Timoshenko systems with fading memory

TIMOSHENKO SYSTEMS WITH FADING MEMORY

arXiv:1309.4605v1 [math.AP] 18 Sep 2013

MONICA CONTI, FILIPPO DELL’ORO AND VITTORINO PATA Abstract. The decay properties of the semigroup generated by a linear Timoshenko system with fading memory are discussed. Uniform stability is shown to occur within a necessary and sufficient condition on the memory kernel µ.

1. Introduction Given a real interval I = [0, ℓ], we consider the viscoelastic beam model of Timoshenko type [12]   ρ1 ϕtt − κ(ϕx + ψ)x = 0 Z ∞ (1.1)  µ(s)ψxx (t − s) ds + κ(ϕx + ψ) = 0 ρ2 ψtt − bψxx + 0

in the unknowns ϕ, ψ : (x, t) ∈ I × [0, ∞) 7→ R, where the strictly positive constants ρ1 , ρ2 , κ, b satisfy the relation ρ1 ρ2 (1.2) = κ b while the memory kernel µ is a (nonnegative) nonincreasing absolutely continuous function on [0, ∞) such that Z ∞ a := b − m > 0 where m := µ(s) ds > 0. 0

The system is complemented with the Dirichlet boundary conditions ϕ(0, t) = ϕ(ℓ, t) = ψ(0, t) = ψ(ℓ, t) = 0,

but our arguments can be used to prove analogous results for other kind of boundary conditions as well, such as Dirichlet/Neumann or Neumann/Dirichlet. Following [7], by rephrasing the problem within the history framework of Dafermos [3], system (1.1) is shown to generate a contraction semigroup S(t) of solutions acting on a suitable Hilbert space H accounting for the presence of the memory. The aim of this work is to establish a necessary and sufficient condition on the memory kernel µ (within the class of kernels considered above) in order for S(t) to be exponentially stable on H, namely, kS(t)zkH ≤ Ke−ωt kzkH , ∀z ∈ H, for some K ≥ 1 and ω > 0. Our main theorem reads as follows. Key words and phrases. Timoshenko system, fading memory, contraction semigroup, exponential stability. 1

2

M. CONTI, F. DELL’ORO AND V. PATA

Theorem 1. The semigroup S(t) is exponentially stable if and only if there exist C ≥ 1 and δ > 0 such that (1.3) for every σ ≥ 0 and s > 0.

µ(σ + s) ≤ Ce−δσ µ(s)

The decay properties of S(t) have been previously studied in the papers [6, 7]. We will discuss and compare those results in the next Sections 4 and 5, where we will also provide the proofs of the two directions of Theorem 1. Condition (1.3) appears for the first time in connection with systems with memory in [2], whereas (1.2), which basically says that the two hyperbolic equations share the same propagation speed, is used in [1] for the same model but with a convolution integral of Volterra type. In that work, the failure of (1.2) is shown to prevent the possibility of any uniform decay of the solutions. Actually, the same phenomenon pops up if the convolution integral, which contains the whole dissipation of the system, is replaced by an instantaneous damping term, such as ψt (see [11]).

Remark. The existence of the semigroup S(t) can be actually established under weaker conditions on µ, which can be only piecewise absolutely continuous with (infinitely many) discontinuity points. For the proof of the necessity part of Theorem 1 nothing more is needed. In particular, no use is made of (1.2). Concerning sufficiency, besides (1.2)-(1.3), one has to require that the set where µ′ < 0 has positive measure, automatically satisfied if µ is absolutely continuous (as in our hypotheses). 2. Functional Setting and Notation In what follows, h·, ·i and k · k are the inner product and norm on the (real) Hilbert space L2 (I). We will also consider the Sobolev space H01 (I) endowed with the gradient norm, due to the Poincar´e inequality, along with the L2 -weighted space of H01 -valued functions on R+ = (0, ∞) Z ∞ 2 + 1 M = L (R ; H0 (I)), hη, ξiM = µ(s)hηx (s), ξx (s)i ds. 0

We define the linear operator T on M by  T η = −Dη, D(T ) = η ∈ M : Dη ∈ M, η(0) = 0 ,

where D stands for weak derivative. The operator T is the infinitesimal generator of the right-translation semigroup Σ(t) on M, acting as ( 0 s ≤ t, [Σ(t)η](s) = η(s − t) s > t.

