Yannick Privat

Emmanuel Trélat

Enrique Zuazua

Optimal location of controllers for the one-dimensional wave equation

Keywords: wave equation, exact controllability, HUM method, shape optimization, relaxation, optimal control, Pontryagin Maximum Principle AMS classification: 49J20, 35L05, 49J15, 49Q10, 49K35

In this paper, we consider the homogeneous one-dimensional wave equation defined on (0, π). For every subset ω ⊂ [0, π] of positive measure, every T 2π, and all initial data, there exists a unique control of minimal norm in L 2 (0, T ; L 2 (ω)) steering the system exactly to zero. In this article we consider two optimal design problems. Let L ∈ (0, 1). The first problem is to determine the optimal shape and position of ω in order to minimize the norm of the control for given initial data, over all possible measurable subsets ω of [0, π] of Lebesgue measure Lπ. The second problem is to minimize the norm of the control operator, over all such subsets. Considering a relaxed version of these optimal design problems, we show and characterize the emergence of different phenomena for the first problem depending on the choice of the initial data: existence of optimal sets having a finite or an infinite number of connected components, or nonexistence of an optimal set (relaxation phenomenon). The second problem does not admit any optimal solution except for L = 1/2. Moreover, we provide an interpretation of these problems in terms of a classical optimal control problem for an infinite number of controlled ordinary differential equations. This new interpretation permits in turn to study modal approximations of the two problems and leads to new numerical algorithms. Their efficiency will be exhibited by several experiments and simulations.

1 Introduction

The optimal design problems

Let T be a positive real number. Consider the one-dimensional controlled wave equation on [0, π] with Dirichlet boundary conditions ∂ tt y(t, x) -∂ xx y(t, x) = h ω (t, x), (t, x) ∈ (0, T ) × (0, π), y(t, 0) = y(t, π) = 0, t ∈ [0, T ], y(0, x) = y 0 (x), ∂ t y(0, x) = y 1 (x), x ∈ [0, π],

(1) * IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman, 35170 Bruz, France (Yannick.Privat@bretagne.ens-cachan.fr).

† Université Pierre et Marie Curie (Univ. Paris 6) and Institut Universitaire de France, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France (emmanuel.trelat@upmc.fr).

‡ BCAM -Basque Center for Applied Mathematics, Mazarredo, 14 E-48009 Bilbao-Basque Country-Spain § Ikerbasque, Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, E-48011, Bilbao-Basque Country-Spain (zuazua@bcamath.org).

where h ω is a control supported in [0, T ] × ω and ω is a measurable subset of [0, π]. For all initial data (y 0 , y 1 ) ∈ H 1 0 (0, π) ×L 2 (0, π) and every h ω ∈ L 2 ((0, T ) ×(0, π)), there exists a unique solution y ∈ C 0 (0, T ; H 1 0 (0, π)) ∩ C 1 (0, T ; L 2 (0, π)) ∩ C 2 (0, T ; H -1 (0, π)) of the Cauchy problem [START_REF] Akhiezer | Elements of the theory of elliptic functions, Translations of Mathematical Monographs 79[END_REF]. The exact null controllability problem settled in these spaces consists of finding a control h ω steering the control system [START_REF] Akhiezer | Elements of the theory of elliptic functions, Translations of Mathematical Monographs 79[END_REF] to

y(T, •) = ∂ t y(T, •) = 0. (2)
It is well-known that, for every subset ω of [0, π] of positive Lebesgue measure, the exact null controllability problem has a solution whenever T 2π (see [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF]). The Hilbert Uniqueness Method (HUM; see [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]) permits to design such a control, achieving moreover the null controllability in minimal L 2 ((0, T ) × (0, π)) norm. This (unique) control is referred to as the HUM control and is defined as follows. Using the observability inequality

C (φ 0 , φ 1 ) 2 L 2 (0,π)×H -1 (0,π) T 0 ω φ(t, x) 2 dxdt, (3)
where C is a positive constant (only depending on T and ω), valuable for every solution φ of the adjoint system ∂ tt φ(t, x) -∂ xx φ(t, x) = 0, (t, x) ∈ (0, T ) × (0, π), φ(t, 0) = φ(t, π) = 0, t ∈ [0, T ], φ(0, x) = φ 0 (x), ∂ t φ(0, x) = φ 1 (x), x ∈ [0, π], [START_REF] Cerf | Continuation from a flat to a round Earth model in the coplanar orbit transfer problem[END_REF] and every T 2π, the functional

J ω (φ 0 , φ 1 ) = 1 2 T 0 ω φ(t, x) 2 dxdt -φ 1 , y 0 H -1 ,H 1 0 + φ 0 , y 1 L 2 , (5)
has a unique minimizer (still denoted (φ 0 , φ 1 )) in the space L 2 (0, π) × H -1 (0, π), for all (y 0 , y 1 ) ∈ H 1 0 (0, π)×L 2 (0, π). In [START_REF] Cox | The rate at which energy decays in a damped string[END_REF] the notation •, • H -1 ,H 1 0 stands for the duality bracket between H -1 (0, π) and H 1 0 (0, π), and the notation •, • L 2 stands for the usual scalar product of L 2 (0, π). The HUM control h ω steering (y 0 , y 1 ) to (0, 0) in time T is then given by h ω (t, x) = χ ω (x)φ(t, x), [START_REF] Fourer | AMPL: A modeling language for mathematical programming[END_REF] for almost all (t, x) ∈ (0, T )×(0, π), where χ ω denotes the characteristic function of the measurable set ω and φ is the solution of (4) with initial data (φ 0 , φ 1 ) minimizing J ω .

In this article we are interested in the problem of optimizing the shape and position of the control support ω, over all possible measurable subsets of [0, π] of given Lebesgue measure. Throughout the article, let L ∈ (0, 1) be fixed. We define the set

U L = {χ ω | ω is a measurable subset of [0, π] such that |ω| = Lπ}, (7)
where |ω| denotes the Lebesgue measure of ω. We consider the two following problems.

First problem. Let (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π) be some fixed initial data. We investigate the problem of minimizing

h ω 2 L 2 ((0,T )×(0,π)) = T 0 ω φ(t, x) 2 dxdt, (8)
where h ω is the HUM control [START_REF] Fourer | AMPL: A modeling language for mathematical programming[END_REF] steering (y 0 , y 1 ) to (0, 0) in time T , over all possible subsets ω of [0, π] such that χ ω ∈ U L .

This problem is the most simple among all possible problems of optimal location of actuators for vibration models. In this minimization problem, the optimal set ω, whenever it exists, depends on the initial data (y 0 , y 1 ). Although this problem has some mathematical interest and provides some insight in such shape optimization problems, it is not really relevant for practical purposes since the knowledge of the initial data of (1) is needed. To discard this dependence and improve the robustness of the cost function, we consider the following second problem. Recall that, for every subset ω of [0, π] of positive measure, the so-called HUM operator Γ ω is defined by Γ ω : H 1 0 (0, π) × L 2 (0, π) -→ L 2 ((0, T ) × (0, π)) (y 0 , y 1 ) -→ h ω , where h ω is the HUM control ( 6) steering (y 0 , y 1 ) to (0, 0) in time T .

Second problem. We investigate the problem of minimizing the norm of the operator Γ ω Γ ω = sup h ω L 2 ((0,T )×(0,π)) (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π), (y 0 , y 1 ) H 1 0 (0,π)×L 2 (0,π) = 1 [START_REF] Henrot | Variation et optimisation de formes[END_REF] over the set U L .

A brief state of the art

Although the literature on such kinds of shape optimization problems is quite abundant in engineering applications (see e.g. [START_REF] Morris | Linear-quadratic optimal actuator location[END_REF][START_REF] Padula | Optimization strategies for sensor and actuator placement[END_REF][START_REF] Sigmund | Systematic design of phononic band-gap materials and structures by topology optimization[END_REF][START_REF] Van De Wal | A review of methods for input/output selection[END_REF]), very few theoretical results and mathematical proofs do exist. In [START_REF] Hébrard | Optimal shape and position of the actuators for the stabilization of a string[END_REF][START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF], similar questions were investigated for optimal stabilization issues of the one-dimensional wave equation. These two articles are important because they were the first (and to our knowledge, up to now the only ones) to provide mathematical arguments and proofs to characterize the optimal set whenever it exists. They have inspired our own works such as the recent one [START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF] in which we investigate similar questions for the optimal observability of the wave equation (with practical issues for the optimal placement of sensors). Concerning the problem investigated in the present article of determining an optimal control domain, we quote the article [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF] whose contribution is to prove the existence of a solution for a relaxed version of our first problem above. Note that we provide in Theorem 1 (Section 2.1) a more precise result with a new shorter proof of [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF]Theorem 2.1] using the frequential analysis approach used throughout our paper. We also quote [START_REF] Münch | Optimal location of the support of the control for the 1-D wave equation: numerical investigations[END_REF] in which the author investigates numerically this relaxed first problem (not only in dimension one), using shape and topological derivatives of the functional under consideration, and gives numerical simulations providing evidence that, for a specific initial data, the optimal solution of the relaxed problem is the characteristic function of some subset ω of [0, π] and thus is solution of the (initial, non relaxed) first problem. In [START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF] the authors investigate the optimal location of the support of the HUM control for the one-dimensional heat equation, with fixed initial data. They give a first-order optimality condition for a relaxed version and then provide some numerical simulations. We stress that in these contributions only the first problem is addressed, from the numerical point of view. As said above, this first problem leads to interesting mathematical properties but is of little practical interest for practical purposes since it depends on the initial data.

Anyway the study of the first problem is interesting since it provides a better insight in such shape optimization problems and is a benchmark for studying other classes of such problems. But the second problem is still to be addressed in the context of the heat equation.

Short description of our contributions

In this article we provide a complete mathematical analysis of the two shape optimization problems settled above, and in turn we obtain efficient algorithms for computing the optimal domains. The article is structured as follows.

Section 2 is devoted to solving the first problem (8) (that is, the problem with fixed initial data). First of all, using a Fourier series expansion of the adjoint φ, we prove existence and uniqueness of a solution of a relaxed version of the first problem (Theorem 1). Here, the relaxation procedure consists of considering the convex closure of the set U L in L ∞ weak star topology, that is

U L = {a ∈ L ∞ ([0, π], [0, 1]) | π 0 a(x) dx = Lπ}.
If the minimizer a ∈ U L of the relaxed first problem belongs to U L , then it is the characteristic function of a subset ω of [0, π] of Lebesgue measure Lπ, and then ω is a solution of the (initial) first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF]. At the opposite, if a ∈ U L \ U L then the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] does not have any solution, and we speak of a relaxation phenomenon. Throughout Section 2 we provide a complete characterization of these phenomena.

More precisely, in Section 2.4 we first provide several possible sufficient conditions ensuring that the first problem (8) has a solution. For instance we will prove the following result (see the more precise statement of Theorem 2).

