How to estimate observability constants of one-dimensional wave equations? Propagation versus spectral methods

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1 How to estimate observability constants of one-dimensional wave equations? Propagation versus spectral methods Alain Haraux, hibault Liard, Yannick Privat o cite this version: Alain Haraux, hibault Liard, Yannick Privat. How to estimate observability constants of onedimensional wave equations? Propagation versus spectral methods. Journal of Evolution Equations, Springer Verlag, 16, <1.17/s y>. <hal v> HAL Id: hal Submitted on 7 Apr 16 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. he documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 How to estimate observability constants of one-dimensional wave equations? Propagation versus Spectral methods Alain Haraux,, hibault Liard,, and Yannick Privat, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-755, Paris, France. Sorbonne Universités, UPMC Univ. Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-755, Paris, France. Abstract For a given bounded connected domain in IR n, the issue of computing the observability constant associated to a wave operator, an observation time and a generic observation subdomain constitutes in general a hard task, even for one-dimensional problems. In this work, we introduce and describe two methods to provide precise and even sharp in some cases) estimates of observability constants for general one dimensional wave equations: the first one uses a spectral decomposition of the solution of the wave equation whereas the second one is based on a propagation argument along the characteristics. Both methods are extensively described and we then comment on the advantages and drawbacks of each one. he discussion is illustrated by several examples and numerical simulations. As a byproduct, we deduce from the main results estimates of the cost of control resp. the decay rate of the energy) for several controlled resp. damped) wave equations. Keywords: wave equation, characteristics method, Sturm-Liouville problems, eigenvalues, Ingham s inequality. AMS classification: 35L5, 93B7, 35Q93, 35B35. 1 Introduction and main results 1.1 Motivations and framework In control theory, well-posedness issues often come down to showing that a given observability constant is positive. Nevertheless, from a practical point of view, if the considered observability constant is close to zero, the cost of control may be huge, leading to numerical instability phenomena for instance. For this reason, a precise estimate of the observability constant brings in many cases an interesting and tractable information. In the context of inverse problems, the observability constant can be interpreted as a quantitative measure of the well-posed character of the problem. In control theory, it is directly related to the cost of control. When realizing experiments, it may also arise, due to several imprecisions on the operating conditions or on the measures, that one only knows partial informations on the parameters of the his work is supported by ANR AVENURES - ANR-1-BLAN-BS1-1-1) 1

3 related inverse/control problem. For instance, one could mention the example of hermoacoustic tomography where the intensity of measures is often very weak and the physical model is in general simplified before exploiting the measures. In that context, it is interesting to obtain estimates of the observability constant depending only on some parameters of the experiment. In what follows, we concentrate on the observation of the wave equation with a zero-order potential function. As physical parameters, we choose the observability time, possibly the Lebesgue measure of the observation subset and some L bounds on the potential function. In what follows, we will then derive estimates of the observability constant depending only on such parameters. More precisely, this work is devoted to providing explicit lower bounds of observability constants for one-dimensional wave equations with potential using two different methods and comparing the results. We choose in the sequel potentials that only depend on the space variable since it is often more relevant in view of physical applications. A typical case is the consideration of geophysics waves influenced by the earth rotation. Let us make the frame of our study more precise. Let denote a positive constant standing for the observability time. We consider the one dimensional wave equation with potential tt ϕt, x) xx ϕt, x) + ax)ϕt, x) = t, x), ), π), ϕt, ) = ϕt, π) = t [, ], ϕ, x) = ϕ x), t ϕ, x) = ϕ 1 x) x [, π], 1) where the potential a ) is a nonnegative function belonging to L, π). It is well known that for every initial data ϕ, ϕ 1 ) H 1, π) L, π), there exists a unique solution ϕ C, ; H 1, π)) C 1, ; L, π)) of the Cauchy problem 1). Notice that, defining the energy function E ϕ by there holds E ϕ : [, ] IR + t π tϕt, x) + x ϕt, x) + ax)ϕt, x) dx, E ϕ ) = E ϕ t) ) for every t [, ] and every solution ϕ of 1). Let be a given measurable subset of, π) of positive Lebesgue measure. he equation 1) is said to be observable on in time if there exists a positive constant c such that, t ϕt, x) dxdt ce ϕ ), 3) for all ϕ, ϕ 1 ) H 1, π) L, π), where E ϕ ) = π ϕ1 x) + ϕ x) + ax)ϕ x) ) dx.. Note that in the case where a ) =, for every subset of [, π] of positive measure, it is well known that the observability inequality 3) is satisfied whenever π see [3]). If 1) is observable on in time, we denote by c, a, ) the largest constant in 3), that is c, a, ) = inf ϕ,ϕ 1) H 1,π) L,π) ϕ,ϕ 1),) tϕt, x) dxdt π ϕ 1x) + ϕ x) + ax)ϕ x) ) dx. 4) Even in the simple case a =, it is not obvious to determine the constant c, a, ) for arbitrary choices of. Indeed, a spectral expansion of the solution ϕ on a spectral basis shows the emergence of nontrivial crossed terms. Reformulating this question in terms of the Fourier coefficients of the initial data ϕ, ϕ 1 ), the quantity c, a, ) is seen as the optimal value of a quadratic functional

