ABSTRACT
We consider a transmission problem for a string composed by two components: one of them is a viscoelastic material (with viscoelasticity of memory type), and the other is an elastic material (without dissipation effective over this component). Additionally, we consider that in one end is attached a tip load. The main result is that the model is exponentially stable if and only if the memory effect is effective over the string. When there is no memory effect, then there is a lack of exponential stability, but the tip load produces a polynomial rate of decay. That is, the tip load is not strong enough to stabilize exponentially the system, but produces a polynomial rate of decay.
Keywords:
transmission problems; memory effect; lack of exponential stability; tip load; hybrid system
1 INTRODUCTION
We consider the transmission problem for the damped vibrations of a string, whose left end is rigidly attached and in the other end has an attached hollow-tip body that contains granular material. The string is composed by two components: one of them is a viscoelastic material (with viscoelasticity of memory type) and the other is an elastic material (without dissipation effective over this component).
More precisely, let us denote by U the displacement of the string. That is
where l is the length of the string and l 0 is the transmission point. The model that we consider in this paper is written as follows.
Here, be the relaxation function, and α 1, α 2, ρ 1, ρ 2 are positive constants that reflect physical properties of the string. The boundary conditions are given by
and the transmission conditions are given by
We turn to model the motion of the right end with the attached tip body. We assume that the container is rigidly attached to the end x = l, and that the container and its contents have mass m and a center of mass O′ located at distance d from the end of the string. We assume that the damping effect of the internal granular material can be represented by damping coefficient γ 1, whose precise contributions are described below
Here, the first term is the contribution of the inertia of the container, and the second term represents the damping that the granular material provides, which is assumed to be proportional to the velocity, and so γ 1 is the damping coefficient. Thus, the force balance at the end x = l is
where the parameters γ 1 and γ 2 are non-negative constants. Finally, the initial data are given by
Here, we assume the following hypotheses on the relaxation function g:
Concerning models of motion with the attached tip body, Andrews and Shillor 11 K. Andrews & M. Shillor. Vibrations of a Beam With a Damping Tip Body. Mathematical and Computer Modelling, 35 (2002), 1033-1042. establish the existence and uniqueness of the model and showed the exponential energy decay of the solution provided and extra damping term is present. In 1111 N. Zietsman, L Van Rensburg & A. Van der Merwe. A Timoshenko Beam With Tip Body and Boundary Damping. Wave Motion, 39 (2004), 199-211. Zietsman, Rensburg and Merwe consider the effect of boundary damping on a cantilevered Timoshenko beam with a rigid body attached to the free end. The authors establish the efficiency and accuracy of the finite element method for calculating the eigenvalues and eigenmodes. Although no conclusion is showed with regard to the stabilization of the system, the authors showed interesting phenomena concerning the damped vibration spectrum and the associated eigenmodes. See also the work of Feireisl and O’Dowd 77 E. Feireisl & G. O’Dowd. Stabilisation d’un syste`me hybride par un feedback non linéaire, non monotone. Comptes Rendus de l’Académie des Sciences, 326 (1998), 323-327. where is showed, for an hybrid system composed of a cable with masses at both end, the strong stability for a nonlinear and nonmonotone feedback law applied at one end.
The main result of this paper is to show that the system (1.1)-(1.8) is exponentially stable if and only if the memory effect is effective over the viscoelastic part of the material. This means that the dissipative properties given by the tip load is not enough to produce exponencial rate of decay when the memory effect is not effective. Finally, when g = 0, we prove that the system is not exponentially stable but the dissipation given by the tip load produce polynomial stability. The method we use is based on Prüss Theorem to show exponential stability. The proof of the lack of exponential stability is based on the Weyl invariance Theorem and the proof of the polinomial stability is based on the Borichev and Tomilov result.
2 EXISTENCE AND UNIQUENESS OF SOLUTIONS
To use the semigroup approach we need to rewrite the problem as an autonomous system. For this reason we introduce the history problem, obtained by replacing the equation (1.1) by the following history equation
Following the ideas of Dafermos 44 C.M. Dafermos. On Abstract Volterra Equation with Applications to Linear Viscoleasticity. Differential and Integral Equations, 7 (1970), 554-569.), (33 C.M. Dafermos. Asymptotic Stability in Viscoleasticity. Arch. Rat. Mech. Anal., 37 (1970), 297-308. and Fabrizio 66 M. Fabrizio & A. Morro. “Asymptotic Stability in Viscoleasticity”, volume 12 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1992)., we introduce the notation
with ; whence we consider the system
with u, v and w, satisfying (1.5) and the initial conditions (1.6), (1.7), (1.8) and η verifying
with boundary conditions are given by
The transmission conditions now are given by
We define the total energy of the system as
Let us introduce the following spaces:
We recall that is a Hilbert space when endowed with the inner product given by
With this notations, we consider the phase space
Note that the space ℋ is a Hilbert space with the norm
where .
