Uniform polynomial observability of time-discrete conservative linear systems

In this paper we study time semi-discrete approximations of a class of polynomially observable infinite dimensional systems. By using a method based on the resolvent estimate, we derive uniform polynomial observability inequalities within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. We also present an application of our result to stabilization problems. Observabilité polynomiale uniforme des systèmes linéaires conservatifs semi-discrets en temps Résumé Dans cet article nous étudions la semi-discrétisation en temps des systèmes de dimension infinie qui sont polynomialement observables. En utilisant une méthode basée sur l’estimation de la résolvante, nous obtenons des inégalités d’observabilité polynomiale uniformes pour les solutions filtrées du problème semi-discret en temps. Nous présentons également des applications de notre résultat aux problèmes de stabilisation.


Introduction
Let H be a Hilbert space endowed with the norm .H and let A 0 : D(A 0 ) ⊂ H → H be a self-adjoint positive operator with A −1 0 compact in H.For α ≥ 0, we introduce the Hilbert spaces H α = D(A α 0 ), with the norm z α = A α 0 z H .The space H −α is the dual of H α with respect to the pivot space H. Let us define We consider the following abstract system: ẅ + A 0 w(t) = 0, w(0) = w 0 , ẇ(0) = w 1 . (1.1) Here and henceforth, a dot ( . ) denotes differentiation with respect to the time t.Such systems are often used as models of vibrating systems (e.g., the wave equation).
Assume that U is another Hilbert space equipped with the norm .U .We identify U with its dual.We denote by L(U, H) the space of bounded linear operators from U to H, endowed with the classical operator norm.For all (w 0 , w 1 ) ∈ X 1 , the initial value problem (1.1) admits a unique solution satisfying ) be an observation operator and define the output function φ(t) = B * 0 ẇ(t). (1.2) In order to give a sense to (1.2), we make the assumption that B * 0 is an admissible observation operator in the following sense (see [18]).
Definition 1.1.The operator B * 0 is an admissible observation operator for systems (1.1)-(1.2) if for every T > 0 there exists a constant k T > 0 such that T 0 φ(t) 2  U dt ≤ k T (w 0 , w 1 ) 2 X , ∀(w 0 , w 1 ) ∈ X 1 . (1.3) Note that if B 0 is bounded in H, i.e., if it can be extended such that B 0 ∈ L(U, H), then B 0 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 (see [12]) of the solutions of the evolution equation (1.1).
The polynomial observability property can be formulated as follows: where α > 0.
Moreover, (1.1)-(1.2) is said to be polynomially observable if it is polynomially observable in some time T > 0.
Let us denote by (λ j ) j∈N * the increasing sequence formed by the eigenvalues of A 1 2 0 and by (ϕ j ) j∈N * a corresponding sequence of eigenvectors, forming an orthonormal basis of H.
, then the problem (1.1) and (1.2) may be rewritten as and where A : X 1 ⊂ X → X and B : U → X −1 are defined by We note that A is skew-adjoint on X with compact resolvent (see [17]) and B * ∈ L(X 1 , U ), i. e., there exists a constant C B such that As the exact observability, the polynomial observability holds if a "weakened" Hautus test holds.More precisely, we have the following result (see Appendix for the proof).
Proposition 1.3.The system (1.1)-(1.2) is polynomially observable if there exist constants β, α > 0 such that Our goal in this paper is to prove uniform polynomial observability inequalities 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 ∆t > 0, we denote by z k and y k respectively the approximations of the solution z and the output function y of system (1.1)-(1.2) at time t k = k∆t for k ∈ N. Consider the following implicit midpoint time discretization of system (1.5): (1.8) The output function of (1.8) is given by (1.9) Note that (1.8)-(1.9) is a discrete version of (1.5)- (1.6).Taking into account that A is skew-adjoint, it is easy to show that z k X is conserved in the discrete time variable k ∈ N, i.e., z k X = z 0 X .Consequently the scheme under consideration is stable and its convergence (in the classical sense of numerical analysis) is guaranteed in an appropriate functional setting.