4 over every sequence c = c j ) j Z l C) of Fourier coefficients such that c l C) = 1. his leads to a delicate mathematical problem. As it will be highlighted in the sequel, it is quite similar to the well-known open problem of determining what are the best constants in Ingham s inequalities. his article is devoted to the introduction and description of several methods permiting to determine explicit positive constants c such that the inequality 3) holds for all ϕ, ϕ 1 ) H 1, π) L, π), or equivalently such that c, a, ) c. One also requires that the constant c only depend on, possibly, as well as some L bounds on the potential function a ). his way, as mentioned above, it is possible to deal with experiments where one only knows partial informations on the operating conditions. It is structured as follows: the main results are presented in Section 1., as well as the presentation of each method spectral versus propagation). In Section 3, we present applications of our results to control and stabilization of wave equations. Section 4, is devoted to the illustration of the main results and we provide several numerical simulations to comment on both methods, illustrate and compare them. For the convenience of the reader, most of the proofs are gathered in Section. 1. Main results In this section, we present the estimates of observability constants obtained using each method. Let ϕ C, ; H 1, π)) C 1, ; L, π)) denote the solution of 1) with initial data ϕ, ) = ϕ ) H 1, π) and t ϕ, ) = ϕ 1 ) L, π). In the sequel and for the sake of simplicity, the notations r + or r will respectively denote max x [,π] rx) and min x [,π] rx). he first method makes full use of the spectral decompo- First method: spectral estimates. sition ϕt, x) = + j=1 a j cosλ j t) + b j sinλ j t)) e j x), 5) where {e j } k IN denotes an orthonormal Hilbert basis of L, π) consisting of eigenfunctions of the operator A a = xx + a ) Id with Dirichlet boundary conditions, associated with the positive eigenvalues λ j ) j IN, and a j = π ϕ x)e j x) dx, b j = 1 π ϕ 1 x)e j x) dx, 6) λ j for every j IN. In the following result, we provide an estimate of the observability constant c, a, ) that only depends on the parameters, and some L bounds on the potential a ). It is interesting to note that no assumption is made on the topology of the set. However, as highlighted in Remark 3 and in the discussion ending Section 1., this approach presents some drawbacks, in particular the fact that it can only be used when the potential function a ) is close to a constant function. heorem 1. Consider a function a L, π; IR + ) writing a ) = ā + r ), where ā IR and r > 1. Let be a measurable subset of, π) of positive measure and r) = π where 3 r ++r γr) =. Assume 1 that r < where denotes the unique positive) solution 4+r + 1+r + γr) 1 With the notations of this theorem, one has r = max{ r, r + }. ) π 1 Uniqueness of follows from the fact that the mapping F : IR 8 e 1 + 4π is increasing. 3

5 of the equation sin = 8 e π ) π ). 7) hen, 1) is observable on for all > r) with c, a, ) C a, ) where ) sin 4π r 4 Dr) sin ) π r C a, ) = K I, r) π + 4π ), 8) Dr) + Ar) + πd Ar) = 1, Dr) = e π r Ar) 1, and K I, r) = ) 4π 1 + r π γr). According to 7), there holds C a, ) >. Furthermore, in the particular case where a ) =, one can improve the estimate 8) by setting [ ] C, ) = sin ), 9) π where the bracket notation stands for the integer floor. he following remarks are in order. Remark 1. he assumption r > 1 sufficient when a L, π; IR + ) ensures that the resolvent of the operators A a and A r are compact. Nevertheless, the assumptions of this theorem can be extended to potential functions that are non necessarily nonnegative. More precisely, replacing the assumption r > 1 by min{r, r + ā} > 1 leads to the same conclusion. Remark. Note that the decomposition of the potential as a ) = ā + r ) where ā IR, r r ) r + almost everywhere in, π) with min{r, r + ā} > 1 and r < may be nonunique whenever it holds. As a consequence, it is relevant, at least from a numerical point of view to look for the best decomposition, that is the one maximizing the estimate 8). Remark 3 Smallness of ). he constant appearing in the statement of heorem 1 is quite small. Indeed, by using the inequality e h 1 h holding for all h, it is easy to obtain sin ) ). + 4)π Numerical computation leads to )π <.8.1 and in particular,.8.1. Remark 4 Simplifying the condition 7)). In the statement of heorem 1, the condition r < is fulfilled as soon as r β 1 sin ) ), with β = he value of β can be easily computed by using the inequality e h 1 h + h whenever h 1. his inequality leads to 1 β 4π ) 1 + 4π, 1 1 4

6 where = )π <.8.1 and the conclusion follows. his means that heorem 1 holds only for very small variations around constant potentials. he next theorem provides an estimate holding for a much larger class of potentials. Nevertheless, as illustrated in Section 4, the interest of the estimate given in heorem 1 rests upon the fact that it is sharper whenever it can be applied. In particular, we will see that for particular resonant observability times and the choice a ) =, it coincides with the value of the observability constant c, a, ). Remark 5. he constant K I, a) introduced in the statement of heorem 1 is a so-called Ingham constant, first introduced by Ingham in [1]. Ingham s inequality constitutes a fundamental result in the theory of nonharmonic Fourier series. It asserts that, for every real number γ and every > π γ, there exist two positive constants C 1, γ) and C, γ) such that for every sequence of real numbers µ n ) n IN satisfying there holds C 1, γ) n Z a n n IN µ n+1 µ n γ, a n e iµnt dt C, γ) a n, 1) n Z n Z for every a n ) n IN l C). Denoting by C 1, γ) and C, γ) the optimal constants in 1), several explicit estimates of these constants are provided in [1]. For example, it is proved in this article that C 1, γ) ) 4π π γ and C, γ) min{π, γ }. Notice that, up to our knowledge, the best constants in [1] are not known. In the particular case where µ n = n for every n IN, one shows easily that for every > π, C 1, γ) = [ ] π π and C, γ) = C 1, γ) + 1, the bracket notation standing for the integer floor. In a general way, one could choose K I, r) = C 1, γr)) with > π γr) in the statement of heorem 1. Finally, let us mention that the idea to use Ingham inequalities in control theory is a long story see for instance [1, 7, 11, 1, 14]). Remark 6. Notice that, due to our use of Ingham s inequality, the time r) needed to apply heorem 1 is greater than the minimal time of observability see e.g. [] for the computation of such a time). his restriction is proper to the use of spectral methods. Indeed, even in the very simple case where the potential a ) vanishes, the time r) is equal to π and is then greater than the minimal observability time. It should be possible to decrease the time r) by using only the asymptotic spectral gap see e.g. [11]), but our main interest here concerns the observability constant and the methods relying on asymptotic gaps do not usually provide good estimates. Actually, when approaches the minimal observabiity time, the observability constant tends to. In particular it is not good to be close to the minimal time when we look for a sharp decay estimate of solutions to the equation with dissipative feedback control. Particular examples of application of this theorem to observation, control and stabilization of one dimensional wave equations are provided in Section 4. Second method: a propagation argument. his method makes great use of propagation properties of the wave equation to derive sharp energy estimates and is inspired by [6]. he technique consists in inverting the roles of the time and space variables, and to propagate the 5