Theorem. Let T 2π and L ∈ (0, 1). Assume that the initial data (y 0 , y 1 ) under consideration have a finite number of Fourier components. Then the first problem (8) has a unique solution χ ω ∈ U L , which has a finite number of connected components.

The uniqueness of the solution must be of course understood up to some subset of zero Lebesgue measure. In Theorem 3 we provide a variant of this result, valuable only when T is an integer multiple of 2π: we show that the above sufficient condition can then be weakened to an exponential decrease of the Fourier coefficients of the initial data. Note that this sufficient condition holds if the initial data are analytic or quasi-analytic.

In Section 2.2 we provide further comments on the relaxed version of the Hilbert Uniqueness Method and on its well-posedness, showing strict convexity properties.

Section 2.3 is another significant contribution of our work. In this section we provide an interpretation of the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] in terms of a classical (however infinite-dimensional) optimal control problem, to which we can apply, up to some slight adaptations, the well-known Pontryagin Maximum Principle and derive necessary and sufficient optimality conditions for the relaxed first problem. The change of point of view consists in considering the functions a ∈ U L of the relaxed problem as controls. These results and this interpretation (which is new to the best of our knowledge) have two important consequences.

First, in the case where T is a multiple integer of 2π, these necessary and sufficient conditions enable us to provide a complete characterization of all initial data for which

• the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] has at least one solution,

• the first problem (8) has exactly one solution,

• the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] does not have any solution (in other words, the relaxation phenomenon occurs).

In turn, we establish some connections between the complexity of the optimal set of the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF], whenever it exists, and the regularity of the initial data. As a particular case, we prove the following result, showing the sharpness of the previous sufficient conditions.

Theorem. There exist initial data (y 0 , y 1 ) of class C ∞ such that the first problem (8) has a unique solution ω, which is a fractal set and thus has an infinite number of connected components. There exist initial data (y 0 , y 1 ) of class C ∞ such that the first problem (8) does not have any solution.

We stress that our results stated in Section 2.5 are stronger since we provide a characterization of all possible initial data for which such or such phenomenon occurs, by establishing a precise correspondence with some classes of Fourier series.

The second consequence of the interpretation in terms of an optimal control problem is the fact that we are able to design new and efficient numerical methods in order to compute an approximation of the optimal set whenever it exists. In Section 2.6 we define and study a modal approximation of the first problem consisting of truncating to N modes the infinite dimensional control system corresponding to our problem. We thus get a usual finite dimensional optimal control problem which can be easily analyzed. We prove in Proposition 5 that the truncated first problem has a unique solution (which is therefore a characteristic function). Moreover, we can apply to this finite dimensional optimal control problem the usual numerical methods of optimal control, which result into new numerical approaches to compute approximations of the optimal set. We provide several numerical simulations that show the efficiency of this new approach to optimal design.

We stress however in Section 2.6.2 on the limitations of numerics in optimal design. Indeed, according to our theoretical results, and although we have a nice Γ-convergence result (stated in Proposition 4), such numerical approaches cannot permit to guess a relaxation phenomenon. By the way note that the occurrence of the relaxation phenomenon for smooth initial data infirms a conjecture of [START_REF] Münch | Optimal location of the support of the control for the 1-D wave equation: numerical investigations[END_REF] based on numerical observations. The second problem [START_REF] Henrot | Variation et optimisation de formes[END_REF], consisting of minimizing the norm of the HUM operator, is investigated in Section 3. First, in Section 3.1 we show that this problem can be reduced to the problem of maximizing the observability constant in the observability inequality [START_REF] Bucur | Variational methods in shape optimization problems[END_REF], over all possible subsets of [0, π] of Lebesgue measure Lπ. This problem was investigated and solved in [START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF], and we briefly report on the results that can be derived, obtaining in particular the following result.

Theorem. Assume that T is an integer multiple of 2π. Then the optimal value of ( 9) is 2 LT , and is reached if and only if L = 1/2. Theorems 5 and 6 provide precise results and show that the natural modal truncation of the second problem has a unique solution sharing particular features such as the spillover phenomenon. Section 4 contains a conclusion and some open problems. An appendix is devoted to the proofs of the results.

Optimal location of controllers for fixed initial data

This section is devoted to solving the first problem, that is the problem of minimizing (8) on U L for fixed initial data. Our objective is first to write the functional (8) in a more suitable way for the problem to be further interpreted in terms of an optimal control problem. For that purpose, we use a series expansion of φ in the Hilbertian basis of eigenfunctions of the Dirichlet Laplacian. This choice is also motivated by the fact that, in Section 2.6, we will take advantage of this optimal control formulation to derive efficient numerical methods in order to compute numerically the optimal domains.

Every solution φ ∈ C 0 (0, 2π; L 2 (0, π)) ∩ C 1 (0, 2π; H -1 (0, π)) of ( 4) can be expanded as

φ(t, x) = +∞ j=1
(A j cos(jt) + B j sin(jt)) sin(jx), [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF] where A = (A j ) j∈IN * and B = (B j ) j∈IN * belong to ℓ 2 (IR). By the way, note that

(φ 0 , φ 1 ) 2 L 2 ×H -1 = π 2 +∞ j=1 (A 2 j + B 2 j ). (11)
Let (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π) be fixed initial data. The functional [START_REF] Cox | The rate at which energy decays in a damped string[END_REF] to be minimized in the HUM method, still denoted J ω , is

J ω (A, B) = Q ω (A, B) - +∞ j=1 jB j sin(j•), y 0 L 2 + +∞ j=1 A j sin(j•), y 1 L 2 , (12)
where

Q ω (A, B) = 1 2 T 0 ω   +∞ j=1 (A j cos(jt) + B j sin(jt)) sin(jx)   2 dxdt. (13)
In [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF] the notation sin(j•) stands for the function x → sin(jx). Note that both series in [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF] converge, in particular the first one, since this term is the duality product φ 1 , y 0

H -1 ,H 1 0 , with φ 1 ∈ H -1 (0, π) and y 0 ∈ H 1 0 (0, π).
It can be also seen from the Cauchy-Schwarz inequality, noting the fact that the series of general terms j sin(j•), y 0 L 2 and sin(j•), y 1 L 2 belong to ℓ 2 (IR) because y 0 ∈ H 1 0 (0, π) and y 1 ∈ L 2 (0, π), and moreover,

y 0 2 H 1 0 (0,π) = 2 π +∞ j=1 j 2 sin(j•), y 0 2 L 2 , y 1 2 L 2 (0,π) = 2 π +∞ j=1 sin(j•), y 1 2 L 2 . (14)

Relaxation procedure, existence and uniqueness result

In this section we provide an existence and uniqueness result for a relaxed version of the first optimization problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF]. Recall that the HUM control h ω is determined by [START_REF] Fourer | AMPL: A modeling language for mathematical programming[END_REF], i.e.,

h ω (t, x) = χ ω (x)φ χω (t, x),
for almost all (t, x) ∈ (0, T ) × (0, π), where φ is the solution of (4) with initial data (φ 0 , φ 1 ) that is the unique minimizer of the HUM functional J ω defined by [START_REF] Cox | The rate at which energy decays in a damped string[END_REF] or equivalently by [START_REF] Lions | Exact controllability, stabilizability and perturbations for distributed systems[END_REF]. The first problem (8) consists of minimizing the functional F defined by

F (χ ω ) = T 0 π 0 χ ω (x)φ χω (t, x) 2 dxdt, (15)
over the set U L , where φ χω is the adjoint state associated with the HUM control, that is

φ χω (t, x) = +∞ j=1
A ω j cos(jt) + B ω j sin(jt) sin(jx), where (A ω , B ω ) is the unique minimizer of J ω over (ℓ 2 (IR)) 2 . Since a minimizer of F over U L defined by [START_REF] Cox | The rate at which energy decays in a damped string[END_REF] does not necessarily exist, we carry out a relaxation procedure as in [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF], consisting of convexifying the problem. This procedure is very usual in shape optimization problems (see e.g. [START_REF] Bucur | Variational methods in shape optimization problems[END_REF]). Here, the closure of U L for the weak star topology of L ∞ is the convex set

U L = {a ∈ L ∞ ([0, π], [0, 1]) | π 0 a(x) dx = Lπ}. (16)
The relaxed version of the first problem (8) then consists of minimizing the functional F defined by

F (a) = T 0 π 0 a(x)φ a (t, x) 2 dxdt, (17)
over the set U L , where φ a is the adjoint state solution of ( 4) with (φ a (0, •), ∂ t φ a (0, •)) = (φ 0 a , φ 1 a ), where (φ 0 a , φ 1 a ) is the minimizer of the "relaxed" HUM functional J a defined by

J a (φ 0 , φ 1 ) = 1 2 T 0 π 0 a(x)φ(t, x) 2 dxdt + φ 0 , y 1 L 2 -φ 1 , y 0 H -1 ,H 1 0 , (18)
for every (φ 0 , φ 1 ) ∈ L 2 (0, π) × H -1 (0, π) and every a ∈ U L (this is the natural relaxation of ( 5)), where φ is the adjoint state solution of ( 4) with (φ(0, •), ∂ t φ(0, •)) = (φ 0 , φ 1 ). In this relaxed version of the first problem, the characteristic function of ω is replaced with a density function a. Note that the Hilbert Uniqueness Method can be easily extended to this relaxed case as underlined in the following proposition.

Proposition 1. Let a ∈ U L and T 2π. The functional J a defined by (18) has a unique minimizer

(A a , B a ) ∈ (ℓ 2 (IR)) 2 .
The proof of this proposition is done in Section A.2. This relaxation procedure permits to ensure existence and uniqueness results, as stated in the next theorem whose first part (existence) is the main result of [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF] as already mentioned in the introduction.

Theorem 1. The relaxed first problem of minimizing the functional F defined by (17) over the set U L defined by (16) has a unique solution.

Proof. It follows from the HUM method (minimization of a quadratic functional) that, for every a ∈ U L ,

- 1 2 F (a) = - 1 2 T 0 π 0 a(x)φ 2 a (t, x) dxdt = 1 2 φ 0 a , y 1 L 2 -φ 1 a , y 0 H -1 ,H 1 0 = 1 2 T 0 π 0 a(x)φ 2 a (t, x) dxdt + φ 0 a , y 1 L 2 -φ 1 a , y 0 H -1 ,H 1 0 = min (A,B)∈(ℓ 2 (IR)) 2 J a (A, B).
Therefore, J is convex (as the opposite of the minimum of affine functions) and lower semicontinuous. The existence of a minimizer follows since U L is compact for the L ∞ weak star topology. The uniqueness is due to the fact that F is actually strictly convex (see Lemma 4 of Section 2.2 further).

Remark 1. If the minimizer a ∈ U L of the theorem actually belongs to the set U L and thus is the characteristic function of a subset ω of [0, π] of Lebesgue measure Lπ, then this set ω is a solution of the (initial, non relaxed) first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF]. Otherwise it means that the (initial) first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] has no solution and we speak of a relaxation phenomenon. In what follows we will characterize all initial data for which such a relaxation phenomenon occurs.