Let us introduce the linear unbounded operator 𝒜 in ℋ as follows
with domain
Using the hypotheses on g, a direct computation yields
which means that 𝒜 is a dissipative operator. The system (2.1)-(2.7), (1.5)-(1.8) is equivalent to
where
𝒰(t) = (u(t), v(t),U(t),V(t), η(t), w(t),W(t))T and 𝒰0 = (u 0, v 0, u 1, v 1, η0, w 0, w 1)T .
Under this conditions, we have
Theorem 2.1. The operator 𝒜 is the infinitesimal generator of a C 0 -semigroup of contractions (S(t)) t ≥ 0 on ℋ. Thus, for any initial data 𝒰 0 ∈ ℋ , the problem (2.8) has a unique weak (mild) solution
Moreover, if 𝒰0 ∈ D(𝒜 ), then 𝒰 is a strong solution of (2.8), that is
Proof. It easy to see that 𝒟(𝒜) is dense in ℋ; and, since 𝒜 is a dissipative operator, it is enought to show that . To do that, we will show that for , there exists only one such that 𝒜𝒰 = F. In terms of the components, we have
Indeed, from the equations (2.9) and (2.13), we get that and that
which means that η is uniquely determined. Moreover, using (1.10) and (2.5), we can write, for each T > 0:
from we obtain
which enables us to conclude that . Thus, it remains only to establish the existence and uniqueness of solution for the system
Let us consider the functional T: X → ℂ given by
for all (φ, ψ) ∈ X , where and is a Hilbert space, endowed with the inner product
It’s clear that T ∈ X′; therefore, by the Riesz representation theorem we conclude that there exists only one weak solution to system (P). So we have that 0 ∈ ρ(𝒜 ). □
3 EXPONENTIAL STABILITY
In this section, we show that if hypothesis (1.9)-(1.11) hold, then the corresponding semigroup is exponentially stable. The main tool we use is Prüss’s results 99 J. Prüss. On the Spectrum of C0-semigroups. Trans. AMS, 284 (1984), 847-857., which is summarized in the following theorem.
Theorem 3.2.Let (𝒮 (t))t≥0 be a C 0 -semigroup on a Hilbert space ℋ generated by 𝒜 . Then the semigroup is exponentially stable if and only if
In the next Lemma we show that the imaginary axis is contained in the resolvent set.
Lemma 3.1. Under the hypotheses (1.9)-(1.11), the operator 𝒜 verify
Proof. In the Theorem 2.1, we have already shown that 0 ∈ ρ(𝒜). Moreover, note that we can’t conclude that the spectrum of 𝒜 is formed only by eigenvalues, since 𝒜 −1 is not compact. So, if (3.1) is not true, then there exists λ 0 ∈ ℝ with , such that and . Follow that there exist sequences (λ n )n ⊂ ℝ and , such that
But, (3.4) is equivalent to
Taking the inner product of (3.4) with 𝒰 n in ℋ , we get
Follows from condition (1.10) and (3.12) that
Now, we use (3.3) and (3.5) for conclude that there exist U, u ∈ L 2(0, l 0) and subsequences still denoted by (U n )n and (u n )n , such that
Moreover, using (3.3), (3.7), (3.13) and (3.14), we get
So, from (3.14), (3.15) and (3.5), we can conclude that
Proceeding analogously, we find
From convergences above, remembering w n = v n (l) and W n = V n (l), follow that
The convergences obtained above allow us to pass to the limit in (3.5)-(3.11), obtaining the following system
We conclude that there exists , such that
where (u, v) is precisely the solution of the system given by
when γ 1 > 0; or, in the case that γ 1 = 0, of the system obtained in the above system, replacing the boundary condition v(l) = 0 by v x (l) + α 3 v(l) = 0, where .
However, each one of these two systems has a unique solution, namely, the null solution; from which it follows that 𝒰 = 0. With this, we rewrite (3.16) as
which contradicts (3.3), completing the proof. □
From now, on C will denote a generic constant, that can be different in different places. Let us denote by
We will prove that the solution 𝒰 of the resolvent equation
is uniformly bounded for any take F = ( f 1 , f 2 , ..., f 7)T ∈ ℋ, that is,
In fact, in terms of the components we have
The dissipative properties of 𝒜 implies that there exists a positive constant C such that
The following Lemma will play an important role in the sequel.
