On the observability of time-discrete conservative linear systems

Transcription

1 On the observability of time-discrete conservative linear systems Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Abstract. We consider various time discretization schemes of abstract conservative evolution equations of the form ż = Az, where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair A, B is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given in []. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations. Contents 1. Introduction. The implicit mid-point scheme 7 3. General time-discrete schemes General time-discrete schemes for first order systems The Newmark method for second order in time systems Applications 4.1. Application of Theorem Boundary observation of the Schrödinger equation Boundary observation of the linearized KdV equation Application of Theorem This work was supported by the Grant MTM5-714 and the i-math project of the Spanish MEC, the DOMINO Project CIT in the PROFIT program of the MEC Spain and the SIMUMAT projet of the CAM Spain. This work began while the authors visited the Isaac Newton Institute of Cambridge within the programm Highly Oscillatory Problems and they acknowledge the hospitality and support of the Institute. This work has been completed while the first author visited IMDEA-Matemáticas, and he acknowledges the support of this Institute.

2 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua 4.3. Application of Theorem Fully discrete schemes Main statement Applications The fully discrete wave equation The 1-d string with rapidly oscillating density 3 6. On the admissibility condition The time-continuous setting The time-discrete setting Further comments and open problems 39 References Introduction Let be a Hilbert space endowed with the norm and let A : DA be a skew-adjoint operator with compact resolvent. Let us consider the following abstract system: żt = Azt, z = z. 1.1 Here and henceforth, a dot denotes differentiation with respect to the time t. The element z is called the initial state, and z = zt is the state of the system. Such systems are often used as models of vibrating systems e.g., the wave equation, electromagnetic phenomena Maxwell s equations or in quantum mechanics Schrödinger s equation. Assume that Y is another Hilbert space equipped with the norm Y. We denote by L, Y the space of bounded linear operators from to Y, endowed with the classical operator norm. Let B LDA, Y be an observation operator and define the output function yt = Bzt. 1. In order to give a sense to 1., we make the assumption that B is an admissible observation operator in the following sense see [6]: Definition 1.1. The operator B is an admissible observation operator for system if for every T > there exists a constant K T > such that T yt Y dt K T z, z DA. 1.3 Note that if B is bounded in, i.e. if it can be extended such that B L, Y, then B is obviously an admissible observation operator. However, in applications, this is often not the case, and the admissibility condition is then a consequence of a suitable hidden regularity property of the solutions of the evolution equation 1.1.

3 Observability of time discrete systems. 3 The exact observability property of system can be formulated as follows: Definition 1.. System is exactly observable in time T if there exists k T > such that T k T z yt Y dt, z DA. 1.4 Moreover, is said to be exactly observable if it is exactly observable in some time T >. Note that observability issues arise naturally when dealing with controllability and stabilization properties of linear systems see for instance the textbook [15]. Indeed, controllability and observability are dual notions, and therefore each statement concerning observability has its counterpart in controllability. In the sequel, we mainly focus on the observability properties of It was proved in [] and [17] that system is exactly observable if and only if the following assertion holds: There exist constants M, m > such that M iωi Az + m Bz Y 1.5 z, ω lr, z DA. This spectral condition can be viewed as a Hautus-type test, and generalizes the classical Kalman rank condition, see for instance [5]. To be more precise, if 1.5 holds, then system is exactly observable in any time T > T = πm see [17]. There is an extensive literature providing observability results for wave, plate, Schrödinger and elasticity equations, among other models and by various methods including microlocal analysis, multipliers and Fourier series, etc. Our goal in this paper is to develop a theory allowing to get results for time-discrete systems as a direct consequence of those corresponding to the time-continuous ones. Let us first present a natural discretization of the continuous system. For any >, we denote by z k and y k respectively the approximations of the solution z and the output function y of system at time t k = k for k Z. Consider the following implicit midpoint time discretization of system 1.1: z k+1 z k z k+1 + z k = A z given. The output function of 1.6 is given by, in, k Z, 1.6 y k = Bz k, k Z. 1.7 Note that is a discrete version of Taking into account that the spectrum of A is purely imaginary, it is easy to show that z k is conserved in the discrete time variable k Z, i.e. z k = z. Consequently the scheme under consideration is stable and its convergence

4 4 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua in the classical sense of numerical analysis is guaranteed in an appropriate functional setting. The uniform exact observability problem for system 1.6 is formulated as follows: To find a positive constant k T, independent of, such that the solutions z k of system 1.6 satisfy: k T z y k, 1.8 Y k,t/ for all initial data z in an appropriate class. Clearly, 1.8 is a discrete version of 1.4. Note that this type of observability inequalities appears naturally when dealing with stabilization and controllability problems see, for instance, [15], [5] and [9]. For numerical approximation processes, it is important that these inequalities hold uniformly with respect to the discretization parameters here only to recover uniform stabilization properties or the convergence of discrete controls to the continuous ones. We refer to the review article [9] and the references therein for more precise statements. To our knowledge, there are very few results addressing the observability issues for time semi-discrete schemes. We refer to [18], where the uniform controllability of a fully discrete approximation scheme of the 1-d wave equation is analyzed, and to [7], where a time discretization of the wave equation is analyzed using multiplier techniques. Especially, the results in [7] may be viewed as a particular instance of the abstract models we address here. In the sequel, we are interested in understanding under which assumptions inequality 1.8 holds uniformly on. One expects to do it so that, when letting, one recovers the observability property of the continuous model. It can be done by means of a spectral filtering mechanism. More precisely, since A is skew-adjoint with compact resolvent, its spectrum is discrete and σa = {iµ j : j ln}, where µ j j ln is a sequence of real numbers. Set Φ j j ln an orthonormal basis of eigenvectors of A associated to the eigenvalues iµ j j ln, that is: Moreover, we define AΦ j = iµ j Φ j. 1.9 C s = span {Φ j : the corresponding iµ j satisfies µ j s}. 1.1 We will prove that inequality 1.8 holds uniformly with respect to > in the class C δ/ for any δ > and for T δ large enough, depending on the filtering parameter δ. This result will be obtained as a consequence of the following theorem: Theorem 1.3. Let δ >. Assume that we have a family of vector spaces δ, and a family of unbounded operators A, B depending on the parameter > such that

