Stability of Elastic Multi-Link Structures (SpringerBriefs in Mathematics) 3030863506, 9783030863500

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SPRINGER BRIEFS IN MATHEMATICS

Kaïs Ammari Farhat Shel

Stability of Elastic Multi-Link Structures 1 23

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Kaïs Ammari • Farhat Shel

Stability of Elastic Multi-Link Structures

Kaïs Ammari Department of Mathematics University of Monastir Monastir, Tunisia

Farhat Shel Department of Mathematics University of Monastir Monastir, Tunisia

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-030-86350-0 ISBN 978-3-030-86351-7 (eBook) https://doi.org/10.1007/978-3-030-86351-7 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In this book, we investigate the asymptotic behavior of some PDEs on networks. The structures we consider consist of finitely interconnected flexible elements such as strings and beams or combinations thereof, distributed along a planar network. Such a study is motivated by the need for engineers to eliminate vibrations in some dynamical structures consisting of elastic bodies, coupled in the form of chain or graph such as pipelines, bridges, and some cable networks. There are other complicated examples in the automotive industry, aircraft and space vehicles, containing rather than strings and beams, plates and shells. These multi-body structures are often complicated; moreover, the mathematical models describing their evolution are quite complex. For the sake of simplicity, we consider only 1 − d networks. We use semigroup approach to investigate well-posedness and stability. We write the dynamical system as an abstract linear evolution equation on an appropriate Hilbert space H : du (t) = Au(t), u(0) = u0 , dt

(*)

where A is an unbounded operator on H (state space); the Hille–Yosida theorem will be a principal tool for proving existence, uniqueness, and regularity of the solution of (*). Moreover, in some way, the exponential (or polynomial) stability of the (physical) system is viewed as the exponential (or polynomial) stability of the corresponding semigroup, then our main tool for studying exponential decay is the frequency domain characterization (Theorem 1.24 or precisely Theorem 1.25 in this book) due to Gearhart [28], Pruss [80], and [42]. For proving polynomial stability, we use an analogous theorem (Theorem 1.26 in this book), due to Borichev and Tomilov [14]. In order that this book be self-contained, we start with an introductory chapter (Chap. 1), in which we first present some elementary notions on graph theory, in relation to the study of partial differential equations on networks. Second, we recall

v

vi

Preface

some basic definitions and theorems about semigroup theory, and we give a brief reminder about Sobolev spaces and interpolation spaces. The rest will be divided into two parts: the first, Part I, includes the first two chapters and deals with networks of elastic materials, some of which are thermoelastic. In Chap. 2, we consider a network of strings under two cases: in the first, two elastic edges cannot be adjacent, while in the second, the network is a tree of elastic strings, the leaves of which thermoelastic edges are added. We prove the exponential stability of the C0 -semigroup associated with the initial boundary value problem. In Chap. 3, we consider a network of beams, and we assume that the graph has at least an external node, that every maximal subgraph of elastic edges is a tree, the leaves of which thermoelastic edges are attached, and that every maximal subgraph of thermoelastic edges is not a circuit. It is proved that the corresponding semigroup associated to some initial and boundary conditions is exponentially stable. The second part of this book includes Chaps. 4–6. In Chap. 4, we consider a tree of elastic strings and beams, where only external controls are applied at leaves (boundary feedback), we prove that the whole system is exponentially stable if there is no beam following a string (from the root to the leaves) and polynomially stable if at least a beam follows a string. Chapter 5 is devoted to studying the stability of a model of fluid propagation in a 1 − d tree-shaped network, under some feedbacks applied at exterior nodes, with the presence of point mass at inner nodes. It is proved that under some (algebraic) conditions on lengths of edges not attached to leaves, the corresponding semigroup is exponentially stable. In the last section, we discuss the case where the graph contains a circuit and the case where at least a control is omitted. In Chap. 6, we consider a network of elastic strings with local Kelvin–Voigt damping on some edges, and we prove, under some condition on the regularity of the damping coefficient function, some results of polynomial and exponential stability of the associated semigroup. Monastir, Tunisia

Kaïs Ammari

Monastir, Tunisia November 2020

Farhat Shel

Contents

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Terminology of Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Spectrum and Resolvents of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Hille–Yosida Generation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Generation of Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dissipative Operators and Contraction Semigroups. . . . . . . . . . . . . . . 6 Abstract Cauchy Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Strong Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Polynomial Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Sobolev Spaces in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Definition and First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Compact Embeddings, H01 () Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Some Useful Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4 6 6 7 8 8 10 10 12 12 12 13 14 14

2

Exponential Stability of a Network of Elastic and Thermoelastic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Functional Spaces, Existence, and Uniqueness of Solutions . . . . . . . . . . . . 2 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 First Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Second Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Comment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Comment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 18 23 23 29 30 30 30

vii

viii

Contents

3

Exponential Stability of a Network of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Functional Spaces, Existence, and Uniqueness of Solutions . . . . . . . . . . . . 2 Exponential Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 38 46 54

4

Stability of a Tree-Shaped Network of Strings and Beams . . . . . . . . . . . . . . 1 Abstract Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Polynomial Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Lack of Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Comment 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Comment 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 60 65 65 71 77 83 86 86 88

5

Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction on a Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Two Examples of Non-exponential Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A Circuit (Fig. 5.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 A Star with Two Fixed Endpoints (Fig. 5.3) . . . . . . . . . . . . . . . . . . . . . . . 4 A Chain with Non-equal Mass Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89 93 95 100 100 106 108

6

Stability of a Graph of Strings with Local Kelvin–Voigt Damping . . . . . 115 1 Well-Posedness of the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Chapter 1

