Ghislain Haine
email: ghislain.haine@iecn.u-nancy.fr

Karim Ramdani
email: karim.ramdani@inria.fr

Reconstructing initial data using observers : error analysis of the semi-discrete and fully discrete approximations

Keywords: Control theory, Observers, Inverse problems, Finite element and finite difference discretization, Convergence analysis, Schrödinger equation, Wave equation Mathematics Subject Classification (2000) Primary : 35Q93 Secondary : 35L05, 35J10, 65M22

. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed).

Introduction

The goal of this paper is to present a convergence analysis for the iterative algorithm recently proposed in Ramdani, Tucsnak and Weiss [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF] for solving initial state inverse problems from measurements over a time interval. This algorithm is based on the use back and forth in time of observers (sometimes called Luenberger observers or Kalman observers; see for instance Curtain and Zwart [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]). Inspired by the works of Mathias Fink on time reversal [START_REF] Fink | Time reversal of ultrasonic fields-basic principles[END_REF][START_REF] Fink | Acoustic time reversal mirrors[END_REF], Phung and Zhang [START_REF] Phung | Time reversal focusing of the initial state for kirchhoff plate[END_REF] used this algorithm in the particular case of the Kirchhoff plate equation with distributed observation, while Ito, Ramdani and Weiss [START_REF] Ito | A time reversal based algorithm for solving initial data inverse problems[END_REF] considered more general evolution PDE's with locally distributed observation. Let us mention also Auroux and Blum [START_REF] Auroux | A nudging-based data assimilation method : the back and forth nudging (bfn) algorithm[END_REF] who implemented a similar algorithm in the context of data assimilation. More generally, during the last decade, observers have been designed for linear and nonlinear infinite-dimensional systems in many works, among which we can mention for instance Deguenon, Sallet and Xu [START_REF] Deguenon | Infinite dimensional observers for vibrating systems[END_REF], Guo and Guo [START_REF] Guo | The strong stabilization of a one-dimensional wave equation by noncollocated dynamic boundary feedback control[END_REF], Guo and Shao [START_REF] Guo | Stabilization of an abstract second order system with application to wave equations under non-collocated control and observations[END_REF] in the context of wave-type systems, Lasiecka and Triggiani [START_REF] Lasiecka | Control theory for partial differential equations: continuous and approximation theories. I[END_REF], Smyshlyaev and Krstic [START_REF] Smyshlyaev | Backstepping observers for a class of parabolic PDEs[END_REF] for parabolic systems and Krstic, Magnis and Vazquez [START_REF] Krstic | Nonlinear control of the viscous burgers equation: Trajectory generation, tracking, and observer design[END_REF] for the non linear viscous Burgers equation.

Let us first briefly describe the principle of the reconstruction method proposed in [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF] in the simplified context of skew-adjoint generators and bounded observation operator. We will always work under these assumptions throughout the paper. Given two Hilbert spaces X and Y (called state and output spaces respectively), let A : D (A) → X be skew-adjoint operator generating a C 0 -group T of isometries on X and let C ∈ L(X, Y ) be a bounded observation operator. Consider the infinite dimensional linear system given by ż(t) = Az(t), ∀t 0, y(t) = Cz(t), ∀t ∈ [0, τ ].

(1.1

)
where z is the state and y the output function (throughout the paper, the dot symbol is used to denote the time derivative). Such systems are often used as models of vibrating systems (e.g., the wave equation, the beam equation,...), electromagnetic phenomena (Maxwell's equations) or in quantum mechanics (Schrödinger's equation). The inverse problem considered here is to reconstruct the initial state z 0 = z(0) of system (1.1) knowing (the observation) y(t) on the time interval [0, τ ] (see Fig. 1). Such inverse problems arise in many applications, like thermoacoustic tomography Kuchment and Kunyansky [START_REF] Kuchment | Mathematics of thermoacoustic tomography[END_REF] or data assimilation Puel [START_REF] Puel | A nonstandard approach to a data assimilation problem and Tychonov regularization revisited[END_REF]. To solve this inverse problem, we assume here that it is well-posed, i.e. that (A, C) is exactly observable in time τ > 0, i.e. that there exists kτ > 0 such that τ 0 y(t) 2 dt ≥ k 2 τ z 0 2 , ∀ z 0 ∈ D(A).

For instance, in the case of the wave equation on a bounded domain Ω, this inequality holds provided we observe the state on O × (0, τ ) where O ⊂ Ω and τ are chosen such that the geometric optics condition of Bardos, Lebeau and Rauch [START_REF] Bardos | Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary[END_REF] holds. For similar results related to other equations, see for instance Burq [START_REF] Burq | Contrôle de l'équation des plaques en présence d'obstacles strictement convexes[END_REF], Burq and Lebeau [START_REF] Burq | Micro-local approach to the control for the plates equation[END_REF] and Jaffard [START_REF] Jaffard | Contrôle interne exact des vibrations d'une plaque rectangulaire. (internal exact control for the vibrations of a rectangular plate)[END_REF] and the monograph of Lions [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF].

Following Liu [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]Theorem 2.3.], we know that A + = A -C * C (respectively

A -= -A -C * C
) generate an exponentially stable C 0 -semigroup T + (respectively T -) on X. Then, we introduce the following initial and final Cauchy problems, called respectively forward and backward observers of (1.1)

ż+ (t) = A + z + (t) + C * y(t), ∀t ∈ [0, τ ], z + (0) = 0, (1.2) ż-(t) = -A -z -(t) -C * y(t), ∀t ∈ [0, τ ], z -(τ ) = z + (τ ). (1.3)
Note that the states z + and z -of the forward and backward observers are completely determined by the knowledge of the output y. If we set Lτ = T - τ T + τ , then by [24, Proposition 3.7], we have η := Lτ L(X) < 1 and by [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF]Proposition 3.3], the following remarkable relation holds true

z 0 = (I -Lτ ) -1 z - (0). (1.4)
In particular, one can invert the operator (I -Lτ ) using a Neumann series and get the following expression for the initial state

z 0 = ∞ n=0
L n τ z -(0).

(1.5) Thus, at least theoretically, the reconstruction of the initial state is given by the above formula. Note that the computation of the first term in the above sum requires to solve the two non-homogeneous systems (1.2) and (1.3), while the terms for n ≥ 1 involve the resolution of the two homogeneous systems associated with (1.2) and (1.3) (i.e. for y ≡ 0). In practice, the reconstruction procedure requires the discretization of these two systems and the truncation of the infinite sum in (1.5) to keep only a finite number of back and forth iterations. For instance, if we consider a space semi-discretization corresponding to a mesh size h (typically a finite element approximation), one can only compute

z 0,h = N h n=0 L n h,τ z - h (0), (1.6)
where

-L h,τ = T - h,τ T + h,τ , where T ± h,τ ∈ L(X) are suitable space discretizations of T ± τ , -z - h (0) ∈ X h is an approximation of z -(0) in a suitable finite dimensional subspace X h of X, -N h is a suitable truncation parameter.
Similarly, if a full discretization described by a mesh size h and a time step ∆t is considered, one can compute

z 0,h,∆t = N h,∆t n=0 L n h,∆t,K z - h 0 .
(1.7)

where

-L h,∆t,K = T - h,∆t,K T + h,∆t,K
, where T ± h,∆t,K are suitable space and time discretiza-

tions of T ± τ , -z - h 0 ∈ X h is an approximation of z -(0), -N h,∆t is a suitable truncation parameter.
For the sake of clarity, the precise definition of the spaces and discretizations used will be given later in the paper.

Our objective in this work is to present a convergence analysis of z 0,h and z 0,h,∆t towards z 0 . A particular attention will be devoted to the optimal choice of the truncation parameters N h and N h,∆t for given discretization parameters (mesh size h and time step ∆t). Let us emphasize that our error estimates (see (2.8), (2.27), (3.15) and (3.25)) provide in particular an upper bound for the maximum admissible noise under which convergence of the algorithm is guaranteed. As usually in approximation error theory of PDE's, some regularity assumptions are needed to obtain our error estimates. Namely, our result allows us to reconstruct only initial data contained in some subspace of X (namely D A 2 ).

Let us emphasize that similar error estimates have been recently obtained by Cîndea, Micu and Tucsnak [START_REF] Cîndea | An approximation method for exact controls of vibrating systems[END_REF] in the context of control problems. Using Russel's "stabilizability implies controllability" principle, the authors derived a new approximation method of exact controls for second order wave type systems with bounded input operator. The convergence analysis is carried out in the case of a Galerkin type semi-discretization.

Let us now make some comments on the type of observation for which we have been able to prove convergence results. First of all, we assume throughout the paper that C ∈ L(X, Y ) is a bounded observation operator (locally distributed observation). This assumption is crucially used many times in the proofs and it seems difficult to extend our result to the case of unbounded observation. However, the reconstruction algorithm seems to be still efficient in this case, as it can be seen from the numerical results given in [START_REF] Ramdani | Recovering the initial state of an infinite-dimensional system using observers[END_REF].

In addition to the boundedness of C, we assume that

C * C ∈ L D A 2 ∩L (D (A)).
The appears naturally in our analysis, but not in the one carried out in Cîndea et al. [START_REF] Cîndea | An approximation method for exact controls of vibrating systems[END_REF]. Indeed, this assumption is used to bound a term which does not appear in the context of control problems they considered. Finally, let us point out that these assumptions are in particular satisfied when the locally distributed observation is obtained via a smooth cut-off function.