The phase space of our problem will be1 normed by

H = H01 (I) × L2 (I) × H01 (I) × L2 (I) × M

˜ 2 + kηk2 . ˜ η)k2 = κkϕx + ψk2 + ρ1 kϕk ˜ 2 + akψx k2 + ρ2 kψk k(ϕ, ϕ, ˜ ψ, ψ, M H 1In

the case of Neumann boundary conditions, one has to work in spaces of zero-mean functions.

TIMOSHENKO SYSTEMS WITH FADING MEMORY

3

Finally, we recall [2, Theorem 3.3]. Theorem 2. The right-translation semigroup Σ(t) acting on M is exponentially stable if and only if (1.3) holds. 3. The Contraction Semigroup We formally define the auxiliary variable η = η t (s) as (the dependence on x is omitted) η t (s) = ψ(t) − ψ(t − s). Then, (1.1) turns into the system in the unknowns ϕ = ϕ(t), ψ = ψ(t) and η = η t  ρ1 ϕtt − κ(ϕx + ψ)x = 0,    Z ∞  ρ2 ψtt − aψxx − µ(s)ηxx (s) ds + κ(ϕx + ψ) = 0, (3.1)  0    ηt = T η + ψt . Introducing the state vector

˜ Z(t) = (ϕ(t), ϕ(t), ˜ ψ(t), ψ(t), η t ), system (3.1) can be clearly written as a linear ODE in H of the form (3.2)

d Z(t) = AZ(t), dt

where the domain D(A) of the linear operator A, whose action can be easily deduced ˜ η) ∈ H such that from (3.1), is made by all the vectors (ϕ, ϕ, ˜ ψ, ψ, Z ∞ 2 1 ˜ ϕ, ψ ∈ H (I), ϕ, ˜ ψ ∈ H0 (I), η ∈ D(T ), µ(s)η(s) ds ∈ H 2 (I). 0

According to [7], the operator A is the infinitesimal generator of a contraction semigroup S(t) = etA : H → H.

˜ η) ∈ D(A), In particular, A is dissipative. Indeed, for every z = (ϕ, ϕ, ˜ ψ, ψ, Z 1 ∞ ′ (3.3) hAz, ziH = hT η, ηiM = µ (s)kηx (s)k2 ds ≤ 0. 2 0

Thus, for every initial datum z = (ϕ0 , ϕ˜0 , ψ0 , ψ˜0 , η0 ) ∈ H given at time t = 0, the unique solution at time t > 0 to (3.2) reads Z(t) = (ϕ(t), ϕt (t), ψ(t), ψt (t), η t ) = S(t)z, where η t fulfills the explicit representation formula (see [5]) ( ψ(t) − ψ(t − s) s ≤ t, (3.4) η t (s) = η0 (s − t) + ψ(t) − ψ0 s > t.

4

M. CONTI, F. DELL’ORO AND V. PATA

4. Theorem 1 (Necessity) The proof of the necessity part of Theorem 1 is essentially the same of [2, Theorem 3.2], dealing with a linearly viscoelastic equation. For the reader’s convenience, we report here the short argument. Proof of Theorem 1 (Necessity). Suppose S(t) exponentially stable on H. Then, for any initial datum z ∈ H of the form z = (0, 0, 0, 0, η0) we have max{kψx (t)k, kη t kM } ≤ max{1/a, 1}kS(t)zkH ≤ Ke−ωt kη0 kM

for some positive K, ω. On the other hand, exploiting the representation formula (3.4), Z ∞ t 2 2kη kM ≥ 2 µ(s)kη0x (s − t) + ψx (t)k2 ds Z ∞t ≥ µ(s)kη0x (s − t)k2 ds − 2mkψx (t)k2 t

We conclude that

≥ kΣ(t)η0 k2M − 2mK 2 e−2ωt kη0 k2M .

p kΣ(t)η0 kM ≤ K 2(1 + m) e−ωt kη0 kM , and the claim is a consequence of Theorem 2.