The uniform polynomial observability problem for system (1.8) is formulated as follows: To find a positive constant c T , independent of ∆t, such that the solutions z k of system (1.8) satisfy: for all initial data z 0 in an appropriate class.We recall that y k is given by (1.9) and z k is the solution of (1.8).
Remark that (1.10) is a discrete version of (1.4).Note that this type of observability inequalities appears naturally when dealing with stabilization problems.For numerical approximation processes, it is important that these inequalities hold uniformly with respect to the discretization parameter ∆t to recover uniform stabilization properties.
The numerical approximation of observability has been intensively studied in the literature (see, for instance, [7] and the references therein).It is by now well-known that discretization processes may create high frequency spurious solutions which might lead to non-uniform observability properties.Several remedies have been proposed to overcome this difficulty: Tychonoff regularization in [9] and filtering of high frequencies in [11].We refer to the paper [8] for more details and extensive references.For stability results this phenomenon was underlined, for example, in [15] where a viscous finite-difference space semi-discretization of a damped wave equation has been studied.Let us mention the works [6,5] based on properties of the continuous system where convergent variational approximations of an exact control are build.For fully discrete approximations schemes, we mention the work [14] , where the uniform controllability of a fully discrete approximation scheme of the 1d wave equation is analyzed, and also the recent work [7], where exact observability issues were discussed for abstract models.Let us also cite the paper [8], where exponential stabilization properties were studied.For time semi-discrete approximations of polynomial observability, the only work we are aware of is [10], and the present work that seems to be the first one deriving uniform polynomial observability in the case of unbounded observation.We note that this issue is still open in the space semi-discrete case.
In the sequel, we are interested in understanding under which assumptions inequality (1.10) holds uniformly with respect to ∆t.One expects to do it so that, when letting ∆t → 0, one recovers the observability property of the continuous model.
Though our paper is inspired from [7].But here, we only need a weaker version of observability in which the observed norm is weaker than .X .
We first need to introduce some notations.
Notation.Since A is skew-adjoint with compact resolvent, its spectrum is discrete and σ(A) = {iµ j : j ∈ Z * }, where If we set ϕ −j = ϕ j , for all j ∈ N * , then an orthonormal family of eigenvectors (Φ j ) j∈Z * of A is given by Moreover, we define The main result of this paper reads as follows: Then, for any δ > 0, there exists a time T δ such that for any T > T δ , there exists a positive constant C = C T,δ , independent of ∆t, such that for ∆t small enough, the solution z k of (1.8) satisfies The outline of this paper is as follows.
In Section 2, we prove Theorem 1.4.Our proof is mainly based on the resolvent estimate (1.7).In Section 3 we present some consequences of Theorem 1.4 to stabilization of time semi-discrete damped models.We end the paper by giving some applications of our main result.