Preliminaries

1 Introduction In this chapter, we summarize some definitions and results used in this book or serve to better understand the content in such a way that this book will be self-contained. We first introduce some definitions and notations on graphs or networks, as introduced in [1, 34, 68], or [8] (see also [18, 98]). Second, we concentrate on some results about spectrum, resolvent set, and resolvents of (unbounded) linear operators on Hilbert spaces. We give the basics about strongly continuous semigroups of operators on Hilbert spaces. Then, we introduce the Hille–Yosida theorem giving a complete characterization of operators that generate strongly continuous semigroup and the well-known Lumer–Phillips theorem (for contraction semigroups). Next we give a positive answer about the well-posedness of an abstract Cauchy problem ⎧ ⎨ du (t) = Au(t) for t > 0, dt ⎩ u(0) = x,

(1.1)

where A : D(A) ⊂ X → X is a linear operator generating a strongly continuous semigroup (T (t))t≥0 on a Hilbert space X, and x ∈ X the initial value. For all this, we can cite [23, 93], and the references therein, in particular [15, 19, 20, 30, 41, 79, 103]. Afterward, we focus on the asymptotic stability of the semigroup (T (t))t≥0 . We present different concepts of stability (strong, exponential, and polynomial), and we give a characterization of each one in terms of its generator and its resolvent. For more details, see [13, 14, 23, 28, 42, 101]. We end this chapter with a brief reminder on Sobolev spaces in one dimension, in particular, Sobolev embedding theorems and Gagliardo–Nirenberg and Poincaré inequalities. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Ammari, F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-86351-7_1

1

2

1 Preliminaries

Throughout this chapter, X is a real or complex Hilbert space with the inner product ., . and the induced norm .. We also use the notation . to denote the (induced) norm in L(X) := L(X, X), the space of bounded linear operators in X.

2 Terminology of Networks We first introduce some notations needed to formulate the problems under consideration (see, for example, [8, 18, 34] for notations used in this book). Let G be a planar metric connected graph embedded in Rm , m ∈ N∗ := N \ {0}, with N edges e1 , . . . , eN , N ≥ 1 and p vertices a1 , . . . , ap , p ≥ 2. When the graph is a tree, it will be denoted by T instead of G, and note that in this case p = N + 1. By degree (or valency) of a vertex of G we mean the number of edges incident at the vertex. If the degree is equal to one, the vertex is called exterior; otherwise, it is said to be interior. Denote the set of boundary vertices by Vext and the set of vertices with degree more than one by Vint . Moreover, we denote by Iint and Iext , respectively, the sets of indices of interior and exterior vertices, and then I := {1, . . . , p} = Iint ∪ Iext is the set of indices of all vertices. We set J := {1, . . . , N }, and for k ∈ I, we will denote by Jk the set of indices of edges adjacent to the vertex ak . If k ∈ Iext , then the unique element of Jk will be denoted by jk . Finally, the notation s(i, k) = s(k, i) denotes the index of the edge connecting ai and ak . The length of the edge ej is denoted by j . Then, ej may be parametrized by its arc length by means of the functions πj : [0, j ] −→ ej , x −→ πj (x). Sometimes, we identify ej with the interval [0, j ]. For a function u : G −→ C, we set uj = u ◦ πj its restriction to the edge ej . For simplicity, we will write u = (u1 , . . . , uN ), and we will use the abbreviation uj (x) = u(πj (x)) for any x ∈ (0, j ). For a node a in the graph G, which is an end of an edge ej , we will often write uj (a) instead of uj (πj−1 (a)) and uj,x (a) instead of uj,x (πj−1 (a)). Moreover, uj (a) will be sometimes written simply as u(a) if u is continuous at a, etc. The orientation of the graph G is given by the incidence matrix D = (dkj )p×N , defined by,

dkj

⎧ ⎨ 1 if πj (j ) = ak , = −1 if πj (0) = ak , ⎩ 0 otherwise.

Define the adjacency matrix E = (eik )p×p of G by  eik =

1 if ai and ak are adjacent, 0 otherwise.

3 Spectrum and Resolvents of an Operator

3

The Hadamard product of two matrices A = (aj k ) and B = (bj k ) of the same size is defined as A ∗ B = (aj k bj k ), and for any function Q : R −→ R, we define the matrix Q(A) = (qik )p×p by  qik =

Q(aik ) if eik = 1, 0 otherwise.

In particular, we write A(r) = Q(A) if Q(x) = x r . Furthermore, denote by L the matrix L = (ik )p×p , where  ik =

s(i,k) if eik = 1, 0 otherwise,

where s(i, k) = s(k, i) is the index of the edge connecting ai and ak .

3 Spectrum and Resolvents of an Operator Definition 1.1 A linear operator A : D(A) ⊂ X → X is called closed if its graph, defined by G(A) = {(x, Ax) | x ∈ D(A)}, is closed in X × X. Clearly, A is closed if and only if for any sequence xn in D(A) such that xn → x in X and Axn → y in X, we have x ∈ D(A) and y = Ax. Note that if A : D(A) ⊂ X → X is closed and B ∈ L(X), then also A + B is closed (the domain of A + B is D(A)). Definition 1.2 Let A : D(A) ⊂ X → X be a linear operator. • The resolvent set of A, denoted by ρ(A), is the set of those points λ ∈ C, for which the operator λI − A : D(A) → X is invertible and (λI − A)−1 ∈ L(X). The spectrum of A, denoted by σ (A), is the complement of ρ(A) in C. For λ ∈ ρ(A), R(λ, A) := (λI − A)−1 is called a resolvent of A. • λ ∈ C is called an eigenvalue of A if the operator λI − A : D(A) → X is not injective. In this case, a zλ ∈ D(A), zλ = 0, satisfying Azλ = λzλ is called an eigenvector of A corresponding to λ. The set of all the eigenvalues of A is called the point spectrum of A, and it is denoted by σp (A). Note that if ρ(A) is not empty, then A is closed. Indeed, let λ ∈ ρ(A), and let xn be a sequence in D(A) such that xn → x in X and (A − λI )xn → y in X. Since (A − λI )−1 is bounded, we have that xn = (A − λI )−1 (A − λI )xn converges to (A − λI )−1 y, which implies that (A − λI )−1 y = x, and hence x ∈ D(A) and (A − λI )x = y.