7 information from the observation domain to other ones. Although the result presented in the next theorem appears a bit technical, the approach used here is robust and holds for very large choices of potential functions a ), as it will be commented in Section 4. Note that a close but non-quantitative result has been obtained in [4, heorem 4] for semilinear wave equations. In the next result, we keep track of the constants in this method, trying to improve each step by choosing the best possible parameters see for instance Lemma 5 or Remark 11 in the proof). In the next result and unlike the framework of heorem 1, the estimate of the observability constant c, a, ) only depends on, a and the precise knowledge of. heorem. Let =, β) with < < β < π and = max{, π β}. Define for η > and k, the quantity ) e Cη) kπ β+3η) e 4kη +e kη e k+η) kη + 1 if k > Kη,, β, k) = ) Cη) π β+ 11) η + 3 if k = { } where Cη) = 1 η + max 1, 1 η + max{1, k }. hen, 1) is observable on for all > with c, a, ) C,,a,λ, β) where if, ), then C,,a,λ, β) = { and A,β,, = max γ,η) A,β,, γ, η) γ, 4 ) γ sup sinλ j γ) j IN λ j ) and η, min 4 γ) 8 + )γkη,, β, a 1/ { })} 16 γ 8, β 4, ) 1) if, then [ ] C,,a,λ, β) = { and A,β,, = max γ,η) A,β,, γ, η) γ, 4 ) γ sup sinλ j γ) j IN λ j ) and η, min + { })} 16 γ 8, β 4. γ )γkη,, β, a 1/ ) 13) Remark 7. We stress the fact that, unlike the statement of heorem 1, no restriction on the L norm of the function a ) is needed in the statement of heorem. Remark 8. he constant C,a,λ, β) given by 1) or 13) writes as the maximum of a two variables function. Due to the presence of highly nonlinear terms in its expression, the maximum cannot be computed explicitly in general, but is nevertheless easy to compute numerically. It will be illustrated in Section 4. Remark 9. Notice that, in the case a ) =, we have λ j = j for every j IN and the quantity γ sup sinλ jγ) j IN λ j simplifies to γ sinγ). In the general case and to avoid to use the knowledge of the whole spectrum, one can simplify 1) and 13) by noting that sinλ j γ) sup γ j IN λ j sup {x λ 1γ} sin x. x Remark 1 Key ingredient of the proof.). he proof of heorem derives benefit from the propagation properties of Equation 1) along the characteristics. his is illustrated on Figure 1, 6

8 representing the propagation of wavefronts in the time-space domain, π), ) in the case where the observation domain is =, β). Recall that every point x, π) generates two characteristics: one is going to the left and the other one to the right, in a symmetrical way. Roughly speaking, the solution ϕ of the wave equation 1) is known on the light-gray rectangle domain, and the propagation properties of the wave equation allow to recover ϕ on the the deepgray domain, provided that the observation time be large enough. β δ β π Figure 1: Propagation zones along the characteristics As a corollary of heorem, we have the following result, extending the estimate of the observability constant to those subsets that are the finite union of open intervals. Corollary 1. Let = n i=1 i, β i ) with < 1 < β 1 < < n < β n < π and = max 1 i n { i β i 1, i+1 β i }, with the convention that β = and n+1 = π. Define ) e Cη i ) k i+1 β i +3η i ) e 4kη i +e kη i e kβ i 1 i +η i ) kη i + 1 if k > K η i, k) = Cη i ) i+1 β i+ i β i 1 η i + 3 ) if k = for i {1,..., n}, η i > and k. hen, we have c, a, ) min 1 i n C,,a,λ i, β i ), where the quantity Kη i, i, β i, k) defined by 11) has been replaced by the quantity K η i, k) defined by 14) in the definition of the constant C,,a,λ i, β i ). In heorem and Corollary 1, the time coincides with the smallest observability time. he following corollary provides a simpler estimate of the observability constant, but is only valid for potentially larger values of the observation time and provided that we have a precise knowledge of the low frequencies λ j ) j IN associated to the operator xx +a ) Id, introduced at the beginning of this section. he proof is based on Ingham inequalities [1]. Corollary. Let = n i=1 i, β i ) with < 1 < β 1 < < n < β n < π. Let j = and introduce the positive real numbers 1+j a = π and γ j = { } γ 1+j j min min j {1...j 1} λ j+1 λ j, a j +1+ j + a j +1+ j + a if j = 1 14) [ ] a 1 +1 if j > 1. 7

9 hen, we have c, a, ) C,,a,λ) = )γ j 4π ) min{π, γ j, } γ j 4π ) min{π, γ j, } + πγ j Λ,,a ) 15) with Λ,,a) = min 1 i n max { { 1 Kη, i, β i, a 1/ ), < η < min, β } } i i Short discussion on the main results. We stress the fact that the main ingredients in the proofs of heorem 1 on the first hand, and heorem, Corollaries 1 and are of different natures. Indeed, heorem 1 is based on a purely spectral argument, and is somehow limited by the misreading of the low frequencies of the spectrum. his explains the restrictions on the norm of the potential a. By the way, notice that the knowledge of the value of a spectral gap γ in the sense made precise in Remark 5) would permit to avoid the technicalities in Lemmas, 3 and 4 and to get directly a simple estimate instead of the one of heorem. he results in heorem, Corollaries 1 and are in some sense more robust since we do not need to make assumptions on the smallness of the difference between the maximal and minimal values of the potential a ). Moreover, on the contrary to the spectral method, propagation ones provide a lower bound of the observability constant working when the observation time is close to the minimal observation time when is an interval. Recall also that, independently of the aforementioned drawbacks, in the context of an inverse problem where there is some indetermination on some parameters of the problem, heorem 1 can be used when the topological nature in particular the number of connected components) of is not known whereas heorem assumes that is known and writes as a finite union of intervals. In each case, one can assume that only L bounds are known on the potential a ). o sum-up, we gather this discussion under the following condensed form. Spectral method Propagation method requires the knowledge of Advantage Drawback - sharp estimate,,, L bounds on a ) - is only assumed to be measurable,, L bounds on a ) no restriction on the potentials works for almost constant potentials - estimate not so accurate - writes as a finite union of intervals Section 4 is devoted to the numerical comparison and results obtained using each method. Proofs of the main results.1 Proof of heorem 1 Let us begin by proving the second part of the statement of heorem 1, the general case using strongly the estimate obtained in this simple case. he case a ) =. Assume that a ) =. According to 5) and since the eigenfunctions of the Dirichlet-Laplacian operator on Ω =, π) are given by e j x) = π sinjx) for every j IN it 8