In the proof of Theorem 1, we only use the lower semicontinuity of F , but note that F is actually continuous for the weak star topology of L ∞ , as mentioned in [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF].

Lemma 1. The map a → F (a) defined on U L is continuous for the L ∞ weak star topology.

In the sequel, we will prove that there exist some initial data (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π) for which a relaxation phenomenon occurs, in other words, the unique solution of the problem of minimizing F defined by [START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF] belongs to U L \ U L . Thus, we will be led to consider a sequence (χ ωn ) n∈IN of U L such that F (χ ωn ) → F (a) as n → +∞, that we will call a minimizing sequence. The following question is then natural: denote by (y n ) n∈IN a sequence of solutions of system (1) associated with the controls

h ωn (x, t) = χ ωn (x)φ χω n (t, x),
where φ χω n denotes the adjoint state provided by the Hilbert Uniqueness Method (with the fixed initial data (y 0 , y 1 )) and control domain ω n . Denote also by y a the solution of (1) associated with the control h a (x, t) = a(x)φ a (t, x). Does the sequence (y n ) n∈IN converge in some sense to y a ? The following proposition provides a positive answer to this question.

Proposition 2. The sequence (φ χω n ) n∈IN converges to φ a strongly in L 2 (0, T ; H -1 (0, π)) and weakly in L 2 (0, T ; L 2 (0, π))∩H 1 (0, T ; H -1 (0, π)), and the sequence (y n ) n∈IN converges to y a strongly in L 2 (0, T ; L 2 (0, π)) and weakly in L 2 (0, T ; H 1 0 (0, π)) ∩ H 1 (0, T ; L 2 (0, π)). The proof of this proposition is done in Section A.3.

Further comments on the relaxed control problem

Recall that

F (a) = T 0 π 0 a(x)φ a (t, x) 2 dxdt, (19)
with

φ a (t, x) = +∞ j=1 A a j cos(jt) + B a j sin(jt) sin(jx), (20)
where (A a , B a ) ∈ (ℓ 2 (IR)) 2 is the minimizer of the quadratic functional J a defined by [START_REF] Padula | Optimization strategies for sensor and actuator placement[END_REF]. Since J a is convex, the first order optimality conditions for the problem of minimizing J a over (ℓ 2 (IR)) 2 are necessary and sufficient. They are written as

Λ a (A, B) = C, (21)
where

Λ a : (ℓ 2 (IR)) 2 → (ℓ 2 (IR)) 2 is defined by Λ a (A, B) j = T 0 π 0 a(x) +∞ k=1 (A k cos(kt) + B k sin(kt)) sin(kx) sin(jx) cos(jt) sin(jt) dxdt = +∞ k=1 T 0 (A k cos(kt) + B k sin(kt)) π 0 a(x) sin(kx) sin(jx)dx cos(jt) sin(jt) dt for every j ∈ IN * , with the notation Λ a (A, B) = (Λ a (A, B) j ) j∈IN * and Λ a (A, B) j ∈ IR 2
, and where

C j = -sin(j•), y 1 L 2 ,L 2 j sin(j•), y 0 H -1 ,H 1 0 ( 22
)
for every j ∈ IN * . The fact that Λ a (A, B) ∈ (ℓ 2 (IR)) 2 for every (A, B) ∈ (ℓ 2 (IR)) 2 can be seen as follows. There holds

Λ a (A, B), (A, B) (ℓ 2 (IR)) 2 = T 0 π 0 a(x) +∞ k=1 (A k cos(kt) + B k sin(kt)) sin(kx) sin(jx) 2 dxdt,
for every (A, B) ∈ (ℓ 2 (IR)) 2 . Using Lemmas 6 and 7, we get

T 2π (πL -sin(πL)) 2 Λ a (A, B), (A, B) (ℓ 2 (IR)) 2 (A, B) 2 (ℓ 2 (IR)) 2 T 2π + 1 (πL + sin(πL)) 2 , (23)
which ensures that Λ a (•), • (ℓ 2 (IR)) 2 is a coercive quadratic form in (ℓ 2 (IR)) 2 . By duality, we immediately deduce the following lemma.

Lemma 2. The operator Λ a is a continuous invertible (and symmetric) operator from

(ℓ 2 (IR)) 2 to (ℓ 2 (IR)) 2 .
Note that Λ a is nothing else but the frequential representation of the Gramian of the relaxed Hilbert Uniqueness Method.

As a result, denoting by (A a , B a ) the unique solution of ( 21), one gets

F (a) = 1 2 Λ a (A a , B a ), (A a , B a ) (ℓ 2 (IR)) 2 = 1 2 C, Λ -1 a (C) (ℓ 2 (IR)) 2 .
An interesting consequence of this expression is that it becomes easier to compute the derivative of F in an admissible direction h. We have indeed the following result. For every a ∈ U L , we denote by T a,U L the tangent cone to the set U L at a, that is the set of functions h ∈ L ∞ (0, π) such that, for any sequence of positive real numbers ε n decreasing to 0, there exists a sequence of functions h n ∈ L ∞ (0, π) converging to h as n → +∞, and a + ε n h n ∈ U L for every n ∈ IN (see for instance [9, chapter 7]).

Lemma 3. Let a ∈ U L and h ∈ T a,U L . The function F is two times Fréchet-differentiable at a in the direction h and one has

dF (a).h = - 1 2 Λ h (A a , B a ), (A a , B a ) (ℓ 2 (IR)) 2 , ( 24
)
d 2 F (a).(h, h) = Λ -1 a (Λ h (A a , B a )), (Λ h (A a , B a )) (ℓ 2 (IR)) 2 . ( 25
)
Proof. Note that the mapping a ∈ U L → Λ a ∈ L((ℓ 2 (IR)) 2 , (ℓ 2 (IR)) 2 ) is linear and hence obviously differentiable. Since the quadratic form Λ a (•),

• (ℓ 2 (IR)) 2 is coercive in (ℓ 2 (IR)) 2
, with a constant that is uniform with respect to a, it follows that the mapping a → Λ -1 a ∈ L((ℓ 2 (IR)) 2 , (ℓ 2 (IR)) 2 ) is differentiable. By composition of differentiable functions, F is differentiable with respect to a, and one has

dF (a), h = - 1 2 Λ -1 a Λ h Λ -1 a (C), C (ℓ 2 (IR)) 2 = - 1 2 Λ h (A a , B a ), (A a , B a ) (ℓ 2 (IR)) 2 = - 1 2 π 0 h(x) T 0 φ a (t, x) 2 dtdx, for every h ∈ L ∞ (0, π).
The second derivative is easily obtained as well from a similar computation. This computation will be important to apply the Pontryagin Maximum Principle in the sequel. The following lemma is a direct consequence of all previous remarks. Lemma 4. The function F is strictly convex on U L .

Proof. Using Lemmas 6 and 7 stated in Section A.1, we get, for every (A, B)

∈ (ℓ 2 (IR)) 2 , Λ a (A, B), (A, B) (ℓ 2 (IR)) 2 T 2π + 1 (πL + sin(πL)) 2 (A, B) 2 (ℓ 2 (IR)) 2 ,
and hence it follows that Λ -1 a is coercive as well in (ℓ 2 (IR)) 2 . Thus, using Lemma 3, the Hessian of F is coercive and hence F is strictly convex.

Interpretation in terms of optimal control

We now give an interpretation of the relaxed first problem [START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF] in terms of a classical optimal control problem, to which we will apply the Pontryagin Maximum Principle. For that purpose, functions a of U L are now considered as controls, and we consider the control system

z ′ 0 (x) = a(x), z ′ j,k (x) = a(x) sin(jx) sin(kx), (j, k) ∈ (IN * ) 2 , (26)
for almost every x ∈ π, with initial conditions

z 0 (0) = 0, z j,k (0) = 0, (j, k) ∈ (IN * ) 2 . ( 27
)
Note the important change of point of view in this presentation. Indeed, the spatial profile of the control of the wave equation becomes a dynamical control of an infinite dimensional differential system where the space variable of the wave equation becomes a pseudo-time.

The relaxed first problem of minimizing ( 17) over U L is then equivalent to the optimal control problem of determining a control a ∈ U L steering the infinite dimensional control system (26) from the initial conditions [START_REF] Trélat | Optimal control and applications to aerospace: some results and challenges, to appear in[END_REF] to the final condition

z 0 (π) = Lπ, (28)
and minimizing the functional F defined by [START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF], written here as

F (a) = 1 2 C, Λ -1 a (C) (ℓ 2 (IR)) 2 , ( 29
)
where Λ a : (ℓ 2 (IR)) 2 → (ℓ 2 (IR)) 2 is the mapping defined as before by Λ a (A, B) = (Λ a (A, B) j ) j∈IN * and Λ a (A, B) j ∈ IR 2 , with

Λ a (A, B) j = T 0 +∞ k=1 (A k cos(kt) + B k sin(kt))z j,k (π) cos(jt) sin(jt) dt,
for every j ∈ IN * . This change of point of view happens to be relevant to solve the problem, both theoretically and numerically. For the theoretical part, we will next see that the application of the Pontryagin Maximum Principle to this optimal control problem leads to a complete characterization of minimizers and of all cases for which the relaxation phenomenon occurs. The optimal control point of view will also permit to derive time-efficient algorithms of numerical computation. Versions of the Pontryagin Maximum Principle in the infinite dimensional setting can be found e.g. in [START_REF] Li | Necessary conditions for optimal control of distributed parameter systems[END_REF]. These versions suffer however from two severe limitations: the first of which is that the functional state space is required to be a strictly convex Banach space, and the second is that the final state must satisfy a finite codimension assumption. In our case here, the second requirement is obviously fulfilled, however the first one is not satisfied a priori. We are however able to find an appropriate equivalent formulation of our problem, which is suitable for the application of the Pontryagin Maximum Principle. This leads to the following result for the unique solution a ∈ U L of the relaxed first problem (stated in Theorem 1), interpreted here as well as the unique solution of the optimal control problem ( 26)-( 27)-( 28)- [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF]. Proposition 3. Let a ∈ U L be the (unique) solution of the relaxed first problem (as stated in Theorem 1). Then there exists a real number p 0 such that

a(x) = 1 if ϕ(x) > 0, 0 if ϕ(x) < 0, (30)
for almost every x ∈ [0, π], where the function ϕ, called switching function, is defined by

ϕ(x) = p 0 + +∞ j,k=1 p j,k sin(jx) sin(kx). ( 31
)
and

p j,k = - T 0 (A a j cos(jt) + B a j sin(jt))(A a k cos(kt) + B a k sin(kt))dt, ( 32
)
for every (j, k) ∈ (IN * ) 2 .
The proof of Proposition 3 is given in Section A.4.

