Appendix 2: Dynamical programming
3.4 The multi-dimensional wave equation
3.5.4 Fourier Filtering
The convergence property of the numerical scheme only guarantees that the group velocity is correct for low frequency wave packets.16 The negative results we have mentioned above are a new reading of well-known pathologies of finite differences scheme for the wave equation.
The careful analysis of this negative example is extremely useful when de- signing possible remedies, i.e., to determine how one could modify the numeri- cal scheme in order to reestablish the uniform observability inequality, since we have only found two obstacles and both happen at high frequencies. The first remedy is very natural: To cut off the high frequencies or, in other words, to ignore the high frequency components of the numerical solutions. This method is analyzed in the following section.
to the sharp spectral results of the previous section we shall use a classical result due to Ingham in the theory of non-harmonic Fourier series (see Ingham [114]
and Young [225]).
Ingham’s Theorem. Let {µk}k∈Z be a sequence of real numbers such that µk+1−µk≥γ >0,∀k∈Z. (3.55) Then, for anyT >2π/γ there exists a positive constantC(T, γ)>0 such that
1 C(T, γ)
X
k∈Z
|ak|2≤ Z T
0
X
k∈Z
akeiµkt
2
dt≤C(T, γ)X
k∈Z
|ak |2 (3.56)
for all sequences of complex numbers {ak} ∈`2.
Remark 3.5.2 Ingham’s inequality can be viewed as a generalization of the or- thogonality property of trigonometric functions we used to prove the observabil- ity of the 1Dwave equation in Section 3.3. Indeed, assume thatµk =kγ, k∈Z for someγ > 0. Then (3.55) holds with equality for allk. We setT = 2π/γ.
Then
Z 2π/γ 0
X
k∈Z
akeiγkt
2
dt=2π γ
X
k∈Z
|ak |2. (3.57) Note that under the weaker gap condition (3.55) we obtain upper and lower bounds instead of identity (3.57). Observe also that Ingham’s inequality does not apply at the minimal time 2π/γ. This fact is also sharp [225].
In the previous section we have seen that, in the absence of spectral gap (or, when the group velocity vanishes) the uniform observability inequality fails. Ingham’s inequality provides the positive counterpart, showing that, as soon as the gap condition is satisfied, there is uniform observability provided the time is large enough. Note that the observability time (T > 2π/γ) is inversely proportional to the gap, and this is once more in agreement with the interpretation of the previous section.
All these facts confirm that a suitable cutoff or filtering of the spurious numerical high frequencies may be a cure for these pathologies.
Let us now describe the basic Fourier filtering mechanism. We recall that solutions of (3.35) can be developed in Fourier series as follows:
~ u=
N
X
k=1
akcos q
λhkt
+ bk q
λhk sin
q λhkt
w~kh (3.58)
whereak, bk are the Fourier coefficients of the initial data, i.e.,
~u0=
N
X
k=1
akw~hk, ~u1=
N
X
k=1
bkw~kh.
Given 0< δ <1, we introduce the following classes of solutions of (3.35):
Cδ(h) = 8
<
:
~
usol. of (3.35) s.t. ~u=
[δ/h]
X
k=1
akcos
„q λhkt
« + bk
pλhksin
„q λhkt
«!
~ wkh
9
=
; . (3.59) Note that in the classCδ(h) the high frequencies corresponding to the indices j >[δ(N+ 1)] have been cut off. We have the following result:
Theorem 3.5.2 ([112],[113]) For any δ >0 there exists T(δ)>0 such that for allT > T(δ)there existsC=C(T, δ)>0 such that
1
CEh(0)≤ Z T
0
uN(t) h
2
dt≤CEh(0) (3.60)
for every solution uof (3.35)in the class Cδ(h), and for allh >0. Moreover, the minimal time T(δ) for which (3.60)holds is such that T(δ)→2 asδ→0 andT(δ)→ ∞asδ→1.
