Time discrete wave equations: Boundary observability and control

In this paper we study the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. We prove that the projection of the solution in an appropriate filtered space is exactly controllable with uniformly bounded cost with respect to the time-step. In this way, the well-known exact-controllability property of the wave equation can be reproduced as the limit, as the time step h → 0, of the controllability of projections of the time-discrete one. By duality these results are equivalent to deriving uniform observability estimates (with respect to h → 0) within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. The later is established by means of a time-discrete version of the classical multiplier technique. The optimality of the order of the filtering parameter is also established, although a careful analysis of the expected velocity of propagation of time-discrete waves indicates that its actual value could be improved.


Introduction
Let Ω be a nonempty open bounded domain in lR d (d ∈ lN) with C 2 boundary Γ, Γ 0 be a nonempty open subset of Γ, and T > 0 be a given time duration.
(1. 6) Inequality (1.4) asserts that the total energy of any solution of (1.3) can be observed in terms of the energy concentrated on Γ 0 in the time interval (0, T ). It is well-known that there are typically two classes of conditions on (T, Ω, Γ 0 ) guaranteeing (1.4).
i) The first one is given by the classical multiplier condition. Fix some x 0 ∈ lR d , put where ν(x) is the unit outward normal vector of Ω at x ∈ Γ. Then (1.4) holds for Γ 0 as in (1.7) provided T > 2R. This is the typical situation one encounters when applying the multiplier technique ( [8]), and Carleman inequalities (e.g. [13]) to deduce (1.4), which can also be applied to many other models.
ii) The second one is when (T, Ω, Γ 0 ) satisfy the Geometric Control Condition (GCC, for short) introduced in [1], which asserts that all rays of geometric optics in Ω intersect the subset of the boundary Γ 0 in a uniform time T . In this case, (1.4) is established by means of tools from micro-local analysis ( [1]). This condition is optimal.
In this paper, we are interested in the time semi-discretization of systems (1.1) and (1.3). We are thus replacing the continuous dynamics (1.1) and (1.3) by time-discrete ones and analyze their controllability/observability properties. Here we take the point of view of numerical analysis and, therefore, we analyze the limit behavior as the time-step tends to zero.
For this purpose, we set the time step h by h = T /K, where K > 1 is a given integer. Denote by y k and u k respectively the approximations of the solution y and the control u of (1.1) at time t k = kh for any k = 0, · · · , K. We then introduce the following trapezoidal time semi-discretization of (1.1): in Ω, k = 1, · · · , K − 1 y k+1 + y k−1 2 = u k χ Γ 0 , on Γ, k = 1, · · · , K − 1 y 0 = y 0 , y 1 = y 0 + hy 1 , in Ω. (1.8) Here (y 0 , y 1 ) ∈ L 2 (Ω) × H −1 (Ω) are the data given in system (1.1) that allow determining the initial data for the time-discrete system too. We refer to Theorem 4.2 below for the well-posedness of system (1.8) by means of the transposition method. 3 The controllability problem for system (1.8) may be formulated as follows: For any (y 0 , y 1 ) ∈ L 2 (Ω) × H −1 (Ω), to find a control {u k ∈ L 2 (Γ 0 )} k=1,··· ,K−1 such that the solution {y k } k=0,··· ,K of (1.8) satisfies: in Ω. (1.9) Note that (1.9) is equivalent to the condition y K−1 = (y K − y K−1 )/h = 0 that is a natural discrete version of (1.2). As in the context of the above continuous wave equation, we also consider the uncontrolled system in Ω, k = 1, · · · , K − 1 In particular, to guarantee the convergence of the solutions of (1.10) towards those of (1.3) one considers convergent data such that with hϕ h 1 being bounded in H 1 0 (Ω). Obviously because of the density of H 1 0 (Ω) in L 2 (Ω) this choice is always possible. Remark 1.1 Note that the choice of the values of ϕ K and ϕ K−1 in (1.10) is motivated by the transposition arguments that are needed to define the solution of the time-discrete non-homogenous problem (1.8), as we will see in Section 8. The energy of system (1.10) is given by which is a discrete counterpart of the continuous energy E(t) in (1.5). Multiplying the first equation of system (1.10) by (ϕ k+1 − ϕ k−1 )/2 and integrating it in Ω, using integration by parts, it is easy to show the following property of conservation of energy: Consequently the scheme under consideration is stable and its convergence (in the classical sense of numerical analysis) is guaranteed in an appropriate functional setting (in particular in the finite-energy space H 1 0 (Ω) × L 2 (Ω), under the condition (1.11)).