Remark 2. Note that the switching function ϕ defined by (31) can be as well written as

ϕ(x) = p 0 - T 0 φ a (t, x) 2 dt. (33)
Since (A a , B a ) ∈ (ℓ 2 (IR)) 2 , it follows easily from (32) (or directly from ( 20) and (33)) that (p j,k ) j∈IN * ∈ ℓ 1 (IR). Therefore the switching function ϕ is continuous on [0, π].

Remark 3. If the switching function ϕ vanishes identically on a subset of positive measure then the value of a(x) remains undetermined on this subset (this is a usual fact in the application of the Pontryagin Maximum Principle in optimal control). If this situation does not occur then a is completely determined by (30), and in this case, it can be noted that, due to the strict convexity of F proved in Lemma 4, the first-order necessary conditions (30)-( 31)-(32) inferred from the Pontryagin Maximum Principle are also sufficient.

At this step, we realize that the application of the Pontryagin Maximum Principle leads to a simple characterization of the minimizer a of Theorem 1 in terms of the switching function ϕ defined by (31) (or (33)). If ϕ does not vanish identically on any subset of [0, π] of positive measure then the undetermined case does not occur, the optimal control a (which is said bang-bang) is completely determined by (30), and in that case a is the characteristic function of some subset ω, which is the (unique) optimal solution of the (non relaxed) first problem [START_REF] Morris | Linear-quadratic optimal actuator location[END_REF]. In the next subsection we provide simple sufficient conditions on the coefficients implying this conclusion. At the opposite, if the switching function vanishes identically on some subset of [0, π] of positive measure (degenerate case), then the optimal control a is not bang-bang and thus is not a characteristic function. This means that in that case the (non relaxed) first problem [START_REF] Morris | Linear-quadratic optimal actuator location[END_REF] does not admit any optimal solution. This is what we call a relaxation phenomenon. In Section 2.4.2 we characterize all possible initial data for which the relaxation phenomenon occurs.

Sufficient conditions for the first problem (8)

In this section we prove that, under suitable assumptions on the regularity of the initial data (y 0 , y 1 ), the first problem has a unique solution. The main idea is to prove that if the initial data are regular enough then the (unique) minimizer of the relaxed first problem (17) characterized previously is the characteristic function of a subset ω and therefore ω is the solution of the initial (non relaxed) first problem (8).

A first sufficient condition

Theorem 2. Assume that the initial data y 0 ∈ H1 0 (0, π) and y 1 ∈ L 2 (0, π) satisfy the following property: there exists N ∈ IN * such that π 0 y 0 (x) sin(jx)dx = π 0 y 1 (x) sin(jx)dx = 0, for every j > N . Then the first problem (8) has a unique 1 solution χ ω , where ω is a measurable subset of [0, π] of Lebesgue measure Lπ. Moreover,

• ω has at most N connected components.

• there exists η > 0 such that ω ⊂ [η, π -η].
Proof. Since y 0 and y 1 have only a finite number of nonzero components in their Fourier series, it follows that C j = 0 for j > N , where C j is defined by [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]. Then, clearly, the unique solution (A a , B a ) of ( 21) is as well of compact support. Hence, only a finite number of the coefficients p j,k are nonzero. Therefore, the function ϕ is analytic and hence cannot vanish identically on a subset of positive measure. Then the optimal control a is bang-bang, determined by (30), and is the characteristic function of some subset ω of [0, π]. Moreover, the switching function ϕ(x) is written as

ϕ(x) = p 0 + N j,k=1 p j,k sin(jx) sin(kx) = p 0 + 1 2 N -1 u=1-N   u+N j=u+1 p j,j-u   cos(ux) - 1 2 2N u=2   u-1 j=u-N p j,u-j   cos(ux),
for every x ∈ [0, π]. Since ϕ can be written as a linear combination of the 2N first Tchebychev polynomials, it follows that ϕ has at most 2N zeros. Finally, since ϕ(0) = ϕ(π) = p 0 and ϕ(x) p 0 for every x ∈ [0, π], the existence of η > 0 follows easily.

A natural question arises when considering modal approximations (y 0,N , y 1,N ) of the initial data (y 0 , y 1 ), defined by

y 0,N (x) y 1,N (x) = N k=1 y 0 , sin(j•) L 2 y 1 , sin(j•) L 2 sin(jx).
Consider the optimal set ω N ∈ U L solution of the first problem with the initial data (y 0,N , y 1,N ) (as claimed in Theorem 2). Let a ∈ U L be the solution of the relaxed first problem with the initial data (y 0 , y 1 ). According to Proposition 3, let p N 0 and p N j,k , for (j, k) ∈ (IN * ) 2 , the coefficients of the switching function ϕ N for the truncated optimal control problem ( 26)-( 27)-( 28)-( 29) with the initial data (y 0,N , y 1,N ). Note that p N j,k = 0 for j > N or k > N . The following Γ-convergence result follows easily from the previous optimal control considerations. Proposition 4. The sequence of finite dimensional optimal control problems (26)-( 27)-( 28)- [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF] with truncated initial data (y 0,N , y 1,N ) Γ-converges to the infinite dimensional optimal control problem (26)-( 27)-( 28)- [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF] with initial data (y 0 , y 1 ), in the sense that, when N tends to +∞, χ ω N converges to a for the weak star topology of L ∞ , p N 0 converges to p 0 , p N j,k converges to p j,k for every (j, k) ∈ (IN * ) 2 , and F N (χ ω N ) converges to F (a).

Sufficient condition when T is an integer multiple of 2π

Throughout this section we assume that T = 2π, the case T = 2pπ with p ∈ IN * being obviously deduced from this case. Note that T = 2π is a very particular time since the orthogonality property leads to a strong simplification of the expression of the norm of the control as explained next. Let us expand φ(t, x) as in [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF]. Then the "relaxed" HUM functional defined by ( 18) is given by

J a (φ 0 , φ 1 ) = +∞ j=1 π 2 (A 2 j + B 2 j ) π 0 a(x) sin 2 (jx) dx -A j π 0 y 1 (x) sin(jx) dx + jB j π 0 y 0 (x) sin(jx) dx ,
and for given initial data (y 0 , y 1 ), the minimization of J a obviously leads to

A a j = 1 π π 0 y 1 (x) sin(jx) dx π 0 a(x) sin 2 (jx) dx
, and B a j = -

j π π 0 y 0 (x) sin(jx) dx π 0 a(x) sin 2 (jx) dx . ( 34
)
Denote

ρ j = π 0 y 1 (x) sin(jx) dx 2 + j π 0 y 0 (x) sin(jx) dx 2 1/2 , ( 35
)
for every j ∈ IN * . Note that, since (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π), it follows that the sequence (ρ j ) j∈IN * belongs to ℓ 2 (IR).

Using these particular and more precise expressions for T = 2π, we immediately derive the following generalization of Theorem 2.

Theorem 3. Assume that T = 2π, and that there exist M > 0 and δ > 0 such that

|ρ j | M e -δj , (36)
for every j ∈ IN * . Then the first problem (15) has a unique solution χ ω , where ω is a measurable subset of [0, π] of Lebesgue measure Lπ. Moreover,

• ω is symmetric with respect to π/2,

• there exists η > 0 such that ω ⊂ [η, π -η],

• ω has a finite number of connected components.

Remark 4.

The assumption (36) holds for instance as soon as y 0 and y 1 are analytic functions on [0, π]. Actually, the assumption (36) guarantees the analyticity of the function ϕ, and can be slightly weakened into

|ρ j | M e -w(j) ,
for every j ∈ IN * , where w is a positive differentiable function such that t → tw ′ (t) increases to +∞ and such that +∞ 1 w(t) t 2 dt = +∞. Indeed, under this weakened assumption the function ϕ enjoys the following unique continuation property (see [START_REF] Mandelbrojt | Quasi-analycité des séries de Fourier[END_REF]): if ϕ is constant on a subset of [0, π] of positive measure then ϕ is constant on [0, π]. Hence, the statement of the theorem holds under this slightened property. More generally this property is related to quasi-analyticity but up to our knowkedge there is no simple necessary and sufficient condition on the coefficients α j ensuring that unique continuation property.

Remark 5. Given a sequence (ρ j ) j∈IN * ∈ ℓ 2 (IR) there exist an infinite number of possible initial data (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π) satisfying ( 35), and all of them lead to the same solution of the first problem (it is indeed a consequence from the formula (39) in the proof).

Proof of Theorem 3. For this particular case where T = 2π, all previously considered expansions do not involve any crossed terms. In particular, when interpreting the first problem as an optimal control problem, the control system (26) reduces here to

z ′ 0 (x) = a(x), z ′ j (x) = a(x) sin 2 (jx), j ∈ IN * , (37)
for almost every x ∈ π, with initial conditions

z 0 (0) = 0, z j (0) = 0, j ∈ IN * . ( 38
)
Using the Parseval identity, we get

F (a) = 2π 0 π 0 a(x)φ 2 a (t, x) dxdt = π 2 +∞ j=1 (A a j 2 + B a j 2 ) π 0 a(x) sin 2 (jx) dx = 1 2π +∞ j=1 ρ 2 j π 0 a(x) sin 2 (jx) dx = 1 2π +∞ j=1 ρ 2 j z j (π) . (39)
The optimal control problem under consideration consists of steering the control system (37) from the initial conditions (38) to the final condition z 0 (π) = Lπ, minimizing the functional F defined by (39) with controls a ∈ U L . Note that in order to apply the Pontryagin Maximum Principle in a strict convex Banach space one has to use the identity sin 2 (jx) = 1 2 -1 2 cos(2jx) and hence to consider the control system given by

w ′ 0 (x) = a(x), w ′ j (x) = 1 2 a(x) cos(2jx), j ∈ IN * , (40)
with initial conditions

w 0 (0) = 0, w j (0) = 0, j ∈ IN * . ( 41
)
The control system (37) is then inferred from the above one by setting

z 0 (x) = w 0 (x), z j,k (x) = 1 2 (w |j-k| (x) -w j+k (x)), (42)
for almost every x ∈ [0, π] and for all integers j and k.

With these notations, one has

F (a) = 1 2π +∞ j=1 ρ 2 j Lπ 2 -w j (π)
, and, skipping the details, the application of the Pontryagin maximum Principle leads to the characterization of the (unique) optimal control

a(x) = 1 if ϕ(x) > 0, 0 if ϕ(x) < 0, (43)
for almost every x ∈ [0, π], where the switching function ϕ is defined by

ϕ(x) = λ 0 + 1 2 +∞ j=1 λ j cos(2jx), (44)
with

λ j = - 1 2π ρ 2 j Lπ 2 -w j (π) 2 = - 1 2π ρ 2 j π 0 a(x) sin 2 (jx) dx 2 , (45)
for every j ∈ IN * . Note that

ϕ(x) = λ 0 - T 0 φ a (t, x) 2 dt = λ 0 - 1 2π +∞ j=1 ρ 2 j π 0 a(s) sin 2 (js)ds 2 sin 2 (jx), (46)
for every x ∈ [0, π]. Under the assumption (36), it follows from (45) and from Lemma 6 that the decay of the coefficients λ j is exponential, and hence that the switching function ϕ defined by (44) is analytic (see e.g. [1, Chapter 11, §63]) and thus ϕ cannot vanish identically on a subset of positive measure. Therefore, the optimal control a is bang-bang, determined by (43), and thus is the characteristic function of some subset ω of [0, π]. The symmetry property follows from the fact that ϕ(π -x) = ϕ(x) for every x ∈ [0, π]. The existence of η comes from the fact that ϕ(0) = ϕ(π) is the minimum of ϕ (since λ j < 0 for every j ∈ IN * ).