10 follows that for all initial data ϕ, ϕ 1 ) H 1, π) L, π), the solution ϕ C, ; H 1, π)) C 1, ; L, π)) of 1) can be expanded as ϕt, x) = + j=1 a j cosjt) + b j sinjt)) sinjx), 16) where the sequences ja j ) j IN and jb j ) j IN belong to l IR) and are determined in function of the initial data ϕ, ϕ 1 ) by a j = π for every j IN. By the way, note that and furthermore, one has π t ϕt, x) dxdt ϕ x) sinjx) dx, b j = jπ ϕ, ϕ 1 ) H 1 L = π = π[ π ] [ ] π π + j=1 π t ϕt, x) dtdx + j=1 [ ] = π j a j + b π j) j=1 he following lemma is a crucial ingredient to conclude. ϕ 1 x) sinjx) dx, 17) j a j + b j), 18) j a j sinjt) + b j cosjt)) sinjx) sin jx) dx. Lemma 1. Let j IN. For every measurable subset of, π), one has sinjx) sin dx. his lemma is noticed as well in [18, ] and used for controllability purposes in [15]. Even though it is well known, we provide at the end of this paragraph an elementary and new proof of this result using the Schwarz symmetrization. As a consequence, one gets t ϕt, x) dxdt π Combining this inequality with 18), we infer [ ] sin ) π whence the estimate 9). π [ ] sin ) π j a j + b j). j=1 ϕ1 x) + ϕ x) ) dx t ϕt, x) dxdt, dtdx 9

11 Proof of Lemma 1. Let us first estimate the quantity { } δ j L) = sup f j x) dx, measurable subset of, π) such that = Lπ, where f j x) = sinjx) and L, 1) is fixed. Let be a measurable subset of, π) such that = Lπ. Denote by f j,s the Schwarz rearrangement 3 of the function f j. According to the Hardy-Littlewood inequality see e.g. [9, 13]), one has π π f j x) dx = χ x)f j x) dx χ S x)f j,s x) dx = f j,s x) dx. S Moreover, notice that f j,s = f 1 for every j IN. Indeed, introducing for every µ, 1) and j IN the set Ω j µ) = {x, π) f j x) µ}, a simple computation ensures that Ω j µ) = j k=1 { x, π) k 1)π x j k 1)π j arcsin } µ), j so that, Ω j µ) = π arcsin µ) = Ω 1 µ). he expected result follows easily. As a consequence and since S = 1 L)π/, 1 + L)π/), one has f j x) dx f 1 x) dx = S Lπ + sinlπ). and then δ j L) Lπ+sinLπ) for every j IN. Finally, denoting by c the complement set of in, π), we get the expected inequality by writing inf measurable =Lπ We thus infer that sinjx) dx = π sup c measurable c =1 L)π for every measurable subset of, π). c sinjx) dx π 1 L)π + sin1 L)π) sinjx) dx sin ) = Lπ sinlπ). he case ā =. We start with some elementary lemmas that play an important role. Lemma. Assume that r > 1 and r + r < 3. hen, for every j IN, one has λ j+1 λ j 3 r + + r 4 + r r +. 3 For every subset U of Ω, we denote by U S the ball centered at π having the same Lebesgue measure as U. We recall that, for every nonnegative Lebesgue measurable function u defined on Ω and vanishing on its boundary, denoting by Ωc) = {x Ω ux) c} its level sets, the Schwarz rearrangement of u is the function u S defined on Ω S by u S x) = sup{c x Ωc)) S }. he function u S is built from u by rearranging the level sets of u into balls having the same Lebesgue measure see, e.g., [9, Chapter ]). 1

12 Proof. he Courant-Fischer minimax principle writes λ j = min V H 1,π) dim V =j max u V \{} π u x) + rx)ux) )dx π. ux) dx Using that r rx) r + for almost every x, π) yields j + r λ j j + r +, 19) for every j IN. It suffices indeed to compare λ j operator with constant coefficients. We infer with the j-th eigenvalue of a Sturm-Liouville λ j+1 λ j j + 1) + r j + r +, = 1 + j r + + r j + 1) + r + j + r +, for every j IN. he sequence j j + 1) + r j + r + being increasing, the expected estimate follows. Notice that similar computations on the asymptotic spectral gap of Sturm-Liouville operators may be found in [19]. Lemma 3. Let j IN, r L, π) such that r > 1 and be a measurable subset of, π). he j-th eigenfunction e j ) of the operator A r = xx + r ) Id satisfies e j x) C j sinλj x) + λ j h j x) sinλ j x) ) ) for every x [, π], where C j = π + π 1 + r ) e λ τr)π 1 ) + 1 j 4 + π1 + r ) e τr)π 1 ) ), 1) λ j h j L, π) is such that h j 1+r )eτr)π 1) λ 4 j and τr) = r 1+r. Proof. he estimate ) is obtained by using a shooting method that we roughly describe. Introduce ψ j : x sinλjx) λ j. he function φ j = ej ) e j ) solves the following Cauchy system φ j x) + rx)φ jx) = λ j φ jx) φ j ) =, φ j ) = 1, x, π), whereas the function h j = φ j ψ j solves the Cauchy system h j x) + rx) λ j )h jx) = rx)ψ j x) x, π), h j ) =, h j ) =. ) Let us denote by V λj the energy function defined by V λj x) = 1 h j x) + λ j h jx). 11

13 Using ), one gets the estimate for every x [, π], with τr) = As a consequence, and since e j ) = π φ j x) dx = π V λ j x) τr)v λj x) + r τr)λ, j r 1+r. According to the Gronwall lemma, we infer h j 1 + r λ 4 j φ j ) π φjx) dx ) e τr)π 1., one gets ) sin λ j x) λ + h j x) sinλ jx) + h j x) dx C j j λ j λ, j where C j is defined by 1). he expected estimate then follows. Lemma 4. Let r L, π) such that r > 1. hen, one has sin λ j x) dx sin π r, j for every j IN. Proof. In accordance with 19), let us write λ j = j + lj j with r l j r + for every j IN. Using Lemma 1, we get sin λ j x)dx = 1 cosλ j x) dx, = sin jx) + l x ) j j cosjx) slj sin ds + 1 )) j sinjx) sin lj x dx, j sin π l j. j Since l j r, sin λ j x)dx sin π r. j We now prove heorem 1. Combining ) with the Cauchy-Schwarz inequality, one gets ) e j x) dx sin λ j x) dx λ j h j x) C dx sin λ j x) dx j for every j IN. he positivity of the right hand side in the inequality above is equivalent to the condition sin λ j x) dx > 4λ j h jx) dx. Moreover, according to Lemma 3, one shows easily that this condition holds whenever r <, where denotes the unique solution of 7) with 1