4

1 Preliminaries

Note also that it can be proved that ρ(A) is open and then σ (A) is closed. In the following proposition, we show that for every λ ∈ ρ(A) there is a canonical relation, called the spectral mapping theorem, between the unbounded operator A and the spectrum of the bounded operator R(λ, A) = (λI − A)−1 . Proposition 1.3 Let A : D(A) ⊂ X → X be a linear operator with non-empty resolvent set ρ(A) (in particular, A is closed). Then,   1 (i) σ ((λI − A)−1 ) \ {0} = (λ − σ (A))−1 := λ−μ , μ ∈ σ (A) . (ii) Analogous statements hold for the point spectrum of A and (λI − A)−1 . An important case is when the resolvent (λI − A)−1 is compact, for some λ ∈ ρ(A) (we say that A has compact resolvent), we then have the following theorem [45]: Theorem 1.4 Let A : D(A) ⊂ X → X be an operator such that (λI − A)−1 is compact, for some λ ∈ ρ(A). Then, σ (A) is discrete and formed only of eigenvalues of finite multiplicity; in particular, we have σ (A) = σp (A). Moreover, (sI − A)−1 is compact for every s ∈ ρ(A). For concrete operators, the following characterization is quite useful. Proposition 1.5 Let A : D(A) ⊂ X → X be an operator with ρ(A) = ∅ and take X1 := (D(A), .A ). Then, the following assertions are equivalent: (a) The operator A has compact resolvent. (b) The canonical injection i : X1 → X is compact.

4 Semigroups Definition 1.6 A family (T (t))t≥0 of bounded linear operators in the Hilbert space X is called strongly continuous semigroup or C0 -semigroup (or just semigroup) if the following conditions are fulfilled: (a) T (0) = I , and we have T (t + s) = T (t)T (s) for all t, s ≥ 0. (b) For each x ∈ X, the orbit, defined as the map T (·)x : R+ → X, t → T (t)x, is continuous. Property (a) is called the semigroup law and (b) is the strong continuity, and it can be replaced by lim T (t)x = x.

t→0+

4 Semigroups

5

We said that (T (t))t0 is uniformly continuous if the mapping t → T (t) is continuous for the operator norm. It is proved that (T (t))t≥0 is uniformly continuous if and only if there exists a bounded operator A on X such that ∀ t ≥ 0, T (t) = etA =

∞  (tA)k k=0

k!

.

The operator A is given by 1 (T (t) − I ) . t

A = lim

t→0+

For a semigroup (T (t))t≥0 , we define an operator A by  1 D(A) = x ∈ X | the limit lim (T (t)x − x) exists in X , t→0+ t and for x ∈ D(A), 1 (T (t)x − x) . t

Ax = lim

t→0+

Then, A is called the infinitesimal generator (or shortly, the generator) of the semigroup (T (t))t≥0 . We also say that A generates (T (t))t≥0 , and sometimes we denote T (t) by eAt . Some elementary properties are collected in the following proposition: Proposition 1.7 Let A be the generator of a C0 -semigroup (T (t))t≥0 on X. The following properties hold: (i) The map A : D(A) ⊂ X → X is effectively a linear operator. (ii) If x ∈ D(A), then T (t)x ∈ D(A) and d T (t)x = T (t)Ax = AT (t)x, for all t ≥ 0. dt (iii) For every t ≥ 0, one has

t

T (t)x − x = A

=

T (t)xds if x ∈ X,

0 t

T (t)Axds if x ∈ D(A).

0

Theorem 1.8 The generator of a C0 -semigroup is closed and densely defined linear operator that determines the semigroup uniquely.

6

1 Preliminaries

We observe that every C0 -semigroup is exponentially bounded. Lemma 1.9 Let (T (t))t≥0 be a C0 -semigroup on X. There exist M ≥ 1 and ω ∈ R such that ∀ t ≥ 0, T (t) ≤ Meωt . Moreover, a semigroup (T (t))t0 is called bounded if we can take ω = 0 and contractive (or semigroup of contractions) if ω = 0 and M = 1 is possible, that is, T (t) ≤ 1 for all t ≥ 0. Definition 1.10 Let (T (t))t≥0 , a C0 -semigroup with generator A. Then,  ω0 (T ) := ω0 (A) := inf ω ∈ R | ∃Mω ≥ 1 : T (t) ≤ Mω eωt for all t ≥ 0  = inf ω ∈ R | t → e−ωt T (t) is bounded on R+ is called the growth bound of (T (t))t≥0 . Note that ω0 (T ) ∈ [−∞, +∞). Proposition 1.11 Let (T (t))t≥0 be a C0 -semigroup with generator A. For every w > ω0 (T ), there exists Mω ∈ [1, +∞) such that T (t) ≤ Mω eωt for all t ≥ 0.

5 Hille–Yosida Generation Theorems 5.1 Generation of Semigroups We want to characterize those linear operators that are the generators of some C0 semigroup. First, for every ω ∈ R, we define the right half-plane Cω := {λ ∈ C with Reλ > ω}, in particular, C0 := {λ ∈ C with Reλ > 0}. Theorem 1.12 (Hille–Yosida Theorem, Contraction Case, Hille, Yosida 1948 [40, 102]) Let A be a linear operator on X. The following assertions are equivalent: (i) The operator A is the generator of a C0 -semigroup of contractions. (ii) A is closed, D(A) is dense in X, (0, ∞) ⊂ ρ(A), and (λI − A)−1  ≤

1 for λ > 0. λ

5 Hille–Yosida Generation Theorems

7

(iii) A is closed, D(A) is dense in X, C0 ⊂ ρ(A), and (λI − A)−1  ≤

1 for λ ∈ C0 . Reλ

The above result can be generalized to any C0 -semigroup as follows: Theorem 1.13 (Hille–Yosida Theorem) Let A be a linear operator on X. The following assertions are equivalent: (i) A is the generator of a C0 -semigroup (T (t))t≥0 with T (t) ≤ Meωt , for t ≥ 0, for some ω ∈ R and some M ≥ 1. (ii) A is closed, D(A) is dense in X, (ω, ∞) ⊂ ρ(A), and (λI − A)−1  ≤