Necessary and sufficient conditions when T is an integer multiple of 2π

Theorems 2 and 3 stated previously provide sufficient conditions for the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] to have a (unique) solution. In the case where T is an integer multiple of 2π, we are actually able to provide a complete characterization (however not so tractable in practice) of all initial data for which this problem has a (unique) solution, that is, for which the optimal control solution of the relaxed first problem is actually a solution of the non relaxed one. This characterization is based on the fact that the necessary conditions for optimality are also sufficient because of the strict convexity of F on U L , as already mentioned above. The following result is a particular consequence of the more precise considerations further.

Theorem 4. Assume that T is an integer multiple of 2π and let L ∈ (0, 1) be fixed.

1. There exist initial data (y 0 , y 1 ) ∈ (C ∞ (0, π)) 2 for which the first problem (8) has a solution χ ω ∈ U L , with ω a fractal domain of Cantor type.

2. There exist initial data (y 0 , y 1 ) ∈ (C ∞ (0, π)) 2 for which the first problem (8) has no solution, but its relaxed version has a solution a ∈ U L \ U L .

Let us assume without loss of generality that T = 2π. Theorem 4 is a direct consequence of the following characterization of all initial data for which either the first problem has a solution or it has no solution (relaxation phenomenon).

Note first that (λ j ) j∈IN * ∈ ℓ 1 (IR), where λ j is defined by (45). Therefore, more precisely than in Remark 2, the switching function ϕ defined by (44) belongs to the set A(0, π) of all functions φ integrable on [0, π] and having a Fourier series expansion of the form

φ(x) = a 0 + 1 2 +∞ j=1 a j cos(2jx), (47)
with a 0 ∈ IR and nonpositive coefficients a j satisfying (a j ) j∈IN * ∈ ℓ 1 (IR). Such considerations are classical in harmonic analysis. In particular, ϕ is continuous, but the above property is stronger.

We are now in a position to derive the following characterizations, mainly based on the fact that the optimal solution a of the above optimal control problem is the solution of the initial first problem [START_REF] Morris | Linear-quadratic optimal actuator location[END_REF] if and only if the switching function ϕ defined by (44) does not vanish identically on any subset of [0, π] of positive measure.

Characterization of all initial data for which the first problem has a solution. Consider any function ϕ ∈ A(0, π) such that ϕ(x) = 0 for almost every x ∈ [0, π], written as

ϕ(x) = λ 0 + 1 2 +∞ j=1 λ j cos(2jx),
with nonpositive coefficients λ j satisfying (λ j ) j∈IN * ∈ ℓ 1 (IR). Let a be the bang-bang control defined by (30), and let ρ j 0 be defined by (45) for every j ∈ IN * , that is,

ρ j = -2πλ j π 0 a(x) sin 2 (jx) dx.
Using Lemma 6, it is clear that (ρ j ) j∈IN * ∈ ℓ 2 (IR). Such a sequence (ρ j ) j∈IN * characterizes a set of initial data (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π) (see Remark 5): these are all initial data for which (35) holds for every j ∈ IN * . The control a, defined in such a way, satisfies all necessary and sufficient conditions of the Pontryagin Maximum Principle and hence is the optimal control of the first problem associated with the initial data (y 0 , y 1 ), and with the constant L defined by π 0 a(x) dx = Lπ (note that L may be equal to 0 or 1, depending on the value of λ 0 , but these cases are of course not interesting). Since a is bang-bang, it is the characteristic function of a subset ω of [0, π], which is the solution of the first problem. Note that, in order to realize a specific value of L, it suffices to tune adequately the constant λ 0 of the Fourier expansion of ϕ so that π 0 a(x) dx = Lπ.

The construction above establishes a correspondence between the optimal sets of the first problem and the zero level sets of functions of A(0, π) that do not vanish almost everywhere. Such level sets may have an intricate structure, for instance may have a fractal structure. We refer to [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF] for the construction of an explicit example where ω is of Cantor type.

We are able as well to characterize precisely the relaxation phenomenon.

Characterization of the relaxation phenomenon. The data (y 0 , y 1 ) for which the relaxation phenomenon occurs are those whose associated coefficients ρ j defined by (35) are such that the corresponding switching function ϕ vanishes identically on a subset I of positive measure (note that this set must be symmetric with respect to π 2 ). The more precise construction goes as follows. Consider any nontrivial function ϕ ∈ A(0, π) vanishing identically on a subset I of [0, π] of positive measure, with Fourier coefficients (λ j ) j∈IN * as before. Let a be a control function defined by (30) on [0, π] \ I and taking arbitrary values on I, and satisfying however π 0 a(x) dx = Lπ. Let (ρ j ) j∈IN * ∈ ℓ2 (IR) be defined by (45), as previously. Such a sequence characterizes a set of initial data (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π). As before the control a defined in such a way is the optimal control of the relaxed version of the first problem associated with the initial data (y 0 , y 1 ). Since a is not bang-bang (it is not a characteristic function on I, by construction), this means that the first problem does not have any solution and that the relaxation phenomenon occurs. Remark 6. This set is indeed non empty, and is even, in some sense, a very large set. To get convinced, it suffices to note the following fact. Consider any nontrivial function ψ of class C ∞ on [0, π], symmetric with respect to π 2 , and whose support is contained in [ π 2 -α, π 2 + α] for some α > 0 small. The C 1 regularity ensures that its Fourier coefficients are summable. To ensure the nonpositivity of its Fourier coefficients, it suffices to consider the function ϕ defined by the convolution ϕ(x)

= IR ψ(y + π 2 )ψ(x -y)dy.
Indeed, the function ϕ defined in such a way is of class

C ∞ on [0, π], of support contained in [ π 2 -2α, π 2 + 2α]
, and all its Fourier coefficients are nonpositive (by the way note that all functions whose Fourier coefficients are nonpositive are of this type).

Another explicit example is easily built as follows. Consider a triangle function 2 defined on [ π 2 -α, π 2 + α], for some α > 0, equal to 0 outside. Its Fourier coefficients are the values on integers of the Fourier transform of the triangle function, hence are positive and summable. The rest of the construction is obvious.

To illustrate this correspondence, we yield a concrete example of construction of initial data for which we observe a relaxation phenomenon.

Example 1. We set T = 2π (for example). Consider the π-periodic switching function φ, symmetric with respect to π/2 defined by

φ(x) = -4 π x + 1 on [0, π/4] 0 on [π/4, π/2].
Thus, φ has a Fourier series expansion of the form (47) with a 0 = 1 4 and a j = 4(1-(-1) j ) j 2 0 for every j 1. Introduce now the function a(•), π-periodic and symmetric with respect to π/2, such that for every x ∈ [0, π],

a(x) = 0 on [0, π/4] 2L + 8L π x -π 2 on [π/4, π/2].
Using the correspondence detailed before, we obtain that a is the unique solution of Problem [START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF], provided that the initial data (y 0 , y 1 ) are such that the coefficients (ρ j ) j∈IN * defined by (35) verify

ρ j = -2πa j π 0 a(x) sin 2 (jx) dx for every j ∈ IN * .
Remark 7. It is not clear whether the above characterizations can be generalized or not to the case where T is not an integer multiple of 2π, in particular because the relation between the coefficients (p j,k ) j∈IN * of the switching function and the coefficients A a j , B a j is not easily invertible (note however that the infinite dimensional symmetric matrix whose coefficients are the p j,k is a Gramian).

Modal approximation and numerical simulations

Case T = 2π

We propose in this section some numerical simulations in the case where T = 2π. The method we use here works exactly in the same way in the case where T 2π is not an integer multiple of 2π except that it is then required to invert the gramian operator Λ a defined in Section 2.3. In view of numerical simulations, it is natural to truncate the control system (40) at some order N ∈ IN * , by considering the optimal control problem of steering the finite dimensional control system

w ′ 0 (x) = a(x), w ′ j (x) = 1 2 a(x) cos(2jx), j ∈ {1, . . . , N }, (48)
from the initial conditions

w 0 (0) = 0, w j (0) = 0, j ∈ {1, . . . , N } (49)
to the final condition (28) (y(π) = Lπ), and minimizing

F N (a) = 1 2π N j=1 ρ 2 j Lπ 2 -w j (π) . ( 50
)
As in Theorem 1, this optimal control problem has a unique solution a N ∈ U L . The usual Pontryagin Maximum Principle of [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF], applied to this finite dimensional optimal control problem, implies that there exist λ 0N 0 and

(λ N 0 , λ N 1 , . . . , λ N N ) ∈ IR N +1 , called costates 3 , with (λ 0 , λ N 0 , λ N 1 , . . . , λ N N ) = (0, . . . , 0), such that a N (x) = 1 if ϕ N (x) > 0, 0 if ϕ N (x) < 0, ( 51
)
for almost every x ∈ [0, π], where the function ϕ N , called switching function, is defined by

ϕ N (x) = λ N 0 + 1 2 N j=1 λ N j cos(2jx), (52)
the control a N being undetermined whenever φ N vanishes identically on some subinterval. Moreover, one has the transversality assumptions

λ N j = λ 0N ρ 2 j Lπ 2 -z N j (π) 2 , (53)
for every j ∈ {1, . . . , N }. As before, there must hold λ 0N = 0 and we assume next that λ 0N = -1.

Since ϕ N can be written as a linear combination of the 2N first Tchebychev polynomials, it follows that ϕ N has at most 2N zeros and cannot vanish identically on any subinterval. We finally get the following result.

Proposition 5. The optimal control a N is the characteristic function of a subset ω N of [0, π] of Lebesgue measure Lπ, which has at most N connected components, is symmetric with respect to π/2, and is such that there exists η > 0 such that ω

N ⊂ [η, π -η].
We provide hereafter several numerical simulations based on this modal approximation. This new numerical approach is based on the interpretation as an optimal control problem. Some results are provided on Figure 1. These simulations were obtained with a direct method applied to the optimal control problem described previously (see e.g. [START_REF] Trélat | Optimal control] Théorie & Applications[END_REF] for a description of possible numerical approaches), consisting in discretizing the underlying differential equations, the control, and to reduce the optimal control problem to some minimization problem with constraints. We used the code IPOPT (see [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF]) combined with AMPL (see [START_REF] Fourer | AMPL: A modeling language for mathematical programming[END_REF]) on a standard desktop machine. The resulting code works out the solution very quickly (for instance, within 3 seconds for N = 50) and is far more efficient than methods based on gradient methods using topological derivatives. It must however be noted that the problem is one-dimensional and that we benefit of the interpretation as an optimal control problem, using spectral considerations.