Remark 1 Using an implicit Euler method preserves the dissipative properties of the high frequency part of the solution (see (2.30) and (3.30)). This is the main reason for which we did not use an explicit or midpoint Euler scheme, but we do not know if this restriction is only technical or not.

The paper is organized as follows: in Section 2 we provide a convergence analysis of the algorithm for an abstract Schrödinger type system, by considering successively the semi-discretization (Subsection 2.1) and the full discretization (Subsection 2.2). In Section 3, similar results are given for an abstract wave system. Once again, we tackle successively the semi-discretization (Subsection 3.1) and the full discretization (Subsection 3.2). However, since the proofs are very similar to those of the Schrödinger case, they will not be given with full details. Finally, the Appendix is devoted to the proof of two technical lemmas which are used several times throughout the paper.

Throughout the paper, we denote by M a constant independent of τ , of the initial state z 0 and of the discretization parameters h and ∆t, but which may differ from line to line in the computations.

Schrödinger equation

Let X be a Hilbert space endowed with the inner product •, • . Let A 0 : D (A 0 ) → X be a strictly positive self-adjoint operator and C ∈ L(X, Y ) a bounded observation operator, where Y is an other Hilbert space. The norm in D(A α 0 ) will be denoted by • α. We assume that there exists some τ > 0 such that (iA 0 , C) is exactly observable in time τ . Thus by Liu [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]Theorem 2.3.],

A + = iA 0 -C * C (resp. A -= -iA 0 -C * C)
is the generator of an exponentially stable C 0 -semigroup T + (resp. T -). We want to reconstruct the initial value z 0 of the following system ż(t) = iA 0 z(t), ∀t 0, y(t) = Cz(t), ∀t ∈ [0, τ ].

(2.1)

Throughout this section we always assume that z 0 ∈ D A 2 0 . Thus by applying Theorem 4.1.6 of Tucsnak and Weiss [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF], we have

z ∈ C [0, τ ], D A 2 0 ∩ C 1 ([0, τ ], D (A 0 )) .
The forward and backward observers (1.2) and (1.3) read then as follows

ż+ (t) = iA 0 z + (t) -C * Cz + (t) + C * y(t), ∀t ∈ [0, τ ], z + (0) = 0, (2.2) ż-(t) = iA 0 z -(t) + C * Cz -(t) -C * y(t), ∀t ∈ [0, τ ], z -(τ ) = z + (τ ). (2.3)
Clearly, the above systems can be rewritten in the general form of an initial value Cauchy problem (simply by using a time reversal for the second system)

q(t) = ±iA 0 q(t) -C * Cq(t) + F (t), ∀t ∈ [0, τ ], q(0) = q 0 , (2.4)
where we have set for the forward observer (2.2) : F (t) = C * y(t) = C * Cz(t) and q 0 = 0, for the backward observer (2.3) :

F (t) = C * y(τ -t) = C * Cz(τ -t) and q 0 = z + (τ ) ∈ D A 2 0 .
2.1 Space Semi-Discretization

Statement of the main result

We use a Galerkin method to approximate system (2.4). More precisely, consider a family (X h ) h>0 of finite-dimensional subspaces of D A 1 2 0 endowed with the norm in X. We denote π h the orthogonal projection from D A 1 2 0 onto X h . We assume that there exist M > 0, θ > 0 and h * > 0 such that we have for all h ∈ (0, h * )

π h ϕ -ϕ ≤ M h θ ϕ 1 2 , ∀ϕ ∈ D A 1 2
0 .

(2.5)

Given q 0 ∈ D A 2 0 , the variational formulation of (2.4) reads for all t ∈ [0, τ ] and all ϕ ∈ D A

1 2 0 as follows q(t), ϕ = ±i q(t), ϕ 1 2 -C * Cq(t), ϕ + F (t), ϕ , q(0) = q 0 .
(2.6)

Suppose that q 0,h ∈ X h and F h are given approximations of q 0 and F respectively in the spaces X and L 1 ([0, τ ], X). For all t ∈ [0, τ ], we define q h (t) ∈ X h as the unique solution of the variational problem

qh (t), ϕ h = ±i q h (t), ϕ h 1 2 -C * Cq h (t), ϕ h + F h (t), ϕ h , q h (0) = q 0,h . (2.7)
for all ϕ h ∈ X h . The above approximation procedure leads in particular to the definition of the semi-discretized versions T ± h of the semigroups T ± that we will use. Indeed, we simply set

T + t q 0 T + h,t q 0 = q h (t)
T - t q 0 T - h,t q 0 = q h (τ -t) where q h is the solution of equation (2.7) with the corresponding sign and for F h = 0 and q 0,h = π h q 0 . The approximation of Lτ = T - τ T + τ follows immediately by setting

L h,τ = T - h,τ T + h,τ .
Assume that y h is an approximation of the output y in L 1 ([0, τ ], Y ) and let z + h and z - h denote the Galerkin approximations of the solutions of systems (2.2) and (2.3), satisfying for all t ∈ [0, τ ] and all ϕ h ∈ X h

ż+ h (t), ϕ h = i z + h (t), ϕ h 1 2 -C * Cz + h (t), ϕ h + C * y h (t), ϕ h , z + h (0) = 0. ż- h (t), ϕ h = i z - h (t), ϕ h 1 2 + C * Cz - h (t), ϕ h -C * y h (t), ϕ h , z - h (τ ) = z + h (τ )
. Thus, our main result in this subsection reads as follows.

Theorem 1 Let A 0 : D (A 0 ) → X be a strictly positive self-adjoint operator and

C ∈ L(X, Y ) such that C * C ∈ L D A 2 0 ∩ L (D (A 0 )).
Assume that the pair (iA 0 , C) is exactly observable in time τ > 0 and set η := Lτ L(X) < 1. Let z 0 ∈ D A 2 0 be the initial value of (2.1) and z 0,h be defined by (1.6).

Then there exist M > 0 and h * > 0 such that for all h ∈ (0, h * )

z 0 -z 0,h ≤ M η N h +1 1 -η + h θ τ N 2 h z 0 2 + N h τ 0 C * (y(s) -y h (s)) ds .
A particular choice of N h leads to an explicit error estimate (with respect to h) as shown in the next Corollary (the proof is left to the reader because of its simplicity)

Corollary 1 Under the assumptions of Theorem 1, we set

N h = θ ln h ln η .
Then, there exist Mτ > 0 and h * > 0 such that for all h ∈ (0, h * )

z 0 -z 0,h ≤ Mτ h θ ln 2 h z 0 2 + | ln h| τ 0 C * (y(s) -y h (s)) ds . (2.8)
Remark 2 In fact, Theorem 1 still holds true for z 0 ∈ D A 3 2

0

(with the same proofs and slightly adapting the spaces). Nevertheless, we have not been able to carry out this analysis for the fully discrete approximation in this case. This is why we restricted our analysis to the case of an initial data z 0 ∈ D A 2 0 .

Proof of Theorem 1

Before proving Theorem 1, we first need to prove some auxiliary results. The next Proposition, which constitutes one of the main ingredients of the proof, provides the error estimate for the approximation in space of the initial value problem (2.6) by using the Galerkin scheme (2.7).

Proposition 1 Given q 0 ∈ D A 2 0 and q 0,h ∈ X h , let q and q h be the solutions of (2.6) and (2.7) respectively. Assume that C * C ∈ L (D (A 0 )). Then, there exist M > 0 and h * > 0 such that for all t ∈ [0, τ ] and all h ∈ (0, h * )

π h q(t) -q h (t) ≤ π h q 0 -q 0,h + M h θ t q 0 2 + F 1,∞ + t 2 F 2,∞ + t 0 F (s) -F h (s) ds, where F α,∞ = sup t∈[0,τ ] F (t) α.
Proof First, we substract (2.7) from (2.6) and obtain (we omit the time dependence for the sake of clarity) for all

ϕ h ∈ X h q -qh , ϕ h = ±i q -q h , ϕ h 1 2 -C * C(q -q h ), ϕ h + F -F h , ϕ h .
Noting that π h q -q, ϕ h 1 2 = 0 for all ϕ h ∈ X h and that π h q makes sense by the regularity of q (see (4.1)), we obtain from the above equality that for all ϕ h ∈ X h

π h q -qh , ϕ h = π h q -q, ϕ h ± i π h q -q h , ϕ h 1 2 -C * C (q -q h ) , ϕ h + F -F h , ϕ h . (2.9)
On the other hand, setting

E h = 1 2 π h q -q h 2 ,
we have Ėh = Re π h q -qh , π h q -q h . Applying (2.9) with ϕ h = π h q -q h and substituting the result in the above relation, we obtain by using Cauchy-Schwarz inequality and the boundedness of C that there exists

M > 0 such that Ėh ≤ ( π h q -q + M π h q -q + F -F h ) π h q -q h = √ 2E h . Since Ėh √ 2E h = d dt √ 2E h
, the integration of the above inequality from 0 to t yields

π h q(t) -q h (t) ≤ π h q 0 -q 0,h + t 0 ( π h q(s) -q(s) + M π h q(s) -q(s) ) ds + t 0 F (s) -F h (s) ds. (2.10)
Thus, it remains to bound π h q(t) -q(t) and π h q(t) -q(t) for all t ∈ [0, τ ]. Using (2.5) and the classical continuous embedding from D(A α ) to D(A β ) for α > β, we get that

π h q(t) -q(t) ≤ M h θ q(t) 1 2 ≤ M h θ q(t) 1 , π h q(t) -q(t) ≤ M h θ q(t) 1 2 ≤ M h θ q(t) 2 , ∀t ∈ [0, τ ], h ∈ (0, h * ).
Using relations (4.2) and (4.3) proved in Lemma 2 of the Appendix, we get for all t ∈ [0, τ ] and all h ∈ (0, h * )

π h q(t) -q(t) + π h q(t) -q(t) ≤ M h θ q 0 2 + t F 2,∞ + F 1,∞ .
Substituting the above inequality in (2.10), we get the result.

