EXPONENTIAL STABILIZATION OF A STRUCTURE WITH INTERFACIAL SLIP

. Two exponential stabilization results are proved for a vibrating structure subject to an interfacial slip. More precisely, the structure consists of two identical beams of Timoshenko type and clamped together but allowing for a longitudinal movement between the layers. We will stabilize the system through a transverse friction and also through a viscoelastic damping.

1. Introduction. The structure under investigation is formed by two identical beams attached together on top of each other. It is subject to transversal and rotational vibrations like the ones produced by transverse bending and torsion. In principle there is always some slip between the layers in a vibrating structure. In many cases this slip is ignored. In some other cases, this slip is annihilated by fastening the layers very tightly (by using bolts, for instance) in order to avoid negative effects such as corrosion. However, fastening very tightly the components may alter negatively the accomplishments of the structure. This structure is known under the name 'laminated beams' and are widely used in many fields of engineering. The beams are continuously clamped together but allowing for a longitudinal displacement (slip) together with the transversal and rotational vibrations. The slip between the components is sometimes intentionally meant in order to produce a significant amount of damping. This damping is capable of bringing back the structure to rest.
The authors concluded that in case (a) and (b) the model reduces to a simple one providing a good approximation in case of harmonic motions with small frequencies.
In [32] the authors transformed the system into via the change of variable ξ = 3s − ψ (the effective rotation angle) subjected to the boundary conditions They succeeded in proving an exponential stabilization result through the boundary control u 1 (t) = k 1 w t (1, t), u 2 (t) = −k 2 ξ t (1, t) using spectral theory provided that r 1 := G ρ = D Iρ =: r 2 and k i = r i , i = 1, 2. It is worth noting here that it has been shown that the slip is able to stabilize asymptotically the structure but it cannot stabilize it exponentially (see Corollary 2.3 and Note 2.1 in [32]).
In [3], the same system is considered under the boundary conditions The authors established the exponential stabilization of the system through the boundary control provided that the 'dominant' part of the closed loop system is itself exponentially stable.
In case s ≡ 0, then the system reduces to the well known Timoshenko model [1,8,11,16,17,[20][21][22][23][24][25][29][30][31]33,34]. Therefore the system gives prominence to the dynamics of the slip described by the third equation. We would like to revive this issue as the slip is usually unavoidable and in many cases it cannot be simply ignored.
We discuss here two models. In the first one we stabilize the system by a frictional damping acting on the transverse displacement. Namely, we will consider the system for x ∈ (0, 1), t > 0, some a > 0 (frictional damping coefficient), with the boundary conditions w(0, t) = ψ(0, t) = s(0, t) = 0, t ≥ 0, In the second one we shall be concerned with the adoption of viscoelastic dampings. These kinds of dampings may be caused by the utilization of a special material capable of driving the structure to rest in a very fast manner without any boundary or other types of control (having in mind that viscoelastic material are sensitive to changes in frequency and temperature [2]). We will investigate the system h for some β 0 , β 1 positive (guaranteeing the hyperbolicity of the system) with the boundary conditions The well-posedness of our system has been discussed in part in [32,3] (see also references in [10]). It suffices to combine these results with the ones in [4,5,7,9,18] to cope with the (nonlocal) viscoelastic term. We have weak solutions in V 1 * × L 2 3 and strong solutions in : v(0) = 0 , k = 1, 2. We shall focus here on the asymptotic behavior of solutions and in particular with the exponential stabilization of the system. Note that, for the first problem, the last two equations imply that Finally, summing up (9) and (10) we get Although from this lemma we see that the energy is uniformly bounded and decreasing, it is not clear how to prove exponential decay from this functional. Our objective is to modify it in a suitable manner so as to obtain an equivalent functional L which satisfies a differential inequality of the form d dt for some positive constant κ. To this end, for δ i , i = 1, ..., 4 positive to be chosen, define and respectively, The Cauchy-Schwarz inequality and the Poincaré inequality allow us to prove easily the equivalence between E 1 (t) and L(t). It remains to prove the differential inequality. To this end we proceed to prepare several lemmas which will be of great help in proving our result later. Lemma 1. The derivative of H 1 (t) along solutions of (1)-(2) is estimated by for t > 0 and some ε 0 > 0.
Proof. From the definition of H 1 (t), the first and the third equations in the system (1) we find and as the term we infer that Now from the definition (11) it is clear that W 2 ≤ s 2 and W t 2 ≤ s t 2 . Consequently, for t > 0 Lemma 2. The derivative of H 2 (t) along solutions of (1)-(2) satisfies for t > 0 where ε 0 > 0.
Proof. Using the second equation of (1), we find for t ≥ 0 Next, in view of where ε 0 > 0, we infer that Lemma 3. We have the estimate Proof. We infer from the first and third equations in (1) and the definition of H 3 (t) that Observe that and for ε 0 > 0. Therefore where we have used (5) i.e.
obtained by summing three times the third equation with the second in the system (1).
The expression in (14) may be estimated as follows That is