M for λ > ω. λ−ω

(iii) A is closed, D(A) is dense in X, Cω ⊂ ρ(A), and (λI − A)−1  ≤

M for λ ∈ Cω . Reλ − ω

5.2 Dissipative Operators and Contraction Semigroups Definition 1.14 An operator A : D(A) ⊂ X → X is said to be dissipative if for any x ∈ D(A), Re(Ax, x) ≤ 0. An operator A is said to be m-dissipative (or maximal dissipative) if it is dissipative and there is a λ0 > 0 such that Ran(λ0 I − A) = X. For more details about the following theorem, see [79]. Theorem 1.15 (Lumer-Phillips) For any linear operator A : D(A) ⊂ X → X, the following statements are equivalent: (i) A is the generator of a C0 -semigroup of contractions. (ii) A is m-dissipative. As a corollary of the above theorem (using that ρ(A), the resolvent set of A, is open in C), the following result will be frequently used in this book. Theorem 1.16 Let A : D(A) ⊂ X → X be a linear operator on X. If A is dissipative and 0 ∈ ρ(A), then A is the generator of a C0 -semigroup of contractions.

8

1 Preliminaries

6 Abstract Cauchy Problems We consider the abstract Cauchy problem ⎧ ⎨ du(t) = Au(t) for t > 0, dt ⎩ u(0) = x,

(1.2)

where the independent variable t represents time, u(.) is a function with values in the Hilbert space X, A : D(A) ⊂ X → X a linear operator, and x ∈ X the initial value. A continuous function u : R+ → X is called a (strong) solution of (1.2) if u is continuously differentiable with respect to X, u(t) ∈ D(A) for all t ≥ 0, and (1.2) holds. a (mild) solution of (1.2) if

t A continuous function u : R+ → X is called t u(s)ds ∈ D(A) for all t ≥ 0 and u(t) = A u(s)ds + x. 0 0 The following theorem ensures the existence and uniqueness of a solution of the initial value problem (1.2). Theorem 1.17 If the operator A generates a C0 -semigroup, then for any initial datum x ∈ X, there exists a unique (mild) solution u of the abstract Cauchy problem (1.2). u satisfies u ∈ C([0, ∞), X). Moreover, for x ∈ D(A), there exists a unique (strong) solution u of the abstract Cauchy problem (1.2). u satisfies u ∈ C([0, ∞), D(A)) ∩ C 1 ([0, ∞), X). In the two cases, u(t) = T (t)x, where (T (t))t≥0 is the C0 -semigroup generated by A.

7 Stability We assume that the operator A in (1.2) generates a C0 -semigroup (T (t))t0 . Then, in general, we define the energy E of the solution u of the abstract Cauchy problem (1.2) as E(t) =

1 1 u(t)2 = T (t)x2 , 2 2

and in fact, it is the case in all the dynamic systems considered in this book.

7 Stability

9

In studying a dynamic system, one important thing is to study the asymptotic behavior of its semigroup T (t) or, equivalently, its energy (for large t > 0). In this book, we focus on the stability, by this we mean that the energy (or the semigroup) converges to zero as t → ∞. We distinguish different concepts of convergences. Definition 1.18 We assume that A generates a C0 -semigroup (T (t))t≥0 on X. (a) We say that (T (t))t≥0 is strongly (or asymptotically) stable if lim T (t)x = 0 for all x ∈ X.

t→∞

(1.3)

(b) We say that (T (t))t≥0 is exponentially (or uniformly) stable if there exist constants α > 0 and M ≥ 1 such that T (t) ≤ Me−αt for all t ≥ 0.

(1.4)

Remark 1.19 Property (b) is equivalent to (b ) lim T (t) = 0

t→∞

. Indeed, the implication (b) ⇒ (b ) is trivial. Conversely, suppose that (b ) holds, there exists t0 > 0 such that q := T (t0 ) < 1. We set M = sup0≤s≤t0 T (s), which exists, since t → T (t) is continuous. If we decompose t = kt0 + s ∈ R+ with k ∈ N and s ∈ [0, t0 ), we obtain T (t) ≤ T (s) · T (kt0 ) ≤ MT (t0 )k  ≤ Mq k = Mek ln(q) ≤

M −α e t, q

where α := − ln(q) t0 . Thus, (b) holds. Note also that exponential stability implies strong stability. In (a), we will also say that the system (1.2) is strongly stable, and it also means that lim E(t) = 0 for all x ∈ X,

t→∞

where E(t) := 12 T (t)x2 . In (b), condition (1.4) can be rewritten as E(t) ≤ CE(0)e−δt for all t > 0,

10

1 Preliminaries

where δ and C are positive constants (not depending on the initial condition). We will also say that the system (1.2) (or the solution or the energy of the system (1.2)) is exponentially stable. In the literature, there is another concept of stability “between” the previous two, called polynomial stability, which is often written for the energy of the following form: E(t) ≤

C x2D(A) , tα

for all x ∈ D(A) and for all t > 0,

where C and α are positive constants and where xD(A) = x+Ax. Recall here that E(t) is the energy of the solution u satisfying the initial condition u(0) = x. This translates in a more precise way as follows: Definition 1.20 We assume that A generates a C0 -semigroup (T (t))t≥0 on X. We say that (T (t))t≥0 is polynomially stable if there exist two positive constants α and M such that, for every x ∈ D(A), T (t)x ≤

M xD(A) for all t > 0. tα

(1.5)

For more details on polynomial stability, one can see, for example, [14, 55], and the references therein. In this book, we choose to characterize the stability of a semigroup in terms of its generator and its resolvent.