Using the last result, we derive an error approximation for the semigroups T ± and for the operator L t = T - t T + t . Proposition 2 Under the assumptions of Proposition 1, the following assertions hold true 1. There exist M > 0 and h * > 0 such that for all t ∈ (0, τ ) and all h ∈ (0, h * )

π h T + t q 0 -T + h,t q 0 ≤ M th θ q 0 2 . (2.11) π h T - t q 0 -T - h,t q 0 ≤ M (τ -t)h θ q 0 2 . (2.

12)

2. There exist M > 0 and h * > 0 such that for all n ∈ N, all t ∈ [0, τ ] and all h ∈ (0, h * ), we have

L n t q 0 -L n h,t q 0 ≤ M (1 + nτ )h θ q 0 2 . (2.13) Proof 1.
It suffices to take F = F h = 0 and q 0,h = π h q 0 in Proposition 1.

2. We first note that

L n t q 0 -L n h,t q 0 ≤ L n t q 0 -π h L n t q 0 + π h L n t q 0 -L n h,t q 0 . (2.14)
Using (2.5) and the fact that L t L(D(A)) ≤ 1 proved in Lemma 1 of the Appendix, the first term in the above relation can be estimated as follows

L n t q 0 -π h L n t q 0 ≤ M h θ q 0 2 , ∀h ∈ (0, h * ). (2.15)
For the second term in (2.14), we prove by induction that for all n ∈ N

π h L n t q 0 -L n h,t q 0 ≤ M nτ h θ q 0 2 , ∀h ∈ (0, h * ). (2.16)
By definition, we have

π h L t q 0 -L h,t q 0 = π h T - t T + t q 0 -T - h,t T + h,t q 0 , ≤ π h T - t T + t q 0 -T - h,t T + t q 0 + T - h,t (T + t q 0 -T + h,t q 0 ) .
By Lemma 1 of the Appendix and equation (2.12), we get

π h T - t T + t q 0 -T - h,t T + t q 0 ≤ M (τ -t)h θ q 0 2 , ∀h ∈ (0, h * ).
Obviously T - h L(X) is uniformly bounded with respect to h (this follows for example from (2.12)), and thus by (2.5) and equation (2.11), we have

T - h,t (T + t q 0 -T + h,t q 0 ) ≤ T + t q 0 -π h T + t q 0 + π h T + t q 0 -T + h,t q 0 ≤ M th θ q 0 2 , ∀h ∈ (0, h * ). Consequently π h L t q 0 -L h,t q 0 ≤ M τ h θ q 0 2 , ∀h ∈ (0, h * ),
(2.17) which shows that (2.16) holds for n = 1. Suppose now that for a given n ≥ 2, there holds

π h L n-1 t q 0 -L n-1 h,t q 0 ≤ M (n -1)τ h θ q 0 2 .
(2.18)

We write

π h L n t q 0 -L n h,t q 0 ≤ π h L t L n-1 t q 0 -L h,t L n-1 t q 0 + L h,t (L n-1 t q 0 -L n-1 h,t q 0 ) .
Thanks to Lemma 1 of the Appendix and to the uniform boundedness of L h,t L(X) with respect to h (which follows from the uniform boundedness of T ± h,t ) and using (2.17) and (2.18), we obtain

π h L n t q 0 -L n h,t q 0 ≤ M (τ + (n -1)τ )h θ q 0 2 ,
which is exactly (2.16). Substituting (2.15) and (2.16) in (2.14), we obtain the result.

We are now able to prove Theorem 1.

Proof (of Theorem 1) Introducing the term

N h n=0 L n h,τ z -(0), we rewrite z 0 -z 0,h in the following form z 0 -z 0,h = ∞ n=0 L n τ z -(0) - N h n=0 L n h,τ z - h (0), = n>N h L n τ z -(0) + N h n=0 L n τ -L n h,τ z -(0) + N h n=0 L n h,τ z -(0) -z - h (0) .
Therefore, we have

z 0 -z 0,h ≤ S 1 + S 2 + S 3 , (2.19)
where we have set

                   S 1 = n>N h L n τ z -(0) , S 2 = N h n=0 L n τ -L n h,τ z -(0) , S 3 = N h n=0 L n h,τ L(X) z -(0) -z - h (0) .
Note that the term S 1 is the truncation error of the tail of the infinite sum (1.5), the term S 2 represents the cumulated error due to the approximation of the semigroups T ± while the term S 3 comes from the approximation of the first iterate z -(0) of the algorithm.

Since η = Lτ L(X) < 1, using relation (1.4), the first term can be estimated very easily

S 1 ≤ M η N h +1 1 -η z 0 2 . (2.20)
The term S 2 can be estimated using the estimate (2.13) from Proposition 2

S 2 ≤ M N h n=0 (1 + nτ ) h θ z -(0) 2 , ∀h ∈ (0, h * ).
Therefore, using (1.4) and the fact that Lτ D(A 2 ) < 1 in the above relation, we finally get that

S 2 ≤ M 1 + (1 + τ )N h + N 2 h τ h θ z 0 2 , ∀h ∈ (0, h * ). (2.21)
It remains to estimate the term S 3 . As η = Lτ L(X) < 1, (2.13) implies that L h,τ L(X) is also uniformly with respect to h bounded by 1, provided h is small enough. Hence, we have

S 3 ≤ M N h z -(0) -z - h (0) ≤ M N h z -(0) -π h z -(0) + π h z -(0) -z - h (0) . (2.22)
By using (2.5) and (1.4), we immediately obtain that

z -(0) -π h z -(0) ≤ M h θ z 0 2 . (2.23)
To estimate the second term π h z -(0) -z - h (0), we apply twice Proposition 1 first for the time reversed backward observer z -(τ -•) and then for the forward observer z + (the time reversal step is introduced as in the formulation of Proposition 1, only initial value Cauchy problems can be considered). After straightforward calculation we obtain that for all h ∈ (0, h * )

π h z -(0) -z - h (0) ≤ M h θ τ ( z + (τ ) 2 + C * y 1,∞ ) + τ 2 C * y 2,∞ + τ 0 C * (y(τ -s) -y h (τ -s)) ds + τ 0 C * (y(s) -y h (s)) ds. (2.24)
Applying (4.2) of Lemma 2 of the Appendix with zero initial data, we obtain that

z + (τ ) 2 ≤ τ C * y 2,∞ .
Therefore (2.24) also reads

π h z -(0) -z - h (0) ≤ M h θ (τ + τ 2 ) C * y 2,∞ + 2 τ 0 C * (y(s) -y h (s)) ds. As C * C ∈ L D A 2 0 ∩ L (D (A 0 )) and z 2,∞ = z 0 2 (since iA 0 is skew-adjoint), the last relation becomes π h z -(0) -z - h (0) ≤ M h θ (τ + τ 2 ) z 0 2 + 2 τ 0 C * (y(s) -y h (s)) ds.
Substituting the above relation and (2.23) in (2.22), we get

S 3 ≤ M N h h θ (1 + τ + τ 2 ) z 0 2 + τ 0 C * (y(s) -y h (s)) ds . ( 2
z 0 -z 0,h ≤ M η N h +1 1 -η + h θ 1 + (1 + τ + τ 2 )N h + τ N 2 h z 0 2 + N h τ 0 C * (y(s) -y h (s)) ds ,
which leads to the result (with possibly reducing the value of h * ).