Proof of Theorem 1.4
Before getting into the proof of Theorem 1.4, we recall some properties of the discrete Fourier transform at scale ∆t (see [16]), that will be used in the sequel.Definition 2.1.Given any sequence (u k ) ∈ l 2 (∆tZ), we define its Fourier transform as : For any function v ∈ L 2 ((−π/∆t, π/∆t)), we define the inverse Fourier transform at scale ∆t > 0: According to this definition, and the Parseval identity holds 1 2π Proof of Theorem 1.4.Taking the initial data z 0 = Φ j , then simple formal calculations give where e iα j ∆t = We get By (2.2), we have α j ∆t ∈]0, π 2 [.Consequently a j e iα j k∆t Φ j .
We have, for any k ∈ N and By using (2.3), we obtain that for any k, Since |µ j | ≤ δ/∆t, the last equality gives The proof of Theorem 1.4 is based on the following lemmas.

Lemma 2.3 shows that the coefficient of (z
when ∆t → 0. Note that K T,δ,ϕ is an increasing function of T tending to −∞ when T → 0 + and to +∞ when T → +∞.Let T δ,ϕ be the unique positive solution of K T,δ,ϕ = 0.Then, for any time T > T δ,ϕ , choosing a positive K T,δ such that 0 < K T,δ < K T,δ,ϕ , there exists ∆t 0 > 0 such that for any ∆t < ∆t 0 the following holds: Besides, using again Lemma 2.3, we have This last term tends to T as ∆t → 0.

Since lim
T →+∞ H T,δ,φ = 0, then there exists a positive constant M such that where c is a very small positive constant.
Consequently, there exists ∆t 1 > 0 such that for any ∆t < ∆t 1 , we get for any ∆t < ∆t 1 .This inequality combined with (2.18) give Using (2.5) and Lemma 2.2, we get the existence of a positive constant C T,δ such that which yileds (1.12).
Remark 2.4.Theorem 1.4 improves the result in [10] where the same result was stated but for some class of operators A. More precisely the uniform polynomial observability holds for time discrete systems, when there exists a positive constant γ > 0 such that the eigenvalues (µ j ) j satisfy the gap condition:

Stabilization properties
This section is mainly based on the article [10], in which stabilization properties are derived for abstract linear damped systems.Below, we assume that A 0 is self-adjoint, positive operator with A −1 0 compact, and that B 0 ∈ L(U, H).

The continuous setting
Consider the following damped wave type equations: 1) The energy of solutions of (3.1) is defined by System (3.1) is said to be polynomially stable if there exists positive constants C and γ such that for all t > 0 and for all (w 0 , w 1 ) ∈ X 1 we have It is by now well-known (see [3]) that this property holds if the observability inequality (1.4) holds for solutions of (1.1), or if (1.7) is satisfied.

The time semi-discrete setting
We now assume that system (1.1)-(1.2) is observable in the sense of (1.4), or that (1.7) holds for some α, β > 0.Then, combining Theorem 1.4 and the results in [10], we get: Theorem 3.1.Let B 0 be a bounded operator in L(U, H), and assume that (1.7) is satisfied.Then the time semi-discrete systems are polynomially stable, uniformly with respect to the time discretization parameter ∆t > 0: there exist two positive constants C 1 and γ 1 independent of ∆t > 0 such that for any ∆t > 0 we have Notice that z, A and B are the same defined in the introduction.We refer to [10] and the references therein for more precise statements.
It is well known that this system is not exactly observable since the "geometric optics" condition is violated [4].Now, we verify (1.7) for α = 1.
It is easy to see that the normalized eigenvectors of A for m, l = 1, 2, ..., form an orthonormal basis for X.The corresponding eigenvalue for Φ m,l (x 1 , x 2 ) is Taking z = Φ m,l (x 1 , x 2 ) for fixed m and l.A simple calculations show that , and .
Notice that (L 2 (Ω)) is the dual of L 2 (Ω) with respect to the pivot space Due to the orthogonality of the families (sin(mx 1 )) m≥1 and (sin(lx 2 )) l≥1 in L 2 (0, π), we get and As above, (1.7) is verified with β = 1 π .Consequently, the system (4.1)-(4.2) is polynomially observable.Then, we introduce the following time semi-discrete approximation scheme: Now, according to Theorem 1.4 we have: Proposition 4.1.For any δ > 0, there exists a time T δ such that for any T > T δ , there exists a positive constant C = C T,δ , independent of ∆t, such that for ∆t small enough, the solution z k of (4.3) satisfies
Hence it is written in the form (1.5)-(1.6)with the following choices: In this setting, A is a skew-adjoint unbounded operator with compact resolvent on the Hilbert space X, and the operator B * is admissible (see [1]).As above, this system is not exactly observable.Now, we verify (1.7) for α = 1.
It is easy to see that the normalized eigenvectors of A .