunity to develop a method for the computation of the observability constant for observation domains which are exactly the union of elementary squares associated with a given subdivision S N , for a fixed integer N > 0.
We start by enumerating the elementary squares composing the observation domain q. In Table 1, we list, for i, j ∈ Z * , the elementary squares C 4 (i,j) included in q and the values of the indices j 4 (i) and −j 4 (j), allowing to compute the Laplacian matrix A 4 (q) associated with the corresponding graph G 4 (q).  Table 1. Elementary squares associated with S 4 and belonging to C 4 (q).   The Laplacian matrix associated with the graph G 4 (q) is given by The spectrum of A 4 (q) can be explicitly computed: Sp(A 4 (q)) = {0, 4, 4, 4, 4, 4, 14, 16}.
It confirms that the kernel of A 4 (q) is one-dimensional -therefore G 4 (q) is connected -and implies that the smallest non-zero eigenvalue of A 4 (q) is λ 4 (q) = 4. If we replace the subdivision S 4 by the subdivision S 4p for any p ∈ N * , then the Laplacian matrix associated with the graph G p 4 (q) is the following one: According to Lemma 2.17, the smallest non-zero eigenvalue of 1 p A p 4 (q) is given by Consequently, the observability constant associated with the observation domain q depicted in Figure 3 is given by

A corollary
We prove a uniform observability inequality for the observation domains q γ defined in (1.9), with γ ∈ G ad (see (1.10)), which will be used in the next section.
Corollary 2.19. Let T ≥ 2. There exists a constant C obs > 0 such that for every γ ∈ G ad , where ϕ is the solution of the wave equation (1.3) associated with the initial condition (ϕ 0 , ϕ 1 ).

Optimization of the shape of the control domain
In this section, we study the problem of finding the optimal shape and position of the control domain, for a given initial condition (y 0 , y 1 ) ∈ V.

Existence of an optimal domain
In order to obtain a well-posedness result, we consider a variant of the optimal problem (1.11) and replace the characteristic function 1 q in (1.1) by a regular function in space. More precisely, we fix δ ∈ (0, δ 0 ) and, for every γ ∈ G ad , we define χ γ (x, t) = χ(x − γ(t)), with χ : R → [0, 1] a C 1 even function such that In the sequel, we also use the function χ γ defined by χ γ (x, t) = χ (x − γ(t)). In this new setting, the HUM control now lives in the weighted space Moreover, we can adapt the uniform observability inequality given in Corollary 2.19. For T ≥ 2, there exists a constant C obs > 0 such that for every γ ∈ G ad , where ϕ is the solution of (1.3) associated with the initial condition (ϕ 0 , ϕ 1 ).
Then, our optimization problem reads as follows: for a given initial datum (y 0 , y 1 ) ∈ V, solve inf γ∈G ad where v is the control of minimal L 2 χ -norm distributed over q γ ⊂ Q T , and ϕ is the associated adjoint state such that v = ϕ| qγ . This adjoint state can be characterized using the HUM method, it is the solution of (1.3) associated with the minimum (ϕ 0 , ϕ 1 ) of the conjugate functional To show the well-posedness of (3.3), we follow the steps of Theorem 2.1 in [29]. We start with a convergence result on the function χ γ .
Proof. It is a direct consequence of the inequality We then have that the following continuity result. Proof. Let (γ n ) n≥0 ⊂ G ad and γ ∈ G ad such that γ n → γ in L ∞ (0, T ) as n → ∞.
From the continuous dependence of the solution of the wave equation with respect to the initial condition, it follows where ϕ is the solution of (1.3) associated with (ϕ 0 , ϕ 1 ). Let ψ ∈ L 2 (0, T ; L 2 (Ω)). We then have Indeed, we can take the weak limit in the first term because ψχ γ ∈ L 2 (0, T ; L 2 (Ω)). Using Lemma 3.1 and the boundedness of (ϕ n ) n≥0 in L 2 (0, T ; L 2 (Ω)), the second term converges to 0 in view of the estimate Consequently, we obtain the convergence ϕ n χ γn ϕχ γ weakly in L 2 (0, T ; L 2 (Ω)).