Full Discretization

Statement of the main result

In order to approximate (2.6), we use an implicit Euler scheme in time combined with the previous Galerkin approximation in space. In others words, we discretize the time interval [0, τ ] using a time step ∆t > 0. We obtain a discretization t k = k∆t, where 0 ≤ k ≤ K and where we assumed, without loss of generality, that τ = K∆t. Given a continuously differentiable function of time f , we approximate its derivative at time t k by the formula

f (t k ) D t f (t k ) := f (t k ) -f (t k-1 ) ∆t .
We suppose that q 0,h ∈ X h and F k h , for 0 ≤ k ≤ K, are given approximations of q 0 and F (t k ) in the space X. We define (q k h ), for 0 ≤ k ≤ K, as the solution of the following problem: for all ϕ h ∈ X h :

   D t q k h , ϕ h = ±i q k h , ϕ h 1 2 -C * Cq k h , ϕ h + F k h , ϕ h , q 0 h = q 0,h .
(2.26)

Note that the above procedure leads to a natural approximation T ± h,∆t,k of the continuous semigroup T ± t k by setting

T + t k q 0 T + h,∆t,k q 0 := q k h , T - t k q 0 T - h,∆t,k q 0 := q K-k h ,
where q k h solves (2.26) with F k h = 0 for all 0 ≤ k ≤ K and for q 0,h = π h q 0 . Obviously, this also leads to an approximation of

Lτ = T - τ T + τ by setting L h,∆t,K = T - h,∆t,K T + h,∆t,K .
Assume that for all 0 ≤ k ≤ K, y k h is a given approximation of y(t k ) in Y and let z + h k and z - h k be respectively the approximations of (2.2) and (2.3) obtained via (2.26) as follows:

-For all 0 ≤ k ≤ K, z + h k = q k h where q k h solves (2.26) with

F k h = C * y k h and q 0 h = 0, -For all 0 ≤ k ≤ K, z - h k = q K-k h where q k h solves (2.26) with F k h = C * y K-k h and q 0 h = (z + h ) K .
Then, our main result (which is the fully discrete counterpart of Theorem 1) reads as follows Theorem 2 Let A 0 : D (A 0 ) → X be a strictly positive self-adjoint operator and

C ∈ L(X, Y ) such that C * C ∈ L D A 2 0 ∩ L (D (A 0 )).
We assume that the pair (iA 0 , C) is exactly observable in time τ > 0. Let z 0 ∈ D A 2 0 be the initial value of (2.1). With the above notation, let z 0,h,∆t be defined by (1.7) and denote η := Lτ L(X) < 1.

Then there exist M > 0, h * > 0 and ∆t * > 0 such that for all h ∈ (0, h * ) and all ∆t ∈ (0, ∆t * ) we have

z 0 -z 0,h,∆t ≤ M η N h,∆t +1 1 -η + (h θ + ∆t)(1 + τ )N 2 h,∆t z 0 2 + N h,∆t ∆t K =0 C * (y(t ) -y h ) .
Corollary 2 Under the assumptions of Theorem 2, we set

N h,∆t = ln(h θ + ∆t) ln η
Then, there exist Mτ > 0, h * > 0 and ∆t * > 0 such that for all h ∈ (0, h * ) and ∆t ∈ (0, ∆t * )

z 0 -z 0,h,∆t ≤ Mτ (h θ + ∆t) ln 2 (h θ + ∆t) z 0 2 + ln(h θ + ∆t) ∆t K =0 C * (y(t ) -y h ) . (2.27)
Remark 3 Contrarily to the semi-discrete case, we have not been able to extend our results for z 0 in a larger space than D A 2 0 .

Remark 4 Let us emphasize that our results hold without assuming a CFL type condition.

Proof of Theorem 2

The proof of Theorem 2 goes along the same lines as the one of Theorem 1 in the semidiscrete case and uses energy estimates similar to those developed in Fujita and Suzuki [11, p. 865]. The main ingredient for the convergence analysis is the following result (the counterpart of Proposition 1) which gives the error estimate for the approximation (in space and time) of system (2.6) by (2.26).

Proposition 3 Given initial states q 0 ∈ D A 2 0 and q 0,h ∈ X h , let q and q k h , for 0 ≤ k ≤ K, be respectively the solutions of (2.6) and (2.26). Assume that C * C ∈ L (D (A 0 )). Then, there exist M > 0, h * > 0 and ∆t * > 0 such that for all h ∈ (0, h * ), all ∆t ∈ (0, ∆t * ) and all 0 ≤ k ≤ K:

π h q(t k ) -q k h ≤ π h q 0 -q 0,h + M ∆t k =1 F (t ) -F h + h θ + ∆t t k q 0 2 + F 1,∞ + Ḟ ∞ + t 2 k F 2,∞ .
Proof Let r 1 (t k ) denote the residual term in the first order Taylor expansion of q around t k-1 , so that

q(t k ) = q(t k ) -q(t k-1 ) ∆t - 1 ∆t r 1 (t k ) = D t q(t k ) - 1 ∆t r 1 (t k ), (2.28)
Subtracting (2.26) from the continuous weak formulation (2.6) applied for t = t k and for an arbitrary test function ϕ = ϕ h ∈ X h , we immediately get by using (2.28) that for all 1

≤ k ≤ K D t q(t k ) -q k h , ϕ h = ±i π h q(t k ), ϕ h 1 2 -C * C q(t k ) -q k h , ϕ h + 1 ∆t r 1 (t k ), ϕ h + F (t k ) -F k h , ϕ h .
The above relation implies that

D t π h q(t k ) -q k h , ϕ h = D t (π h q(t k ) -q(t k )) , ϕ h ± i π h q(t k ) -q k h , ϕ h 1 2 -C * C q(t k ) -q k h , ϕ h + 1 ∆t r 1 (t k ), ϕ h + F (t k ) -F k h , ϕ h . (2.29)
Now, for all 1 ≤ k ≤ K, let

E k h = 1 2 π h q(t k ) -q k h 2 .
Using the identity

1 2 u 2 -v 2 + u -v 2 = Re u -v, u , ∀u, v ∈ X,
one easily obtains that for all 1

≤ k ≤ K D t E k h ≤ Re D t π h q(t k ) -q k h , π h q(t k ) -q k h .
(2.30) Substituting (2.29) with ϕ h = π h q(t k ) -q k h in the above inequality and using the boundedness of C, we obtain the existence of M > 0 such that for all 1

≤ k ≤ K D t E k h ≤ D t (π h q(t k ) -q(t k )) + M π h q(t k ) -q(t k ) + 1 ∆t r 1 (t k ) + F (t k ) -F k h π h q(t k ) -q k h . (2.31)
Using the straightforward relations

D t E k h = D t E k h E k h + E k-1 h , (2.32)
and

π h q(t k ) -q k h ≤ √ 2 E k h + E k-1 h , (2.33)
we obtain from (2.5) and (2.31) that for all h ∈ (0, h * )

D t E k h ≤ M h θ D t q(t k ) 1 2 + q(t k ) 1 2 + 1 ∆t r 1 (t k ) + F (t k ) -F k h .
By (2.28) and relations (4.2) and (4.3) in Lemma 2 of the Appendix, the last estimate yields

D t E k h ≤ M h θ q 0 2 + t k F 2,∞ + F 1,∞ + F (t k ) -F k h + h θ ∆t r 1 (t k ) 1 2 + 1 ∆t r 1 (t k ) . (2.34)
To conclude, it remains to bound the two last terms in the above estimate. By definition of r 1 , we have

r 1 (t k ) = q(t k-1 ) -q(t k ) + ∆t q(t k ), in D A 1 2
0 , and thus by the mean value theorem, we get

r 1 (t k ) 1 2 ≤ ∆t sup s∈[t k-1 ,t k ] q(s) 1 2 + ∆t q(t k ) 1 2 .
Using once again (4.3), we obtain that there exists M > 0 such that

r 1 (t k ) 1 2 ≤ M ∆t q 0 2 + t k F 2,∞ + F 1,∞ . (2.35)
Now by the regularity of q (see Lemma 2), the residual r 1 can be expressed via the integral

r 1 (t k ) = t k t k-1 q(s) (t k-1 -s) ds,
in X, and thus

r 1 (t k ) ≤ ∆t 2 sup s∈[t k-1 ,t k ]
q(s) .

Using equation (2.4) verified by q and the boundedness of C, we have

q(t) = d q dt (t) = d dt ± iA 0 q(t) -C * Cq(t) + F (t) , ≤ q(t) 1 + M q(t) + Ḟ (t) .
Hence, once again by (4.3), we get

r 1 (t k ) ≤ ∆t 2 q 0 2 + t k F 2,∞ + F 1,∞ + Ḟ ∞ . ( 2
D t E k h = E k h -E k-1 h ∆t
, for k = 1, . . . , K, that can be added together to get the desired inequality (since π h q(t k ) -

q k h = 2E k h ).
Using this Proposition, we can derive an error estimate for the semigroup T ± t k (for all 1 ≤ k ≤ K) and for the operator Lτ = T - τ T + τ (the counterpart of Proposition 2).

Proposition 4 Under the assumptions of Proposition 3, the following assertions hold true 1. There exist M > 0, h * > 0 and ∆t * > 0 such that for all h ∈ (0, h * ), all ∆t ∈ (0, ∆t * ) and all 0 ≤ k ≤ K π h T + t k q 0 -T + h,∆t,k q 0 ≤ M t k (h θ + ∆t) q 0 2 .

(2.37)

π h T - t k q 0 -T - h,∆t,k q 0 ≤ M (τ -t k )(h θ + ∆t) q 0 2 . ( 2

.38)

2. There exist M > 0, h * > 0 and ∆t * > 0 such that for all n ∈ N, all h ∈ (0, h * ), all ∆t ∈ (0, ∆t * ) and all 0 ≤ k ≤ K

(L n t k -L n h,∆t,k )q 0 ≤ M h θ + nτ h θ + ∆t q 0 2 .
(2.39) Proof 1. It suffices to apply Proposition 3 with F (t k ) = F k h = 0 for all 0 ≤ k ≤ K and q 0,h,∆t = π h q 0 .