14 ā =. Finally, the constant C j being defined by 1), one finally gets using the estimate 19) an upper bound on it, uniform with respect to j, so that ) sin 4π r 4 Dr) e j x) sin ) π r dx π + 4π ), 3) Dr) + Ar) + πdr) for every j IN, using the notations introduced in the statement of heorem 1. By decomposing the solution ϕ of 1) in the spectral basis {e j } j IN as in 5)-6), one gets using the so-called Ingham inequality t ϕt, x) dxdt = 1 i sgnk)λ k a k + b k ei sgnk)λ k t θ k ) e k x) dtdx, k Z K I, a) a k + b k )λ k e k x) dx k Z + = K I, a) a j + b j)λ j e j x) dx, j=1 where θ j ) j IN denotes the sequence defined by e iθj = aj+ibj for every j IN, and K a j +b I, a) is j the Ingham constant introduced in the statement of heorem 1 and whose choice is commented in Remark 5. In the sequel and for the sake of simplicity, the notations ϕ x or ϕ t will respectively denote the partial derivatives x ϕ and t ϕ. he energy identity π holding for almost every t [, ], we infer ϕ t t, x) + ϕ xt, x) + ax)ϕ t, x) ) + dx = λ ja j + b j), t ϕt, x) dxdt K I, a) inf e j x) dx j IN π j=1 ϕ t t, x) + ϕ xt, x) + ax)ϕ t, x) ) dx. Finally, the combination of this inequality with 3) provides the desired result. Furthermore, by construction, the right-hand side of the obtained inequality, denoted C a, ) is positive. he general case. he general estimates are obtained by mimicking the proof in the case where ā =. Indeed, note that λ j is an eigenvalue of the operator A a if, and only if µ j = λ j ā is an eigenvalue of the operator A r. Since µ j 1 + r >, one has µ j+1 µ j = λ j+1 λ j 3 r + r ) λ j+1 λ ā + j ā 4 + r r + for every j IN. We obtain successively the same estimates as those in the statements of Lemmas 3 and 4, replacing the quantity λ j by µ j. he expected conclusion follows. 13

15 . Proof of heorem Let us denote by ϕ the unique solution of the wave equation 1) with a given potential a ) L, π) and with initial data ϕ, ), t ϕ, )) = ϕ, ϕ 1 ) H 1, π) L, π). Using the notations introduced in Section 1, let us define the function E a by E a : [, ] [, π] IR + t, x) t ϕt, x) + x ϕt, x) + ax)ϕt, x), and the function E k by E k : [, ] [, π] IR + t, x) t ϕt, x) + x ϕt, x) + k ϕt, x), where k = a. his proof is divided into several steps and is illustrated on Figure. Let us comment on this figure: in a nutshell, the main ingredients of the proof are contained in the statements of Lemmas 6 and 7, that take advantage of the propagation properties of the wave equation 1) to derive energy estimates on the zone P 1 respectively P ) from the observation on the zone Q 1 respectively Q ). Finally, since we will establish estimates along the characteristics, the space and time variables will play symmetric roles in the algebraic computations that follows. δ P Q Q 1 P 1 δ β η η β π Figure : Scheme of the proof Propagation method) Let us start with several instrumental lemmas. he first one states an observability result in the case where = Ω =, π). Lemma 5. Let γ, /). hus, inf ϕ,ϕ 1) H 1,π) L,π) γ Ω tϕt, x) dxdt Ω ϕ 1x) + ϕ x) + ax)ϕ x) ) dx = γ sup sinλ j γ), 4) j IN λ j where ϕ denotes the unique solution of the wave equation 1) with initial data ϕ, ), t ϕ, )) = ϕ, ϕ 1 ) H 1, π) L, π)and λ j ) j IN the sequence of eigenvalues introduced in Section

16 Proof. Denote by LHS the left hand side in 4). Decomposing ϕ as in 5)-6) yields LHS = inf λ ja j,λ jb j) j IN l IR)) j=1 λ j γa j sinλ j t) b j cosλ j t)) dt j=1 λ j a j + b j ). Next, setting λ j a j = ρ j cosθ j ), λ j b j = ρ j sinθ j ) with ρ j ) j IN l IR) and θ j for all j IN, we get using a homogeneity argument LHS = inf inf ρ j=1 ρ j =1 j θ j) j N j=1 + = γ sup sup j=1 ρ j =1 θ j) j N j=1 sin λ j t θ j ) dt, γ ρ sinλ j γ) j cosλ j γ) θ j ). λ j o reach the maximum, it suffices to choose θ j such that cosλ j γ) θ j ) = sgnsinλ j γ)) for all j IN and ρ j ) j IN as a Kronecker delta, and we thus get 4). In the following lemma, pointed out as a main ingredient of the proof, we establish an energy estimate on η, η) + η, β η) for η, β ). Lemma 6. Let η, β ) and η. he energy estimate η β η η +η holds with Cη) = 1 η + max { 1, 1 η } + max{1, k }. Proof. Multiplying the identity β E k t, x) dxdt Cη) ϕt t, x) + ϕt, x) ) dxdt. 5) ϕϕ t ) t t, x) ϕϕ x ) x t, x) ϕ t t, x) + ϕ x t, x) + ax)ϕt, x) = valid for almost every t, x), ), π), by any smooth function ζ Cc, ), β)) such that ζ on, ), β) and ζ = 1 in [η, η] [ + η, β η], we get after integration on, ), π), β η η +η η ϕ x t, x) ζt, x) dtdx = β β β + ϕ x t, x) ζt, x) dtdx, ϕ t t, x) ax)ϕt, x) )ζt, x)dtdx ϕϕ x ) x t, x) ϕϕ t ) t t, x))ζt, x)dtdx. We now estimate the right hand side in this inequality. Integrating by parts, one gets and β β ϕϕ t ) t t, x)ζt, x) dxdt ζ t β ϕϕ x ) x t, x)ζt, x)dxdt ζ xx ϕ t t, x) + ϕ t, x)) dxdt β ϕ t, x)dxdt. 15