Let now (ψ 0 , ψ 1 ) ∈ W and ψ the corresponding solution of (1.3). Taking the weak limit in the optimality condition This means that (ϕ 0 , ϕ 1 ) is the minimum of J γ . Besides, we remark that this property uniquely characterizes the weak limit of any subsequence of (ϕ n 0 , ϕ n 1 ). This implies that the whole sequence (ϕ n 0 , ϕ n 1 ) weakly converges. The continuity of J is finally obtained by taking the weak limit in the optimality condition The continuity of J then allows to show that the extremal problem (3.3) is well-posed. Proof. The cost J being bounded from below, there exists a minimizing sequence (γ n ) n≥0 ⊂ G ad . By definition of G ad , this sequence is bounded in W 1,∞ (0, T ). Moreover, since W 1,∞ (0, T ) is compactly embedded in L ∞ (0, T ), there exists a curve γ ∈ L ∞ (0, T ) such that, up to a subsequence, γ n → γ in L ∞ (0, T ). From the definition of G ad , all the curves γ n are M -Lipschitz, with M independent of n. So, taking the pointwise limit in the expressions we notice that γ ∈ G ad . Finally, using Proposition 3.2, we obtain J(γ n ) → J(γ) = inf G ad J which means that γ is a minimum of J over G ad .
It follows in particular from Proposition 3.2 that the optimal cost J(γ ) = inf γ∈G ad J(γ) is stable with respect to the initial datum (y 0 , y 1 ) ∈ V in the following sense: J(γ ) ≤ C obs (y 0 , y 1 ) 2 V , where C obs is the observability constant appearing in Corollary 2.19. However, we highlight that the optimal curve γ does not vary continuously with respect to (y 0 , y 1 ) (see Sect. 4.1).
Remark 3.4. We emphasize that the functional J may have several global minima. Indeed, if the initial datum (y 0 , y 1 ) ∈ V is such that y 0 (x) = −y 0 (1 − x), y 1 (x) = 0 for x ∈ Ω, there are examples (see [29] and example (EX1) in Sect. 4.1) where the optimal domain is not the cylinder centered at x = 1 2 . Hence, if γ is an optimal curve for J corresponding to this initial datum, from symmetry arguments, the curve 1 − γ is also optimal.

First directional derivative of the cost J
We now give the expression of the directional derivative of the cost J.
Proof. For η > 0 small enough, we denote by (ϕ η 0 , ϕ η 1 ) the minimum of J γη , and ϕ η the corresponding solution of (1.3). Similarly, we denote by (ϕ 0 , ϕ 1 ) the minimum of J γ , and ϕ the corresponding solution of (1.3). Using the optimality conditions of J γη and J γ , we can write Arguing as in the proof of Proposition 3.2, we can show that ϕ η ϕ weakly in L 2 (0, T ; L 2 (Ω)). As a result, we have Indeed, we can take the weak limit in the first term since ϕγχ γ ∈ L 2 (0, T ; L 2 (Ω)). Moreover, using Lemma 3.6 and the boundedness of (ϕ η ) η>0 in L 2 (0, T ; L 2 (Ω)), the second term converges to 0 in view of the estimate . Remark 3.8. We emphasize that the expression of the directional derivative dJ(γ; γ) does not involve the solution of an adjoint boundary value problem. This is due to the fact that the curve γ, argument of the cost J, is associated with the control of minimal L 2 (q γ )-norm. We refer to the proof of Theorem 2.3 in [23] for more details.