2. First, we note that

L n t k q 0 -L n h,∆t,k q 0 ≤ L n t k q 0 -π h L n t k q 0 + π h L n t k q 0 -L n h,∆t,k q 0 .
(2.40) Using (2.5), the fact that L n t L(D(A)) ≤ 1 (proved in Lemma 1 of the Appendix), the first term in the above relation can be estimated as follows

L n t k q 0 -π h L n t k q 0 ≤ M h θ q 0 2 , ∀h ∈ (0, h * ).
(2.41)

For the second term in (2.40), we prove by induction that for all n ∈ N, all h ∈ (0, h * ) and all ∆t ∈ (0, ∆t * ) (for some ∆t * > 0)

π h L n t k q 0 -L n h,∆t,k q 0 ≤ M nτ h θ + ∆t q 0 2 . (2.42)
By definition, we have

π h L t k q 0 -L h,∆t,k q 0 = π h T - t k T + t k q 0 -T - h,∆t,k T + h,∆t,k q 0 , ≤ π h T - t k -T - h,∆t,k π h T + t k q 0 + T - h,∆t,k π h T + t k -T + h,∆t,k q 0 .
Using (2.38) and Lemma 1, we get

π h T - t k -T - h,∆t,k π h T + t k q 0 ≤ M (τ -t k ) h θ + ∆t q 0 2 .
Obviously T - h,∆t,k L(X) is uniformly bounded (with respect to h and ∆t), and thus again by (2.37) we have

T - h,∆t,k π h T + t k -T + h,∆t,k q 0 ≤ M t k h θ + ∆t q 0 2 .
So, by adding the two last inequalities, we obtain that

π h L t k q 0 -L h,∆t,k q 0 ≤ M τ h θ + ∆t q 0 2 , (2.43)
showing that (2.42) holds for n = 1. Suppose now that for some n ≥ 2

π h L n-1 t k q 0 -L n-1 h,∆t,k q 0 ≤ M (n -1)τ h θ + ∆t q 0 2 . (2.44) Writing π h L n t k q 0 -L n h,∆t,k q 0 ≤ π h L t k L n-1 t k q 0 -L h,∆t,k π h L n-1 t k q 0 + L h,∆t,k (π h L n-1 t k q 0 -L n-1 h,∆t,k q 0 ) ,
we get by using Lemma 1, the uniform boundedness of L h,∆t,k L(X) with respect to h and ∆t, (2.43) and (2.44) that

π h L n t k q 0 -L n h,∆t,k q 0 ≤ M (1 + (n -1))τ h θ + ∆t q 0 2 ,
which is exactly (2.42). Substituting (2.41) and (2.42) in (2.40), we obtain the result.

We are now able to prove Theorem 2.

Proof (of Theorem 2)

We first introduce the term

N h,∆t n=0
L n h,∆t,K z -(0) to rewrite the approximation error z 0 -z 0,h,∆t in the following form:

z 0 -z 0,h,∆t = ∞ n=0 L n τ z -(0) - N h,∆t n=0 L n h,∆t,K z - h 0 = n>N h,∆t L n τ z -(0) + N h,∆t n=0 L n τ -L n h,∆t,K z -(0) + N h,∆t n=0 L n h,∆t,K z -(0) -z - h 0 .
Therefore, we have

z 0 -z 0,h,∆t ≤ S 1 + S 2 + S 3 , (2.45)
where we have set

                       S 1 = n>N h,∆t L n τ z -(0) , S 2 = N h,∆t n=0 L n τ -L n h,∆t,K z -(0) , S 3 =   N h,∆t n=0 L n h,∆t,K L(X)   z -(0) -z - h 0 .
Since η = Lτ L(X) < 1, the first term can be estimated very easily

S 1 ≤ M η N h,∆t +1 1 -η z 0 2 . (2.46)
The second term S 2 can be estimated using the estimate (2.39) from Proposition 4

S 2 ≤ M N h,∆t n=0 h θ + nτ (h θ + ∆t) z -(0) 2 , ∀h ∈ (0, h * ), ∆t ∈ (0, ∆t * ).
Therefore, using (1.4), the fact that Lτ D(A 2 ) < 1 (see Lemma 1) in the above relation, we get that for all h ∈ (0, h * ) and ∆t ∈ (0, ∆t * )

S 2 ≤ M 1 + (1 + τ )N h,∆t + (1 + τ )N 2 h,∆t h θ + ∆t z 0 2 . (2.47)
It remains to estimate the term S 3 . As for the semi-discrete case, on can easily show that L h,∆t,K L(X) is uniformly bounded by 1 (with respect to h and ∆t), and thus we have

S 3 ≤ M N h,∆t z -(0) -(z - h ) 0 ≤ M N h,∆t z -(0) -π h z -(0) + π h z -(0) -(z - h ) 0 .
(2.48) By using (2.5) and (1.4), we immediately obtain that

z -(0) -π h z -(0) ≤ M h θ z 0 2 . (2.49)
To estimate the second term π h z -(0) -(z - h ) 0 , we apply twice Proposition 3 first for the time reversed backward observer z -(τ -•) and then for the forward observer z + (the time reversal step is introduced simply because Proposition 3 is written for initial (and not final) value Cauchy problems). After straightforward calculation we obtain that for all h ∈ (0, h * ) and all ∆t ∈ (0, ∆t * )

π h z -(0) -(z - h ) 0 ≤ M (h θ + ∆t) τ ( z + (τ ) 2 + C * y 1,∞ + C * ẏ ∞) + τ 2 C * y 2,∞ + ∆t K =1 C * y(τ -t ) -y K- h + ∆t K =1 C * y(t ) -y h .
(2.50) Applying (4.2) of Lemma 2 of the Appendix with zero initial data, we obtain that

z + (τ ) 2 ≤ τ C * y 2,∞ . As C * C ∈ L D A 2 0 ∩ L (D (A 0 )) and z 2,∞ = z 0 2 (since iA 0 is skew-adjoint), (2.50) also reads π h z -(0) -(z - h ) 0 ≤ M (h θ + ∆t)(τ + τ 2 ) z 0 2 + 2∆t K =0 C * y(t ) -y h .
Substituting the above relation and (2.49) in (2.48), we get

S 3 ≤ M N h,∆t (h θ + ∆t)(1 + τ + τ 2 ) z 0 2 + ∆t K =0 C * y(t ) -y h . (2.
z 0 -z 0,h,∆t ≤ M N h,∆t ∆t K =0 C * y(t ) -y h + η N h,∆t +1 1 -η z 0 2 +(h θ + ∆t) 1 + (1 + τ + τ 2 )N h,∆t + (1 + τ )N 2 h,∆t z 0 2 ,
which leads to the result (with possibly reducing the value of h * and ∆t * ).

The wave equation

Let H be a Hilbert space endowed with the inner product •, • . The corresponding norm of H is denoted by • . Let A 0 : D (A 0 ) → H be a strictly positive self-adjoint operator and C 0 ∈ L(H, Y ) a bounded observation operator, where Y is an other Hilbert space. The norm in D(A α 0 ) will be denoted by • α. Given τ > 0, we deal with the general wave type system ẅ(t) + A 0 w(t) = 0, ∀t 0,

y(t) = C 0 ẇ(t), ∀t ∈ [0, τ ], (3.1)
and we want to reconstruct the initial value (w 0 , w 1 ) = (w(0), ẇ(0)) of (3.1) knowing y(t) for t ∈ [0, τ ]. In order to use the general iterative algorithm described in the introduction, we first rewrite (3.1) as a first order system of the form (1.1). To achieve this, it suffices to introduce the following notation:

z(t) = w(t) ẇ(t) , X = D A 1 2 0 × H, A = 0 I -A 0 0 , D (A) = D (A 0 ) × D A 1 2 0 , (3.2) C ∈ L(X, Y ), C = 0 C 0 . (3.3)
The space X is endowed with the norm

z = z 1 2 1 2 + z 2 2 , ∀ z = z 1 z 2 ∈ X.
Note that the operator iA is selfadjoint but has no sign so that the problem studied here does not fit into the framework of Section 2. We assume that the pair (A, C) is exactly observable in time τ > 0. Thus, according to Liu [21, Theorem 2.3.],

A + = A -C * C (resp. A -= -A -C * C
) is the generator of an exponentially stable C 0 -semigroup T + (resp. T -). We set as usually

Lτ = T - τ T + τ .
Throughout this section we always assume that (w 0 , w 1 )

∈ D A 2 = D A 3 2
0 ×D (A 0 ). Thus by applying Theorem 4.1.6 of Tucsnak and Weiss [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF], we have

w ∈ C [0, τ ], D A 3 2 0 ∩ C 1 ([0, τ ], D (A 0 )) ∩ C 2 [0, τ ], D A 1 2 0 .
The forward and backward observers (1.2) and (1.3) read then as follows (as secondorder systems)

ẅ+ (t) + A 0 w + (t) + C * 0 C 0 ẇ+ (t) = C * 0 y(t), ∀t ∈ [0, τ ], w + (0) = 0, ẇ+ (0) = 0, (3.4) ẅ-(t) + A 0 w -(t) -C * 0 C 0 ẇ-(t) = -C * 0 y(t), ∀t ∈ [0, τ ], w -(τ ) = w + (τ ), ẇ-(τ ) = ẇ+ (τ ). (3.5)
Clearly, the above two systems can be written as a general initial value Cauchy problem of the same form (simply by using a time reversal for the second system)

p(t) + A 0 p(t) + C * 0 C 0 ṗ(t) = f (t), ∀t ∈ [0, τ ], p(0) = p 0 , ṗ(0) = p 1 (3.6)
where we have set for the forward observer (3.4) :

f (t) = C * 0 y(t) = C * 0 C 0 ẇ(t)
and (p 0 , p 1 ) = (0, 0), for the backward observer (3.5) :

f (t) = -C * 0 y(τ -t) = -C * 0 C 0 ẇ(τ -t) and (p 0 , p 1 ) = (w + (τ ), -ẇ+ (τ )) ∈ D A 2 = D A 3 2 0 × D (A 0 ).
Let us emphasize that with these notation, the semigroups T ± are given by the relations

T + t p 0 p 1 = p(t) ṗ(t) T - t p 0 p 1 = p(τ -t) -ṗ(τ -t) (3.7)
where p solves (3.6) with f = 0.