7.1 Strong Stability First, we give some sufficient conditions of strong stability of C0 -semigroups. Theorem 1.21 (Arendt and Batty [13]) Let (T (t))t≥0 be a C0 -semigroup of contractions, with generator A on the Hilbert space X. Denote by σ (A) the spectrum of A. If σ (A) ∩ iR is at most countable and no eigenvalue of A lies on the imaginary axis, then (T (t))t≥0 is strongly stable. Corollary 1.22 Let (T (t))t≥0 be a C0 -semigroup of contractions, with generator A on the Hilbert space X. If iR ⊂ ρ(A), then (T (t))t≥0 is strongly stable.

7.2 Exponential Stability First, it is clear that a C0 -semigroup, generated by an operator A, is exponentially stable if and only if ω0 (A) := ω0 (T ) < 0.

7 Stability

11

Next, as mentioned below, we want to characterize the exponential stability of the semigroup in terms of its generator. For this purpose, define the spectral bound of an operator A as follows: s(A) := sup{Reλ, λ ∈ σ (A)}. We know that in the case of finite dimension, the C0 -semigroup etA is exponentially stable if and only if s(A) < 0.

(1.6)

This condition, although necessary, is not sufficient in general to have exponential stability in infinite dimension, nevertheless it is necessary and sufficient for a class of semigroups (see [23]). Proposition 1.23 For a C0 -semigroup with generator A, one has −∞ ≤ s(A) ≤ ω0 (A) < +∞. In particular, if s(A) = 0, then the C0 -semigroup with generator A is not exponentially stable. Now we introduce the main result used for proving exponential stability of C0 semigroups in this book [28, 42, 80]. Theorem 1.24 (Gearhart [28], Prüss [80], and Huang [42]) Let (T (t))t≥0 be a C0 -semigroup on a Hilbert space X, with generator A. Then, (T (t))t≥0 is exponentially stable if and only if {λ ∈ C, Reλ > 0} ⊂ ρ(A),

(1.7)

sup (λI − A)−1 L(X) < ∞.

(1.8)

and Reλ>0

Note that condition (1.7) can be replaced by s(A) := sup{Reλ, λ ∈ σ (A)} < 0. Moreover, the inequality Reλ > 0 in (1.8) can be large. The following invariant of the result is due to Gearhart [28]. Theorem 1.25 Let (T (t))t≥0 be a C0 -semigroup of contractions on a Hilbert space X, with generator A. Then, (T (t))t≥0 is exponentially stable if and only if iR ⊂ ρ(A) and lim sup β∈R,|β|→+∞

(1.9)

    (iβI − A)−1 

L(X)

< ∞.

Note that condition (1.10) can be replaced by sup (iwI − A)−1 L(X) < ∞. w∈R

(1.10)

12

1 Preliminaries

7.3 Polynomial Stability The following characterization of polynomial stability of a C0 -semigroup of contractions is due to Borichev and Tomilov [14]. Theorem 1.26 A C0 -semigroup of contraction (etA )t≥0 on the Hilbert space X such that iR ⊂ ρ(A) satisfies, for every x ∈ D(A), etA x ≤

M xD(A) for all t > 0, tα

(1.11)

for some constant M > 0 and for α > 0 if and only if  1    − A)−1  < ∞. 1 (iβI L(X) β∈R,|β|→+∞ |β| α lim sup

(1.12)

8 Sobolev Spaces in One Dimension In this section, we collect some basic results on function spaces, which will be used frequently in this book. For more details, we refer the reader to [2, 15, 52], and the first Chapter in [57]. See also [25, 31, 74] and in particular the Appendix in [93] and the references therein.

8.1 Definition and First Properties First, we briefly recall the definition and main results of the Sobolev spaces H m (), where  =]a, b[ such that a, b ∈ R and a < b. We denote by C0∞ () or D() the set of all ϕ in C ∞ (), which have compact support contained in . We define the space   H 1 () := u ∈ L2 () | ∂x u ∈ L2 () . In other words, a function u ∈ L2 () is in H 1 () if and only if there exists v ∈ L2 () such that



u∂x ϕdx ¯ =− v ϕdx ¯ 



for every ϕ in C0∞ (). The function v is denoted by ∂x u.

8 Sobolev Spaces in One Dimension

13

With the inner product defined by



uvdx ¯ +

u, v =

∂x u∂x vdx, ¯





H 1 () is a Hilbert space. In the same way, we define the Sobolev spaces H m (), where m ∈ N by   H m () := u ∈ L2 () | ∂xα u ∈ L2 (), α ∈ N, α ≤ m . From the above definition, it clearly follows that H 0 () = L2 (). Equipped with the inner product: u, v =

m



∂xα u∂xα vdx, ¯

α=0 

H m () is a Hilbert space. ¯ is dense in H m (). Theorem 1.27 (Density Theorem) For m ≥ 1, C m () Theorem 1.28 For m >

1 2

¯ + k, H m () ⊂ C k ().

Remark 1.29 Any function u ∈ H 1 () admits a unique representative continuous ¯ still denoted by u, and we have, for every x, y ∈ , ¯ on ,

u(y) = u(x) +

y

∂x u(t)dt. x

Proposition 1.30 (Integration by Parts) If u, v ∈ H 1 (), then uv ∈ H 1 () and ∂x (uv) = ∂x u · v + u∂x v. In particular, for all [x, y] ⊂ [a, b],

[x,y]

(∂x u · v + u∂x v) = (uv)(y) − (uv)(x).

8.2 Compact Embeddings, H01 () Space Theorem 1.31 For 0 ≤ m1 < m2 , H m2 () ⊂ H m1 () with compact embedding. We define the space H01 () as the closure of C0∞ () in H 1 (). We have   H01 () := u ∈ H 1 () | u(a) = u(b) = 0 . Theorem 1.32 The embedding operator H01 () → L2 () is compact.