Regularization and Gradient algorithm
At the practical level, in order to solve the optimal problem (3.3) numerically, we need to handle the Lipschitz constraint included in G ad . In this respect, we add a regularizing term to the cost J in order to keep the derivative of γ uniformly bounded. The optimization problem is now the following one: for > 0 fixed, solve The regularization parameter , which can be compared to the Lipschitz constant M in the definition of G ad , controls the speed of variation of the curves γ ∈ W 1,∞ (0, T ). We fix γ ∈ W 1,∞ (0, T ) such that δ 0 ≤ γ ≤ 1 − δ 0 . Using Proposition 3.7, for any admissible perturbation γ ∈ W 1,∞ (0, T ), a direct calculation provides the expression of the directional derivative of J dJ (γ; γ) = dJ(γ; γ) In the expression of j γ , the function ϕ is the solution of (1.3) associated with the minimum (ϕ 0 , ϕ 1 ) of J γ . Consequently, a minimizing sequence (γ n ) n∈N for J is defined as follows: where P [δ0,1−δ0] is the pointwise projection in the interval [δ 0 , 1 − δ 0 ], ρ > 0 a descent step-size and j γn ∈ H 1 (0, T ) is the solution of the variational formulation j γn , γ L 2 (0,T ) + j γn , γ L 2 (0,T ) = j γn , γ L 2 (0,T ) + γ n , γ L 2 (0,T ) , ∀ γ ∈ H 1 (0, T ). (3.5) This implies that dJ (γ n ; j γn ) = j γn 2 L 2 (0,T ) + j γn 2 L 2 (0,T ) ≥ 0. Remark 3.9. For simplicity, we have relaxed here the constraint γ L ∞ (0,T ) ≤ M appearing in the set G ad (see (1.10)) through the regularization term 2 γ 2 L 2 (0,T ) , with > 0. We may instead consider the term 2 γ 2 Our numerical experiments -some of them discussed in the next section -suggest that this later possibility is not necessary.

Numerical experiments
Before presenting some numerical experiments, let us briefly mention some aspects of the resolution of the underlying discretized problem.
-The discretization of the curve γ is performed as follows. For any fixed integer N > 0, we denote δt = T /N and define the uniform subdivision {t i } i=0,··· ,N of [0, T ] with t i = iδt. We then approximate the curve γ in the space of dimension N + 1 P δt For any γ ∈ P δt is the usual Lagrange basis. Consequently, γ is defined by the N + 1 points (γ i , t i ) ∈ Ω × [0, T ]. The knowledge of the initial curve γ 0 ∈ P δt 1 such that δ 0 ≤ γ 0 ≤ 1 − δ 0 determines such points and then a triangular mesh of Q T . At each iteration n ≥ 0, these points are updated along the x-axis according to the pointwise time descent direction j γn ∈ H 1 (0, T ) (see (3.5)) as follows: We emphasize that a re-meshing of Q T is performed at each iteration n according to the set of points (x n i , t n i ) i=0,··· ,N +1 .
Remark that, for a fixed value of N , the curves γ ∈ P δt 1 are uniformly Lipschitz with constant M = 1 δt , as required in G ad .
-Each iteration of the algorithm requires the numerical approximation of the control of minimal L 2 (q γn )-norm for the initial datum (y 0 , y 1 ) ∈ V. We use the space-time method described in Sections 3-4 of [6], which is very well-adapted to the description of γ embedded in a space-time mesh of Q T . The minimization of the conjugate functional J γn (see (3.4)) with respect to (ϕ 0 , ϕ 1 ) ∈ W is replaced by the search of the unique saddle-point of the Lagrangian L : Z × L 2 (0, T ; H 1 0 (Ω)) → R defined by . The corresponding mixed formulation is solved with a conformal space-time finite element method, while a direct method is used to invert the discrete matrix. The interesting feature of the method, for which the adaptation of the mesh is very simple to handle, is that only a small part of the matrix -corresponding to the term ϕ 2 L 2 (qγ n ) -is modified between two consecutive iterations n and n + 1. Moreover, this space-time framework leads to strong convergent approximations of the control as the fineness of the mesh goes to zero. We refer to [6] for more details.

Numerical illustrations
We discuss several experiments performed with FreeFEM++ (see [14]) for various initial data and control domains. We notably use an UMFPACK type solver.