Our Theorem 1 (Necessity), seems to contradict the following result established by Messaoudi and Said-Houari: Theorem 3 (Theorem 2.1 in [6]). Let µ satisfy the differential inequality (4.1)

µ′ (s) + δ[µ(s)]p ≤ 0

for some δ > 0 and some p ∈ (1, 23 ). Then, for every initial datum z ∈ H, the inequality E(t) := 12 kS(t)zk2H ≤

M (1 + t)1/(p−1)

holds for some M > 0 depending on z. Let us observe that the correct conclusion of Theorem 3 should have been that the energy decays exponentially, for lack of exponential stability prevents the existence of uniform decay patterns. Indeed, the thesis of Theorem 3 implies that, for every z ∈ H, k(1 + t)1/(2p−2) S(t)zkH ≤ Qz

for some Qz > 0, and a direct application of the Uniform Boundedness Principle yields Q kS(t)zkH ≤ kzkH , (1 + t)1/(2p−2) where Q > 0 is now independent of z. Hence the operator norm of S(t) goes below one for large values of t, and exponential stability readily follows. At the same time, S(t) cannot have a uniform decay if, for instance, 1 , µ(s) = (1 + s)1/(p−1) which complies with (4.1) but clearly violates (1.3).

TIMOSHENKO SYSTEMS WITH FADING MEMORY

5

5. Theorem 1 (Sufficiency) The exponential stability of S(t) has been proved in [7] within the hypotheses µ′ (s) + k1 µ(s) ≥ 0,

(5.1)

|µ′′ (s)| ≤ k2 µ(s),

for some k1 , k2 > 0, and µ′ (s) + δµ(s) ≤ 0,

(5.2)

for some δ > 0. Let aside (5.1), which is only technical, condition (5.2) is equivalent to (1.3) with C = 1. Nonetheless, (1.3) with C > 1 turns out to be much more general than (5.2). For instance, any compactly supported µ (in the class of kernels considered in the present paper) satisfies (1.3), but cannot comply with (5.2) if it has flat zones, or even horizontal inflection points. Besides, (1.3) with C > 1 makes no assumptions at all on the derivative µ′ . Analogously to [7], the proof of the sufficiency part of Theorem 1 is based on the following abstract result from [10] (see also [4] for the precise statement used here). Lemma 4. The contraction semigroup S(t) on H is exponentially stable if and only if there exists ε > 0 such that inf kiλz − AzkH ≥ εkzkH ,

λ∈R

∀z ∈ D(A),

where A and H are understood to be the complexifications of the original A and H. Proof of Theorem 1 (Sufficiency). Within hypothesis (1.3), suppose S(t) be not exponentially stable. Then, Lemma 4 ensures the existence of sequences λn ∈ R and zn = (ϕn , ϕ˜n , ψn , ψ˜n , ηn ) ∈ D(A) with kzn k2H = κkϕnx + ψn k2 + ρ1 kϕ˜n k2 + akψnx k2 + ρ2 kψ˜n k2 + kηn k2M = 1

satisfying the relation (5.3)

iλn zn − Azn → 0 in H.

Componentwise, (5.3) reads (5.4)

iλn ϕn − ϕ˜n → 0 in H01 ,

(5.5)

iλn ρ1 ϕ˜n − κ(ϕnx + ψn )x → 0 in L2 ,

(5.6)

iλn ψn − ψ˜n → 0 in H01 , Z ∞ ˜ iλn ρ2 ψn − aψnxx − µ(s)ηnxx (s) ds + κ(ϕnx + ψn ) → 0 in L2 ,

(5.7)

0

(5.8)

iλn ηn − T ηn − ψ˜n → 0 in M.

We assume λn 6→ 0 (the case λn → 0 is much simpler and left to the reader). Accordingly, up to a subsequence (uts in the sequel), λn → λ⋆ ∈ [−∞, ∞] \ {0}.

6

M. CONTI, F. DELL’ORO AND V. PATA

We will reach a contradiction by showing that every single component of zn goes to zero in its norm uts.2 The first part of the proof borrows some ideas from [9]. Since µ is an absolutely continuous function vanishing at infinity, the set  S = s ∈ R+ : Kµ′ (s) + µ(s) < 0 has positive measure for some K > 0 large enough. Let us define the space S = L2µ (S; H01 (I)). We need three preliminary lemmas. We will lean several times (without explicit mention) on the boundedness in H of zn . Lemma 5. We have the convergence kηn kS → 0. Proof. By means of (5.3), Re hiλn zn − Azn , zn iH = −Re hAzn , zn iH → 0, and using (3.3) we are led to Z Z ′ 2 (5.9) 0 ≤ − µ (s)kηnx (s)k ds ≤ −

∞

0

S

µ′ (s)kηnx (s)k2 ds → 0.