4
By means of classical duality arguments, it is easy to show that the above controllability property (1.9) is equivalent to the following boundary observability property for solutions {ϕ k } k=0,··· ,K of (1.10): (1.14) The analysis of controllability and/or observability properties of numerical approximation schemes for the wave equation has been the object of intensive studies. However most analytical results concern the case of space semi-discretizations (see [16] and the references cited therein). In practical applications, fully discrete schemes need to be used. The most typical example is the classical central scheme which converges under a suitable CFL condition ( [4,5,11]). However, in the present setting in which the Laplacian ∆ is kept continuous, without discretizing it, this scheme is unsuitable since it is unstable. To see this, we choose {µ 2 j } j≥1 to be eigenvalues of the Dirichlet Laplacian and {Φ j } ∞ j=1 ⊂ H 1 0 (Ω) the corresponding eigenvectors (constituting an orthonormal basis of L 2 (Ω)), i.e., Since {µ 2 j } j≥1 tends to infinity, it is easy to check that the central scheme is unstable. Indeed, the stability of (1.16) would be equivalent to the stability of the scheme for all values of µ 2 j , j ≥ 1. But this stability property fails clearly, regardless how small h is, when µ 2 j is large enough. Hence, we choose the trapezoidal scheme (1.10) for the time-discrete problem, which is stable (due to the property of conservation of energy), as mentioned before.
The first result of this paper is of negative nature. Indeed, as we shall see in Theorem 5.1, the observability inequality (1.14) (resp. the controllability property (1.9)) fails for system (1.10) (resp. (1.8)) without filtering. From the proof of Theorem 5.1 below, it will be obvious that these negative results of observability and controllability are related to the fact that the spaces in which the solutions evolve are infinite dimensional; while the number of time-steps is finite. Accordingly, to make the observability inequality possible one has to restrict the class of solutions of the adjoint system (1.10) under consideration by filtering the high frequency components. Similarly, since the property of exact controllability of system (1.8) fails, the final requirement (1.9) has to be relaxed by considering only low frequency projections of the solutions. Controlling such a projection can be viewed as a partial controllability problem.
This filtering method has been applied successfully in the context of controllability of time discrete heat equations ( [14]) and space semi-discrete schemes for wave equations ( [2,7,15,16]).
As far as we know, the subject of control and observation of the time-discrete wave equation under consideration has not been addressed before. In this paper we shall develop a discrete version of the classical multiplier approach which allows to view the time discrete wave equation as an evolution process with its own dynamics.
As in the continuous case, the multiplier technique we use here applies mainly to the case when the controller/observer Γ 0 is given in (1.7) and some variants ( [10]), but does not work when (T, Ω, Γ 0 ) is assumed to satisfy the GCC. As we shall see, the main advantage of our multiplier approach is that the filtering parameter we use has the optimal scaling in what concerns the frequency of observed/controlled solutions with respect to h.
It is important to note that this kind of results can not be obtained by standard perturbation arguments that rely simply on measuring the distance between solutions of the time-discrete and continuous wave equations. Indeed, when proceeding that way, one needs much stronger filtering requirements. In other words, the optimal filtering can only be obtained by a careful analysis of the time evolution of the system under consideration. This is already well-known in the context of space semi-discretizations (see [16]). Our discrete multiplier approach can also be extended to other PDEs of conservative nature, and in particular to the Schrödinger, plate, Maxwell's equations, among others.
The rest of the paper is organized as follows. In Section 2, we collect some preliminary results which are useful in what follows. In Section 3, we present two fundamental identities by means of discrete multipliers, which will play an important role in the sequel. In Section 4 we discuss the hidden regularity property of solutions of (1.10) and the uniform well-posedness property of system (1.8). Section 5 is devoted to show the lack of controllability/observability of systems (1.8) and (1.10) without filtering. The uniform observability result for (1.10) is presented in Section 6. In Section 7 we show the optimality of the filtering parameter in the uniform observability result. Moreover, we give a heuristic explanation of the necessity of the filtering in terms of the group velocity of propagation of waves. Section 8 is devoted to the uniform controllability of system (1.8) and the convergence of the controls and solutions.