We denote in this section by J n and dJ n a numerical approximation of J (γ n ) and dJ (γ n ) respectively. We fix δ 0 = 0.15 and δ = δ 0 /4. Moreover, according to (3.1), we define the function χ ∈ C 2 (R) in [δ 0 − δ, δ 0 ] as the unique polynomial of degree 5 such that Concerning the stopping criterion for the descent algorithm, we observed that the usual one based on the relative quantity |J n − J n−1 |/J 0 is inefficient because too noisy. This is due to the uncertainty on the numerical computation of the adjoint state ϕ n and the perturbation j γn . Consequently, in order to better capture the variations of the sequence (J n ) n∈N , we replace J n and J n−1 by the right and left p-point average respectively leading to the stopping criterion In the sequel, we fix p = 10 and η = 10 −3 . Furthermore, in order to measure the gain obtained by using non-cylindrical control domains rather than cylindrical ones, we introduce a performance index associated with each optimal curves γ opt : identifying any constant curve γ ≡ x 0 with its value x 0 ∈ [δ 0 , 1 − δ 0 ], we compute the minimal cost min x0 J (x 0 ) for cylindrical domains; the performance index of γ opt is then defined as follows: In the sequel, in practice, the minimum of J with respect to x 0 is searched among 13 distinct values equi-distributed between 0.2 and 0.8.
-We first consider the regular initial datum (y 0 , y 1 ) given by and T = 2, = 10 −2 , ρ = 10 −4 . We initialize the descent algorithm with the following three initial curves: The corresponding initial and optimal domains are depicted in Figure 5 together with typical space-time meshes. The numbers of iterations until convergence, the values of the functional J ε evaluated at the optimal curve γ opt and the performance indices of γ opt are listed in Table 2.
In Figure 5, we observe that the optimal domain computed by the algorithm depends on the chosen initial domain. This indicates that the functional J ε have several local minima. Moreover, one can show that, among the cylindrical domains, there are two optimal values, x 0 = 1/4 and x 0 = 3/4, leading to J ε (x 0 ) ≈ 46.94. These values correspond to the extrema of the function x → sin(2πx), x ∈ [0, 1]. The simulations associated with the initial curves γ 1 0 and γ 2 0 are in agreement with this fact. On the other hand, the worst cylindrical domain corresponds to x 0 = 1/2 (see Fig. 7-Left).
Eventually, the adjoint states ϕ (from which we obtain the control v = ϕ| q n γ ) computed for the optimal domains in Figure 5-Bottom, are displayed in Figure 6.  -We now consider the initial datum (y 0 , y 1 ) given by for x ∈ (0, 1). (EX2) As can be seen in Figure 9−3, this initial condition, plotted in Figure 8, generates a travelling wave. For T = 2, = 10 −2 and ρ = 10 −4 , we initialize the descent algorithm with the curve γ 0 ≡ 1/2. The convergence is reached after 68 iterations and the optimal cost is J (γ opt ) ≈ 48.70. Moreover, the minimal cost  for cylindrical domains is min x0 J (x 0 ) ≈ 179.22 leading to a performance index Π(γ opt ) ≈ 72.83%. The noncylindrical setup is in that case much more efficient that the cylindrical one. This is due to the fact that the domains we consider can follow very closely the propagation of the travelling wave. This can be noticed in Figure 9, where we display the optimal control domain, the corresponding adjoint state ϕ, the uncontrolled and controlled solutions over the optimal domain. The evolution of the cost J n and the derivative dJ n with respect to n are displayed in Figure 10. Figure 7-Right depicts the values of the functional J for the constant curves γ ≡ x 0 used to determine the best cylindrical domain and highlights the low variation of the cost with respect to the position of such domains.
