Non uniform observability

Remark 3.5.1 Note that the construction above applies to any sequence of eigenvaluesλ^h_j(h).

It is important to note that the solution we have used in the proof of this theorem is not the only impediment for the uniform observability inequality to hold.

Indeed, let us consider the following solution of the semi-discrete system (3.35), constituted by the last two eigenvectors:

~ u= 1

√λN

exp(ip

λNt)w~N−exp(ip

λ_N−1t)~wN−1

. (3.52)

This solution is a simple superposition of two monochromatic semi-discrete waves corresponding to the last two eigenfrequencies of the system. The total energy of this solution is of the order 1 (because each of both components has been normalized in the energy norm and the eigenvectors are orthogonal one to each other). However, the trace of its discrete normal derivative is of the order ofhinL²(0, T). This is due to two facts.

• First, the trace of the discrete normal derivative of each eigenvector is very small compared to its total energy.

• Second and more important, the gap between√

λN andp

λN−1is of the order ofh, as it is shown in Figure 4. This wave packet has then a group velocity of the order ofh.

Thus, by Taylor expansion, the difference between the two time-dependent complex exponentials exp(i√

λ_Nt) and exp(ip

λ_N−1t) is of the orderT h.

Thus, in order for it to be of the order of 1 inL²(0, T), we need a timeT of the order of 1/h. In fact, by drawing the graph of the function in (3.52) above one can immediately see that it corresponds to a wave propagating at a velocity of the order ofh(Figure 5).

This construction makes it possible to show that, whatever the time T is, the observability constantC_h(T) in the semi-discrete system is at least of order 1/h. In fact, this idea can be used to show that the observability constant has to blow-up at infinite order. To do this it is sufficient to proceed as above but combining an increasing number of eigenfrequencies. This argument allows one to show that the observability constant has to blow-up as an arbitrary negative power of h. Actually, S. Micu in [163] proved that the constantC_h(T) blows up exponentially¹⁵by means of a careful analysis of the biorthogonal sequences to the family of exponentials{exp(ip

λjt)}j=1,...,N as h→0.

15According to [163] we know that the observability constantCh(T) necessarily blows-up exponentially ash→0. On the other hand, it is known that the observability inequality

Figure 3.4: Square roots of the eigenvalues in the continuous and discrete case.

The gaps between these numbers are clearly independent ofkin the continuous case and of orderhfor large kin the discrete one.

All these high frequency pathologies are in fact very closely related with the notion of group velocity. We refer to [224], [217] for an in depth analysis of this notion that we discuss briefly in the context of this example.

According to the fact that the eigenvector w~j is a sinusoidal function (see (3.43)) we see that these functions can also be written as linear combinations

is true ifCh(T) is large enough. To see this we can apply in this semi-discrete system the classical method of sidewise energy estimates for 1Dwave equations (see [233]). Recall that solutions of the semi-discrete system vanish at the boundary pointx= 1, i.e.,uN+1 ≡0.

On the other hand, the right hand side of the observability inequality provides an estimate ofuNinL²(0, T). The semi-discrete equation at the nodej=Nreads as follows:

uN−1=h²u⁰⁰_N+ 2uN, 0< t < T. (3.53) This provides an estimate ofuN−1 inL²(0, T). Indeed, in principle, in view of (3.53), one should lose two time derivatives when doing this. However, this can be compensated by the fact that we are dealing with a finite-dimensional model in which two time derivatives ofu are of the order ofAhu,Ahbeing the matrix in (3.37), which is of norm 4/h². Iterating this argument we can end up getting an estimate inL²(0, T) for alluj withj = 1, ..., N. But, taking into account thatN∼1/h, the constant in the bound will necessarily be exponential in 1/h. The problem of obtaining sharp asymptotic (ash→0) estimates on the observability constant is open.

of complex exponentials (in space-time):

exp

±ijπ pλ_j

jπ t−x

.

In view of this, we see that each monochromatic wave propagates at a speed pλj

jπ =2sin(jπh/2)

jπh = ωh(ξ) ξ

{ξ=jπh}

=ch(ξ) {ξ=jπh}

, (3.54) with

ω_h(ξ) = 2sin(ξ/2).

This is the so called phase velocity. The velocity of propagation of monochro- matic semi-discrete waves (3.54) turns out to be bounded above and below by positive constants, independently ofh, i. e.

0< α≤ch(ξ)≤β <∞, ∀h >0,∀ξ∈[0, π].

Note that [0, π] is the relevant range of frequencies. Indeed, ξ = jπh and j= 1, ..., N andN h= 1−h.

However, it is well known that, even though the velocity of propagation of each eigenmode is bounded above and below, wave packets may travel at a different speed because of the cancellation phenomena we have exhibited above (see (3.52)). The corresponding speed for those semi-discrete wave packets accumulating is given by the derivative of ωh(·) (see [217]). At the high fre- quencies (j∼N) the derivative ofωh(ξ) atξ=N πh=π(1−h), is of the order ofhand therefore the wave packet (3.52) propagates with velocity of the order ofh.

Note that the fact that this group velocity is of the order ofhis equivalent to the fact that the gap betweenp

λN−1 and√

λN is of order h.

Indeed, the definition of group velocity as the derivative of ωh is a natural consequence of the classical properties of the superposition of linear harmonic oscillators with close but not identical phases (see [56]). The group velocity is thus, simply, the derivative of the curve in the dispersion diagram of Figure 4 describing the velocity of propagation of monocromatic waves, as a function of frequency. Taking into account that this curve is constituted by the square roots of the eigenvalues of the numerical scheme, we see that there is a one-to- one correspondence between group velocity and spectral gap. In particular, the group velocity decreases when the gap between consecutive eigenvalues does it.

According to this analysis, the fact that group velocity is bounded below is a necessary condition for the uniform observability inequality to hold. Moreover, this is equivalent to an uniform spectral gap condition.

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.