17 We then infer β η η +η where cη) = ζt + max η β η η +η η { β ϕ x t, x) dtdx cη) ϕt t, x) + ϕt, x) ) dxdt, ζ, ζxx }, for every η, β ). he energy estimate β E k t, x) dxdt cη) + max{1, a }) ϕt t, x) + ϕt, x) ) dxdt is thus a consequence of the previous inequalities. Finally, choosing a particular function ζ enjoying the symmetry properties ζ t, x) = ζt, x), ζt, + y) = ζt, β y) for every t [, ] and y [, β ], and defined by ζt, x) = t η to the expected result. x ) η on [, η] [, + η] leads Remark 11. We stress that the particular choice of function ζ used above is motivated by the facts that the function f : t t/η solves the problem inf{ f f W 1,, η), f) = and fη) = 1} and the function g : x x /η solves the problem inf{ g g W,, η), g) = and gη) = 1}. With the notations of Figure, propagations properties of the wave equation are used in the following lemma to derive an energy estimate from Q 1 to P 1 and from Q to P. Lemma 7. Let > with = max{, π β} and η, β 4 ). Recall that Cη) is the constant defined in the statement of Lemma 6. i) Introduce the positive number δ such that π β = δ. hen, one has +δ π δ β η E k t, x) dxdt Cη) kη for every η, min{ β 4, δ 4 }). e kπ β+3η) e 4kη) ii) Introduce the positive number δ such that = δ. hen, one has +δ δ +η E k t, x)dxdt Cη) kη e kη e k+η) ) for every η, min{ β 4, δ 4 }). β β ϕ t t, σ) + ϕ t, σ)) dσdt, 6) ϕ t t, σ) + ϕ t, σ)) dσdt, 7) 16

18 Proof. Let ξ, β η) and x ξ, min{ + ξ, π}). Using an integration by parts 4, one gets d dx x+ξ x ξ We thus infer that E k t, x)dt = ϕ t ϕ x ) + ξ x, x) ϕ t + ϕ x ) x ξ, x) d dx 4 x+ξ x ξ k ϕ + ξ x, x) + ϕ x ξ, x) ) + 4 x ξ) x ξ k ϕ x ϕ dt. E k t, x)dt k x+ξ x ξ x ξ) x ξ k ϕ x ϕ dt, E k t, x)dt. 8) By noting that the function x ξ, π) e kx ξ η) x+ξ E x ξ k t, x)dt is non increasing, it follows that +δ x+ξ η E k t, x) dt E k t, x) dt e kx ξ η) E k t, ξ + η) dt, δ x ξ η for every η, δ 4 ), ξ [β 4η, β η] and x [β η, π]. Integrating this inequality with respect to ξ on [β 4η, β 3η] yields +δ E k t, x) dt ekx β+3η) δ η η β η η β 3η E k t, σ) dtdσ, 9) for every x [β η, π]. Notice that β 3η, β η) + η, β η) since η β 4. Hence, one deduces the expected result by combining the last inequality with the conclusion of Lemma 6. It remains now to prove the second energy estimate. It suffices to mimic the first part of the proof, noting first that d dx ξ+x ξ x E k t, x) dt k ξ+x ξ x E k t, x) dt, 3) for all ξ +η, β) and x [, ξ], and second that the function x, ξ) e kx ξ+η) ξ+x E ξ x k t, x) dt is non decreasing. We now prove heorem. With the notations of Lemma 7, introduce δ = min{δ, δ } = 4. Combining 5), 6) et 7) yields +δ δ π E k t, x) dxdt Kη,, β, k) β where Kη,, β, k) is defined by 11). According to ), we infer ) π ) δ ϕ t, δ E a, x dx Kη,, β, k) L,π) ϕ t + ϕ )t, x) dxdt, 31) β 4 We also use that, for any function f regular enough, the following identity d x+ξ x+ξ f ft, x)dt = f + ξ x, x) fx ξ, x) + t, x)dt dx x ξ x ξ x holds for every x ξ, min{ + ξ, π}). ϕ t t, x) + ϕ t, x)) dxdt, 3) 17

19 for every η, min{ δ 4, β 4 }) and >. It remains now to compare the right hand in 3) with the observation term. Let γ > such that γ δ and let τ γ, ). Introduce the function ψ defined by ψ :, τ), π) IR t, x) t τ ϕ ts, x) + ϕ t τ s, x)) ds. Notice that ψ satisfies the main equation of 1) on, τ) Ω and that ψ t τ, ) = ϕ t τ, ). Furthermore, the function ψ clearly satisfies the inequality 3). Since τ > /, we claim ψ t τ, ) L,π) Kη,, β, k) µ τ β ψ t + ψ )t, x) dxdt, for every η, min{ µ 4, β 4 }), where µ = min{µ, µ } with µ = τ π+β, and µ = τ. We thus infer that 8µ Kη,, β, k) ϕ tτ, ) L,π) τ β τ β + ϕ t t, x) + ϕ t τ t, x)) dxdt τ t ϕ t s, x) + ϕ t τ s, x) ds) dxdt. Introducing R τ = τ t ϕ t s, x) + ϕ t τ s, x))ds) dt, τ one gets using the Cauchy-Schwarz inequality R τ and eventually τ τ + τ τ t) τ t) t τ τ τ ϕ t s, x) ds, t ϕ t s, x) dsdt + τ ϕ t τ s, x) dsdt + τ t τ) τ τ t τ ϕ t s, x) dsdt, t τ) t τ ϕ t τ s, x) dsdt, ϕ t τ, ) L,π) Kη,, β, k) µ 1 + τ ) τ β ϕ t t, x) dxdt. Let us now provide an estimate of the right-hand side in the last inequality that is uniform with respect to τ. Using that τ γ, ), µ γ c + β, µ γ and µ 4 γ and that the expression of Kη,, β, k) above makes sense for η, min{ 16 γ 8, β 4 }), we get ϕ t τ, ) γkη,, β, k) γ L,π) dτ 1 + ) β ϕ t t, x) dxdt. γ 8 18