In the next two subsections, we present a convergence analysis of semi-discretized and fully discretized approximation schemes for the forward and backward observers (3.4) and (3.5). Our proof is based on the convergence analysis of the semi and fully discretizations of (3.6). For the sake of clarity, we dropped in the proofs some of the details which are very close to the ones given in the Schrödinger. As far as we know, the existing literature on the convergence analysis of full discretizations of wavetype systems concern only the particular cases of conservative systems (i.e. without damping), see e.g. Raviart and Thomas [25, p. 197] or Dautray and Lions [7, p. 921] and systems with constant damping coefficients Geveci and Kok [START_REF] Geveci | The convergence of Galerkin approximation schemes for second-order hyperbolic equations with dissipation[END_REF]. For a recent review of numerical approximation issues related to the control and the observation of waves, we refer the reader to the review paper of Zuazua [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF].

Space Semi-Discretization

Statement of the main result

We use a Galerkin method to approximate system (3.6). More precisely, consider a family (H h ) h>0 of finite-dimensional subspaces of D A 1 2 0 endowed with the norm in H. We denote π h the orthogonal projection from D A 1 2 0 onto H h . We assume that there exist M > 0, θ > 0 and h * > 0 such that we have for all h ∈ (0, h * )

π h ϕ -ϕ ≤ M h θ ϕ 1 2 , ∀ϕ ∈ D A 1 2 0 . (3.8)
Given (p 0 , p 1 ) ∈ D A 2 , the variational formulation of (3.6) reads for all t ∈ [0, τ ] and

all ϕ ∈ D A 1 2 0 as follows p(t), ϕ + p(t), ϕ 1 2 + C * 0 C 0 ṗ(t), ϕ = f (t), ϕ , ∀t ∈ [0, τ ], p(0) = p 0 , ṗ(0) = p 1 . (3.9)
Suppose that (p 0,h , p 1,h ) ∈ H h × H h and f h are given approximations of (p 0 , p 1 ) and f respectively in the spaces X and L 1 ([0, τ ], H). We define p h (t) as the solution of the variational problem

ph (t), ϕ h + p h (t), ϕ h 1 2 + C * 0 C 0 ṗh (t), ϕ h = f h (t), ϕ h , ∀t ∈ [0, τ ], p h (0) = p 0,h , ṗh (0) = p 1,h . (3.10)
for all t ∈ [0, τ ] and all ϕ h ∈ H h .

The above approximation procedure leads in particular to the definition of the semi-discretized versions T ± h of the semigroups T ± that we will use. Indeed, we simply set

T + h,t p 0 p 1 = p h (t) ṗh (t) T - h,t p 0 p 1 = p h (τ -t) -ṗh (τ -t) (3.11)
where p h solves (3.10) for f h = 0 and (p 0,h , p 1,h ) = (π h p 0 , π h p 1 ). The semi-discretized

counterpart of Lτ = T - τ T + τ is then given by L h,τ = T - h,τ T + h,τ .
Assume that y h is an approximation of the output y in L 1 ([0, τ ], Y ) and let w + h and w - h denote the Galerkin approximations of the solutions of systems (3.4) and (3.5), satisfying for all t ∈ [0, τ ] and all

ϕ h ∈ H h ẅ+ h (t), ϕ h + w + h (t), ϕ h 1 2 + C * 0 C 0 ẇ+ h (t), ϕ h = C * 0 y h (t), ϕ h , w + h (0) = 0, ẇ+ h (0) = 0, (3.12)
ẅ-

h (t), ϕ h + w - h (t), ϕ h 1 2 -C * 0 C 0 ẇ- h (t), ϕ h = -C * 0 y h (t), ϕ h , w - h (τ ) = w + h (τ ), ẇ- h (τ ) = ẇ+ h (τ ). (3.13)
With the above notation, the main result of this section reads as follows.

Theorem 3 Let A 0 : D (A 0 ) → H be a strictly positive self-adjoint operator and

C 0 ∈ L(H, Y ) such that C * 0 C 0 ∈ L D A 3 2 0 ∩ L D A 1 2 0 . Define (A, C) by (3.2)
and (3.3). Assume that the pair (A, C) is exactly observable in time τ > 0 and set

η := Lτ L(X) < 1. Let (w 0 , w 1 ) ∈ D A 3 2
0 × D (A 0 ) be the initial value of (3.1) and let (w 0,h , w 1,h ) be defined by

w 0,h w 1,h = N h n=0 L n h,τ w - h (0) ẇ- h (0) . (3.14)
Then there exist M > 0 and h * > 0 such that for all h ∈ (0, h * )

w 0 -w 0,h 1 2 + w 1 -w 1,h ≤ M η N h +1 1 -η + h θ τ N 2 h w 0 3 2 + w 1 1 + N h τ 0 C * 0 (y(s) -y h (s)) ds .
Corollary 3 Under the assumptions of Theorem 3, we set

N h = θ ln h ln η .
Then, there exist Mτ > 0 and h * > 0 such that for all h ∈ (0, h * )

w 0 -w 0,h 1 2 + w 1 -w 1,h ≤ Mτ h θ ln 2 h w 0 3 2 + w 1 1 + | ln h| τ 0 C * 0 (y(s) -y h (s)) ds . (3.15)

Proof of Theorem 3

The next Proposition provides the error estimate for the approximation of (3.9) by using the Galerkin scheme (3.10).

Proposition 5 Given (p 0 , p 1 )

∈ D A 3 2 0 × D (A 0 ) and (p 0,h , p 1,h ) ∈ H h × H h , let
p and p h be the solutions of (3.9) and (3.10) respectively. Assume that

C * 0 C 0 ∈ L D A 1 2

0

. Then, there exist M > 0 and h * > 0 such that for all t ∈ [0, τ ] and all h ∈ (0, h * )

π h p(t) -p h (t) 1 2 + π h ṗ(t) -ṗh (t) ≤ M π h p 0 -p 0,h 1 2 + π h p 1 -p 1,h + h θ t p 0 3 2 + p 1 1 + f 1 2 ,∞ + t 2 f 1,∞ + t 0 f (s) -f h (s) ds.
Proof First, we substract (3.10) from (3.9) to obtain (we omit the time dependence for the sake of clarity) for all

ϕ h ∈ H h p -ph , ϕ h + p -p h , ϕ h 1 2 + C * 0 C 0 ( ṗ -ṗh ) , ϕ h = f -f h , ϕ h .
Noting that π h p -p, ϕ h 1 2 = 0 for all ϕ h ∈ H h and that π h p makes sense by the regularity of p (this is a direct consequence of relation (4.1) from Lemma 2 used with q = p ṗ ), we obtain from the above equality that for all

ϕ h ∈ H h π h p -ph , ϕ h + π h p -p h , ϕ h 1 2 = π h p -p, ϕ h + C * 0 C 0 ( ṗh -ṗ) , ϕ h + f -f h , ϕ h . (3.16)
On the other hand, setting

E h = 1 2 π h ṗ -ṗh 2 + 1 2 π h p -p h 2 1 2
,

we have Ėh = π h p -ph , π h ṗ -ṗh + π h p -p h , π h ṗ -ṗh 1 2 .
Applying (3.16) with ϕ h = π h ṗ -ṗh and substituting the result in the above relation, we obtain by using Cauchy-Schwarz inequality and the boundedness of C 0 that there exists M > 0 such that

Ėh ≤ π h p -p + M π h ṗ -ṗ + f -f h π h ṗ -ṗh ≤ √ 2E h . Since Ėh √ 2E h = d dt √ 2E h
, the integration of the above inequality from 0 to t yields

π h p(t) -p h (t) 1 2 + π h ṗ(t) -ṗh (t) ≤ M π h p 0 -p 0,h 1 2 + π h p 1 -p 1,h + t 0 ( π h p(s) -p(s) + π h ṗ(s) -ṗ(s) ) ds + t 0 f (s) -f h (s) ds . (3.17)
Thus, it remains to bound π h p(t) -p(t) and π h ṗ(t) -ṗ(t) for all t ∈ [0, τ ]. Using (3.8) and the classical continuous embedding from D(A α ) to D(A β ) for α > β, we get that

π h p(t) -p(t) ≤ M h θ p(t) 1 2 , π h ṗ(t) -ṗ(t) ≤ M h θ ṗ(t) 1 2 ≤ M h θ ṗ(t) 1 , ∀t ∈ [0, τ ], h ∈ (0, h * ).
Using relations (4.3) proved in Lemma 2 of the Appendix for the first order unknown q = p ṗ and the right-hand side F = 0 f , we get for all t ∈ [0, τ ] and all h ∈ (0, h * )

π h p(t) -p(t) + π h ṗ(t) -ṗ(t) ≤ M h θ p 0 3 2 + p 1 1 + t f 1,∞ + f 1 2 ,∞ .
Substituting the above inequality in (3.17), we get the result.