Preliminaries
In this section, we collect some preliminary results that will be used in the sequel.
Next, we claim that the solution of system (1.10) can be expressed explicitly by means of Fourier series. Indeed, we have . Then the solution of system (1.10) is given by

15)
where Proof of Lemma 2.2: By Lemma 2.1, it suffices to find a solution ϕ k of the form The characteristic polynomial of (2.18) (which is a difference equation) reads The roots n j and m j of p(λ) are as follows (2.21) Noting the definition of ω j in (2.16), by (2.20), it follows Therefore, the r k j 's given by (2.21) can be re-written as Finally, combining (2.17) and (2.22), we conclude the desired formula (2.15).
The third one is a classical multiplier identity for the Dirichlet Laplacian: Identity (2.23) can be easily proved multiplying ∆ψ by · ∇ψ where · stands for the scalar product in lR d . We refer to [8, identity (1.25)] or to [13,Lemma 3.3] for the details.

Identities via multipliers
This section is addressed to establish two fundamental identities by means of discrete multipliers. First, we show the following one: (3.5) Proof of Lemma 3.1: Multiplying (2.1) by · ∇(θ k+1 + θ k−1 )/2 (which is a discrete version of the multiplier · ∇θ for the wave equation), integrating it in Ω, summing it for First, we analyze the first term in the left hand side of (3.6): (3.7) However, where U and V 1 are defined respectively by (3.2) and (3.3). Next, we analyze the second term in the left hand side of (3.6). Applying Lemma 2.3 (with ψ replaced by (θ k+1 + θ k−1 )/2), we find where V 2 is defined by (3.4). Further, using integration by parts and noting (3.11) Finally, by (3.6), (3.9)-(3.11) and recalling the definition of W in (3.5), we conclude the desired identity (3.1).
As we shall see in the next section, Lemma 3.1 is the basis to provide an important hidden regularity property of solutions of system (2.1), and via which the well-posedness of system (1.8) follows. Meanwhile, as a consequence of Lemma 3.1, we now show the following identity for the solutions of (1.10), which will play a crucial role in the proof of Theorem 6.1:

Lemma 3.2 For any h > 0 and any solution {ϕ
Remark 3.1 Identity (3.12) is a time discrete analogue of the well known identity for the wave equation (1.10) obtained by multipliers, which reads (see [8]): Here, There are clear analogies between (3.12) and (3.16). In fact the only major differences are that, in the discrete version (3.12), two extra reminder terms (Y and Z) appear, which are due to the time discretization. It is easy to see, formally, that Y and Z tend to zero as h → 0. But this convergence does not hold uniformly for all solutions. Consequently, these added terms impose the need of using filtering of the high frequencies to obtain observability inequalities and also modify the observability time, as we shall see.
For V 1 defined in (3.3) (with θ k replaced by ϕ k ), noting div = d and using the first equation in (1.10), one has For V 2 defined in (3.4), noting i x j = δ i j (the Kronecker delta) and using the elementary identity (a + b) 2 Now, by (3.1) in Lemma 3.1, recalling the definition of U in (3.2) (with θ k replaced by ϕ k ), noting W = 0 and (3.19), we conclude that where Y is defined in (3.14).
On the other hand, multiplying the first equation of (1.10) by ϕ k (which is a discrete version of the multiplier ϕ in the time-continuous setting, that leads to the identity of equipartition of energy), integrating it in Ω, summing it for k = 1, · · · , K − 1 and using integration by parts, we obtain: Combining (3.21) and (3.22), we end up with the following equipartition of energy identity for the time semi-discrete system (1.10):

Hidden regularity and well-posedness
This section is devoted to show a hidden regularity property of solutions of system (2.1) and to establish the well-posedness of system (1.8).
We begin with the following hidden regularity property of solutions of system (2.1) (recall (2.2) for the definition of E k h ): (4.1)

Remark 4.1
When h tends to zero, the limit of the system (2.1) is Inequality (4.1) is a time discrete analogue of the following boundary estimate of (4.2): Proof of Theorem 4.1: As in [8], we choose a vector ∈ C 1 (Ω; lR d ) so that = ν on the boundary Γ. Then, the desired estimate (4.1) follows immediately from Lemma 3.1 and Lemma 2.1.
We now show the following well-posedness result for this system: . (4.11) ii) When K is even, .