Another possible method is to implement a shooting method, based on the optimality conditions provided by the Pontryagin Maximum Principle. Here, it consists of numerically determining the costates (λ N 0 , λ N 1 , . . . , λ N N ) such that ( 28) and (32) are satisfied. The shooting method is a combination of a Newton type method and of some integration method of differential equations.

The main difficulty is the initialization of this method but it can be done using a direct method. We used the code COTCOT (see [START_REF] Bonnard | Second order optimality conditions in the smooth case and applications in optimal control[END_REF]). The convergence is then obtained instantaneously and can permit, when combined with a continuation method, to reach very large values of N (see [START_REF] Trélat | Optimal control and applications to aerospace: some results and challenges, to appear in[END_REF] for a survey on these numerical approaches).

Complexity of the optimal set and limitations of numerics

The graphs presented in the last paragraph represent the optimal set for fixed initial data that have a finite number of Fourier components. It follows from Theorem 4 that numerical simulations cannot permit in general to guess what happens when the initial data have an infinite number of components in their Fourier series expansion. Indeed, note that in [START_REF] Münch | Optimal location of the support of the control for the 1-D wave equation: numerical investigations[END_REF][START_REF] Münch | Optimal distribution of the internal null control for the 1D heat equation[END_REF] the authors conjecture the existence of a relaxation phenomenon with smooth initial data, based on numerical simulations. Although such an intuition seems reasonable, according to our previous theoretical results this is not the case. In Section 2.5 we have shown a precise correspondence between the regularity of the optimal set and the regularity of the initial data. We have also characterized all initial data for which the relaxation phenomenon occurs, leading to Theorem 4. Using Theorem 4, there exist smooth initial data for which the optimal set has a fractal structure, and thus an infinite number of connected components but for which relaxation does not occur. Of course this particular feature cannot be guessed from numerical simulations. To summarize, considering as in the previous section a truncated version of the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] with the N first Fourier components, two different phenomena arise at the limit, depending on the initial data under consideration: a relaxation phenomenon, or the emergence of complex (possibly fractal) optimal sets. In particular the occurence of relaxation cannot be illustrated on numerical simulations.

Additional comments on the case T = 2π: study of the case N = 2

In this section, we study the particular case N = 2 of the modal truncature of the first problem. This means that the only nonzero terms of the sequence (ρ j ) j∈IN * are ρ 1 and ρ 2 . In this case, the functional (50) is written as

F 2 (χ ω ) = ρ 2 1 π 0 χ ω (x) sin 2 x dx + ρ 2 2 π 0 χ ω (x) sin 2 (2x) dx . Setting t = ρ 2 1 ρ 2 1 +ρ 2 2
, the minimization problem reduces to the problem of minimizing

t π 0 χ ω (x) sin 2 x dx + 1 -t π 0 χ ω (x) sin 2 (2x) dx (54)
over the set U L . The following proposition yields a precise characterization of the minimizers in the case N = 2 and describes the variations of the optimal set with respect to L. Proposition 6. Let χ ω 2 be the solution of the minimization problem (54). Then,

• ω 2 is symmetric with respect to π/2;

• the set ω 2 ∩ [0, π/2] has only one connected component α t , α t + πL 2 ; • α 0 = π 4 (1 -L) and α 1 = π 2 (1 -L); • there exists t 0 (L) ∈ (0, 1) such that t → α t is increasing on [0, t 0 (L)], and is constant equal to α 1 on [t 0 (L), 1].
The proof of this proposition, based on lengthy and tedious computations, is provided in Appendix A.5. Figure 2 illustrates this proposition. This section is devoted to solving the second problem [START_REF] Henrot | Variation et optimisation de formes[END_REF], that is the problem of minimizing the norm of the HUM operator over U L .

! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ! !"' # #"' $ $"' % ,-./0123145 $ 16740189!"# (a) L = 0.1 ! !"# !"$ !"% !"& !"' !"( !") !"* !"+ # ! !"' # #"' $ $"' % ,-./0123145 $ 16740189!"& (b) L = 0.4

Reduction of the problem

For every measurable subset ω of [0, π], set

W (χ ω ) = sup h ω 2 L 2 ((0,T )×(0,π)) (y 0 , y 1 ) 2 H 1 0 (0,π)×L 2 (0,π) (y 0 , y 1 ) ∈ H 1 0 (0, π) × L 2 (0, π) \ {(0, 0)}
Recall that the second problem consists of minimizing W (χ ω ) over all possible subsets ω of [0, π] of measure equal to Lπ.

Lemma 5. For every measurable subset ω of [0, π], there holds

W (χ ω ) = 1 C T (χ ω )
,

where C T (χ ω ) is the largest observability constant in the inequality (3), associated with the wave equation (4), that is

C T (χ ω ) = inf T 0 ω φ(t, x) 2 dxdt (φ 0 , φ 1 ) 2 L 2 (0,π)×H -1 (0,π) (φ 0 , φ 1 ) ∈ L 2 (0, π) × H -1 (0, π) \ {(0, 0)} ,
where φ is the solution of (4). Remark 8. Expanding φ in Fourier series as in [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF], it is easy to see that

C T (χ ω ) = inf (A,B)∈(ℓ 2 (IR)) 2 P +∞ j=1 (A 2 j +B 2 j )=1      T 0 ω   +∞ j=1 (A j cos(jt) + B j sin(jt)) sin(jx)   2 dxdt      .
This lemma illustrates the classical duality between controllability and observability (see [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF]), however we provide a short proof hereafter.

Proof. Let us expand φ in Fourier series as in [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF], and use the operator Λ a defined by [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF] to rewrite W (χ ω ). One has

W (χ ω ) = sup C∈(ℓ 2 (IR)) 2 Λ -1 χω (C), C (ℓ 2 (IR)) 2 C 2 (ℓ 2 (IR)) 2 = sup C∈(ℓ 2 (IR)) 2 Λ -1/2 χω (C) 2 (ℓ 2 (IR)) 2 C 2 (ℓ 2 (IR)) 2
, where Λ -1/2 χω denotes the square root of the operator Λ -1 χω . Introducing ϕ = Λ -1/2 χω (C), one computes

W (χ ω ) = sup ϕ∈(ℓ 2 (IR)) 2 ϕ 2 (ℓ 2 (IR)) 2 Λ 1/2 χω (ϕ) 2 (ℓ 2 (IR)) 2 = 1 inf Λ 1/2 χω (ϕ) 2 (ℓ 2 (IR)) 2 ϕ 2 (ℓ 2 (IR)) 2 | ϕ ∈ (ℓ 2 (IR)) 2 = 1 inf Λχ ω (ϕ),ϕ (ℓ 2 (IR)) 2 ϕ 2 (ℓ 2 (IR)) 2 | ϕ ∈ (ℓ 2 (IR)) 2 = 1 C T (χ ω )
.

The lemma is proved.

As a consequence, we have inf

χω∈U L W (χ) = inf χω∈U L 1 C T (χ ω ) = 1 sup χω∈U L C T (χ ω ) ,
and therefore the second problem ( 9) is equivalent to the new optimization problem

sup χω∈U L C T (χ ω ), (55)
that is, the problem of maximizing the observability constant over all possible subsets of [0, π] of Lebesgue measure Lπ. This problem was investigated in [START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF], and solved in the case where T is an integer multiple of 2π (the general case being still open). The problem of maximizing the functional lim

T →+∞ C T (χ ω )/T over U L was also solved.

Case where T is an integer multiple of 2π

In this case, according to the computations of Section 2.4.2, the second problem ( 9) is exactly equivalent to

sup χω∈U L inf j∈IN * ω sin 2 (jx) dx. (56)
This kind of problem was addressed in [START_REF] Hébrard | Optimal shape and position of the actuators for the stabilization of a string[END_REF][START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF], where a different optimal design problem was studied, concerning stabilization issues for the one-dimensional wave equation. More precisely, they consider the damped wave equation

∂ tt y -∂ xx y + 2kχ ω ∂ t y = 0 (t, x) ∈ (0, T ) × (0, π), y(t, 0) = y(t, π) = 0 t ∈ [0, T ], y(0, x) = y 0 (x), ∂ t y(0, x) = y 1 (x) x ∈ [0, π],
where k > 0, and investigate the problem of determining the best possible subset ω of [0, π] maximizing the decay rate of the total energy of the system, among all possible subsets of [0, π] of Lebesgue measure Lπ and having a finite number of connected components (note that this last restriction is discarded in [START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF]). This corresponds to choose in (1) the control h ω = -2kχ ω ∂ t y, which does not coincide with the HUM control. The overdamping phenomenon is underlined in [START_REF] Hébrard | Optimal shape and position of the actuators for the stabilization of a string[END_REF] (see also [START_REF] Cox | The rate at which energy decays in a damped string[END_REF]), meaning that if k is too large then the decay rate tends to zero. It is moreover explained that, if k is small enough then the decay rate is equivalent to k inf j∈IN * ω sin 2 (jx) dx. This explains their motivation to investigate the optimization problem (56).

It is proved in [START_REF] Hébrard | Optimal shape and position of the actuators for the stabilization of a string[END_REF][START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF][START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF] that the optimization problem (56) does not have any solution except for L = 1/2. In other words, the supremum is not reached. In particular, as an immediate consequence of [20, Theorem 1], we have the following result.

Theorem 5. For every L ∈ (0, 1), there holds

inf χω∈U L W (χ ω ) = 2 LT , (57)
and the infimum is reached if and only if L = 1/2. Moreover, if L = 1/2 then the problem has an infinite number of solutions, consisting of all measurable subsets ω ⊂ [0, π] of measure π/2 such that ω and its symmetric ω ′ = π -ω are disjoint and complementary in [0, π].

In particular if L = 1/2 then the second problem (9) does not have any optimal set. Since it is more realistic from an engineering point of view to take into consideration only a finite number of modes, the authors of [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] consider a truncated version of (56) involving only the first N modes, for a given N ∈ IN * , and investigate the optimization problem

sup χω∈U L min 1 j N π 0 χ ω (x) sin 2 (jx) dx. (58)
The following theorem was proved in [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] but with an erroneous proof, corrected in [START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF].

Theorem 6. For every N ∈ IN * , the problem (58) has a unique4 solution χ ω N , where ω N is a subset of [0, π] of measure Lπ that is the union of at most N intervals and is symmetric with respect to π/2. Moreover there exists L N ∈ (0, 1] such that, for every L ∈ (0, L N ], the optimal domain ω N satisfies

ω N sin 2 x dx = ω N sin 2 (2x) dx = • • • = ω N sin 2 (N x) dx. (59)
It is explained in [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] that (59) permits to show that the optimal domain ω N concentrates around the nodes kπ N +1 , k = 1, . . . , N . This implies the well-known spillover phenomenon, according to which the optimal domain ω N solution of (58) with the N first modes is the worst possible domain for the problem with the N + 1 first modes.

