OBSERVABILITY OF A STRING-BEAMS NETWORK WITH MANY BEAMS

. We prove the direct and inverse observability inequality for a network connecting one string with inﬁnitely many beams, at a common point, in the case where the lengths of the beams are all equal. The observation is at the exterior node of the string and at the exterior nodes of all the beams except one. The proof is based on a careful analysis of the asymptotic behavior of the underlying eigenvalues and eigenfunctions, and on the use of a Ingham type theorem with weakened gap condition [C. Baiocchi, V. Komornik and P. Loreti, Acta Math. Hung. 97 (2002) 55–95.]. On the one hand, the proof of the crucial gap condition already observed in the case where there is only one beam [K. Ammari, M. Jellouli and M. Mehrenberger, Networks Heterogeneous Media 4 (2009) 2009.] is new and based on elementary monotonicity arguments. On the other hand, we are able to handle both the complication arising with the appearance of eigenvalues with unbounded multiplicity, due to the many beams case, and the terms coming from the weakened gap condition, arising when at least 2 beams are present.

associated to the observability conditions. Uniformity in the constants indeed allows to cope with perturbations of the topology of the network, for instance by adding or removing a beam. The extension from "one string-one beam" system to a "one string-many beams" system comes with some technical difficulties. This is related to the appearance of eigenvalues with unbounded multiplicity, and of new terms in the observable associated to the weakened gap condition. Future perspectives include the generalization of the present approach to different boundary conditions and networks. We remark that in [12] an application of a Minkowski theorem on Diophantine approximations seems to prevent the observability of infinite strings: however the discussion of string-beam structures with many strings remains an open problem.
The paper is organized as follows. In Section 2 we introduce the main results of the paper. They are: Theorem 2.1, dealing with the well posedness of the system under exam; Theorem 2.2, providing new norm estimates for the initial energy of a "one string-one beam" system; and Theorem 2.3, stating some sufficient exact observability conditions for a "one string-(infinitely) many beams" system. In Section 3 we perform a spectral analysis of the system, we characterize the eigenvalues and related eigenspaces, and we express the solutions in terms of Fourier series, so to prove Theorem 2.1. Section 4 is devoted to the proofs of Theorem 2.2 and Theorem 2.3. More precisely, in Section 4.1 we establish a generalized gap condition on the eigenvalues; Section 4.2 contains the proof of Theorem 2.2 and some estimates of Ingham type with weakened gap conditions and Section 4.3 is devoted to the proof of Theorem 2.3. Finally, in Section 5 we present some numerical simulations.
In the remaining part of the present introduction, we present the system under exam. We use the following notation N * := N \ {0} and Z * := Z \ {0}. Now, let J ∈ N * ∪ {∞}, > 0 and α 1 , . . . , α J > 0, such that A := J j=1 α j < ∞. We consider the following system (see [13], e.g. pages 80-81, for the model) coupling one string with J beams of same length : x ∈ (0, 1), (∂ 2 t u j + ∂ 4 x u j )(t, x) = 0, x ∈ (0, ), t > 0, u 0 (t, 0) = 0, u j (t, 0) = 0, ∂ 2 x u j (t, 0) = 0, ∂ 2 x u j (t, ) = 0, t > 0, u 0 (t, 1) = u j (t, ), ∂ x u 0 (t, 1) = J k=1 α k ∂ 3 x u k (t, ), t > 0, u 0 (0, x) = u 0 0 (x), ∂ t u 0 (0, x) = u 1 0 (x), x ∈ (0, 1), u j (0, x) = u 0 j (x), ∂ t u j (0, x) = u 1 j (x), x ∈ (0, ), (1.1) for j = 1, . . . , J. A stabilization problem corresponding to such system has been studied in [1] for J = 1, in [3] for arbitrary finite J and in [5] in a more general framework. In particular, for J = 1, an observability inequality is obtained thanks to Ingham's theorem, using a gap condition of the underlying eigenvalues. For arbitrary J, however, the gap does no longer hold and an Ingham type approach is more challenging, as also pointed in [4], where another generalization of the case J = 1 has been performed. Some numerical illustrations have been performed in [3] to suggest that such observability inequality still holds, but the proof remains to be done. The origin of this work dates back to [2], where the model was introduced, but the study was incomplete. We are here able to give a complete theoretical study and also to analyse the asymptotic behavior, where the number of beams becomes large, which was not considered in [2]. In this present case, the analysis is simplified, as the lengths of the beams are all equal, and we let the case of infinitely many beams with arbitrary length as open problem. To the best of our knowledge, the case of infinitely many beams was first considered in [12], in the different context of simultaneous observability; there, a very different behavior was observed between strings and beams.