(4.12)
Furthermore, the constant C > 0 in the estimates (4.11) and (4.12) is independent of the time-step h.

Lack of controllability/observability without filtering
This section is devoted to prove the following negative controllability/observability result: Proof of Theorem 5.1: We emphasize that, in this proof, h is fixed so that the system under consideration involves only a finite number of time-steps while it is a distributed parameter system (infinite-dimensional one) in space. This is precisely the main reason for the lack of observability results. The proof is divided into two steps.
On the other hand, (5.2) can be rewritten as This, combined with the standard regularity theory for elliptic equations of second order, gives One can also re-write (5.2) as Therefore, using again the standard elliptic regularity theory, we conclude that for any τ ≤ 2, it holds ).

Uniform observability under filtering
In this section, we shall establish uniform observability estimates for system (1.10) (with respect to the time step h) after filtering the spurious high frequency components.

Statement of the uniform observability result
As mentioned in Introduction, due to the negative results stated in Theorem 5.1, we need to introduce suitable filtering spaces in which the solutions of system (1.10) evolve. Recalling the definition of Φ j and µ j , for any s > 0, define and subspaces of H 1 0 (Ω), L 2 (Ω) and H −1 (Ω), respectively, with the induced topologies. It is clear that in H −1 (Ω). Denote by π 1,s , π 0,s and π −1,s the projection operators from H 1 0 (Ω), L 2 (Ω) and H −1 (Ω) to C 1,s , C 0,s and C −1,s , respectively. The space C −1,s and the projector π −1,s will not be used in this section but we will need them later.
Our uniform observability result for system (1.10) is stated as follows: Then there exist two constants h 0 > 0 and δ > 0, depending only on d, T and R, such that for all (ϕ h 0 , ϕ h 1 ) ∈ C 1,δh −2 × C 0,δh −2 , the corresponding solution {ϕ k } k=0,··· ,K of (1.10) satisfies Remark 6.1 In the proof we see that δ depends only on d, T and R. In particular it indicates that δ decreases as T decreases. This is natural since, as T decreases, less and less time-step iterations are involved in system (1.10) and, consequently, less Fourier components of the solutions may be observed. Further, δ tends to zero as T tends to 2R. This is natural too since our proof of (6.4) is based on the method of multipliers which works at the continuous level for all T > 2R but that, at the time-discrete level, due to the added dispersive effects, may hardly work when T is very close to 2R, except if the filtering is strong enough.
Remark 6.2 In view of the hidden regularity result of Theorem 4.1, the right hand side term of (6.4) is finite.

Remark 6.3
In the observability result of Theorem 6.1, the filtering parameter has been taken to be of the order of h −2 . This is the optimal order for the filtering parameter since for higher frequencies there are solutions for which the observability constant blows-up, as Theorem 7.1 in the next section shows. However, as we shall see, the necessity of the filtering parameter δ to be small is not completely justified. In fact, our analysis of the velocity of propagation of solutions in section 7 supports that, whatever δ > 0 is, observability could be expected to hold for large enough values of time T .

A technical result
As mentioned before, the key point in the proof of Theorem 6.1 is Lemma 3.2. We need to estimate first the term X and the error terms Y and Z in (3.12).

Proof of the uniform observability result
We are now in a position to prove the uniform observability result, i.e., Theorem 6.1.
Proof of Theorem 6.1: Combining (3.12) in Lemma 3.2 and (6.5) in Lemma 6.1, recalling the definition of Γ 0 in (1.7), we deduce that For this inequality to yield an estimate on E 0 h we need to choose s = δh −2 with h small enough such that or, more precisely, Once this is done, for h ∈ (0, h 0 ), T has to be chosen such that Hence, (6.4) holds for h ∈ (0, h 0 ]. Conversely, for any T > 2R one can always choose h 0 and δ small enough so that (6.24) and (6.25) hold and guaranteeing the uniform observability inequality.

Optimality of the filtering parameter
This section is addressed to analyze the optimality of the filtering mechanism introduced in Theorem 6.1.