Note that, although (56) has no solution, the optimal solution χ ω N of (58) converges for the weak star topology of L ∞ to the function a ∈ U L that is the solution of the relaxed problem max

a∈U L inf j∈IN * π 0 a(x) sin 2 (jx) dx, (60)
as expected, with a convergence of the optimal values.

Conclusion and open problems

In this article we have described a frequential approach based on Fourier series expansions, permitting to derive some existence and uniqueness results and to interpret these problems in terms of optimal control. The use of the Pontryagin Maximum Principle led to a complete characterization of optimal solutions permitting in particular to characterize all initial data for which the first problem has a solution or not. Efficient numerical algorithms have been derived from this approach and numerical simulations were given on a modal approximation of the problem. The second problem was shown to be equivalent to the problem of maximizing the observability constant over the class of subsets under consideration. The optimal value of the second problem was given, although, except for L = 1/2, there does not exist any optimal set. In accordance with this nonexistence result, the modal approximation of that problem leads to the spillover phenomenon.

We end this article indicating several open directions for future investigation.

In the present article we restricted the study of the second problem to the case where T is an integer multiple of 2π. Indeed for more general values of T we do not know how to handle the general expressions of quadratic functional like [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]. The problem happens to be related to the one of determining what are the best constants in Ingham's inequality (see [START_REF] Privat | Optimal observation of the one-dimensional wave equation[END_REF] for detailed comments on this issue).

For every subset ω of [0, π] of positive measure and T 2π, the observability inequality (3) is satisfied. However 2π is not the smallest possible time for a specific choice of ω. For instance if ω is a subinterval of [0, π] then the smallest such time is 2 diam((0, π) \ ω). This question is nontrivial if, instead of an interval, the set ω is chosen to be a fractal measurable set, for example of Cantor kind. To our knowledge, given L ∈ (0, 1), the question of determining the existence or not of a time T L ∈ (0, 2π) such that for every ω of measure Lπ, the system (1) is controllable in any time T T L , is open.

In the same spirit, another interesting possibility consists of maximizing the criterion (χ ω , T ) → inf (y 0 ,y 1 )∈H 1 0 (0,π)×L 2 (0,π)

1 T T 0 π 0 h ω (t, x) 2 (y 0 , y 1 ) 2 H 1 0 ×L 2 dxdt
over the set U L × (0, +∞). The same questions arise when the control domain belongs to a larger class in which it is not necessarily cylindrical but is rather a measurable space-time set of measure 2πL.

For T 2π that is not an integer multiple of 2π, the problem of determining the set of all initial data (y 0 , y 1 ) for which the HUM control h ω is analytic, is open. The answer to this question would permit to make Theorem 2 more precise since the analyticity of the switching function implies the existence of a solution of the first problem [START_REF] Hébrard | A spillover phenomenon in the optimal location of actuators[END_REF] having moreover a finite number of connected components.

In the present article truncations of the functionals were considered with the N first modes. It would be interesting to consider similar optimal design problems for other classes of initial data, for instance initial data whose Fourier coefficients satisfy an uniform exponential decreasing property. We could also truncate the Fourier series and keep only the modes whose index is between two integers N and M .

Our analysis can be extended to the multi-dimensional case, and for wave equations with more general nonconstant coefficients, but such an analysis requires other considerations related to ergodicity features (see [START_REF] Privat | Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains[END_REF]) that are beyond the scope of the present article. Here for the onedimensional wave equation we used many times the fact that the Hilbertian basis of eigenfunctions of the Dirichlet Laplacian consists of sine functions, but our approach may be led with more general eigenfunctions, although it may be more technical and not so precise. For example the derivation of such a precise identity like (59) is probably not easily provable in general.

Finally, it is an open question to investigate the shape and position of the support of the HUM control of the one-dimensional heat equation or more general parabolic systems. This question will be investigated in forthcoming works.

A Proofs of Section 2 A.1 Preliminaries

We start with some elementary lemmas that play an important role. The following very simple lemma, noticed as well in [START_REF] Periago | Optimal shape and position of the support for the internal exact control of a string[END_REF], is useful.

Lemma 6. Let j ∈ IN * .
1. The problem of minimizing the functional

K j (a) = π 0 a(x) sin 2 (jx) dx (61)
over the set U L has a unique solution χ ω inf j ∈ U L that is the characteristic function of the measurable set ω inf j given by

ω inf j = 0, Lπ 2j j-1 k=1 kπ j - Lπ 2j , kπ j + Lπ 2j π - Lπ 2j , π ,
and the value of the minimum is Lπ 2 -sin Lπ 2 > 0.

2. The problem of maximizing the functional K j over the set U L has a unique solution χ ω sup j ∈ U L that is the characteristic function of the measurable set ω sup j given by Remark 9. Note that the extremal values of K j , given in this lemma, do not depend on j.

ω sup j = j k=1 (2k -1)π 2j - Lπ 2j , ( 2k
Proof. It suffices to note that optimal sets exist and are clearly characterized in terms of level sets of the function x → sin 2 (jx), for every j ∈ IN * . Using the symmetry properties of this function and the fact that its minimum on [0, π] is equal to 0 and is reached at every x k = kπ j , k ∈ {0, 1, . . . , j}, the formula on ω inf j follows, and the value of the minimum is given by

ω inf j sin 2 (jx) dx = 2j πL/2j 0 sin 2 jx dx = 2 πL/2 0 sin 2 u du = 1 2 (Lπ -sin(Lπ)).
The proof of the second part is similar.

The next lemma makes precise a well-known result due to Ingham (see [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF]) in the very particular context of harmonic Fourier series. The bracket notation stands for the integer floor value. Lemma 7. ( [START_REF] Ingham | Some trigonometrical inequalities with applications to the theory of series[END_REF]) For every 2π-periodic complex valued function f on IR such that f ∈ L 2 (0, 2π; C), for every T 2π, there holds

M 1 (T ) 2π 0 |f (t)| 2 dt T 0 |f (t)| 2 dt M 2 (T ) 2π 0 |f (t)| 2 dt, with M 1 (T ) = T 2π and M 2 (T ) = M 1 (T ) + 1.
Moreover, these constants are sharp. Proof. The existence of two positive constants M 1 (T ) and M 2 (T ) is a direct consequence of Ingham's Lemma (see [10, Theorems 1 and 2]). We define

M 1 (T ) = inf T 0 |f (t)| 2 dt 2π 0 |f (t)| 2 dt f ∈ L 2 (0, 2π; C) \ {0} ,
and M 2 (T ) is defined in the same way as a supremum. Notice that given f ∈ L 2 (0, 2π; C), one has

T 0 |f (t)| 2 dt = 2π[ T 2π ] 0 |f (t)| 2 dt + T -2π[ T 2π ] 0 |f (t)| 2 dt = T 2π 2π 0 |f (t)| 2 dt + T -2π[ T 2π ] 0 |f (t)| 2 dt.
The conclusion of the lemma is then obvious. The sharpness of the constants is obtained by considering either the function g vanishing on 0, T -2π T 2π and equal to 1 on T -2π T 2π , 2π or the function 1 -g.

A.2 Proof of Proposition 1

Using Lemmas 6 and 7, ( 14) and the Cauchy-Schwarz inequality, one has

J a (A, B) π 2 T 2π (πL -sin(πL)) 2 A 2 ℓ 2 (IR) + B 2 ℓ 2 (IR) - π 2 y 0 H 1 0 (0,π) B ℓ 2 (IR) - π 2 y 1 L 2 (0,π) A ℓ 2 (IR) . for every (A, B) ∈ ℓ 2 (IR) 2 . It follows easily that δ = inf{J a (A, B) | (A, B) ∈ (ℓ 2 (IR)) 2 } is finite. Consider now a minimizing sequence (A n , B n ) n∈IN * of J ω such that J a (A n , B n ) δ + 1 n 2 ,
for every n ∈ IN * . Fix temporarily n and m, two nonzero integers. Any convex combination

t(A n , B n ) + (1 -t)(A m , B m ), with t ∈ [0, 1] and (m, n) ∈ (IN * ) 2 is admissible and thus satisfies J a (t(A n , B n ) + (1 -t)(A m , B m )) δ. Moreover, easy computations show that (J a (t(A n , B n ) + (1 -t)(A m , B m )) -δ) + t(1 -t)Q a ((A n , B n ) -(A m , B m )) = (1 -t) (J a (A m , B m ) -δ) + t (J a (A n , B n ) -δ) . (62)
This implies

0 t(1 -t)Q a ((A n , B n ) -(A m , B m )) 1 -t m 2 + t n 2 .
From Lemmas 6 and 7, and using the observability inequality (3), the quadratic form Q a induces a norm on ℓ 2 (IR) that is equivalent to the standard hilbertian norm of ℓ 2 (IR). Hence, (A n , B n ) n∈IN * is a Cauchy sequence in (ℓ 2 (IR)) 2 and converges to some (A a , B a ) ∈ (ℓ 2 (IR)) 2 , which is clearly a minimizer of J a . The uniqueness of the minimizer follows from the strict convexity of J a which is proved in Lemma 4.

A.3 Proof of Proposition 2

By weak compactness of U L , up to a subsequence the sequence (χ ωn ) n∈IN converges to some a ∈ U L for the L ∞ weak star topology. Since F is lower semi-continuous for the L ∞ weak star topology (see Theorem 1), we have

F ( a) lim n→+∞ F (χ ωn ) = F (a),
and necessarily a = a by uniqueness of the minimizer (see Theorem 1). Using Lemmas 6 and 7, one has

F (χ ωn ) M 1 (T ) (πL -sin(πL)) 2 (φ χω n (0, •), ∂ t φ χω n (0, •)) 2 L 2 ×H -1 ,
for every n ∈ IN and since the sequence (F (χ ωn )) n∈IN * is bounded, it follows that the sequence (φ χω n (0, •), ∂ t φ χω n (0, •)) n∈IN * is bounded in L 2 (0, π) × H -1 (0, π). Since (φ χω n ) n∈IN is a solution of the wave equation ( 4), it follows immediately that the sequence (φ χω n ) n∈IN is bounded in L 2 (0, T ; L 2 (0, π)) and (∂ t φ χω n ) n∈IN is bounded in L 2 (0, T ; H -1 (0, π)). Then by Aubin's Lemma (see e.g. [START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF]), it follows that, up to a subsequence, the sequence (φ χω n ) n∈IN converges weakly in L 2 (0, T ; L 2 (0, π)) ∩ H 1 (0, T ; H -1 (0, π)) and strongly in L 2 (0, T ; L 2 (0, π)) to some φ. Since (χ ωn ) n∈IN converges L ∞ weakly star to a, this sequence converges also weakly in L 2 . From this fact and the uniqueness of the minimizer of the HUM functional J a , we deduce that the sequence (φ χω n ) n∈IN converges weakly in L 2 (0, T ; L 2 (0, π)) ∩ H 1 (0, T ; H -1 (0, π)) and strongly in L 2 (0, T ; L 2 (0, π)) to φ a . Since for n ∈ IN, y n is solution of (1) with right hand side h χω n , it follows that (y n ) n∈IN is bounded in L 2 (0, T ; H 1 0 (0, π)) and (∂ t y n ) n∈IN is bounded in L 2 (0, T ; L 2 (0, π)). Then by Aubin's Lemma it follows that, up to a subsequence, (y n ) n∈IN converges strongly to y a in L 2 (0, T ; L 2 (0, π)) and weakly in L 2 (0, T ; H 1 0 (0, π)) ∩ H 1 (0, T ; L 2 (0, π)). In the previous reasonings, since there is only one closure point at the limit, then the convergence is not only up to subsequence, but for the whole sequence. The proposition follows.