5 Observability of time discrete systems. 5 H1 For each >, the operator A is skew-adjoint on δ,, and the vector space δ, is globally invariant by A. Moreover, A z δ z, z δ,, > H There exists a positive constant C B such that B z Y C B A z, z δ,, >. 1.1 H3 There exist two positive constants M and m such that M A iωiz + m B z Y z, z δ, DA, ω lr, > Then there exists a time T δ such that for all time T > T δ, there exists a constant k T,δ such that for small enough, the solution of z k+1 z k z k+1 + z k = A, in δ,, k Z, with initial data z δ, satisfies k T,δ z k,t/ B z k Y, z δ, Moreover, T δ can be taken to be such that M T δ = π [1 + δ + m C δ 4 ] 1/, B where C B is as in.1. As we shall see in Theorem.1, taking A = A, B = B and δ/ = C δ/, Theorem 1.3 provides an observability result within the class C δ/ for system , as a consequence of assumption 1.5 and since B LDA, Y. Theorem 1.3 is also useful to address observability issues for more general time-discretization schemes of than 1.6. For instance, one can consider time semi-discrete schemes of the form z k+1 = T z k, y k = Bz k, 1.17 where T is a linear operator with the same eigenvectors as the operator A. We will prove that, under some general assumptions on T, inequality 1.8 holds uniformly on for solutions of 1.17 when the initial data are taken in the class C δ/, as we shall see in Theorem 3.1. We can also consider second order in time systems such as üt + A ut =, u = u, u = v, 1.18 where A is a positive self-adjoint operator. Of course, such systems can be written in the same first-order form as 1.1. However, there are time-discretization schemes such as the Newmark method which cannot be put in the form 1.17.

6 6 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Hence we present a specific analysis of the Newmark method for 1.18, still based on Theorem 1.3. One of the interesting applications of our results is that it allows us to develop a two-step strategy to study the observability of fully discrete approximation schemes of First, one uses the observability properties for space semidiscrete approximation schemes, uniformly with respect to the space mesh-size parameter, as it has already been done in many cases see [3], [6], [7], [1], [19], [], [8] and [9] for more references. Second, from the results of this paper on time discretizations, the uniform observability with respect to both the time and space mesh-sizes for the fully discrete approximation schemes is derived. To our knowledge, the observability issues for fully discrete approximation schemes have been studied only in [18], in the very particular case of the 1-d wave equation. The results we present here can be applied to a much wider class of systems and time-discretization schemes. To complete our analysis of the discretizations of system , we also analyze admissibility properties for the time semi-discrete systems introduced throughout this paper. They are useful when deriving controllability results out of the observability ones. More precisely, it allows proving controllability results by means of duality arguments combined with observability and admissibility results see for instance the textbook [15] and the survey article [9]. In particular, we prove that the admissibility inequality 1.3 can be interpreted in terms of the behavior of wave packets. From this wave packet estimate, we will deduce admissibility inequalities for the time semi-discrete schemes. This part can be read independently from the rest of the article. The outline of this paper is as follows. In Section we prove Theorem 1.3, from which we deduce the uniform observability property 1.8 for system , assuming that the initial data are taken in some subspace of filtered data C δ/ for arbitrary δ >. Our proof of Theorem 1.3 is mainly based on the resolvent estimate 1.13, combined with standard Fourier arguments adapted to the time-discrete setting. In Section 3, we show how to apply Theorem 1.3 to obtain similar results for time semi-discrete approximation schemes such as 1.17 and the Newmark approximation schemes, for which we prove that a uniform observability inequality holds as well, provided the initial data belong to C δ/. In Section 4, we give some applications to the observability of some classical conservative equations, such as the Schrödinger equation or the linearized KdV equation, etc. In Section 5, we give some applications of our main results to fully discrete schemes for skew-adjoint systems as 1.1. In Section 6, we present admissibility results similar to 1.3 for the time semi-discrete schemes used along the article. We end the paper by stating some further comments and open problems.

7 . The implicit mid-point scheme Observability of time discrete systems. 7 In this section we show the uniform observability of system , which can be seen as a direct consequence of Theorem 1.3. In other words, its proof is a simplified version of the one of Theorem 1.3. To avoid the duplication of the process, we only give the proof of the latter one, which is more general. Let us first introduce some notations and definitions. The Hilbert space DA is endowed with the norm of the graph of A, which is equivalent to A since A has a compact resolvent. It follows that B LDA, Y implies Bz Y C B Az, z DA..1 We are now in position to claim the following theorem based on the resolvent estimate 1.5: Theorem.1. Assume that A, B satisfy 1.5 and that B LDA, Y. Then, for any δ >, there exist T δ and > such that for any T > T δ and,, there exists a positive constant k T,δ, independent of, such that the solution z k of 1.6 satisfies k T,δ z Bz k, Y z C δ/.. k,t/ Moreover, T δ can be taken to be such that T δ = π [M 1 + δ + m C δ 4 ] 1/, B where C B is as in.1. Remark.. If we filter at a scale smaller than, for instance in the class C δ/ α, with α < 1, then δ in.3 vanishes as tends to zero. In that case the uniform observability time T we obtain is T = πm, which coincides with the time obtained by the resolvent estimate 1.5 in the continuous setting see [17]. Note that, however, even in the continuous setting, in general πm is not the optimal observability time. Proof of Theorem.1. Theorem.1 can be seen as a direct consequence of Theorem 1.3, which will be proved below. Indeed, one can easily verify that H1 H3 hold by taking A = A, B = B and δ, = C δ/. Before getting into the proof of Theorem 1.3, let us first introduce the discrete Fourier transform at scale, which is one of the main ingredients of the proof of Theorem 1.3. Definition.3. Given any sequence u k l Z, we define its Fourier transform as: ûτ = k Z u k exp iτk, τ π, π]..4

8 8 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua For any function v L π/, π/, we define the inverse Fourier transform at scale > : ṽ k = 1 π/ vτ expiτk dτ, k Z..5 π π/ According to Definition.3, and the Parseval identity holds These properties will be used in the sequel. û = u, ˆṽ = v,.6 1 π/ ûτ dτ = u k..7 π π/ k Z Proof of Theorem 1.3. The proof is split into three parts. Step 1: Estimates in the class δ,. Let us take z δ,. Then the solution of 1.14 has constant norm since A is skew-adjoint see H1. Indeed, I + z k+1 = A I A where the operator T is obviously unitary. Further, since we get that for any k, z k + z k+1 = 1 z + z 1 I + T z k = = z k + z k+1 z k := T z k, I I A δ z k, 1 z, as a consequence of 1.11 and the skew-adjointness assumption H1 of A. Step : The resolvent estimate. Set χ H 1 lr and χ k = χk. Let g k = χ k z k, and f k = gk+1 g k g k+1 + g k A..9