Since

kηn k2S the conclusion follows.

=

Z

S

2

µ(s)kηnx (s)k ds ≤ −K

Z

µ′ (s)kηnx (s)k2 ds,

S



Lemma 6. The sequence |λn |kψ˜n k∗ is bounded, where k · k∗ is the norm in H −1 (I). Proof. By the triangle inequality, Z ∞



˜ ˜ ρ2 |λn |kψn k∗ ≤ iλn ρ2 ψn − aψnxx − µ(s)ηnxx (s) ds + κ(ϕnx + ψn ) ∗ 0 Z ∞



+ aψnxx + µ(s)ηnxx (s) ds − κ(ϕnx + ψn ) . 0

∗

Due to (5.7) and the continuous embedding L2 (I) ⊂ H −1(I), the first term in the righthand side goes to zero, whereas the second one is dominated by Z ∞ akψnx k + µ(s)kηnx (s)k ds + κkϕn k + κkψn k∗ , 0

which is bounded uniformly with respect to n ∈ N.

Lemma 7. Within (1.3), for any ξ ∈ M we have the estimate Z ∞ Z s √ kξkM . µ(s) kξx (σ)k dσds ≤ 4Cm δ 0

2It

0

is understood that passing to a subsequence means to refine the former one.



TIMOSHENKO SYSTEMS WITH FADING MEMORY

7

Proof. Exploiting (1.3) and the H¨older inequality, Z ∞ Z s √ √ Z ∞p µ(s) G(s) ≤ CmkGkL2 (R+ ) , µ(s) kξx (σ)k dσds ≤ C 0

0

0

having set

G(s) =

Z

s

δ

e− 2 (s−σ) 0

p

µ(σ) kξx (σ)k dσds.

By a well-known result of measure theory,

√

kGkL2 (R+ ) ≤ 2 µ kξx k δ

L2 (R+ )

= 2δ kξkM ,

which completes the argument.



We are now ready to prove the norm-decay of every single component of zn .

Lemma 8. The convergence kψ˜n k → 0 holds uts.

Proof. Introducing Ψn such that −Ψnxx = ψ˜n , we infer from Lemmas 5 and 6 that Z √ ˜ |iλn hηn , Ψn iS | ≤ |λn |kψn k∗ µ(s)kηnx (s)k ds ≤ m |λn |kψ˜n k∗ kηn kS → 0. S

On the other hand, calling

ξn = iλn ηn − T ηn − ψ˜n ,

we find the explicit expression Z s 1 −iλn s ˜ (1 − e )ψn + e−iλn (s−σ) ξn (σ) dσ. (5.10) ηn (s) = iλn 0 Hence, iλn hηn , Ψn iS = αn kψ˜n k2 + βn → 0, where we put Z αn = µ(s)(1 − e−iλn s ) ds, S Z Z s βn = iλn µ(s) e−iλn (s−σ) hξnx (σ), Ψnx i dσds. 0

S

The conclusion follows by showing that βn → 0 whereas αn remains away from zero for large n. Indeed, by Lemmas 6-7 and the convergence (5.8) we get |βn | ≤

√

4Cm |λn |kψ˜n k∗ kξn kM δ

→ 0.

Concerning αn , we have two possibilities. If λ⋆ ∈ {−∞, ∞}, the Riemann-Lebesgue lemma yields the convergence Z αn → µ(s) ds > 0, S

whereas, if λ⋆ ∈ R \ {0},

Re αn →

Z

S

µ(s)(1 − cos λ⋆ s) ds > 0.

In either case Re αn has positive limit (again, uts).



8

M. CONTI, F. DELL’ORO AND V. PATA

Lemma 9. The convergence kψnx k → 0 holds uts. Proof. Define

1 (1 − e−iλn s )(ψ˜n − iλn ψn ). iλn By means of (5.6), it is apparent that ζn → 0 in M and, thanks to (5.10), Z s −iλn s (5.11) ηn (s) = (1 − e )ψn + e−iλn (s−σ) ξn (σ) dσ + ζn (s), ζn (s) =

0

which, on account of Lemma 5, entails

with αn as above and

hηn , ψn iS − hζn , ψn iS = αn kψnx k2 + γn → 0, γn =

Z

µ(s)

S

An application of Lemma 7 gives

Z

s

0

|γn | ≤

√

e−iλn (s−σ) hξnx (σ), ψnx i dσds. 4mC kψnx kkξn kM δ

→ 0.