Main results
We first consider an assumption on the length of the beam(s). So we define We state the well posedness of the system. To this end, recall the definition A := J j=1 α j and consider the function f given by We prove in Lemma 3.2 that the positive zeros of f form a strictly increasing sequence that we denote by (z n ) n∈N * . Also we use the notation u = (u 0 , u 1 , . . . , u J ) and we consider the space

and the Hilbert space
Our first result is a series representation for the solution to (1.1).
We focus now on observability results. We also use the notation a b, meaning that there exists constants Let us first consider the case where J = 1. Note that such case was already studied in [1], but, there, the observation was at the junction point with a more restrictive assumption on the length . We define the initial energy of the system (1.1) as We get the following result, when observing on the exterior node of the string: Theorem 2.2. Let ∈ L and J = 1. Let (a n ) n∈Z * ∈ C be a sequence with a finite number non zero elements and assume that the initial data of (1.1) are and Then, the corresponding solution u of (1.1) satisfies for T > 2, where the related constants are independent of the initial data.
We then have , and the underlying constants do not depend on the initial data nor on J.

Spectral analysis
The spectral analysis of such system has already been performed in [1,3], for J = 1 or finite J. We suppose here that 1 ≤ J ≤ ∞. We use the notation w = (w 0 , w 1 , . . . , w J ).
In order to characterize the eigenvalues of A we shall need some notation. Recall the function f given by and rewrite S as a (unique) strictly increasing sequence (s n ) n∈N . We prove below that the positive zeros of f form a strictly increasing sequence that we denote by (z n ) n∈N * . We finally consider and we arrange the elements of Λ in a strictly increasing sequence (ω m ) m∈Z * . We define the upper density D + (X) of a set X ⊂ R as where n + (X, r) denotes the largest number of elements of X contained in an interval of length r.
Lemma 3.2. We suppose that ∈ L.
(i) f (z) is well defined if and only if z 2 ∈ S.
(ii) f is strictly decreasing for √ s n < z < √ s n+1 , for each n ∈ N.
(iii) The strictly positive zeros of f form a strictly increasing, diverging sequence (z n ) n∈N * , with s n−1 < z 2 n < s n , for all n ∈ N * . (iv) The upper density of Λ 1 is 1 π , the upper density of Λ 2 is 0 and the upper density of Λ is equal to 1 π .
Proof. (i) It suffices to remark that z 2 ∈ S if and only if either sin(z 2 ) = 0 or sin(z ) = 0. In particular z 2 ∈ S if and only if f (z) is well defined and C ∞ . (ii) Let g(z) = z(cot(z) − coth(z)) and note that f (z) = 2 cot(z 2 ) + Ag(z )/ . As cot(·) is a strictly decreasing function and > 0, it suffices to prove that g(z) is decreasing too. We show in particular that g (z) ≤ 0. Indeed for all positive z in the domain of g. (iii) Note that lim z→ √ sn ± f (z) = ∓∞ for all n ∈ N. We apply the mean value theorem for the existence and the strict monotonicity of f for unicity, and use the fact that s 0 = 0.
(iv) On an interval [p 1 π, (p 1 + r 1 )π], we have at least r 1é lements of Λ 1 and at most r 1 + 1, if r 1 π < δ k := (k+1) 2 π 2 2 − k 2 π 2 2 . As the size δ k tends to infinity, when k goes to infinity, we deduce that D + (Λ 1 ) = 1 π . Also we have D + (Λ 2 ) = 0 and then D + (Λ) = 1 π . Now, we want to give a more precise description of the eigenvectors and eigenvalues. For convenience, in the sequel, we define z −n = −z n , n ∈ N * . Note that f is an even function, then f (z −n ) = f (−z n ) = f (z n ) = 0, since by definition z n are the positive zeros of f .
We finally distinguish the cases C = 0 and C = 0.
Once we have the eigenvalues and eigenvectors, we can readily express the solution of (1.1) as a nonharmonic Fourier series, proving Theorem 2.1.
Proof of Theorem 2.1. Let u = (u 0 , . . . , u J ) be as in the statement. The claim readily follows by Proposition 3.3, characterizing the eigenvalues and the eigenspaces of the operator A, and by Proposition 3.1, implying that u ∈ H. Remark 3.4. Note that the solution belongs also to C([0, ∞), D(A)) ∩ C 1 ([0, ∞), H), since the initial data are in D(A). By density arguments, the well posedness extends to initial data of the form of the previous theorem, but with infinite number of coefficients. For J = 1 we can consider such initial data satisfying that E 0,J < +∞, which, as we shall see, gives the explicit condition n∈Z * |z n | 4 |a n | 2 < ∞.
We can consider a basis e q j , with only a finite number of indices j such that e q j = 0 for a given q, giving non ambiguous convergence of the series The property of the completeness of a basis of eigenvectors is not considered here; we refer to [3] for example, for the proof of the property that A is a skew adjoint operator with compact resolvant for finite J, leading to the completeness of a basis of eigenvectors. We note that the resolvant is no more compact for J = ∞, since we have then eigenvalues with geometrical infinite multiplicity. The completeness of a basis of eigenvectors permits to have well posedness for arbitrary initial data in H (E 0,J is then finite), with solution in C([0, ∞), H).