9 Observability of time discrete systems. 9 One can easily check that z k+1 + z k + χk+1 + χ k z k+1 z k χ k+1 + χ k z k+1 + z k A + χk+1 χ k z k+1 z k = χk+1 χ k z k + z k+1 z k+1 z k A 4 χ k+1 χ k = I z A k + z k+1 4 f k = χk+1 χ k Especially, recalling.8 and 1.11,.1 implies f k χ k+1 χ k z + z δ In particular, f k l Z;. Taking the Fourier transform of.9, for all τ π/, π/, we get ˆfτ = k Z f k exp ikτ exp ikτ = g k+1 g k g k+1 + g k A k Z = expiτ 1 expiτ + 1 A k Z = i τ I tan A ĝτ exp i τ cos We claim the following Lemma: Lemma.4. The solution z k in 1.14 satisfies g k exp ikτ τ αm χ k + χ k+1 z k + z k+1 B k Z Y z + z 1 [ a 1 χ k + χ k+1 a ] χ k+1 χ k,.13 k Z k Z with a 1 = 1 1, β a = M 1 + δ 4 + m CB δ 4 α δ β 1, for any α > and β > 1, where C B, M, m are as in

10 1 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Proof of Lemma.4. Let Gτ = ĝτ expi τ cosτ..15 By its definition and the fact that z k δ,, it is obvious that Gτ δ,. In view of.1, applying the resolvent estimate 1.13 to Gτ, integrating on τ from π/ to π/, it holds π/ M ˆfτ π/ dτ + m B Gτ Y dτ π/ π/ π/ π/ Applying Parseval s identity.7 to.16, and noticing that G k = gk + g k+1 gk + g, i.e. Gτ = k+1 τ, we get M f k + m g k B + g k+1 k Z k Z Y g k + g k+1 k Z Gτ dτ Now we estimate the three terms in.17. The first term can be bounded above in view of.11. Second, since g k+1 + g k = χk+1 + χ k z k+1 + z k + χ k+1 χ k z k+1 z k,.18 using a + b 1 + α a b, α we deduce that g k+1 B + g k χ k+1 + χ k z k+1 + z k 1 + α B Y Y χ k+1 χ k z k+1 z k B α 16 Y χ k+1 + χ k z k+1 + z k 1 + α B α δ 4 16 C B Y χ k+1 χ k z + z 1..19

11 Observability of time discrete systems. 11 In.19 we use the fact that recalling 1.11 and 1.1 z k B + z k+1 A z C B A k + z k+1 δ C B z + z 1. Y Finally, for any β > 1, recalling.8, 1.11 and.18, we get g k+1 + g k 1 1 χ k+1 + χ k z k+1 + z k β χ k+1 χ k z k+1 z k β χ k+1 + χ k z + z 1 β 4 χ k+1 χ k z + z 1 β 1 A, 1 1 χ k+1 + χ k z + z 1 β δ χ k+1 χ k z + z 1 β 1 4. where we used a + b 1 1 a β 1 b. β Applying.11,.19 and. to.17, we complete the proof of Lemma.4. Step 3: The observability estimate. This step is aimed to derive the observability estimate 1.15 stated in Theorem 1.3 from Lemma.4 with explicit estimates on the optimal time T δ. First of all, let us recall the following classical Lemma on Riemann sums: Lemma.5. Let χt = φt/t with φ H H 1, 1, extended by zero outside, T. Recalling that χ k = χk, the following estimates hold: χ k + χ k+1 T φ L,1 T φ L,1 φ, L,1 k Z χ k+1 χ k 1 T φ L,1 T φ φ.1. L,1 L,1 k Z Sketch of the proof of Lemma.5. It is easy to show that for all f = ft C 1, T and sequence τ k [k, k + 1], it holds T ftdt fτ k fs ds dt [k,k+1] k,t/ k,t/ T f dt..

12 1 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Replacing f by φ we get the first inequality.1. Similarly, replacing f by φ, the second one can be proved too. Taking Lemma.4 and.5 into account, the coefficient of z + z 1 / in.13 tends to k T,δ,α,β,φ = 1 [ m 1 1 T φ L 1 + α β,1 M 1 + δ + m C 4 B δ 4 1 α 16 T φ L,1 when. Note that k T,δ,α,β,φ is an increasing function of T tending to when T + and to + when T. Let T δ,α,β,φ be the unique positive solution of k T,δ,α,β,φ =. Then, for any time T > T δ,α,β,φ, choosing a positive k T,δ such that < k T,δ < k T,δ,α,β,φ, there exists > such that for any <, the following holds: z + z 1 k T,δ B z k..3 Y k,t/ This combined with.8 yields This construction yields the following estimate on the time T δ in Theorem 1.3. Namely, for any α >, β > 1 and smooth function φ, compactly supported in [, 1]: φ [ T δ L φ L β β 1 ] 1/ [M 1 + δ + m C 4 B α We optimize in α, β and φ by choosing α =, β = and ß sinπt, t, 1 φt =, elsewhere, which is well-known to minimize the ratio φ L. φ L δ 4 16 ] 1/. ],.4 For this choice of φ, this quotient equals π, and thus we recover the estimate This completes the proof of Theorem 1.3. Theorem.1 has many applications. Indeed, it roughly says that, for any continuous conservative system, which is observable in finite time, there exists a time semi-discretization which uniformly preserves the observability property in finite time, provided the initial data are filtered at a scale 1/. Later, using formally some microlocal tools, we will explain why this filtering scale is the optimal