A.4 Proof of Proposition 3

Due to technical requirements, the application of the Pontryagin Maximum Principle in the infinite dimensional setting requires to work within a strictly convex Banach space (see [START_REF] Li | Necessary conditions for optimal control of distributed parameter systems[END_REF]). The optimal control problem settled above does not fit exactly (a priori) in this framework since the diagonal coefficients z j,j (π) are bounded below and above by positive constants (see Lemma 6), and hence a priori the adapted state space is ℓ ∞ (IR), which is not strictly convex. To overcome this difficulty, it suffices to write the control system [START_REF] Trélat | Optimal control] Théorie & Applications[END_REF] in another way so that the new control system is settled in a strictly convex Banach space. This reduction relies on the identity sin(jx) sin(kx) = 1 2 (cos((j -k)x) -cos((j + k)x)).

(63)

We consider the control system

w ′ m (x) = a(x) cos(mx), m ∈ IN, (64)
with initial conditions w m (0) = 0 for every m ∈ IN. The control system ( 26) is then inferred from the above one by setting z 0 (x) = w 0 (x), z j,k (x) = 1 2 (w |j-k| (x) -w j+k (x)), (65)

for almost every x ∈ [0, π] and for all integers j and k. Then the optimal control problem ( 26)-( 27)-( 28)-( 29) is equivalent to the reduced problem of determining a control a ∈ U L steering the infinite dimensional control system (64) from the zero initial conditions to the final condition w 0 (π) = Lπ (this is ( 28)), and minimizing the criterion F (a) (easily rewritten in terms of the w p 's). This reduced optimal control problem now perfectly fits the requirement of having a strictly convex Banach space. Indeed, for every x ∈ [0, π] and every a ∈ U L , the sequence (w m (x)) m∈IN * belongs5 to the Hilbert space ℓ 2 (IR), and hence ℓ 2 (IR) can be chosen as the functional state space for this reduced optimal control problem. It enjoys the strict convexity property.

The second main assumption of [START_REF] Li | Necessary conditions for optimal control of distributed parameter systems[END_REF] required to apply the Pontryagin Maximum Principle is, as mentioned above, a finite codimension assumption, which is obvious here since the final states z j,k (π) are not fixed (there is only one constraint, on z 0 (π) = w 0 (π)).

Therefore, the infinite dimensional version of the Pontryagin Maximum Principle stated in [START_REF] Li | Necessary conditions for optimal control of distributed parameter systems[END_REF] can be applied to our problem, and implies the following necessary conditions. There exist p 0 0 and (q m ) m∈IN ∈ ℓ 2 (IR) called costates 6 , with (p 0 , (q m ) m∈IN ) = 0, such that a(x) = 1 if ϕ(x) > 0, 0 if ϕ(x) < 0, for almost every x ∈ [0, π], where the function ϕ, called switching function, is defined by

ϕ(x) = +∞ m=0
q m cos(mx).

The control a is however undetermined from the maximization condition of the Pontryagin Maximum Principle in the case where the switching function ϕ vanishes identically on a subset of positive measure.

To transpose these results to the original optimal control problem ( 26)-( 27)-( 28)-( 29), we use the formulas (65), which indicate how the state coordinates must be changed. To get the corresponding change of costate coordinates, we apply the result of [START_REF] Cerf | Continuation from a flat to a round Earth model in the coplanar orbit transfer problem[END_REF]Appendix], yielding to the relations q 0 = p 0 + 1 2 +∞ j=1 p j,j , q 1 = j∈IN * p j,j+1 , q m = j,k∈IN * j-k=m p j,k -1 2 j,k∈IN * j+k=m p j,k , where p 0 and the p j,k denote the costates of the original optimal control problem. This change of coordinates leads exactly to the expression (31), as expected. Moreover, since the coefficients z j (π) are not fixed, the transversality conditions of the Pontryagin Maximum Principle (see e.g. [START_REF] Trélat | Optimal control] Théorie & Applications[END_REF] for a detailed discussion) imply, using a computation similar to [START_REF] Sigmund | Systematic design of phononic band-gap materials and structures by topology optimization[END_REF], that p j,k = p 0 T 0 (A a j cos(jt) + B a j sin(jt))(A a k cos(kt) + B a k sin(kt))dt, (66) for every (j, k) ∈ (IN * ) 2 . Clearly, the constant p 0 cannot be equal to 0, otherwise (66) would imply that p j,k = 0 for every (j, k) ∈ (IN * ) 2 , and necessarily p 0 = 0, so that the switching function ϕ would be constant, and the optimal control would be constant on [0, π], either equal to 1 or to 0. This would raise a contradiction with (28) since L ∈ (0, 1). Therefore p 0 < 0, and since (p 0 , p 0 , (p j,k ) (j,k)∈(IN * ) 2 ) is defined up to a multiplicative scalar, it is then usual to choose a normalization of the costates so that p 0 = -1. This leads finally to (32). The proposition is proved.

Remark 10. Everything works as if the Pontryagin Maximum Principle could be applied directly to the optimal control problem ( 26)-( 27)-( 28)- [START_REF] Wächter | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming[END_REF]. However as explained previously for technical reasons we had to consider an equivalent formulation with a change of coordinates.

A.5 Proof of Proposition 6

First of all, observe that Theorem 5 implies that the set ω To solve this minimization problem, it is more convenient to use the change of variable ξ = cos 2α + πL 2 so that, after computation of the integrals, the minimization problem reduces to min J t (ξ) = 4t πL -sin πL (67)

Let us study the variations of J t on [-ξ

Figure 1 :

 1 Figure 1: Several numerical simulations

Figure 2 :

 2 Figure 2: Graph of the optimal domain ω 2 in function of t

2 ∩

 2 [0, π/2] has only one connected component. According to the symmetry property proved in Theorem 5 and since {χ ω2 = 1} ∩ [0, π/2] has only one component, we write ω 2 = α, α + πL2and rewrite the minimization problem (54) as a minimization problem with respect to the variable α. The functional to minimize is thenα ∈ 0, π 2 (1 -L) →   t [α,α+ πL 2 ] sin 2 xdx + 1 -t [α,α+ πL 2 ] sin 2 (2x)dx   .

2 (2ξ 2

 22 -1) , ξ ∈ [-ξ 1 , ξ 1 ] where ξ 1 = cos πL 2 > 0.

  1 , ξ 1 ]. One has hence the sign of J ′ t (ξ) is positive on [0, ξ 1 ] and is equal to the sign of G t (ξ) on [-ξ 1 , 0], where The first and second derivative of G t with respect to ξ are

		J ′ t (ξ) =	4t sin πL 2 πL -sin πL 2	ξ	2 +	8(1 -t) sin(πL)ξ πL -sin(πL) 2 (2ξ 2 -1)	2 ,
	G t (ξ) = 4t sin	πL 2	πL -	sin(πL) 2	(2ξ 2 -1)
	G ′ t (ξ) = 8 sin(πL) 2 sin	πL 2	sin(πL)ξ 3 + 3(1 -t) sin 2 πL 2	ξ 2
					+ sin	πL 2	((2πL -sin(πL)t -4πL) ξ + (1 -t)π 2 L 2
	G ′′ t (ξ) = 8 sin(πL) sin	πL 2	6 sin(πL)ξ 2 + 6(1 -t) sin	πL 2	ξ

2 + 8(1 -t) sin(πL)ξ πL -sin πL 2 ξ 2 .

+(2πL -sin(πL))t -4πL

Similarly to the definition of elements of L ∞ , the subset ω is unique within the class of all measurable subsets of [0, π] quotiented by sets of zero measure. It means that if ω is optimal, thus any set ω ∪ N or ω \ N is also optimal, for every measurable subset N of [0, π] of zero Lebesgue measure.

By triangle function, we mean a function whose graph is an equilateral triangle

Note that, since the dynamics of (48) do not depend on the state, it follows that the costates are constant.

Here the uniqueness must be understood up to some subset of zero Lebesgue measure. In other words if ω is optimal then the union of ω with any subset of zero measure is also a solution.

Indeed, writing wm(x) = R 2π 0 χ [0,x] (s)a(s) cos(ms) ds, we see that wm(x) is the scalar product of an element of the usual Fourier basis with the function s → χ [0,x] (s)a(s) which belongs to L 2 (0, 2π) since a(•) ∈ U L .

Note that, since the dynamics of (64) do not depend on the state, it follows that the costates are constant.

Acknowledgment. The authors wish to thank Institut Henri Poincaré (Paris, France) for providing a very stimulating environment during the "Control of Partial and Differential Equations and Applications" program in the Fall 2010. The first author was partially supported by the ANR project GAOS "Geometric analysis of optimal shapes". The third author was partially supported by the Grant MTM2011-29306-C02-00 of the MICINN (Spain), project PI2010-04 of the Basque Government, the ERC Advanced Grant FP7-246775 NUMERIWAVES, the ESF Research Networking Program OPTPDE.

Indeed, the sign of G ′′ t (-ξ 1 ) can be obtained in the following way. Define the function f on [0, π] by f (u) = 3 2 sin(2u) -4t sin u + 2(t -2)u, where u plays the role of πL. One has f ′ (u) = 6 cos 2 u -4t cos u + 2t -7 and f ′′ (u) = 4 sin u(t -3 cos u). Hence f ′ is decreasing on [0, arccos(t/3)] and increasing on [arccos(t/3), π]. Moreover, f ′ (0) = -2t -1 < 0 and f ′ (arccos(t/3)) < 0 since t ∈ [0, 1]. Since f ′ (π) = 6t -1, distinguishing between the two cases t 1 6 and t > 1 6 , we obtain in the first case that f is decreasing on [0, π] and in the second case the existence of

) is continuous, affine, has a positive limit when t tends to 1 and a negative one when t tends to 0, we deduce that

and positive on [-ξ t , 0]; note moreover that -ξ t is solution of the equation G t (ξ) = 0;

where t 0 ∈ [0, 1] is the unique solution of the equation G t (-ξ 1 ) = 0 where the unknown is t.

We deduce from all previous arguments that

and increasing on [-ξ t , ξ 1 ], and thus

Let us study in particular the case t < t 0 . It follows from an implicit function argument that the map t ∈ [0,