ASSANE LO AND NASSER-EDDINE TATAR
Using the previous lemmas we obtain the following proposition.
Theorem 1. For the energy E 1 defined above in (6), there exist two positive constants K and κ 0 such that Proof. In view of Lemmas 1-4, we see that Using the inequalities

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Therefore, in view of Proposition 1, we can write To recover all the terms in E 1 (t) (with negative coefficients) in the right hand side of this last relation we add and subtract µ 3s t − ψ t 2 (with a 'small' coefficient µ to be determined) in this right hand side (notice that we could have done this at an earlier stage). Using the inequality 3s t − ψ t 2 ≤ (1 + ε 0 ) ψ t 2 + 9 1 + 1 ε 0 s t 2 say with ε 0 = 1, we see that the insertion of µ 3s t − ψ t 2 , with positive µ, will give rise to the terms ψ t 2 and s t 2 with small coefficients. These coefficients will be easily absorbed by the corresponding ones in the last relation. Therefore We forget about the first two terms in the right hand side of (15) for a while and we concentrate on the remaining coefficients

EXPONENTIAL STABILIZATION OF A STRUCTURE WITH INTERFACIAL SLIP 6295
and by and note that Again multiplying the third equation of (3) by s t and integrating on (0, 1) we obtain Now adding (20) and (22) taking into account (21), we obtain Next we use the second equation in (3) to get

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The identity Gathering the previous relations (23) and (24), we are lead to Again we find a decreasing energy functional. However the nonnegativity of E 2 (t) is not clear. This difficulty does not arise in the usual viscoelastic problems as there will be always a term of the form G w x 2 which will take of the negative term (G t 0 h(s)ds w x 2 ). We are tempted to use the expression G ((ψ − w x ) x , w t ) in the proof of Proposition 2 to get the derivative of the term G w x 2 but then we will be faced with an even more complicated situation which is the appearance of the term G (ψ x , w t ) . The argument below shows that this functional is indeed nonnegative under a certain relationship between h and G. The user will have a certain freedom in the manner of imposing this condition. Proposition 3. The functional E 2 (t) is nonnegative (and actually equivalent to the same functional without the term in w x 2 ) provided that Proof. We have for η 1 , η 2 > 0

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Our goal is to control −G t 0 h(s)ds w x 2 by the expression G ψ − w x 2 + I ρ (3s − ψ) x 2 + 3 s x 2 which is already in E 2 (t). Clearly we will need To fix ideas let us pick η 1 = ξ 1−ξ to fulfill the first relation. For the other two relations to hold we will need

This condition is valid in case
(note that the term in the right side is smaller than one).
Clearly E 2 (t) ≤ 0 (that is E 2 (t) is nonincreasing as E 1 (t)) but we have lost the term − w t 2 in E 2 (t) (which was in E 1 (t)) we need to recover it by introducing the functionalH Let ε 1 > 0 and let t ≥ t 0 > 0 so that Thereforẽ
Proof. From the definition ofH 1 (t), the first and the third equations in the system we find

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and as we infer that It is easy to see that and Therefore, for t > 0 Let Lemma 7. The derivative ofH 4 (t) along solutions of (3)-(4) is estimated as follows, for ε 0 > 0

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Proof. We havẽ The first term in the right hand side of (25) is equal to and may be estimated as follows ThereforeH