13 Observability of time discrete systems. 13 one. Note that in Theorem 7.1 of [7] this scale was proved to be optimal for a particular time-discretization scheme on the wave equation. Besides, as we will see in Section 3, Theorem 1.3 is a key ingredient to address observability issues. 3. General time-discrete schemes 3.1. General time-discrete schemes for first order systems In this section, we deal with more general time-discretization schemes of the form We will show that, under some appropriate assumptions on the operator T, inequality 1.8 holds uniformly on for solutions of 1.17 when the initial data are taken in the class C δ/. More precisely, we assume that 1.17 is conservative in the sense that there exist real numbers λ j, such that T Φ j = expiλ j, Φ j. 3.1 Moreover, we assume that there is an explicit relation between λ j, and µ j as in 1.9 of the following form: λ j, = 1 hµ j, 3. where h : [ δ, δ] [ π, π] is a smooth strictly increasing function, i.e. hη π, inf{h η, η δ} >. 3.3 Roughly speaking, the first part of 3.3 reflects the fact that one cannot measure frequencies higher than π/ in a mesh of size. The second part is a non-degeneracy condition on the group velocity see [1] of solutions of 1.17 which is necessary to guarantee the propagation of solutions that is required for observability to hold. We also assume hη 1 as η. 3.4 η This guarantees the consistency of the time-discrete scheme with the continuous model 1.1. We have the following Theorem: Theorem 3.1. Assume that A, B satisfy 1.5 and that B LDA, Y. Under assumptions 3.1, 3., 3.3 and 3.4, for any δ >, there exists a time T δ such that for all T > T δ, there exists a constant k T,δ > such that for all small enough, any solution of 1.17 with initial value z C δ/ satisfies k T,δ z z k B + z k k,t/ Y

14 14 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Besides, we have the following estimate on T δ : [ hδ { cos T δ π M 1 + tan 4 hη/ } sup η δ h η where C B is as in.1. + m C B sup η δ { hη } η tan tan 4 hδ ] 1/, 3.6 Proof. The main idea is to use Theorem 1.3. Hence we introduce an operator A such that the solution of 1.17 coincides with the solution of the linear system z k+1 z k z k + z k+1 = A, z = z. 3.7 This can be done defining the action of the operator A on each eigenfunction: where Indeed, if A Φ j = ik µ j Φ j, 3.8 k ω = hω tan. 3.9 z = a j Φ j, then the solution of 1.17 can be written as z k = a j φ j expiλ j k = a j φ j expihµ j k and the definition of A follows naturally. Obviously, when the scheme 1.17 under consideration is the one of Section, that is 1.6, the operator A is precisely the operator A. Then 3.5 would be a straightforward consequence of Theorem 1.3, if we could prove the resolvent estimate for A. We will see in the sequel that a weak form of the resolvent estimate holds, and that this is actually sufficient to get the desired observability inequality. In the sequel, δ is a given positive number, determining the class of filtered data under consideration. Step 1: A weak form of the resolvent estimate. By hypothesis 1.5, M A iωz + m Bz Y z, z DA, ω lr. 3.1 For z C δ/, that is one can easily check that z = µ j δ/ A iωz = a j µ j ω a j φ j, 3.11

15 Observability of time discrete systems. 15 and A iωz = a j k µ j ω. Especially, for any ω lr, this last estimate takes the form A ik ωz = a j k µ j k ω with k as in 3.9. Thus, taking ε >, it follows that for any ω < δ + ε/, { } A ik ωz inf k ω A iωz Hence, setting α,ε = k δ + ε ω δ+ε 1,, C δ,ε = inf{k ω : ω δ + ε} 3.1 which is finite in view of 3.3, we get the following weak resolvent estimate: Cδ,εM A iω z + m Bz Y z, z C δ/, ω α,ε Our purpose is now to show that this is enough to get the time-discrete observability estimate. We emphasize that the main difference between 3.13 and 1.13 is that 1.13 is assumed to hold for all ω lr while 3.13 only holds for ω α,ε. Step : Improving the resolvent estimate Here we prove that 3.13 can be extended to all ω lr. Indeed, consider ω such that ω α,ε and z C δ/ as in Then A iωz µ j δ/ µ j δ/ k µ j k δ + ε δ k Using the explicit expression 3.9 of k, we get a j. δ + ε a k j ε inf ω [δ,δ+ε] k ω z. A iωz ε inf η [δ,δ+ε] {h η} z Therefore, for each ε >, in view of 3.3 and 3.1, there exists ε > such that, for ε Cδ,εM A iω z + m Bz Y z, z C δ/, ω lr. 3.15

16 16 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Step 3: Application of Theorem 1.3. First, one easily checks from that A z δ z, z C δ/, 3.16 with δ = tanhδ/. Second, we check that there exists a constant C B,δ such that Bz Y C B,δ A z, z C δ/, 3.17 where C B is as in.1. Indeed, for z C δ/, { k ω } Az A z ω, and therefore one can take where sup ω δ C B,δ = β δ C B, 3.18 { hη } β δ = sup η δ η tan, which is finite from hypothesis 3.3 and 3.4. Third, the resolvent estimate 3.15 holds. Then Theorem 1.3 can be applied and proves the observability inequality 3.5 for the solutions of 1.17 with initial data in C δ/. Besides, we have the following estimate on the observability time T δ,ε : T δ,ε = π [1 + δ M δ Cδ,ε + m C 4 Bβδ 4 ] 1/. 16 In the limit ε, T δ,ε converges to an observability time T δ. Besides, using the explicit form of the constants C δ,ε, δ and β δ one gets The Newmark method for second order in time systems In this subsection we investigate observability properties for time-discrete schemes for the second order in time evolution equation Let H be a Hilbert space endowed with the norm H and let A : DA H be a self-adjoint positive operator with compact resolvent. We consider the initial value problem 1.18, which can be seen as a generic model for the free vibrations of elastic structures such as strings, beams, membranes, plates or threedimensional elastic bodies. The energy of 1.18 is given by Et = ut A H + 1/ ut, 3.19 H which is constant in time. We consider the output function yt = B 1 ut + B ut, 3.