Lemma 3.2.Under the above notations, for any ε > 0 sufficiently small, there exist a constant C ε > 0 such that, for |λ| large enough, hold
Proof. Multiplying the equation (3.22) by and using (3.24), we get
For each ε > 0, we use (1.10) and (3.27) for obtain
Using (3.27) once more we get
and
Therefore, for ε > 0 sufficiently small, we have
On the other hand, multiplying (3.22) by and using (3.20), we get
Since
using (3.20) we get, for each ε > 0 and for λ ≠ 0:
Therefore, for ε > 0 sufficiently small, we have
From above inequality and (3.28), our conclusion follows. □
The next lemma is crucial to ensure that the exponential decay occurs in the case where γ 1 = 0.
Indeed, it provides an estimate for the term involving |W| 2 that can be obtained from (3.27) only when γ 1 is positive.
Lemma 3.3.There exist C > 0 such that
Proof. Multiplying (3.23) by , using (3.21), and remembering that V (l) = W , we get
Taking the real part, we get the desired inequality. □
Now we are in condition to show the main result of this section.
Theorem 3.3. Let us suppose that (1.9)-(1.11) hold. Then the semigroup e 𝒜t is exponentially stable.
Proof. In view of Proposition 1, we only need to show that there exist C > 0 such that:
Since the resolvent operator is holomorphic, it is enough to prove the above inequality for |λ| large enough. In fact, multiplying (3.23) by and using (3.21), we get
Taking the real part, we get
On the other hand, multiplying (3.22) by , and using (3.20) and (3.24), we get
From Lemma 3.2, we get, for ε small enough and for |λ| large enough, that
Using the transmission conditions, inequality (3.29) can be estimated by (3.30), that is
Moreover from Lemma 3.2 and inequality (3.30), we get
Therefore, Lemma 3.3, equation (3.25), and inequality (3.31) implies
From the three last inequalities and (3.27), we get
which implies in
Then, the semigroup is exponentially stable. □
4 THE LACK OF EXPONENTIAL STABILITY
Now we shall prove that the dissipation given by the memory effect is necessary for exponential stability of the system. Let us consider the problem without memory effect; namely
with boundary conditions
and with transmission conditions
and initial data
where α 1, α 2, ρ 1, ρ 2, ρ 3, γ 2 are as before, and γ 1, now, is a positive constant. Moreover, for this problem, we consider the phase space
The total energy associed with the system is
and it is not difficult to see that, for all 𝒰 ∈ , we have
Let us denote by ℬ the unbounded operator of given by
with domain
It is easy to see that
It is not difficult to see that the operator ℬ is the infinitesimal generator of a C 0-semigroup of contractions over , which we will denote by T(t). This shows that the problem (4.1)-(4.6) is well-posed.
To prove that the system (4.1)-(4.6) is not exponentially stable, the main tool we use is the Weyl’s theorem about the invariance of the essential spectral radius by compact perturbations. To do that, let us consider the following conservative system
verifying the same boundary and transmission conditions and with the same initial data, where α 1, α 2, ρ 1, ρ 2, ρ 3 and γ 2 are as before. That is, with boundary conditions
and transmission conditions
and initial data
The total energy associed with this system is
and it is not difficult to see that
Therefore the system is conservative and there is no decay. Now we are in conditions to show the main result of this section.
Theorem 4.4.The semigroup T(t) associated to system (4.1)-(4.6) is not exponentially stable.
Proof. The main ideia is to prove that T(t) have the same essential spectral radius of the semi- group associated to conservative system (4.8)-(4.13), that we denote as T 0(t). Here, we use the Weyl’s Theorem (Theorem XIII.14, 1010 M. Reed & B. Simon. “Methods of Modern Mathematical Physics”, volume 04. Academic Press Inc., California (1978).; see also Kato’s book 88 T. Kato. “Perturbation Theory for Linear Operators”. Springer-Verlag, New York (1980)., Theorem 5.35, p. 244 for details of the proof), which stablish that if the difference of two operators is compact, then the your essential spectrum radii are equals. More precisely
Theorem 4.5. Let S and T two continuous operator over a Banach space X. If S−T is a compact operator, then S and T have the same essential spectrum radius.
So, we will show that the difference T (t) −T 0(t) is a compact operator; from which we obtain
But since T 0(t) is unitary, then ω ess (T 0) = 0. Denoting by ω(T) and ω σ (ℬ) the type of semi- group T(t) and the spectral upper bound of spectrum σ(ℬ) respectively, we have that (see 55 K.J. Engel & R. Nagel. “One-parameter Semigroups for Linear Evolution Equations”, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York (2000)., Corollary 2.11, p. 258):
This imply that T(t) is not exponentially stable. In fact. Let (u, v, w) and be the solutions of the systems (4.1)-(4.6) and (4.8)-(4.13), respectively. Denoting by
we have that (U,V,W) is solution of the system
with boundary conditions
and transmission conditions
and initial data
The energy associated with the system (4.14)-(4.19) is given by
It is easy to verify that
from where follows
Now, let us denote by a bounded sequence of initial data in the phase space . We will show that the corresponding sequence of solutions has a subsequence that converges strongly in .