Optimality of the order of the filtering parameter
We first show the following result, which indicates that the order h −2 of the filtering parameter that we have chosen in Theorem 6.1 is optimal.
Some remarks are in order.
Remark 7.1 The argument above, based on the use of separated variables monochromatic solutions, shows that the order of filtering µ 2 ≤ Ch −2 is sharp, in the sense that the observability inequality fails to be uniform when we take into account eigenvalues µ 2 such that µ 2 h −2 . Note however that our observability results require to restrict the class of eigenvalues under consideration to µ 2 ≤ δh −2 with δ > 0 small. The discussion above does not justify the optimality of this smallness condition on the filtering constant. Actually, as we shall show in the next section, one may expect that uniform observability and controllability properties hold within classes of filtered solutions of the form µ 2 ≤ Ch −2 with arbitrary C > 0 for a sufficiently large time.
Remark 7.2 In a first look to this problem it might seem to be surprising that the negative result in Theorem 7.1 is related to monochromatic waves. Nevertheless, the lack of uniform observability is related to the fact that the quantity in the right hand side of (7.9) is of the order of cos 2 (ω j 0 ) which tends to zero as h → 0.  . and x 2 (t) = · · · = x d (t) = 0. Thus, x j (t) for j = 2, · · · , d remain constant and . This allows us to show that, as h → 0, there exist rays that remain trapped on a neighborhood of x 0 for time intervals of arbitrarily large length. In order to guarantee the boundary observability these rays have to be cut-off by filtering. This can be done by restricting the Fourier spectrum of the solution to the range |τ | ≤ ρπ/2h with 0 < ρ < 1. This corresponds to |ξ| 2 ≤ 4 sin 2 (ρπ/2) h 2 cos(ρπ/2) , (7.10) for the root of the symbol P h . This is the same scaling of the filtering operators we imposed on Theorems 6.1 and 8.1, namely, µ 2 j ≤ δ/h 2 . Note however that, in (7.10), as ρ → 1, the filtering parameter δ = 4 sin 2 (ρπ/2) cos(ρπ/2) → ∞.
Thus, in principle, as mentioned above, the analysis of the velocity of propagation of bicharacteristic rays does not seem to justify the need of letting the filtering parameter δ small enough as in Theorems 6.1 and 8.1. Thus, this last restriction seems to be imposed by the rigidity of the method of multipliers rather than by the underlying wave propagation phenomena.
We can reach similar conclusions by analyzing the behavior of the so-called group velocity. Indeed, following [12], in 1 − d the group velocity has the form with the graphs as in Figure 1. Obviously, it tends to zero when h 2 ξ 2 tends to infinity. This corresponds precisely to the high frequency bicharacteristic rays constructed above for which the velocity of propagation vanishes. Based on this analysis one can show that, whatever the filtering parameter δ is, uniform observability requires the observation time to be large enough with T (δ) ∞ as δ ∞. This may be done using an explicit construction of solutions concentrated along rays (see, for instance, [9]). The positive counterpart of this result guaranteeing that, for any value of the filtering parameter δ > 0, uniform observability/controllability holds for large enough values of time, is an interesting open problem whose complete solution will require the application of microlocal analysis tools.

Uniform controllability and convergence of the controls
In this section, we present the following uniform partial controllability result for system (1.8) and the convergence result for the controls : ii) There exists a constant C > 0, independent of h, y 0 and y 1 , such that iii) When h → 0, where u is a control of system (1.1), fulfilling (1.2); iv) When h → 0, where y is the solution of system (1.1) with the limit control u as above.
The above theorem contains two results: the uniform partial controllability and the convergence of the controls and states as h → 0. The proof is standard. Indeed, the partial controllability statement follows from Theorem 6.1 and classical duality arguments ( [8]); while for the convergence result, one may use the approach developed in [16]. However, for readers' convenience, we give below a sketch of the proof of Theorem 8.1.
Once the strong convergence of the controls is known, the estimates of Theorem 4.2 allow getting a uniform bound of {y h } h>0 (defined in (8.4)) in C([0, T ]; L 2 (Ω))∩H 1 ([0, T ]; H −1 (Ω)), which yields the desired strong convergence result for the extension {y h } h>0 of the timediscrete solution {y k } k=0,··· ,K of (1.8) to continuous time, as indicated by (8.4). This completes the proof of Theorem 8.1.