17 Observability of time discrete systems. 17 where B 1 and B are two observation operators satisfying B 1 LDA, Y and B LDA 1/, Y. In other words, we assume that there exist two constants C B,1 and C B,, such that A 1/ B 1 u Y C B,1 A u H, B v Y C B,. 3.1 In the sequel, we assume either B 1 = or B =. This assumption is needed for technical reasons, as we shall see in Remark 3.3 and in the proof of Theorem 3.. System can be put in the form Indeed, setting equation 1.18 is equivalent to v z 1 t = u + ia 1/ u, z t = u ia 1/ u, 3. ż = Az, z = Å z1 z ã Ç 1/ ia, A = ia 1/ å, 3.3 for which the energy space is = H H with the domain DA = DA 1/ DA 1/. Moreover, the energy Et given in 3.19 coincides with half of the norm of z in. Note that the spectrum of A is explicitly given by the spectrum of A. Indeed, if µ j j N µ j > is the sequence of eigenvalues of A, i.e. A φ j = µ jφ j, j N, with corresponding eigenvectors φ j, then the eigenvalues of A are ±iµ j, with corresponding eigenvectors Å ã Å ã φj Φ j =, Φ j =, j ln. 3.4 φj Besides, in the new variables 3., the output function is given by yt = Bzt = B 1 A 1/ iz t iz 1 t z1 t + z t + B. 3.5 Recalling the assumptions on B 1 and B in 3.1, the admissible observation B belongs to LDA, Y. In the sequel, we assume that the system is exactly observable. As a consequence of this we obtain that system is exactly observable and therefore the resolvent estimate 1.5 holds. We now introduce the time-discrete schemes we are interested in. For any > and β >, we consider the following Newmark time-discrete scheme for system 1.18: u k+1 + u k 1 u k + A βu k βu k + βu k 1 =, u + u 1, u1 u = u, v DA 1 H. 3.6

18 18 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua The energy of 3.6 is given by E k+1/ = A1/ u k + u k+1 + u k+1 u k + 4β 1 u k+1 u k 4 A1/, k Z, 3.7 which is a discrete counterpart of the continuous energy Multiplying the first equation of 3.6 by u k+1 u k 1 / and using integration by parts, it is easy to show that 3.7 remains constant with respect to k. Furthermore, we assume in the sequel that β 1/4 to guarantee that system 3.6 is unconditionally stable. The output function is given by the following discretization of 3.: u y k+1/ k + u k+1 u k+1 u k = B 1 + B, 3.8 where, as in 3., we assume that either B 1 or B vanishes. For any s >, we define C s as in 1.1. Note that this space is invariant under the actions of the discrete semi-groups associated to the Newmark time-discrete schemes 3.6. We have the following theorem: Theorem 3.. Let β 1/4 and δ >. We assume that either B 1 or B. Then there exists a time T δ such that for all T > T δ, there exists a positive constant k T,δ, such that for small enough, the solution of 3.6 with initial data u, v C δ/ satisfies k T,δ E 1/ y k+1/, 3.9 Y k,t where y k+1/ is defined in 3.8 and B 1, B satisfy 3.1. Besides, T δ can be chosen as T δ,1 = π [1 + βδ 1 + β 1 M 4 δ + m CB,1 δ 4] 1/, if B = and as T δ, = π [1 + βδ 1 + β 1 δ M + m C δ 4 ] 1/, B, if B 1 =. Remark 3.3. This result, and especially the time estimates 3.3 and 3.31 on the observability time need further comments. As in Theorem.1, we see that, if we filter at a scale smaller than, for instance in the class C δ/ α, with α < 1, then the uniform observability time T is given by T = πm, which coincides with the value obtained by the resolvent estimate 1.5 in the continuous setting.

19 Observability of time discrete systems. 19 Note that the estimates 3.3 and 3.31 do not have the same growth in δ when δ goes to. This fact does not seem to be natural because the observability time is expected to depend on the group velocity see [1] and not on the form of the observation operator. By now we could not avoid the assumption that either B 1 or B vanishes, the special case β = 1/4 being excepted. However, we can deal with an observable of the form 1/ u y k+1/ = B 1 I + β 1/4 k + u k+1 u k+1 u k A + B, 3.3 with both non-trivial B 1 and B. Indeed, in this case, the operator B arising in the proof of Theorem 3. does not depend on and therefore the proof works as in the case B 1 =, and yields the time estimate However, this observation operator, which compares to the continuous one 3. when δ, does not seem to be the most natural discretization of 3.5. When β = 1/4, both 3.3 and 3.31 have the same form. Besides, one can easily adapt the proof to show that when β = 1/4, we can deal with a general observation operator B as in 3.. Actually, the Newmark scheme 3.6 with β = 1/4 is equivalent to a midpoint scheme, and therefore Theorem.1 applies. Proof. Step 1. We first transform system 3.6 into a first order time-discrete scheme similar to 3.3. For this, we define Then 3.6 can be rewritten as A, = A [I + β 1/4 A ] u k+1 + u k 1 u k u k 1 + u k + u k+1 + A, = As in 3., using the following change of variables z k+1/ 1 = uk+1 u k + ia 1/, z k+1/ = uk+1 u k ia 1/, system 3.6 and also system 3.34 is equivalent to with A = z k+1/ z k 1/ Ç ia 1/, ia 1/, u k + u k+1 u k + u k+1,, 3.35 z k 1/ + z k+1/ = A, 3.36 å, z k+1/ = Ñ z k+1/ 1 z k+1/ é. 3.37

20 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua Consequently, the observation operator y k+1/ in 3.8 is given by y k+1/ = B 1 A 1/ k+1/ iz iz k+1/ k+1/ 1 z 1 + z k+1/, + B = B z k+1/ Step. We now verify that system satisfies the hypothesis of Theorem 1.3. We first check H1. It is obvious that the eigenvectors of A are the same as those of A see 3.4. Moreover, for any Φ j we compute A Φ j = l j Φ j, with l j = iµ» j β 1/4 µ j In other words, we are close to the situation considered in Subsection 3.1, and the time semi-discrete approximation scheme 3.36 satisfies the hypothesis 3.1, 3., 3.3, 3.3 and 3.4 with the function h defined by η hη = arctan β 1/4η. 3.4 In particular, this implies that 3.16 holds in the class C δ/, and takes the form A z Second, we check hypothesis H: B z Y A z H C B,1 A A 1 δ 1 + β 1/4δ z, z C δ/. 3.41, LCδ/,H A 1/ +C B, A 1/, LCδ/,H A z H 1 + β 1/4δ C B,1 +»1 + β 1/4δ C B, C B,δ A z H. 3.4 The third point is more technical. Following the proof of Theorem 3.1, for any ε >, we obtain the following resolvent estimate: Cδ,εM A iω z + m Bz Y z, z C δ/, ω lr, 3.43 where C δ,ε is given by 3.1, with ω k ω =. 1 + β 1/4ω Straightforward computations show that, actually, 3/. C δ,ε = 1 + β 1/4δ + ε 3.44