Knowing that αn remains away from zero, we conclude that kψnx k → 0.



Lemma 10. The convergence kηn kM → 0 holds uts. Proof. Making use of (5.11), we easily obtain the estimate Z ∞ Z s √ 2 kηn kM ≤ 2 m kψnx kkηn kM + µ(s)kηnx (s)k kξnx (σ)k dσds + kζn kM kηn kM . 0

0

Arguing exactly as in Lemma 7, we see that Z ∞ Z s µ(s)kηnx (s)k kξnx (σ)k dσds ≤ 0

0

Consequently,

√ kηn kM ≤ 2 m kψnx k + on account of Lemma 9.

√ 4C kξn kM δ

√ 4C kηn kM kξn kM . δ

+ kζn kM → 0



At this point, we introduce the sequence of functions Z ∞ Fn (x) = aψnx (x) + µ(s)ηnx (x, s) ds. 0

An asymptotic control of certain boundary terms will be needed. Lemma 11. The convergence Fn (x)ϕnx (x) → 0 holds uts for x = 0 and x = ℓ.

Proof. The argument follows the lines of [7]. Accordingly, choose a real function q ∈ C 1 (I) satisfying q(0) = −q(ℓ) = 1. A multiplication of (5.7) by Qn (x) = q(x)Fn (x)

yields the convergence iλn ρ2 hψ˜n , Qn i − hFnx , Qn i + κhϕnx + ψn , Qn i → 0.

TIMOSHENKO SYSTEMS WITH FADING MEMORY

9

By Lemmas 9 and 10, it is readily seen that Fn → 0 in L2 (I), thus κhϕnx + ψn , Qn i → 0.

We claim that

Re iλn hψ˜n , Qn i → 0.

Indeed, exploiting (5.6) we get iλn hq ψ˜n , aψnx i = −ahq ψ˜n , iλn ψnx i = −ahq ψ˜n , ψ˜nx i + εn

for some complex sequence εn → 0, whereas (5.8) and an integration by parts yield (the boundary terms disappear as in [5]) Z ∞ Z ∞ iλn µ(s)hq ψ˜n , ηnx (s)i ds = − µ(s)hq ψ˜n , iλn ηnx (s)i ds 0 0 Z ∞ =− µ′ (s)hq ψ˜n , ηnx (s)i ds − mhq ψ˜n , ψ˜nx i + νn 0

for some complex sequence νn → 0. Collecting the two identities, we obtain Z ∞ ˜ ˜ ˜ iλn hψn , Qn i = −bhq ψn , ψnx i − µ′ (s)hq ψ˜n , ηnx (s)i ds + εn + νn . 0

Using Lemma 8 and integrating by parts, we infer that Z ℓ d ˜ ˜ 2Re hq ψn , ψnx i = q(x) |ψ˜n (x)|2 dx = −hq ′ ψ˜n , ψ˜n i → 0. dx 0 Besides, invoking (5.9), Z ∞ µ′ (s)hq ψ˜n , ηnx (s)i ds → 0, 0

and the claim is established. Summarizing, we arrive at the convergence Writing

Re hFnx , Qn i → 0.

−2Re hFnx , Qn i = |Fn (ℓ)|2 + |Fn (0)|2 + hq ′ Fn , Fn i, and noting that the last term in the right-hand side vanishes as n → ∞, we reach the conclusion |Fn (ℓ)|2 + |Fn (0)|2 → 0. To finish the proof it is now enough showing that the sequence |ϕnx (ℓ)|2 + |ϕnx (0)|2 = −2Re hqϕnx , ϕnxx i − hq ′ ϕnx , ϕnx i

is bounded. So is certainly the second term in the right-hand side above. Concerning the first one, multiplying (5.5) by qϕnx , and taking advantage of (5.4) and Lemma 9, we have −ρ1 hq ′ ϕ˜n , ϕ˜n i + 2κ Re hqϕnx , ϕnxx i = 2ρ1 Re hq ϕ˜n , ϕ˜nx i + 2κ Re hqϕnx , ϕnxx i → 0.