14

1 Preliminaries

8.3 Some Useful Inequalities The following inequalities are frequently used in this book. The firsts two inequalities can be found in a more general case in [57] (see also [26, 76]). Theorem 1.33 (Some Gagliardo–Nerenberg Inequalities) (1) There are two positive constants C1 and C2 such that for any w in H 1 (), w∞ ≤ C1 ∂x w1/2 w1/2 + C2 w .

(1.13)

(2) There are two positive constants C3 and C4 such that for any w in H 2 (),  1/2   ∂x w ≤ C3 ∂x2 w  w1/2 + C4 w .

(1.14)

The following inequality is known as the Poincaré inequality: Theorem 1.34 There exists a positive constant C, depending only on  such that u ≤ C∂x u,

∀ u ∈ H01 ()

(the constant C = |b − a| is suitable).

9 Comments Stability of PDEs (in particular, stability of beam and string systems) can be found in many books such as Z.-H. Liu et al. [58], J. Oostveen [78], M. Krstic et al. [47], T. Meurer [70], A. Zuyev [108], M. Gugat [32], G. Sklyar et al. [91], and [18]. For interested people, there are some recent developments in the field of differential equations on networks, such as [64, 71, 105], on some fractional boundary value problems on graphs.

Chapter 2

Exponential Stability of a Network of Elastic and Thermoelastic Materials

The wave equation on an elastic body is conservative; to make the system stable, several authors have introduced different types of dissipative mechanisms, for example, a frictional damping [38] or frictional boundary conditions [46, 77]. For the stabilization of a network governed by wave equations, we refer to [4, 10], where the authors considered a star-shaped and tree-shaped networks of elastic strings, and they proved that when a feedback is applied on particular nodes, the system will be polynomially stable but not exponentially stable. We can see, in [75], that the authors considered a network with delay term in the nodal feedbacks. In [12], the authors studied, in particular, the stabilization of a chain of beams and strings; see also [33–35, 94]. Another type of stabilization of an elastic material is to add thermoelastic materials to it. In [61, 62, 81], the authors proved that the system is then exponentially stable; see also [84] where the authors considered the case of beams and proved that the whole system is also exponentially stable. We want to know if this result holds true for a network of elastic and thermoelastic materials. To our knowledge, the asymptotic behavior of such a system has not been studied yet. In this chapter, we consider particular cases of such network that can be partially generalized. In the first case, we suppose that two elastic edges cannot be adjacent (Fig. 2.1). In the second one, we consider a tree of elastic materials, the leaves of which thermoelastic materials are added as follows: the thermoelastic body is related to only one leaf by an end, and the second is free or connects two leaves, with the condition that each leaf is connected to only one thermoelastic body (Fig. 2.2). With the continuity condition for the displacement and the Neumann condition for the temperature at the internal nodes, we prove that the thermal effect is strong enough to stabilize the system. We will use a frequency method as described in the introduction. We consider a network of elastic and thermoelastic bodies, with N edges e1 , . . . , eN and p vertices a1 , . . . , ap , and that coincides with the graph G. We suppose that G contains at least one thermoelastic edge, and we assume that Vext contains at least one element. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Ammari, F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-86351-7_2

15

16

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

Fig. 2.1 Elastic–thermoelastic network

Fig. 2.2 Tree of strings

Let uj = uj (x, t) be the function describing the displacement at time t of the body ej , and in the case where ej is thermoelastic, we further define θj = θj (x, t) be the temperature difference to a fixed reference temperature of ej at time t. Assume the following: • Every thermoelastic edge ej satisfies the following equations: uj,tt − uj,xx + γj θj,x = 0

in (0, j ) × (0, +∞),

θj,t + γj uj,tx − κj θj,xx = 0

in (0, j ) × (0, +∞),

where γj and κj are positive constants, with initial conditions uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x), θj (x, 0) = θj0 (x).

(2.1)

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

17

• Every elastic edge ej satisfies the following equation: uj,tt − uj,xx = 0

in (0, j ) × (0, +∞),

with initial conditions uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x). Denote by Jkte the set of indices of thermoelastic edges incident to ak and by Jke the set of indices of elastic edges incident to ak . Thus, the boundary conditions on the graph are described as follows: • The system satisfies the Dirichlet condition for the displacement and temperature at exterior nodes, uj (ak , t) = 0,

j ∈ Jk , ak ∈ Vext ,

θj (ak , t) = 0,

j ∈ Jkte , ak ∈ Vext .

• The displacement is continuous at every inner node, that is, uj (ak , t) = ul (ak , t)

j, l ∈ Jk , ak ∈ Vint .

• The temperature satisfies the Neumann condition at inner nodes, θj,x (ak , t) = 0,

j ∈ Jkte , ak ∈ Vint .

(2.2)

• The system satisfies the balance condition at every inner node, 

   dkj uj,x (ak , t) − γj θj (ak , t) + dkj uj,x (ak , t) = 0, ak ∈ Vint . j ∈Jke

j ∈Jkte

(2.3) Note that we can replace the Neumann condition (2.2) for the temperature at all ak ∈ Vint , with Jkte containing at least two elements, by the continuity condition [48], θj (ak , t) = θl (ak , t) = 0, with Kirchhoff’s law, 

κj dkj θj,x (ak , t).

j ∈Jkte

Then, we get the same results with almost the same proofs.

18

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

This chapter is organized as follows: in Sect. 1 we shall formulate our problem in a suitable Hilbert space and deduce the existence and uniqueness of solutions. In Sect. 3 we will prove the exponential stability of the whole system in the case where two elastic edges are not adjacent and in the case of a tree of elastic bodies to which we add in leaves thermoelastic materials.