Proof of the observability results
In this section we prove Theorem 2.2 and Theorem 2.3. The proof relies on non-harmonic analysis methods and on Ingham type inequalities. An ad hoc analysis is needed in view of the peculiarity of the system and, in particular, its transmission conditions.
Now it remains to consider the situation where s n = kπ 2 . If n is big enough, we can assume that (a) z n A ≥ 1; (b) s n−1 = pπ, s n+1 = (p + 1)π.

Applying Ingham's theorem
A direct application of the gap condition established above permits to prove Theorem 2.2 by applying Ingham's theorem with weakened gap condition [7,8]. Note that similar Ingham type theorems were also published by Avdonin et al. (see [6] and references therein).
Proof of Theorem 2.2. We first remark that, by Theorem 2.1, the solution (u, ∂ t u) of (1.1) is of the form with the coefficients a n (n ∈ Z * ) identically zero but a finite number of them. In particular ∂ x u 0 (t, x) = ∞ n∈Z * a n z 2 n e n |n| iz 2 n t cos(z 2 n x).
n a n e n |n| iz 2 n t | 2 dt.
Since the sequence ( n |n| z 2 n ) is discrete by Proposition 4.1 and since the underlying upper density is 1 π , from Lemma 3.2, then we can apply Ingham's theorem [10] (see also [9,11]) and obtain for T > 2, Now, to establish the equivalence with the initial energy E 0,1 , we need to express also the latter one in terms of sums of squares of the Fourier coefficients. We set sin(z n ) .
Using this notation, we have Note that the eigenvalue λ n of A associated to the eigenvector Φ n := (φ n , λ n φ n ) is of the form n |n| iz 2 n . By Proposition 3.1, for m, n ∈ Z * , one has Then Φ m , Φ n H = 0 and for all n, m ∈ Z * such that n = m. This implies that e In particular Since c n = c −n , we can assume without loss of generality that n ∈ N * and remark that, by definition, f (z n ) = 0, that is 2 cot(z 2 n ) = α 1 z n (coth(z n ) − cot(z n )). This leads to 2 cos(z 2 n ) where C := 2/(α 1 z 1 ) + coth(z 1 ) > 1. Then leading to c n 1. We finally have In the case J ≥ 2, we have no more a gap condition. We recall that we have a sequence ω m , m ∈ Z, which is a strictly increasing sequence formed by the sets Λ 1 = {±z 2 n , n = 1, . . . } and Λ 2 = {± k 2 π 2 2 , k = 1, . . . }. We remark that a weakened gap condition on (ω m ) is satisfied: there exists γ ∈ (0, min{γ , π/4}), where γ is like in Proposition 4.1, such that ω m+2 − ω m > 2γ for all m ∈ Z.