21 Observability of time discrete systems. 1 Our goal now is to derive from 3.43 the resolvent estimate H3 given in Here, we will handle separately the two cases B 1 = and B =. The case B 1 =. Under this assumption, B = B, and therefore, 3.43 is the resolvent estimate H3 we need. The case B =. In this case, we observe that Ç 1/ A A 1/ å, B z = BR z, where R = A 1/ A 1/, = AA 1. Note that the operator R commutes with A, maps C δ/ into itself, and is invertible. Then, applying 3.43 to R z, we obtain that Cδ,εM R A iω z + m B z Y R z, z C δ/, ω lr We now compute explicitly the norm of R and R 1 in the class C δ/. Since one easily checks that A A 1, = 1 + β 1/4 A, R δ = 1 + β 1/4δ, R 1 δ = 1, 3.46 where δ denotes the operator norm from C δ/ into itself. Applying 3.46 into 3.45, we obtain Cδ,εM 1 + β 1/4δ A iω z + m B z Y z, z C δ/, ω lr Thus, in both cases, we can apply Theorem 1.3, which gives the existence of a time T δ,ε such that for T > T δ,ε, there exists a positive k T,δ such that any solution of 3.36 with initial data z 1/ C δ/ satisfies z 1/ k T,δ T/ B z k+1/ k= Besides, the estimates of Theorem 1.3 allow to estimate the observability time T δ,ε : π [1 + βδ 1 + β 1/4δ + ε β 1/4δ M + m C δ 4] 1/, B,1 16 if B =, T δ,ε = π [1 + βδ 1 + β 1/4δ + ε β 1/4δ M + m CB, δ 4 ] 1/, 16 if B 1 =. Letting ε, we obtain the estimates Y.

22 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua To complete the proof we check that if the initial data z 1/ is taken within the class C δ/, the solution of 3.6 satisfies z 1/ z = k+1/ 1 + β 1/4δ Ek+1/, which can be deduced from the explicit expression of the energy 3.7 and the formula Applications 4.1. Application of Theorem Boundary observation of the Schrödinger equation. The goal of this subsection is to present a straightforward application of Theorem.1 to the observability properties of the Schrödinger equation based on the results in [13]. Let Ω lr n be a smooth bounded domain. Consider the equation ß iut = x u, t, x, T Ω, 4.1 u = u, x Ω, ut, x =, t, x, T Ω. where u L Ω is the initial data. Equation 4.1 obviously has the form 1.1 with A = i x of domain DA = H 1 Ω H Ω. Let Γ Ω be an open subset of Ω and define the output yt = ut. ν Γ Using Sobolev s embedding theorems, one can easily check that this defines a continuous observation operator B from DA to L Γ. Let us assume that Γ satisfies in some time T the Geometric Control Condition GCC introduced in [1], which asserts that all the rays of Geometric Optics in Ω touch the sub-boundary Γ in a time smaller than T. In this case, the following observability result is known [13] : Theorem 4.1. For any T >, there exist positive constants k T > and K T > such that for any u L Ω, the solution of 4.1 satisfies T k T u L Ω ut dγ dt K T u L ν Ω. 4. Γ We introduce the following time semi-discretization of system 4.1: i uk+1 u k u k+1 + u k = x, x Ω, k N u k x =, x Ω, k N u x = u x, x Ω, that we observe through y k = uk. ν Γ 4.3

23 Observability of time discrete systems. 3 Then Theorem.1 implies the following result: Theorem 4.. For any δ >, there exists a time T δ such that for any time T > T δ, there exists a positive constant k T,δ > such that for small enough, the solution of 4.3 satisfies k T u L Ω uk dγ 4.4 ν for any u C δ/. k,t/ Note that we do not know if inequality 4.4 holds in any time T > as in the continuous case see 4.. This quesion is still open. Remark 4.3. Note that in the present section, we do not state any admissibility result for the time-discrete systems under consideration. However, uniform with respect to > admissibility results hold for all the examples presented in this article. These results will be derived in Section 6 using the admissibility property of the continuous system Boundary observation of the linearized KdV equation. We now present an application of Theorem.1 to the boundary observability of the linear KdV equation. We consider the following initial-value boundary problem for the KdV equation: u t + u xxx =, t, x, T, π, ut, = ut, π, t, T, u x t, = u x t, π, t, T, 4.5 u xx t, = u xx t, π, t, T, u, x = u x, x, π. For any integer k we set H k p = { u H k, π; j xu = j xuπ for j k 1 }, 4.6 where H k, π denotes the classical Sobolev spaces on the interval, π. The initial data of 4.5 are taken in the space = Hp, π, endowed with the classical H, π-norm. Let A denote the operator Au = xu 3 with domain DA = Hp. 5 As shown in [3], A is a skew-adjoint operator with compact resolvent. Moreover, its spectrum is given by σa = {iµ j with µ j = j 3, j Z}. The output function yt and the corresponding operator B : DA Y is given by Ñ ut, é yt = But = u x t, u xx t,, with the norm Bu Y = u + u x + u xx. Note that B LH 5 p, lr 3. Γ

24 4 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua The following observability inequality for system 4.5 is well-known Prop.. of []: Lemma 4.4. Let T >. Then there exist positive numbers k T and K T such that for every u Hp, π, T k T u H ut, + u p x t, + u xx t, dt K T u H. p 4.7 We now introduce the following time semi-discretization of system 4.5: u k+1 u k + uk+1 xxx + u k xxx =, x, π, k N u k = u k π, k N u k x = u k 4.8 xπ, k N u k xx = u k xxπ, u x = u x, k N x, π. It is easy to show that the eigenfunctions of A are given by {Φ j = e ijx } j Z with the corresponding eigenvalues {ij 3 } j Z. Hence, for any δ >, we have C δ/ = span {Φ j, j 3 δ/}. 4.9 As a direct consequence of Theorem.1 we have the following uniform observability result for system 4.8: Theorem 4.5. For any δ >, there exists a time T δ such that for any T > T δ, there exists a positive constant k T,δ > such that for small enough, the solution u k of 4.8 satisfies k T,δ u H u k + u k x + u k xx, 4.1 p k,t for any initial data u C δ/. As in Theorem 4., we do not know if the observability estimate 4.1 holds in any time T > as in the continuous case see Lemma Application of Theorem 3.1 Let us present an application of Theorem 3.1 to the so-called fourth order Gauss method discretization of equation 1.1 see for instance [8]-[9]. This fourth order Gauss method is a special case of the Runge-Kutta time approximation schemes, which corresponds to the only conservative scheme within this class. Consider the following discrete system: κ i = A z k + α ij κ i, i = 1,, j=1 z k+1 = z k + κ 1 + κ, z C δ/ given, α ij =