1 Functional Spaces, Existence, and Uniqueness of Solutions Denote by J e the set of indices of the elastic edges and by J te the set of indices of the thermoelastic ones. For j in J e , let γj = 0, κj = 0, Vj = {0}, Vjk = {0}, k = 1, 2, and for j in J te , let Vj = L2 (0, j ) and Vjk = H k (0, j ), k = 1, 2. Thus, the system (2.1)–(2.3) will be equal to the following: uj,tt − uj,xx + γj θj,x = 0, in (0, j ) × (0, ∞),

(2.4)

θj,t + γj uj,tx − κj θj,xx = 0, in (0, j ) × (0, ∞),

(2.5)

where j = 1, . . . , N , with conditions ujk (ak , t) = 0,

∀ak ∈ Vext ,

(2.6)

θjk (ak , t) = 0,

∀ak ∈ Vext ,

(2.7)

uj (ak , t) = ul (ak , t), θj,x (ak , t) = 0, 

∀j, l ∈ Jk , ∀ak ∈ Vint ,

∀j ∈ Jk , ∀ak ∈ Vint ,

dkj (uj,x (ak , t) − γj θj (ak , t)) = 0, ∀ak ∈ Vint ,

(2.8) (2.9) (2.10)

j ∈Jk

and uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x), θj (x, 0) = θj0 (x),

(2.11)

which is similar to the system (10)–(17) of [1], except that, in the last paper, we consider the continuity condition of the temperature at the interior nodes, instead of the Neumann condition. Notice that it follows from (2.5) and condition (2.9) that, for all j ∈ Gint , we

j

j have (γj uj,xt + θj,t )dx = 0, that is, (γj uj,x + θj )dx is conservative all 0 0

j (γj uj,x + θj )dx = 0. the time. Without loss of generality, we assume that 0

1 Functional Spaces, Existence, and Uniqueness of Solutions

Otherwise, we can make the substitution uj = uj −

j c2 (γj (u0j )x + θj0 )dx. θj − 0j 2 , where c0j = 1j 1+γj

19 2 γj c0j

1+γj2

x, vj = uj,t , and θj =

0

Denote L2 (G) =

N 

L2 (0, j ), H k (G) =

j =1

V (G) =

N 

H k (0, j ), k = 1, 2,

j =1

N 

Vj , V k (G) =

j =1

N 

Vjk , k = 1, 2,

j =1

and set   H01 (G) = f = (f1 , . . . , fN ) ∈ H 1 (G) satisfying (2.12) and (2.13) , fj (ak ) = fl (ak ), j, l ∈ Jk , ak ∈ Vint ,

(2.12)

fjk (ak ) = 0,

(2.13)

ak ∈ Vext .

Denote by Gint the set of indices of interior edges (i.e., edges with extremities in Vint ). Then, we define the space H = {(f , g, h) ∈ H01 (G) × L2 (G) × V (G) satisfying (2.14)},

j

(γj ∂x fj + hj )dx = 0 for all j ∈ Gint .

0

Equipped with the usual inner product, y, y ˜ H=

=

N  

     ∂x fj , ∂x f˜j + gj , g˜j + hj , h˜j

j =1 N 



j

∂x fj (x)∂x f˜j (x)dx +

0

j =1



j

+



j

gj (x)g˜j (x)dx

0

 hj (x)h˜j (x)dx ,

0

˜ and the space H is a Hilbert space. where y = (f , g, h) and y˜ = (f˜, g, ˜ h),

(2.14)

20

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

In this Hilbert space, define the operator A : D(A) ⊆ H → H by ⎛

⎞ 0 A1 0 A = ⎝ A2 0 A3 ⎠ , 0 A3 A4

(2.15)

where A1 = diag(I, . . . , I ), A2 = diag(∂xx , . . . , ∂xx ), A3 = diag(−γ1 ∂x , . . . ., −γN ∂x ), A4 = diag(κ1 ∂xx , . . . , κN ∂xx ), and I is the N-unit matrix and with domain    D(A) = (u, v, θ ) ∈ H ∩ H 2 (G) × H01 (G) × V 2 (G) , satisfying (2.16) , where ⎧ ⎪ θ (a ) = 0, ak ∈ Vext , ⎪ ⎨ jk k j ∈ Jk , ak ∈ Vint , ∂x θj (ak ) = 0,   " ⎪ ∂ d u (a ) − γj θ j (ak ) = 0, ak ∈ Vint . ⎪ kj x j k ⎩

(2.16)

j ∈Jk

For y = (u, v, θ ) ∈ D(A), the components of Ay are ⎧ j = 1, . . . , N, ⎨ vj , j = 1, . . . , N, ∂xx uj − γj ∂x θj , ⎩ j = 1, . . . , N. −γj ∂x vj + κj ∂xx θj , Then, the system (2.4)–(2.11) may be rewritten as the first-order evolution equation on H, ⎧ ⎨ d y = Ay, (2.17) dt ⎩ y(0) = y 0 , where y = (u, ut , θ ) and y 0 = (u0 , u1 , θ 0 ). For y = (u, v, θ ) ∈ D(A), a direct calculation gives Re Ay, yH = −

N 

  2  2 κj ∂x θj  = − κj ∂x θj  ≤ 0,

j =1

j ∈J te

which implies that A is a dissipative operator on H. Moreover, we have the following result: Theorem 2.1 Let H and A be defined as before. Then, 1 ∈ ρ(A), the resolvent set of A, and (I − A)−1 is compact.

1 Functional Spaces, Existence, and Uniqueness of Solutions

21

Proof Let z = (f , g, h) ∈ H. We look for an element y = (u, v, θ ) ∈ D(A) such that (I − A)y = z, that is, uj − vj = fj ,

j = 1, . . . , N,

(2.18)

vj − ∂xx uj + γj ∂x θj = gj ,

j = 1, . . . , N,

(2.19)

θj + γj ∂x vj − κj ∂xx θj = hj ,

j = 1, . . . , N.