Since hq ′ ϕ˜n , ϕ˜n i is bounded, the same is true for Re hqϕnx , ϕnxx i.



Remark. Observe that, within the Neumann boundary condition for either ϕ or ψ, Lemma 11 is trivially true. Lemma 12. Within condition (1.2), the convergence kϕnx + ψn k → 0 holds uts.

10

M. CONTI, F. DELL’ORO AND V. PATA

Proof. Multiplying (5.7) by ϕnx + ψn and exploiting (5.5), we obtain ℓ ρ1 κkϕnx + ψn k2 − Fn ϕnx 0 + iλn ρ2 hψ˜n , ϕnx + ψn i − iλn hFn , ϕ˜n i → 0. κ

The boundary term goes to zero by Lemma 11. Besides, multiplying (5.6) by ψ˜n and applying Lemma 8, we learn that iλn ρ2 hψ˜n , ψn i → 0,

and we deduce the convergence

ρ1 κkϕnx + ψn k2 + iλn ρ2 hψ˜n , ϕnx i − iλn hFn , ϕ˜n i → 0. κ Finally, using (1.2), (5.4), (5.6) and (5.8), we have ρ1 iλn ρ2 hψ˜n , ϕnx i − iλn hFn , ϕ˜n i κ Z bρ1 ρ1 ∞ ˜ = −ρ2 hψn , iλn ϕnx i − hiλn ψnx , ϕ˜n i − µ(s)hiλn ηnx (s) − iλn ψnx , ϕ˜n i ds κ κ 0 Z ρ ρ1 ∞ ′ ρ2  ˜ 1 hψn , ϕ˜nx i − µ (s)hηnx (s), ϕ˜n i ds + εn =b − κ b κ 0 Z ρ1 ∞ ′ =− µ (s)hηnx (s), ϕ˜n i ds + εn , κ 0 for some complex sequence εn → 0. Since (5.9) bears Z ∞ µ′ (s)hηnx (s), ϕ˜n i ds → 0, 0

the proof is finished.



On account of Lemmas 8, 9, 10 and 12, the sought contradiction is attained once we prove the convergence kϕ˜n k → 0 uts. To this end, a multiplication of (5.5) by ϕn will do.  References [1] F. Ammar Khodja, A. Benabdallah, J.E. Muoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations 194 (2003), 82–115. [2] V.V. Chepyzhov and V. Pata, Some remarks on stability of semigroups arising from linear viscoelasticity, Asymptot. Anal. 46 (2006), 251–273. [3] C.M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal. 37 (1970), 297– 308. [4] C. Giorgi, M.G. Naso and V. Pata, Exponential stability in linear heat conduction with memory: a semigroup approach, Comm. Appl. Anal. 5 (2001), 121–134. [5] M. Grasselli and V. Pata, Uniform attractors of nonautonomous systems with memory, in “Evolution Equations, Semigroups and Functional Analysis” (A. Lorenzi and B. Ruf, Eds.), pp.155–178, Progr. Nonlinear Differential Equations Appl. no.50, Birkh¨auser, Boston, 2002. [6] S.A. Messaoudi and B. Said-Houari, Uniform decay in a Timoshenko-type system with past memory, J. Math. Anal. Appl. 360 (2009), 459–475. [7] J.E. Mu˜ noz Rivera and H.D. Fern´ andez Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (2008), 482–502.

TIMOSHENKO SYSTEMS WITH FADING MEMORY

11

[8] J.E. Mu˜ noz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems–global existence and exponential stability, J. Math. Anal. Appl. 276 (2002), 248–278. [9] V. Pata, Exponential stability in linear viscoelasticity with almost flat memory kernels, Commun. Pure Appl. Anal. 9 (2010), 721–730. [10] J. Pr¨ uss, On the spectrum of C0 -semigroups, Trans. Amer. Math. Soc. 284 (1984), 847–857. [11] A. Soufyane, Stabilisation de la poutre de Timoshenko, C. R. Acad. Sci. Paris S´er. I Math. 228 (1999), 731–734. [12] S.P. Timoshenko, On the correction for shear of a differential equation for transverse vibrations of prismatic bars, Philos. Magazine 41 (1921), 744–746. Politecnico di Milano - Dipartimento di Matematica “F. Brioschi” Via Bonardi 9, 20133 Milano, Italy E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]