20 Applying Lemma 5 leads to ) β γ sup sinλ j γ) 4 γ) j IN λ j 8 + )γkη,, β, k) ϕ t t, x) dxdt π E, 33) a, x) dx providing hence a lower estimate of the observability constant. For large values of, it is yet possible to improve this estimate. Assuming that, we write β ϕ t t, x) dxdt [ ] β ϕ t t, x) dxdt = [ ] 1 i+1) β and using the time reversibility of the wave equation, the inequality 33) improves into [ ] ) γ sup sinλ j γ) j IN λ j.3 Proof of Corollary 1 i= γ) + )γkη,, β, k) i ϕ t t, x) dxdt β ϕ t t, x) dxdt π E. 34) a, x) dx Notice that, in the proof of heorem, we only made local reasonings around the observation open set, and we never used the Dirichlet boundary conditions. Considering ϕ, the unique solution of the wave equation 1) with initial data ϕ, ), t ϕ, )) = ϕ, ϕ 1 ) H 1, π) L, π). Let us apply the estimate of the observability constant proved for one open interval in Section. on, ) β i 1, i+1 ), we obtain using the notations of heorem and replacing 11) by 14) i+1 βi C,,a,λ i, β i ) E a, x) dx ϕ t t, x) dxdt, β i 1 i for every i = 1,, n. Since π E a, x) dx n i+1 i=1 β i 1 E a, x) dx, it follows that whence the conclusion. min 1 i n C,,a,λ i, β i ).4 Proof of Corollary π E a, x) dx ϕ t t, x) dxdt, Assume in a first time that =, β). We follow the same lines as in the proof of heorem, and propose a different way to conclude from the inequality 31). For this reason, we only provide here some explanations to adapt the proof of heorem to this simpler case. Let us decompose the solution ϕ of 1) in the spectral basis {e j } j IN as in 5)-6). Since j = [ a 1 ] + 1, one gets by adapting the proof of Lemma, 1 + j a λ j+1 λ j j j + a >, for every j j. hus, λ j+1 λ j γ j for every j IN. 19

21 and hus, applying Ingham s inequalities see [1] and Remark 5) leads to ϕt, x) dxdt = 1 a k + b k ei sgnk)λ k t θ k ) e k x) dxdt, k Z 1 + a j + b min{π, γ j } j) e j x) dx j=1 t ϕt, x) dxdt = 1 i sgnk)λ k a k + b k ei sgnk)λ k t θ k ) e k x) dxdt, k Z ) 1 4π + π γj λ ja j + b j) e j x) dx, j=1 for every >. Using 19), one gets ϕt, x) 1πγj dxdt γj 4π ) min{π, γ j } Combining this inequality with 3) yields )γ j 4π ) min{π, γ j, } Kη,, β, a 1/ ) γj 4π ) min{π, γ j, } + πγj ) β t ϕt, x) dxdt. π E a, x) dx ϕ t t, x) dxdt, for every η, min{ 4 }), providing an estimate similar to 33) in this case. he conclusion follows, adapting the proof of Corollary 1 to extend the results to observation domains that are the finite union of open intervals. 3 Applications 16, β 3.1 Extension of the previous results to general wave equations Let l and denote two positive constants. We consider the general one dimensional wave equation with Dirichlet boundary conditions tt φt, z) z bz) z φ)t, z) + az)φt, z) = t, z), ), l), φt, ) = φt, l) = t [, ], φ, z) = φ z), t φ, z) = φ 1 z) z [, l], where it is assumed that b and a denote nonnegative functions in L, l) and that there exist a constant b > such that b b a.e. in, l). Recall that for every initial data φ, φ 1 ) H 1, l) L, l), Equation 35) has a unique solution φ C, ; H 1, l)) C 1, ; L, l)). Moreover, the equation 35) is said to be observable on a measurable subset, l) in time if there exists a positive constant c such that l t φt, z) dz dt c φ1 z) + bz)φ z) + az)φ x) ) dz. 35)

22 Let us denote by c, l, b, a, ) the best constant in this inequality, that is c, l, b, a, ) = inf φ,φ 1) H 1,l) L,l) φ,φ 1),) tφt, z) dzdt l φ 1z) + bz)ϕ z) + az)ϕ z) ) dz. 36) Notice in particular that, with the notations of Section 1, one has c, a, ) = c, π, Id, a, ). he following result highlights the link between the observability of 35) and the observability of 1). For that purpose, let us introduce the standard change of variable for Sturm-Liouville equations X : z π z ds l, bs) with l = l ds. Since b b a.e in, l), the function X is continuous nondecreasing and bs) defines thus a change of variable. Proposition 1. Let us assume that the function b belongs to b W,, l), let l = l = π l. We define the function a ) by for almost every z, l) is nonnegative. hus, ds and bs) ax) = l π az) 1 b z) 16 bz) + 1 ) 4 b z), 37) c, l, b, a, ) = l π c, a, X)). Proof. Using the change of variable t = π l t and introducing φt, ) = φt, ), the main equation of 35) rewrites t t φt, z) z bz) z φ)t, z) + ãz) φt, z) =, t, z), ), l), where bz) = l π bz) and ãz) = l π az). Provided that b b and 37) are verified, φ is solution of 35) if and only if the function ϕ : t, x) bz) 1/4 φt, z) is solution of 1). he result follows, by writing that π tφt, z) dzdt l l φ 1z) + bz)φ z) + az)φ z) ) dz = = l φt, z) + bz) φ z, z) + ãz) φ, ) z) t φt, z) dzdt X) t ϕt, x) dxdt π ϕ t, x) + ϕ x, x) + ax)ϕ, x) ) dx., dz Using the correspondence between the observability of Equations 35) and 1), it is easy to deduce estimates of c, l, b, a, ) from the observability constants estimates in heorems 1,, Corollaries 1 and, provided that the assumptions above be verified. 1

23 3. Evaluating the cost of control for the Hilbert Uniqueness Method. Consider the internally controlled wave equation on, π) with Dirichlet boundary conditions tt yt, x) xx yt, x) + ax)yt, x) = h t, x) t, x), ), π), yt, ) = yt, π) = t [, ], y, x) = y x), t y, x) = y 1 x) x, π), 38) where h is a control supported in [, ] where, π) is Lebesgue measurable. Recall that for all initial data y, y 1 ) H 1, π) L, π) and every h L, ), π)), the problem 38) is well posed and its solution y belongs to C, ; H 1, π)) C 1, ; L, π)) C, ; H 1, π)). In what follows, we will endow the space H 1, π) with the inner product H 1, π)) ϕ, ψ) π ϕ x)ψ x) dx + π ax)ϕx)ψx) dx, instead of the standard inner-product of H 1, π), according to the choice of norm in the energy space made to introduce the inequality 3). herefore, the space H 1, π) will be endowed with the dual norm. he exact null controllability problem settled in these spaces consists of finding a control h steering the control system 38) to y, ) = t y, ) =. It is well known that the Hilbert Uniqueness Method HUM, see [16, 17]) provides a way to design the unique control solving the above exact null controllability problem and having moreover a minimal L, ), π)) norm. Using the observability inequality ϕt, x) dxdt c ϕ, ϕ 1 ) L,π) H 1,π), 39) where c is a positive constant only depending on and ), valuable for every solution ϕ of the adjoint system tt ϕt, x) xx ϕt, x) + ax)ϕt, x) =, t, x), ), π), ϕt, ) = ϕt, π) =, t [, ], ϕ, x) = ϕ x), t ϕ, x) = ϕ 1 x), x [, π], 4) and every π, the functional J ϕ, ϕ 1 ) = 1 ϕt, x) dxdt ϕ 1, y H 1,H 1 + ϕ, y 1 L, 41) has a unique minimizer still denoted ϕ, ϕ 1 )) in the space L, π) H 1, π), for all y, y 1 ) H 1, π) L, π). In 41) the notation, H 1,H 1 stands for the duality bracket between H 1, π) and H 1, π), and the notation, L stands for the usual scalar product of L, π). he HUM control h steering y, y 1 ) to, ) in time is then given by h t, x) = χ x)ϕt, x), 4) for almost all t, x), ), π), where χ denotes the characteristic function of the measurable set and ϕ is the solution of 4) with initial data ϕ, ϕ 1 ) minimizing J. he HUM operator Γ is then defined by Γ : H 1, π) L, π) L, ), π)) y, y 1 ) h