This second example allows to emphasize the sensitivity of the optimal curve with respect to the initial condition. For any δ, consider the initial condition (y δ 0 , y δ 1 ) = (y 0 , y 1 ) + δ(w 0 , w 1 ), where (y 0 , y 1 ) is given by (EX2) and supported in the interval [0.4, 0.6], and where (w 0 , w 1 ) is any initial condition supported for instance in [0.8, 0.9]. For δ = 0 and any T > 0, the optimal curve γ opt follows the travelling wave starting in the interval [0.4, 0.6]. On the other hand, for any δ = 0, the optimal curve γ δ opt will also have to cross the travelling waves generated by the initial condition (w 0 , w 1 ) starting in [0.8, 0.9]. If the controllability time T is small enough, Figure 10. (EX2) -Evolution of the cost J n (left) and the derivative dJ n (right) for the initial curve γ 0 ≡ 1/2. these additional travelling waves remain far from the initial wave. It follows that, for any δ = 0, the resulting optimal curve γ δ opt will be not arbitrarily close to γ opt . -We now consider the initial datum (y 0 , y 1 ) given by This initial condition generates two waves travelling waves in opposite directions, as can be seen in Figure 11.3. For T = 2, = 10 −2 and ρ = 10 −4 , we initialize the algorithm with the initial curve γ 0 ≡ 1/2. The convergence is observed after 111 iterations leading to J (γ opt ) ≈ 41.02. Moreover, the minimal cost for cylindrical domains is min x0 J (x 0 ) ≈ 85.08 leading to a performance index Π(γ opt ) ≈ 51.79%. Once again, our non-cylindrical setup is much more efficient than the cylindrical one. It is still due to the fact that the domains we consider can follow the propagation of the travelling waves. This can be noticed in Figure 11, where we display the optimal control domain, the corresponding adjoint state ϕ, the uncontrolled and the controlled wave over the optimal domain. In order to emphasize the influence of the controllability time on the optimal domain, for = 10 −2 and γ 0 ≡ 1/2, we use the descent algorithm with T = 1 and ρ = 2.5 × 10 −5 , initialized with the curve γ 0 ≡ 1/2. Remark that the corresponding domain satisfies the geometric optic condition. The convergence is observed after 213 iterations and the optimal cost is J (γ opt ) ≈ 94.78. Moreover, the minimal cost for cylindrical domains is min x0 J (x 0 ) ≈ 183.98 and the performance index is Π(γ opt ) ≈ 48.48%. Observe that the cylindrical domains associated with x 0 / ∈ (0.25, 0.75) do not verify the geometric optic condition. This highlights the necessity to use non-cylindrical domains. Compared to the simulation for T = 2, the optimal cost increases by a factor around 2.3. Figure 12 displays the optimal control domain, the corresponding adjoint state ϕ, the uncontrolled and controlled wave over the optimal domain. We remark that the projection of the optimal domain on the x-axis covers the whole domain Ω, in contrast with the domain associated with T = 2.
In Table 3, we illustrate the influence of the controllability time T on the optimal cost J (γ opt ). More exactly, we remark that the cost is of order O(T −1.27 ). Figure 11. (EX3) -From left to right, optimal control domain, isovalues of the corresponding adjoint state ϕ, isovalues of the uncontrolled and controlled wave over the optimal domain, for T = 2, for the initial curve γ 0 ≡ 1/2. Figure 12. (EX3) -From left to right, optimal control domain, isovalues of the corresponding adjoint state ϕ, isovalues of the uncontrolled and controlled wave over the optimal domain, for T = 1, for the initial curve γ 0 ≡ 1/2.
-Eventually, in order to highlight the influence of the regularization parameter on the optimal domain, we now consider the initial datum (y 0 , y 1 ) given by For T = 2 and ρ = 10 −5 , we initialize the descent algorithm with the curve γ ≡ 1/2 and consider = 10 −2 and = 0. The numbers of iterations until convergence, the values of the functional J evaluated at the optimal curve γ opt and the performance indices of γ opt are listed in Table 4. For the initial datum (EX4), the minimal cost for cylindrical domains is min x0 J (x 0 ) ≈ 47.71.