25 Observability of time discrete systems. 5 The scheme is unstable for the eigenfunctions corresponding to the eigenvalues µ j such that µ j 3 [8]-[9]. Thus we immediately impose the following restriction on the filtering parameter : δ < 3. To use Theorem 3.1, we only need to check the behavior of the semi-discrete scheme 4.11 on the eigenvectors. If z = Φ j, an easy computation shows that where z 1 = expil j z, In other words, l j = hµ j, where hη = arctan l j = arctan µ j µ j. 4.1 /6 η η /6 Then, a simple application of Theorem 3.1 gives : Theorem 4.6. Assume that B is an observation operator such that A, B satisfy 1.5 and B LDA, Y. For any δ, 3, there exists a time T δ > such that for any T > T δ, there exists > such that for all < <, there exists a constant k T,δ >, independent of, such that the solutions of system 4.11 satisfy k T,δ z Bz k, Y z C δ/ k,t/ Note that Theorem 3.1 also provides an estimate on T δ by using 3.6. In particular, this provides another possible time-discretization of 4.5, for which the observability inequality holds uniformly in provided the initial data are taken in C δ/, with δ < 3, where C δ/ is as in Application of Theorem 3. There are plenty of applications of Theorem 3.. We present here an application to the boundary observability of the wave equation. Consider a smooth nonempty open bounded domain Ω lr d and let Γ be an open subset of Ω. We consider the following initial boundary value problem: with the output. u tt x u =, x Ω, t, ux, t =, x Ω, t, ux, = u, u t x, = v, x Ω 4.14 yt = u ν Γ

26 6 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua This system is conservative and the energy of 4.14 Et = 1 ] [ u t t, x + ut, x dx, 4.16 Ω remains constant, i.e. Et = E, t [, T ] The boundary observability property for system 4.14 is as follows: For some constant C = CT, Ω, Γ >, solutions of 4.14 satisfy T E C u dγ dt, u, v H 1 Ω L Ω ν Γ Note that this inequality holds true for all triplets T, Ω, Γ satisfying the Geometric Control Condition GCC introduced in [1], see Subsection In this case, 4.18 is established by means of micro-local analysis tools see [1]. From now, we assume this condition to hold. We then introduce the following time semi-discretization of 4.14: u k+1 + u k 1 u k = x βu k βu k + βu, k 1 in Ω Z, u k =, in Ω Z, 4.19 u + u 1, u1 u = u, v H 1 Ω L Ω, where β is a given parameter satisfying β 1 4. The output functions y k are given by y k = uk. 4. ν Γ System or system can be written in the form 1.18 or 3.6 with observation operator 3. by setting: H = L Ω, DA = H Ω H 1 Ω, Y = L Γ, A ϕ = x ϕ ϕ DA, B 1 ϕ = ϕ, ϕ DA. ν Γ One can easily check that A is self-adjoint in H, positive and boundedly invertible and DA 1/ = H 1 Ω, DA 1/ = H 1 Ω. Proposition 4.7. With the above notation, B 1 LDA, Y is an admissible observation operator, i.e. for all T > there exists a constant K T > such that: If u satisfies 4.14 then T u dγ dt K T u H ν 1Ω + v L Ω Γ for all u, v H 1 Ω L Ω.

27 Observability of time discrete systems. 7 The above proposition is classical see, for instance, p. 44 of [15], so we skip the proof. Hence we are in the position to give the following theorem: Theorem 4.8. Set β 1/4. For any δ >, system 4.19 is uniformly observable with u, v C δ/. More precisely, there exists T δ, such that for any T > T δ, there exists a positive constant k T,δ independent of, such that for > small enough, the solutions of system 4.19 satisfy k T,δ u + v uk dγ, 4.1 ν for any u, v C δ/. k,t/ Proof. The scheme proposed here is a Newmark discretization. Hence this result is a direct consequence of Theorem 3.1. Remark 4.9. One can use Fourier analysis and microlocal tools to discuss the optimality of the filtering condition as in [7]. The symbol of the operator in 4.19, that can be obtained by taking the Fourier transform of the differential operator in space-time is of the form see for instance [16] 4 τ sin Γ ξ 1 4β sin τ Note that this symbol is not hyperbolic in the whole range τ, ξ π/, π/ lr n. However, the Fourier transform of any solution of 4.19 is supported in the set of τ, ξ satisfying 1 4β sin τ/ >, where the symbol is hyperbolic. As in the continuous case, one expects the optimal observability time to be the time needed by all the rays to meet Γ. Along the bicharacteristic rays associated to this hamiltonian the following identity holds τ = arctan ξ β 1/4 ξ. These rays are straight lines as in the continuous case, but their velocity is not 1 anymore. Indeed, one can prove that along the rays corresponding to ξ < δ/, the velocity of propagation is given by dx = dt β ξ β 1/4ξ 1 + βδ 1 + β 1/4δ. In other words, in the class C δ/, the velocity of propagation of the rays concentrated in frequency around δ/ is 1 + δ /4 1 times that of the continuous wave equation. Therefore we expect the optimal observability time Tδ in the class C δ/ to be Tδ = T 1 + βδ 1 + β 1 δ 4, 4.