Proof of Theorem 3
We will need the following estimation of the coefficients.
If moreover m ∈ M 2 , we have to show that zn sin(z 2 n ) sin(zn ) 1. We have f (z n ) = 0, that is 2 cot(z 2 n ) = Az n (coth(z n ) − cot(z n )), where A = J j=1 α j . This leads, after a direct computation, to z n sin(z 2 n ) sin(z n ) = 2 A cos(z 2 n ) cos(z n ) − coth(z n ) sin(z n ) .
We need a technical lemma.
Using the previous proposition, we are able to get the following estimates for the observable.
Proposition 4.4. We suppose that ∈ L. Then for T > 2, we have where we have set B := J j=2 β j . The underlying constants do not depend on J. Proof. We first have for T > 2, as in the proof of Theorem 2.1, (4.6) On the other hand, as showed in (4.2), we also have We use the symbols x y and x y to respectively denote x ≤ C 1 y and x ≥ C 2 y for some positive constants C 1 and C 2 . Clearly x y and x y is equivalent to x y. Note that m ∈ M 2 implies |ω m+1 − ω m | ≤ γ and, due to the gap condition, m + 1 ∈ M 1 . This implies By Proposition 4.2, if m ∈ Γ 1 then |c m,j | 2 |z n | 6 sin 2 (z 2 n ) sin 2 (zn ) |a n | 2 . If otherwise m ∈ Γ 2 then |c m,j | 2 = . Then one has the direct inequality To complete the proof we need to prove the inverse inequality. We preliminarly remark that the gap condition ω m+2 − ω m ≥ 2γ (and the fact that Λ 1 and Λ 2 are discrete sets) implies that if m ∈ M 2 ∩ Γ 1 then m + 1 ∈ M 1 ∩ Γ 2 and (4.7) We then apply Lemma 4.3 by deducing that for all β > 0 m∈M2 |c m,j + c m+1,j | 2 = m∈M2∩Γ1 |c m,j + c m+1,j | 2 + m∈M2∩Γ2 |c m,j + c m+1,j | 2 where the last inequality follows from c β − 2β < 0 and (4.7). By Proposition 4.2, |c m,j | 2 |z n | 4 |a n | 2 for all m ∈ M 2 ∩ Γ 1 (where n is such that ω m = n/|n|z 2 n ) and, in particular, |c m,j | 2 ≤Ĉ|z n | 4 |a n | 2 for some positiveĈ (independent from j, because of the definition of c m,j when m ∈ Γ 1 ) and for all j = 2, . . . , J. Similarly, again by Proposition 4.2, |c m,j | 2 ≥D|z n | 6 sin 2 (z 2 n ) sin 2 (zn ) |a n | 2 for some positiveD and for all j = 2, . . . , J. Therefore Now, using (4.6), let K 0 > 0 be such that We choose β ∈ (0, 1/2) sufficiently small to have K 0 − K 1 BĈ √ 2β > 0. Using above estimates and recalling that  Proof of Theorem 2.3. Let (u 0 , u 1 ) ∈ Z and let (a n ) and (b k,q ) be the corresponding coefficients. In view of Proposition 4.4 and Proposition 4.5, for T > 2 we have which concludes the proof.

Numerical results
The code that is used for the numerical results is available here https://github.com/mehrenbe/InghamWaveBeam.
On Figure 3, we represent the gap z 2 n+1 − z 2 n with respect to √ n, for n ∈ {1, . . . , 10 4 }, for different values of > 0 belonging to L, and taking A = 1 . We remark that the gap oscillates between a value γ min 2 and π for large values of n. The gap is almost always around π, but almost periodically, with a period proportional to 1/ , it falls down near γ min . The behavior at the beginning is different. In particular, when is large, the gap becomes very small, even if it still remains strictly positive. These results are in agreement with the gap result in Proposition 4.1.
On Figure 6, we represent now the gap ω m+1 − ω m with respect to √ m. The results are then quite different, and we can see on these numerical results that we no longer have a gap.
In order to have an idea of the constants c 1 (T ), c 2 (T ) such that we look for constants c 1,n,N loc (T ), c 2,n,N loc (T ) such that k )t dt) n+N loc j,k=n . We vary the value of N loc ; the larger it is, the better is the result. We remark that the constant c 1,n,N loc (T ) can be quite small for low values of n, in the case where is large; this is coherent with the previous result, as the gap is very small for n small (low frequencies). The results are then better by increasing the value of T . There are some oscillations in the graphs which are pushed at later n, taking a larger N loc . Finally, we do the same for  We see that c 3,n,N loc (T ) is no more minored by a strictly positive constant, which is coherent, because there is no longer an asymptotic gap. On the other hand, when we express in the basis of divided differences, we observe a minoration by a strictly positive constant (for small n, the value is still very small, especially for big value of , as for the case for (z 2 k ), but it's getting better by increasing the value of T ) which is coherent with the weakened gap condition.
Note that the theoretical part is based on such estimates, for which we now have a numerical illustration of the behavior of its underlying constants. In particular, we observe that time of observation, even if it is enough to take it greater than π, we should take it bigger for having not too small constants, which can occur when taking large.