Thanks to the last result, we are now in position to derive an error approximation for the semigroups T ± and for the operator L t = T - t T + t . This result has been recently proved in [START_REF] Cîndea | An approximation method for exact controls of vibrating systems[END_REF] we refer the interested reader to the proof given there, which is similar to the one of Proposition 2.

Proposition 6

Let Π h = π h 0 0 π h . Under the assumptions of Proposition 5, the following assertions hold true 1. There exist M > 0 and h * > 0 such that for all t ∈ (0, τ ) and all h ∈ (0, h * )

(Π h T + t -T + h,t ) p 0 p 1 ≤ M th θ p 0 3 2 + p 1 1 , (3.18)
(Π h T - t -T - h,t ) p 0 p 1 ≤ M (τ -t)h θ p 0 3 2 + p 1 1 . (3.19)
2. There exist M > 0 and h * > 0 such that for all n ∈ N, all t ∈ [0, τ ] and all h ∈ (0, h * ), we have

(L n t -L n h,t ) p 0 p 1 ≤ M (1 + nτ )h θ p 0 3 2 + p 1 1 . (3.20)
Now, we can turn to the proof of Theorem 3

Proof (of Theorem 3) Introducing the term

N h n=0 L n h,τ w -(0) ẇ-(0)
, we first rewrite the error

term w 0 w 1 - w 0,h w 1,h = ∞ n=0 L n τ w -(0) ẇ-(0) - N h n=0 L n h,τ w - h (0) ẇ- h (0)
in the following form

w 0 w 1 - w 0,h w 1,h = n>N h L n τ w -(0) ẇ-(0) + N h n=0 L n τ -L n h,τ w -(0) ẇ-(0) + N h n=0 L n h,τ w -(0) -w - h (0) ẇ-(0) -ẇ- h (0)
. Therefore, we have

w 0 w 1 - w 0,h w 1,h ≤ S 1 + S 2 + S 3 , (3.21)
where we have set

                     S 1 = n>N h L n τ w -(0) ẇ-(0) , S 2 = N h n=0 L n τ -L n h,τ w -(0) ẇ-(0) , S 3 = N h n=0 L n h,τ L(X)
w -(0) ẇ-(0) .

Following exactly the same way than in the proof of the Schrödinger case, we get the claimed result.

Full Discretization

Statement of the main result

In order to approximate (3.9) in space and time, we use an implicit Euler scheme in time combined with the previous Galerkin approximation in space. We discretize the time interval [0, τ ] using a time step ∆t > 0. We obtain a discretization t k = k∆t, where 0 ≤ k ≤ K and where we assumed, without loss of generality, that τ = K∆t. Given a function of time f of class C 2 , we approximate its first and second derivative at time t k by

f (t k ) D t f (t k ) := f (t k ) -f (t k-1 ) ∆t . f (t k ) D tt f (t k ) := f (t k ) -2f (t k-1 ) + f (t k-2 ) ∆t 2 .
We suppose that (p 0,h,∆t , p 1,h,∆t ) ∈ H h × H h and f k h , for 0 ≤ k ≤ K, are given approximations of (p 0 , p 1 ) and f (t k ) in the space X and H respectively. We define the approximate solution (p k h ) 0≤k≤K of (3.9) as the solution of the following problem:

p k h ∈ H h such that for all ϕ h ∈ H h    D tt p k h , ϕ h + p k h , ϕ h 1 2 + C * 0 C 0 D t p k h , ϕ h = f k h , ϕ h , 2 ≤ k ≤ K p 0 h = p 0,h,∆t , p 1 h = p 0 h + ∆t p 1,h,∆t . (3.22)
Note that the above procedure leads to a natural approximation T ± h,∆t,k of the con-

tinuous operators T ± t k by setting            T + t k p 0 p 1 T + h,∆t,k p 0 p 1 := p k h D t p k h T - t k p 0 p 1 T - h,∆t,k p 0 p 1 := p K-k h -D t p K-k h (3.23)
where p k h solves (3.22) with f k h = 0 for all 0 ≤ k ≤ K and for (p 0,h,∆t , p 1,h,∆t ) = (π h p 0 , π h p 1 ). Obviously, this also leads to a fully discretized approximation of the operator Lτ

= T - τ T + τ by setting L h,∆t,K = T - h,∆t,K T + h,∆t,K .
Assume that for all 0 ≤ k ≤ K, y k h is a given approximation of y(t k ) in Y and let w + h k and w - h k be respectively the approximations of (3.4) and (3.5) obtained via (3.22) as follows:

-For all 0 ≤ k ≤ K, w + h k = p k h where p k h solves (3.22) with f k h = C * 0 y k h and (p 0,h,∆t , p 1,h,∆t ) = (0, 0), -For all 0 ≤ k ≤ K, w - h k = p K-k h where p k h solves (3.22) with f k h = -C * 0 y K-k h and (p 0,h,∆t , p 1,h,∆t ) = ((w + h ) K , -D t (w + h ) K
). Then, our main result (the fully discrete counterpart of Theorem 3) reads as follows Theorem 4 Let A 0 : D (A 0 ) → H be a strictly positive self-adjoint operator and

C 0 ∈ L(H, Y ) such that C * 0 C 0 ∈ L D A 3 2 0 ∩ L D A 1 2 0 . Define (A, C) by (3.2)
and (3.3). Assume that the pair (A, C) is exactly observable in time τ > 0 and set

η := Lτ L(X) < 1. Let (w 0 , w 1 ) ∈ D A 3 2
0 × D (A 0 ) be the initial value of (3.1) and let (w 0,h,∆t , w 1,h,∆t ) be defined by

w 0,h,∆t w 1,h,∆t = N h n=0 L n h,∆t,K (w - h ) 0 D t (w - h ) 1 , (3.24)
where

D t (w - h ) 1 = (w - h ) 1 -(w - h ) 0 ∆t .
Then there exist M > 0, h * > 0 and ∆t * > 0 such that for all h ∈ (0, h * ) and ∆t ∈ (0, ∆t * )

w 0 -w 0,h,∆t 1 2 + w 1 -w 1,h,∆t ≤ M η N h,∆t +1 1 -η + h θ + ∆t (1 + τ ) N 2 h,∆t w 0 3 2 + w 1 1 +N h,∆t ∆t K =0 C * 0 (y(t ) -y h ) .
Corollary 4 Under the assumptions of Theorem 4, we set

N h,∆t = ln(h θ + ∆t) ln η
Then, there exist Mτ > 0, h * > 0 and ∆t * > 0 such that for all h ∈ (0, h * ) and ∆t ∈ (0, ∆t * )

w 0 -w 0,h,∆t 1 2 + w 1 -w 1,h,∆t ≤ Mτ (h θ + ∆t) ln 2 (h θ + ∆t) w 0 3 2 + w 1 1 + ln(h θ + ∆t) ∆t K =0 C * 0 y(t ) -y h . (3.25)

Proof of Theorem 4

As in the semi-discrete case, the main ingredient for the convergence analysis is the following result (the counterpart of Proposition 5) which gives the error estimate for the full approximation of the general system (3.9) by (3.22).

Proposition 7 Given (p 0 , p 1 ) ∈ D A . Then, there exist M > 0, h * > 0 and ∆t * > 0 such that for all 1 ≤ k ≤ K, all h ∈ (0, h * ) and all ∆t ∈ (0, ∆t * )

π h p(t k ) -p k h 1 2 + π h ṗ(t k ) -D t p k h ≤ M π h p 0 -p 0,h,∆t 1 2 + π h p 1 -p 1,h,∆t + h θ + ∆t t k p 0 3 2 + p 1 1 + f 1 2 ,∞ + ḟ ∞ + t 2 k f 1,∞ + ∆t k =1 f (t ) -f h .
Proof Denote by r 1 (t k ) the residual term in the first order Taylor expansion of p around t k-1 . Then

ṗ(t k ) = p(t k ) -p(t k-1 ) ∆t - 1 ∆t r 1 (t k ) = D t p(t k ) - 1 ∆t r 1 (t k ), (3.26)
We have

π h ṗ(t k ) -D t p k h ≤ π h ṗ(t k ) -π h D t p(t k ) + D t (π h p(t k ) -p k h ) ≤ 1 ∆t r 1 (t k ) + D t (π h p(t k ) -p k h )
Therefore, the error we need to bound satisfies

π h p(t k ) -p k h 1 2 + π h ṗ(t k ) -D t p k h ≤ 2 E k h + 1 ∆t r 1 (t k ) (3.27)
where we have set for all 1 ≤ k ≤ K

E k h = 1 2 D t π h p(t k ) -p k h 2 + π h p(t k ) -p k h 2 1 2
.