28 8 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua where T is the optimal observability time for the continuous system. According to this, the estimate T δ, in 3.31 on the time of observability has the good growth rate when δ. Besides, when δ goes to, we have that T δ, πm1 + βδ 1 + β 1 δ Recall that πm = T is the time of observability that the resolvent estimate 1.5 in the continuous setting yields see [17]. The similarity between 4. and 4.3 indicates that the resolvent method accurately measures the group velocity. Note however that πm is not the expected sharp observability time T in 4. in the continuous setting. This is one of the drawbacks of the method based on the resolvent estimates we use in this paper. Even at the continuous level the observability time one gets this way is far from being the optimal one that Geometric Optics yields. 5. Fully discrete schemes 5.1. Main statement In this section, we deal with the observability properties for time-discretization systems such as depending on an extra parameter, for instance the space mesh-size, or the size of the microstructure in homogenization. To this end, it is convenient to introduce the following class of operators: Definition 5.1. For any m, M, C B lr + 3, we define Cm, M, C B as the class of operators A, B satisfying: A1 The operator A is skew-adjoint on some Hilbert space, and has a compact resolvent. A The operator B is defined from DA with values in a Hilbert space Y, and satisfies.1 with C B. A3 The pair of operators A, B satisfies the resolvent estimate 1.5 with constants m and M. In this class, Theorems apply and provide uniform observability results for any of the time semi-discrete approximation schemes , 1.17, and Indeed, this can be deduced by the explicit form of the constants T δ and k T,δ which only depend on m, M and C B. Note that this definition does not depend on the spaces and Y. When considering families of pairs of operators A, B, it is not easy, in general, to show that they belong to the same class Cm, M, C B for some choice of the constants m, M, C B. Indeed, item A3 is not obvious in general. Therefore, in the sequel, we define another class included in some Cm, M, C B and that is easier to handle in practice. Definition 5.. For any C B, T, k T, K T lr + 4, we define DC B, T, k T, K T as the class of operators A, B satisfying A1, A and:

29 Observability of time discrete systems. 9 B1 The admissibility inequality T B exptaz Y dt K T z, 5.1 where expta stands for the semigroup associated to the equation B The observability inequality ż = Az, z = z. 5. k T z T B exptaz Y As we will see below, assumptions B1-B imply A3: dt. 5.3 Lemma 5.3. If the pair A, B belongs to DC B, T, k T, K t, then there exist m and M such that A, B Cm, M, C B. Besides m and M can be chosen as m = T k T, M = T K T k T. 5.4 Proof. We only need to prove A3. This is actually already done in [17] or in [5]. Indeed, it was proved that once the admissibility inequality 1.3 and the observability inequality 1.4 hold for some time T, then the resolvent estimate 1.5 hold with m and M as in 5.4. Note that assumptions B1-B are related to the continuous systems 5.. Now we consider a sequence of operators A p, B p depending on a parameter p P, which are in some L p LDA p, Y p for each p, where p and Y p are Hilbert spaces. We want to address the observability problem for a timediscretization scheme of ż = A p z, z = z p, yt = B p zt Y p. 5.5 In applications, we need the observability to be uniform in both p P and > small enough. The previous analysis and the properties of the class DC B, T, k T, K T suggest the following two-steps strategy: 1. Study the continuous system 5.5 for every parameter p and prove the uniform admissibility 5.1 and observability Apply one of the Theorems.1, 3.1 and 3. to obtain uniform observability estimates 1.8 for the corresponding time-discrete approximation schemes. This allows dealing with fully discrete approximation schemes. In that setting the parameter p is actually the standard parameter h > associated with the space mesh-size. In this way one can use automatically the existing results for space semi-discretizations as, for instance, [3], [6], [7], [1], [19], [], [8] and [9], etc.

30 3 Sylvain Ervedoza, Chuang Zheng and Enrique Zuazua 5.. Applications The fully discrete wave equation. Let us consider the wave equation 4.14 in a -d square. More precisely, let Ω =, π, π lr and Γ be a subset of the boundary of Ω constituted by two consecutive sides, for instance, Γ = {x 1, π : x 1, π} {π, x : x, π} = Γ 1 Γ. As in 4.15, the output function yt = But is given by Bu = u = ux 1, π + uπ, x. ν Γ x Γ1 x 1 Γ Let us first consider the finite-difference semi-discretization of The following can be found in [8]. Given J, K ln we set h 1 = π J + 1, h = π K We denote by u jk t the approximation of the solution u of 4.14 at the point x jk = jh 1, kh. The space semi-discrete approximation scheme of 4.14 is as follows: ü jk u j+1k + u j 1k u jk u jk+1 + u jk 1 u jk = h 1 < t < T, j = 1,, J; k = 1,, K u jk =, < t < T, j =, J + 1; k =, K + 1 u jk = u jk,, u jk = u jk,1, j = 1,, J; k = 1,, K. h 5.7 System 5.7 is a system of JK linear differential equations. Moreover, if we denote the unknown Ut = u 11 t, u 1 t,, u J1 t,, u 1K t, u K t,, u JK t T, then system 5.7 can be rewritten in vector form as follows Üt + A,h Ut =, < t < T. U = U h,, U = U h,1, 5.8 where U h,, U h,1 = u jk,, u jk,1 1 j J,1 k K lr JK are the initial data. The corresponding solution of 5.7 is given by U h, U h = u jk, u jk 1 j J,1 k K. Note that the entries of A,h belonging to M JK lr may be easily deduced from 5.7. As a discretization of the output, we choose ujk ujk B h U =,. 5.9 h j {1,,J} h 1 k {1, K} The corresponding norm for the observation operator B h is given by J B h Ut Y h = h 1 u jkt K + h u Jkt. h 1 j=1 h k=1

31 Observability of time discrete systems. 31 Besides, the energy of the system 5.8 is given by E h t = h 1h J K j= k= u jk t + As in the continuous case, this quantity is constant. u j+1kt u jk t u jk+1 t u jk t h h 1 E h t = E h, < t < T. In order to prove the uniform observability of 5.8, we have to filter the high frequencies. To do that we consider the eigenvalue problem associated with 5.8: A,h ϕ = λ ϕ As in the continuous case, it is easy to show that the eigenvalues λ j,k,h1,h are purely imaginary. Let us denote by ϕ j,k,h1,h the corresponding eigenvectors. Let us now introduce the following classes of solutions of 5.8 for any < γ < 1: Ĉ γ h = span {ϕ j,k,h1,h such that λ j,k,h1,h maxh 1, h γ}. The following Lemma holds see [8]: Lemma 5.4. Let < γ < 1. Then there exist T γ such that for all T > T γ there exist k T,γ > and K T,γ > such that k T,γ E h T B h Ut Y h dt K T,γ E h 5.1 holds for every solution of 5.8 in the class Ĉγh and every h 1, h small enough satisfying sup h 1 γ < h 4 γ. Now we present the time discrete schemes we are interested in. For any >, we consider the following time Newmark approximation scheme of system 5.8: U k+1 + U k 1 U k + A,h βu k βu k + βu k 1 =, U + U 1 with β 1/4., U 1 U = U h,, U h,1, 5.13