24 and the norm of the operator Γ is given by { h L, ),π) Γ = sup y, y 1 ) H 1,π) L,π) y, y 1 ) H 1, π) L, π) \ {, )} }. Recall that, by duality, the best constant in 39) coincides with the constant c, a, ) defined by 4), that is the optimal constant in the observability inequality 3). Proposition. Let > and let be measurable subset of, π). If c, a, ) > then and if c, a, ) =, then Γ = +. Γ = 1 c, a, ), For a proof of this result, we refer for instance to [5, ]. As a consequence, the results in heorems 1,, Corollaries 1 and permit to provide an estimate of the cost of control given by the Hilbert Uniqueness Method. 3.3 Estimating the stabilization rate of the damped wave equation. Consider the damped wave equation on, π) with Dirichlet boundary conditions tt yt, x) xx yt, x) + kχ x) t yt, x) = t, x), ), π), yt, ) = yt, π) = t [, ], y, x) = y x), t y, x) = y 1 x) x, π), with k >. Recall that for all initial data y, y 1 ) H 1, π) L, π), the problem 43) is well posed and its solution y belongs to C, ; H 1, π)) C 1, ; L, π)). he energy associated to System 43) is defined by Et) = π t yt, x) + x yt, x) ) dx. According to [3, 4], if has positive measure, this system is exponentially stable, i.e. its energy is known to obey Et) CE)e δt 44) for t >, where C and δ denote positive constant that do not depend on the initial data. We define the decay rate δk, ) as the largest such δ, in other words δk, ) = sup{ δ C > such that Et) CE)e δt, for every t > and for every solution of 43)}. According to the following proposition, one can provide an estimate of the constants C and δ or δk, )) from the estimates proved in heorems 1,, Corollaries 1 and. Proposition 3. Let be a measurable subset of, π) and denotes the minimal observability time 5. hus, for every y, y 1 ) H 1, π) L, π) and t, with Et) E)e δt, δ = 1 ) ln )c, a, ) 1 + )c,, a, ) the constant c, a, ) denoting the observability constant defined by 4). For a proof of this result, we refer for instance to [8]. 5 For instance, if =, β) one has = max{, π β} 43) 3

25 4 Examples and numerical illustrations We provide here some numerical simulations and illustrations of observability constants estimates using both methods spectral versus propagation) δ a) = 4π δ b) Asymptotic behavior in Figure 3: a.) = and =, δ) π δ, π). Plots of C, ) - -), C,,,λ //) and C,,,λ ) with respect to δ ε x 1 3 a) Small values of ε t ε b) Large values of ε Figure 4: a : x εx, =, 3 ) and = 8π. Plots of C a, ) - -), C,,a,λ //) and C,,a,λ ) with respect to ε he case a ) = is investigated on Figure 3. We chose as observation subset =, δ) 4

26 π δ, π) with δ π. In that case, the observability constant can be explicitly computed 6 whenever is a multiple of π, as well as the limit of c, a, )/. o compute the estimates 1) and 13), we fully use Remark 9. More precisely, on Figure 3a), the observation time is = 4π. For such a choice of time, the estimate C, ) obtained by the spectral method coincides with the value of the observability constant. he graphs of the estimates obtained by each method, namely C,,,λ defined by 13) and C,,,λ defined by 15), are represented with respect to the parameter δ and compared with the observability constant C, ). on Figure 3b), the graphs of the limit of the estimates as goes to + obtained by each method, namely C,,,λ and C,,,λ, are represented with respect to the parameter δ and as goes to +. compared with the limit of the observability constant C, ) In accordance with Figure 3, one can easily show that the quantity C,,a,λ decreases whenever the measure of is large enough, which is a confirmation that this estimate is not sharp. On Figure 4, we consider the potential a : x εx with ε >, the observation set =, 3 ) and the observation time = 8π. More precisely, the graphs of the estimates obtained by each method, namely C a, ) defined by 8), C, defined by 13) and C,a,λ,,a,λ defined by 15), are represented with respect to the parameter ε. On Figure 4b), only the graphs of C,,a,λ propagation method, heorem ) and C,,a,λ propagation method, Corollary ) are plotted because of the smallness of the range of ε for which the spectral method makes sense due to Condition 7)). Moreover, the range of ε for which C,,a,λ is defined is restricted, since we use Ingham s inequalities see the proof of Corollary for more details). According to Figures 3 and 4a), the spectral method seems more accurate than propagation methods, but needs the strong condition 7) on the potential function a ) see the comments and comparison between both methods at the end of the previous section). We provide on the figures 5, 6 and 7 several other examples illustrating each method. 6 Let =, δ) π δ, π) and is a multiple of π. he observability constant is exactly [ ] c, a, ) = C, ) = δ sinδ)), π and one has the asymptotic expansion as +. c, a, ) δ sinδ)). π 5

27 ε x 1 4 a) Small positive values of ε ε b) Large positive values of ε Figure 5: a : x + εx, =, π/3) and = 1π. Plots of C, ) - -), C,,,λ C,,,λ ) with respect to ε //) and ε x 1 4 Figure 6: a : x ε cos x, =, π 3 ) and = 1π. Plots of C a, ) - -), C,,a,λ C,,a,λ ) with respect to ε //) and References [1] J. M. Ball, M. Slemrod, Nonharmonic Fourier series and the stabilization of distributed semilinear control systems, Comm. Pure Appl. Math ), no. 4, [] C. Bardos, G. Lebeau, J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim ), no. 5, [3] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math ), no. 1,

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