Remark 3.5.3 Theorem 3.5.2 guarantees the uniform observability in each classCδ(h), for all 0< δ <1, provided the timeT is larger thanT(δ).
The last statement in the Theorem shows that when the filtering parameter δtends to zero, i.e. when the solutions under consideration contain fewer and fewer frequencies, the time for uniform observability converges toT = 2, which is the corresponding one for the continuous equation. This is in agreement with the observation that the group velocity of the low frequency semi-discrete waves coincides with the velocity of propagation in the continuous model.
By contrast, when the filtering parameter increases, i.e. when the solutions under consideration contain more and more frequencies, the time of uniform control tends to infinity. This is in agreement and explains further the negative showing that, if the absence of filtering, there is no finite timeT for which the uniform observability inequality holds.
The proof of Theorem 3.5.2 below provides an explicit estimate on the minimal observability time in the classCδ(h): T(δ) = 2/cos(πδ/2).
Remark 3.5.4 In the context of the numerical computation of the bound- ary control for the wave equation the need of an appropriate filtering of the high frequencies was observed by R. Glowinski [95]. This issue was further
investigated numerically by M. Asch and G. Lebeau in [4]. There, finite dif- ference schemes were used to test the Geometric Control Condition in various geometrical situations and to analyze the cost of the control as a function of time.
Proof of Theorem 3.5.2 The statement in Theorem 3.5.2 is a consequence of Ingham’s inequality and the gap properties of the semi-discrete spectra. Let us analyze the gap between consecutive eigenvalues. We have
q λhk−q
λhk−1=πcos
π(k−1 +η)h 2
for some 0 < η < 1. Observe that cos ((π(k−1 +η)h)/2) ≥ cos (πkh/2). Therefore
q λhk −q
λhk−1≥πcos (πkh/2).It follows that q
λhk−q
λhk−1≥πcos (πδ/2), fork≤δh−1. (3.61) We are now in the conditions for applying Ingham’s Theorem. We rewrite the solution~u∈ Cδ(h) of (3.35) as
~
u= X
|k|≤δ/h k6=0
ckeiµhktw~hk (3.62)
where
µh−k=−µhk; µhk = q
λhk; w~−k=w~k; ck= ak−ibk/µhk
2 ;c−k=ck. Then,
uN(t) = X
|k|≤δ/h k6=0
ckeiµhktwk,N.
Therefore
Z T 0
uN(t) h
2
dt= Z T
0
X
|k|≤δ/h k6=0
ckeiµhktwk,N
h
2
dt. (3.63)
In view of the gap property (3.61) and, according to Ingham’s inequality, it follows that ifT > T(δ) with
T(δ) = 2/cos(πδ/2) (3.64)
there exists a constantC=C(T, δ)>0 such that 1
C X
|k|≤δ/h k6=0
|ck |2
wk,N h
2
≤ Z T
0
uN(t) h
2
dt≤C X
|k|≤δ/h k6=0
|ck |2
wk,N h
2
(3.65)
for every solution of (3.35) in the classCδ(h). On the other hand, we observe that
λhkh2= 4 sin2 πkh
2
≤4 sin2 πδ
2
(3.66) for allk≤δ/h. Therefore, according to Lemma 3.5.1, it follows that
1 2
wN
h
2
≤h
N
X
j=0
wj+1−wj
h
2
≤ 1
2 cos2(πδ/2)
wN
h
2
(3.67) for all eigenvalues with indexk≤δ/h.
Combining (3.65) and (3.67) we deduce that for all T > T(δ) there exists C >0 such that
1 C
X
|k|≤δ/h k6=0
|ck|2≤ Z T
0
uN(t) h
2
dt≤C X
|k|≤δ/h k6=0
|ck |2. (3.68)
Finally we observe that X
|k|≤δ/h k6=0
|ck |2∼Eh(0).
This concludes the proof of Theorem 3.5.2.