On the other hand, if r 2 (t k ) denote the residual term first order the Taylor expansion of ṗ around t

k-1 , then p(t k ) = D tt p(t k ) -γ k , (3.28)
where

γ k = 1 ∆t 2 (r 1 (t k ) -r 1 (t k-1 )) + 1 ∆t r 2 (t k ).
Using (3.26) and (3.28), and subtracting (3.22) from the variational formulation (3.9) written for t = t k and for an arbitrary test function ϕ = ϕ h ∈ H h , one easily obtains

D tt π h p(t k ) -p k h , ϕ h + π h p(t k ) -p k h , ϕ h 1 2 = D tt (π h p(t k ) -p(t k )) , ϕ h -C * 0 C 0 D t p(t k ) -p k h , ϕ h + γ k , ϕ h + 1 ∆t C * 0 C 0 r 1 (t k ), ϕ h + f (t k ) -f k h , ϕ h . (3.29) Using the identity 1 2 u 2 -v 2 + u -v 2 = Re u -v, u , ∀u, v ∈ H, one easily obtains that for all 2 ≤ k ≤ K D t E k h ≤ D tt π h p(t k ) -p k h , D t π h p(t k ) -p k h + π h p(t k ) -p k h , D t π h p(t k ) -p k h 1 2
. (3.30)

Taking ϕ h = D t π h p(t k ) -p k h in (3.29) and substituting in the above inequality and using the boundedness of C 0 , we obtain the existence of M > 0 such that for all 2

≤ k ≤ K D t E k h ≤ M D tt (π h p(t k ) -p(t k )) + D t (
D t E k h ≤ M h θ p 0 3 2 + p 1 1 + t k f 1,∞ + f 1 2 ,∞ + f (t k ) -f k h + h θ ∆t 2 r 1 (t k ) -r 1 (t k-1 ) 1 2 + h θ ∆t r 1 (t k ) 1 2 + r 2 (t k ) 1 2 + 1 ∆t 2 r 1 (t k ) -r 1 (t k-1 ) + 1 ∆t r 1 (t k ) + r 2 (t k ) . (3.32)
To conclude, it remains to bound the terms including the residuals r 1 and r 2 in the above estimate. By definition of r 2 , the mean value theorem and using once again (4.3), we obtain that there exists M > 0 such that r 2 (t k ) Using this Proposition, we can derive an error estimate for the semigroup T ± t k (for all 0 ≤ k ≤ K) and for the operator Lτ = T - τ T + τ (the counterpart of Proposition 6). We skip the proof, which is nearly the same as the one of Proposition 4.

+ p 1 1 + t k-1 f 1,∞ + f 1 2 ,∞ + ḟ ∞ . ( 3

Proposition 8

Let Π h = π h 0 0 π h . Under the assumptions of Proposition 7, the following assertions hold true 1. There exist M > 0, h * > 0 and ∆t * > 0 such that for all h ∈ (0, h * ), all ∆t ∈ (0, ∆t * ) and all 0 ≤ k ≤ K Lemma 2 Given q 0 ∈ D A 2 and F ∈ C [0, τ ], D A 2 ∩ C 1 ([0, τ ], D (A)), let q denote the solution of the initial value problem q(t) = Aq(t) -C * Cq(t) + F (t), t ∈ (0, τ ), q(0) = q 0 .

(Π h T + t k -T + h,∆t,k ) p 0 p 1 ≤ M t k (h θ + ∆t) p
Then, we have the following statements 1. Regularity:

q ∈ C [0, τ ], D A 2 ∩ C 1 ([0, τ ], D (A)) ∩ C 2 ([0, τ ], X) , (4.1)
2. Bound for q: q(t) α ≤ q 0 α + t F α,∞, for α = 0, 1, 2, (4.2)

3. Bound for q : there exists M > 0 such that q(t) α ≤ M q 0 α+1 + t F α+1,∞ + F α,∞, for α = 0, 1, (

where F α,∞ = sup The last inclusion follows then from the fact that q(t) = (A -C * C) q(t) in D (A).

2. By Duhamel's formula, we have q(t) α = T t q 0 + t 0 T t-s F (s)ds α , ≤ T t q 0 α + t 0 T t-s F (s) α ds,

≤ q 0 α + t F α,∞,
where we have used Lemma 1 of the Appendix for the last inequality.

3. Using the estimate (4.2) obtained for q(t) and the continuity of the embeddings D A 2 → D (A) → X, we easily get q(t) α = (A -C * C) q(t) + F (t) α, ≤ q(t) α+1 + M q(t) α + F α,∞, ≤ M q 0 α+1 + t F α+1,∞ + F α,∞.

Fig. 1

 1 Fig. 1 An initial data inverse problem for evolution PDE's : How to reconstruct the initial state (light grey) for a PDE set on a domain Ω from partial observation on O × [0, τ ] (dark grey)?

  fact that C * C ∈ L (D (A)) ensures that the contraction property for T + and T - is still satisfied when restricted to D (A) and D A 2 (see Lemma 1 of the Appendix). Let us point out that this is proved for the damped wave equation in Cîndea et al. [5, Proposition 2.5]. Moreover, we also have Lτ D(A) < 1 and Lτ D(A 2 ) < 1 (by application of [27, Proposition 2.10.4]). The second technical assumption C * C ∈ L D A 2

3 2 0

 32 × D (A 0 ) and (p 0,h,∆t , p 1,h,∆t ) ∈ H h × H h , let p and (p k h ) k be the solutions of (3.9) and (3.22) respectively. Assume that C * 0

  .38) Substituting (3.33), (3.35), (3.36), (3.37) and (3.38) in relation (3.32) provides estimates for D t E k h = E k h -E k-1 h ∆t , for k = 1, . . . , K. By adding all these inequalities, we immediately get an upper bound for E k h , and thus the desired inequality thanks to (3.27) and (3.36).

0 3 2 + p 1 1 . 1 ≤ 3 2+ p 1 1 .

 21131 M (τ -t k )(h θ + ∆t) p 0 (3.40) Proof As C ∈ L(X, Y ) is bounded, we clearly have D (A) = D (A -C * C). Moreover, C * C ∈ L (D (A)) implies that D A 2 = D (A -C * C) 2 .The result follows then from [27, Proposition 2.10.4].

Proof 1 .

 1 By[START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF] Theorem 4.1.6], we have q ∈ C [0, τ ], D A 2 ∩ C 1 ([0, τ ], D (A)). But since C * C ∈ L (D (A)) and F ∈ C [0, τ ], D A 2 ∩ C 1 ([0, τ ], D (A)), we have A -C * C q(t) ∈ C ([0, τ ], D (A)) ∩ C 1 ([0, τ ], X) .

  π h p(t k ) -p(t k )) + γ k + 1 ∆t r 1 (t k ) + f (t k ) -f k h D t (π h p(t k ) -p k h ) . (3.31) Using relations (2.32) and (2.33), we obtain from (3.8), (3.31), (3.26), (3.28) and relations (4.2) and (4.3) in Lemma 2 of the Appendix for the first order formulation of (3.6) that for all h ∈ (0, h

* )

  Now by the regularity of p (see Lemma 2 applied to the first order formulation of (3.6)), the residual r 2 can be expressed via the integral r 2 (t k ) = Using equation (3.6) verified by p and the boundedness of C 0 , we have Hence, once again by (4.3), we getr 2 (t k ) ≤ M ∆t 2 p 0 3 2 + p 1 1 + t k f 1,∞ + f 1 2 ,∞ + ḟ ∞ .For the term implying r 1 , we note thatr 1 (t k ) = the difference r 1 (t k ) -r 1 (t k-1) on the integral form.Using the above relation, it comes by using once again (4.3)r 1 (t k ) -r 1 (t k-1 ) 1 + p 1 1 + t k-1 f 1,∞ + f 1 2 ,∞ .(3.37) Finally r 1 (t k ) -r 1 (t k-1 ) ≤ ∆t

			t k
				p(s)(t k-1 -s)ds,
			t k-1
	1
	in D A 0 . Hence, by a similar argument and (4.3), 2
	r 1 (t k ) ≤ M r 1 (t k ) 1 2	≤ M ∆t 2 p 0 3 2	+ p 1 1 + t k f 1,∞ + f 1 2 ,∞ . (3.36)
			1
	2 0 Then, we write in D A
			≤ M ∆t 2	sup	p(s) 1	,
		2	s∈(t k-2 ,t k-1 )	2
			≤ M ∆t 2 p 0 3 2
				t k-1 t k-2	s s-∆t	d 3 p dσ 3 (σ) dσ ds,
			≤ M ∆t 3	sup
	1 2	≤ M ∆t p 0 3 2	+ p 1 1 + t k f 1,∞ + f 1 2 ,∞ .	(3.33)
			t k	d 3 p
	t k-1 ds d 3 p dt 3 (t) = dp dt (t) = d dt -A 0 p(t) -C * 0 C 0 ṗ(t) + f (t) ,	(3.34)
			≤ ṗ(t) 1 + M p(t) + ḟ (t) .
					(3.35)

3

(s) (t k-1 -s) ds, in H. s∈(t k-3 ,t k-1 ) d 3 p ds 3 (s) .

Using (3.34) and (

4

.3), we get

r 1 (t k ) -r 1 (t k-1 ) ≤ M ∆t 3 p 0 3 2

2. There exist M > 0, h * > 0 and ∆t * > 0 such that for all n ∈ N, all h ∈ (0, h * ), all ∆t ∈ (0, ∆t * ) and all 0 ≤ k ≤ K

We are now able to prove Theorem 4.

Proof (of Theorem 4) Introducing the term

, we can rewrite

in the following form

.

Therefore, we have

where we have set

Once again, using similar as the ones detailed for the Schrödinger case, we get the claimed result.

Appendix