In Figure 13, we clearly see the regularizing effect of and the need of regularization in this case, as the optimal domain obtained when = 0 is very oscillating. Actually, since the initial position y 0 is supported on the whole domain, the corresponding uncontrolled solution is not localized on a small part of Q T (in contrast for instance to the second example, see (EX2)). The optimal domain is very likely composed of many parts Table 4. (EX4) -Number of iterations, optimal value of the functional J and performance index, for ∈ {0, 10 −2 }, for the initial curve γ 0 ≡ 1/2. distributed in Q T leading an homogenization phenomenon. Consequently, when = 0, the algorithm tends to reach these parts leads to large oscillations of the curve γ. The -term aims to reduces these oscillation by imposing regularity property of the curves γ describing the domains q γ . In this topic, we refer to the seminal work of Chenais [7]. Let us also mention that a well posedness result is achieved in [29] by introducing a relaxation of the problem 1.4, i.e. by replacing characteristic functions by densities (see also [26] in the context of stabilization of the wave equation).

Approximation of the observability constant by an iterative method
In this last part, we formally describe and use an algorithm allowing to approximate the observability constant appearing in (1.2), associated with any domain q ⊂ Q T . The algorithm is based on the following characterization: where Λ q and R are respectively the control operator associated with the domain q and the duality operator between the space W and V: . Figure 14. Most expensive initial data y 0 (left) and y 1 (right) to be controlled.
In the definition of Λ q , ϕ 0 ∈ W is the minimum of the functional J (see (3.4)) associated with y 0 ∈ V. The characterization (4.2) can be obtained by following the steps of Section 2 in [25] and Remark 2.98 in [10].
The main consequence of this characterization is that C obs (q) can be viewed as the largest eigenvalue of the operator RΛ q in V. Consequently, we can formally adapt the power iteration method to our infinite-dimensional setting. The algorithm reads as follows. Let y 0 0 ∈ V be given such that y 0 0 V = 1. For n ≥ 0, using the spacetime finite element method described in Section 3-4 of [6], we compute ϕ n 0 = Λ q y n 0 then set z n 0 = R ϕ n 0 and y n+1 0 = z n 0 / z n 0 V . We finally have C obs (q) = lim n→∞ z n 0 V while y n 0 converges in V to the most expensive initial datum to control. For the control domain of Figure 3, this algorithm initialized with y 0 0 = K(x(1 − x), 0), K such that y 0 0 V = 1, produces the following sequence { z n 0 V } n≥0 = {2.689, 3.829, 3.981, 3.994, 3.997, · · · } converging toward the value 4, in agreement with the result of Section 2.2 based on a graph argument. The most expensive initial datum to be controlled is displayed in Figure 14. Remark that the initial datum solution of (4.2) is not unique.

Conclusion and perspectives
Making use of the d'Alembert formula for the solutions of the one dimensional wave equation, we have shown a uniform observability inequality with respect to a class of non cylindrical observation domains satisfying the geometric optic condition. The proof based on arguments from graph theory allows notably to relate the value of the observability constant to the spectrum of the Laplacian matrix, defined in term of the graph associated to any domain q ⊂ Q T . The uniform observability property then allows to consider and analyze the problem of the control's optimal support associated to fixed initial conditions. For simplicity, the optimization is made over connected domains defined by regular curves. As expected, the optimal domains (approximated within a space-time finite element method) are closely related to the travelling waves generated by the initial conditions. This work may be extended to several directions. First, the characterization of the observability constant in term of a computable eigenvalue problem in Section 4.2 may allow to consider the optimization of such constant with respect to the domain of observation, i.e. inf q∈Q ε ad C obs (q) (as done in [31]). The domain providing the smallest observability constant is nothing else than the trajectory minimizing the cost of control, i.e. the norm of the control corresponding to the worst controllable initial datum. Moreover, from an approximation point of view, we may also consider more general domains (than connected ones) and use, for instance, a level-set method to describe the geometry (as done in [23]). Eventually, this work may be adapted to the case of controls supported on single curves of Q T , using the uniform observability property given in [5]. We refer to [3] for a numerical study in this case. The extension of this work to the multi-dimensional case studied in [17,32] is also a challenge.