To show this, note that T(t)?? 0,n and T 0(t)𝒰 0,n are bounded in . This implies that is bounded in L 2(0, T), for all T > 0. Therefore (4.16) implies that W n,t is bounded in H 1(0, T). Since H 1(0, T) has compact embedding in L 2(0, T), it follows that there exist subsequences, we still denote as (W n )n and such that
and similarly
From the above convergences we have
Using this convergence in (4.20) it follows that converges, which implies that [T(t) − T 0(t)]𝒰 0,n converges strongly in . This means that T(t) − T 0(t) is a compact operator in , and therefore the proof is complete. □
5 POLYNOMIAL DECAY
In this section we show that the solutions of the system (4.1)-(4.6) decays polynomially to zero as t −1/2 . To show this, we use the Borichev and Tomilov’s Theorem (see 22 A. Borichev & Y. Tomilov. Optimal Polynomial Decay of Functions and Operator Semigroups. Math. Ann., 347 (2009), 455-478.):
Theorem 5.6.Let S(t) be a bounded C0-semigroup on a Hilbert space ℋ with generator 𝒜 such that iℝ ⊂ ρ(𝒜). Then
Our starting point is to study the solution of the resolvent equation
which is rewritten, in terms of the components, in
Taking the inner product with 𝒰 and using (4.7) it follows that
Lemma 5.4.For |λ| large enough, there exist C > 0 such that
Proof. Multiplying (5.4) by and using (5.2), we have
On the other hand, using (5.6) we get
Finally, from (5.7)-(5.9) our conclusion follows. □
Lemma 5.5.For |λ| large enough, there exist C > 0 such that
Proof. Multiplying equation (5.3) by , using (5.1) and the transmission conditions, we get
On the other hand, multiplying equation (5.4) by and using (5.2) we get
From this, and from Lemma 5.4 there exist C > 0, such that
Combining (5.10) and (5.11), our conclusion follows. □
Now we are able to stablish the main result of this section.
Theorem 5.7.The semigroup T(t) associated to the transmission problem with load tip (4.1)-(4.6) decays polynomially as t−1/2. Moreover, if 𝒰0 ∈ D(ℬ k ), then
Proof. The proof of iℝ ⊂ ρ(ℬ) is analogous to the proof of Lemma 3.1. And, from Lemmas 5.4 and 5.5, follows that, for |λ| large enough, we have
and this implies that
which gives us, for |λ| large enough
Therefore, from Theorem 5.6 our conclusion follows. □
REFERENCES
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1K. Andrews & M. Shillor. Vibrations of a Beam With a Damping Tip Body. Mathematical and Computer Modelling, 35 (2002), 1033-1042.
-
2A. Borichev & Y. Tomilov. Optimal Polynomial Decay of Functions and Operator Semigroups. Math. Ann., 347 (2009), 455-478.
-
3C.M. Dafermos. Asymptotic Stability in Viscoleasticity. Arch. Rat. Mech. Anal., 37 (1970), 297-308.
-
4C.M. Dafermos. On Abstract Volterra Equation with Applications to Linear Viscoleasticity. Differential and Integral Equations, 7 (1970), 554-569.
-
5K.J. Engel & R. Nagel. “One-parameter Semigroups for Linear Evolution Equations”, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, New York (2000).
-
6M. Fabrizio & A. Morro. “Asymptotic Stability in Viscoleasticity”, volume 12 of SIAM Studies in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1992).
-
7E. Feireisl & G. O’Dowd. Stabilisation d’un syste`me hybride par un feedback non linéaire, non monotone. Comptes Rendus de l’Académie des Sciences, 326 (1998), 323-327.
-
8T. Kato. “Perturbation Theory for Linear Operators”. Springer-Verlag, New York (1980).
-
9J. Prüss. On the Spectrum of C0-semigroups. Trans. AMS, 284 (1984), 847-857.
-
10M. Reed & B. Simon. “Methods of Modern Mathematical Physics”, volume 04. Academic Press Inc., California (1978).
-
11N. Zietsman, L Van Rensburg & A. Van der Merwe. A Timoshenko Beam With Tip Body and Boundary Damping. Wave Motion, 39 (2004), 199-211.
Publication Dates
-
Publication in this collection
29 May 2023 -
Date of issue
Apr-Jun 2023
History
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Received
17 Dec 2021 -
Accepted
11 Oct 2022