(2.20)

Using (2.18) in (2.19) and (2.20), we have uj − ∂xx uj + γj ∂x θj = gj + fj ,

(2.21)

θj + γj ∂x uj − κj ∂xx θj = hj + γj ∂x fj ,

(2.22)

for j in {1, . . . , N }. We consider the space H1 = {(w, ϕ) ∈ H01 (G) × V 1 (G) satisfying (2.23) and (2.24)}, ϕjk (ak ) = 0,



j

ak ∈ Vext ,

(2.23)

(γj ∂x wj + ϕj )dx = 0 for all j ∈ Gint ,

(2.24)

0

equipped with the inner product y, y ˜ H1 =

N        ∂x wj , ∂x w˜j + ϕj , ϕ˜j + κj ∂x ϕj , ∂x ϕ˜j , j =1

where y = (w, ϕ) and y˜ = (w, ˜ ϕ). ˜ Then, H1 is a Hilbert space. Let (w, ϕ) in H1 . Multiplying (2.21) by wj and (2.22) by ϕ j for j = 1, . . . , N and summing, we obtain, by taking into account (2.16), a((u, θ ), (w, ϕ)) = F (w, ϕ),

(2.25)

where a((u, θ), (w, ϕ)) =

#

N j  j =1

+

0

uj wj dx +

#

N j  j =1

0

j 0

θj ϕj dx + γj

∂x uj ∂x wj dx − γj

j 0

$

j

∂x uj ϕj dx + κj

θj ∂x wj dx

0

$

j 0

∂x θj ∂x ϕj dx

22

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

#

N j 

=

j =1

+κj

0

j 0

uj wj dx +

j 0

∂x uj ∂x wj dx + #

∂x θj ∂x ϕj dx + γj

j

0

j 0

∂x uj ϕj dx −

θj ϕj dx $$

j 0

θj ∂x wj dx

and F (w, ϕ) =

N 

 j =1

j

(gj + fj )wj dx +

0

j

 (hj + γj fj )wj dx .

0

It is clear that a is a continuous sesquilinear form on H1 × H1 and that F is a continuous linear form on H1 . Moreover, for every (w, ϕ) ∈ H1 , % %  2 % %   %a(w, ϕ), (w, ϕ))% ≥ (w, ϕ) . H1

Then, by the Lax–Milgram lemma (complex version), (2.25) has a unique solution (u, θ ) ∈ H1 . The classical elliptic theory [51, 57] implies that the solution of (2.21)–(2.22), associated with the conditions uj (ak ) = ul (ak ), j, l ∈ Jk , ak ∈ Vint , ∂x θj (ak ) = 0, j ∈ Jk , ak ∈ Vint , ujk (ak ) = 0, θjk (ak ) = 0,



j

(γj ∂x fj + hj )dx = 0

ak ∈ Vext ,

for all j ∈ Gint ,

0

  dkj ∂x uj (ak ) − γj θj (ak ) = 0, ak ∈ Vint ,

j ∈Jk

belongs to the space H 2 (G) × V 2 (G), and y2H ≤ c z2H , where c is a positive constant independent of y, which proves that (u, v, θ ) ∈ D(A) and (I − A)−1 ∈ L(H), that is, 1 ∈ ρ(A). The Sobolev embedding theorem asserts that (I − A)−1 is a compact operator. This finishes the proof of the theorem.  

2 Exponential Stability

23

Since 1 ∈ ρ(A) and A is dissipative, we have the following result due to Theorem 1.15: Corollary 2.2 The operator A generates a C0 -semigroup of contraction (S(t))t≥0 on the Hilbert space H. Therefore, for an initial datum y 0 ∈ H, there exists a unique solution y ∈ C([0, +∞), H) of the Cauchy problem (2.17). Moreover, if y 0 ∈ D(A), then y ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H).

2 Exponential Stability In this section, we study the asymptotic behavior of the system (2.4)–(2.11). Precisely, we will prove that the C0 -semigroup (S(t))t≥0 is exponentially stable. Recall that a C0 -semigroup (T (t))t≥0 is exponentially stable if and only if there exist constants C ≥ 1 and a > 0 such that T (t) ≤ Ce−at , ∀t ≥ 0. We will use the frequency domain condition due to Gearhard (Theorem 1.25).

2.1 First Case In this section, we consider the case where two elastic edges are not adjacent (Fig. 2.1). Lemma 2.3 Let A be the operator given by (2.15), then condition (1.9) holds for (S(t))t≥0 , that is, iR = {iβ | β ∈ R} ⊆ ρ(A). In particular, by Theorem 1.22, the semigroup (S(t))t≥0 is strongly stable. Proof Suppose that (1.9) is not true. Then, there is a real number β ∈ R such that λ := iβ ∈ σ (A), the spectrum of A. By the fact that (I − A)−1 is compact, the spectrum of A consists of all isolated eigenvalues, that is, σ (A) = σp (A), and then there exists y = (u, v, θ ) ∈ D(A), y = 0, such that Ay = iβy. Then, Re(Ay, yH ) = Re(λy, yH ) = 0.

24

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

This leads to −

N 

 2 κj ∂x θj  = 0,

j =1

which implies that ∂x θj = 0 for j = 1, . . . , N. Then, θj is constant for j = 1, . . . , N, and (u, v) satisfies ⎧ ⎨

j = 1, . . . , N, vj = λuj , j = 1, . . . , N, ∂xx uj − γj ∂x θj = λ2 uj , ⎩ j = 1, . . . , N. −λγj ∂x uj = λθj ,

(2.26)

If λ = 0, then vj = 0 and ∂x uj is constant for j = 1, . . . , N . Multiplying the second equation in (2.26) by uj and summing, we obtain N 

 − j =1

j 0

% % %∂x uj %2 dx + γj





j

θj ∂x uj dx

= 0.

0

Then, using (2.14) and the fact that θj and ∂x uj are constant for j = 1, . . . , N, we deduce that ∂x u = 0 and θ = 0. Now, suppose that λ = 0. For j in J te , the third and second equations of (2.26) imply uj = 0, and then ∂x uj = 0, θj = 0, and vj = 0 in L2 (0, j ). Moreover, uj , ∂x uj , and θj vanish on both ends of ej . For j in J e , by taking into account that u ∈ H01 (G), condition (2.16), and the fact that two elastic edges are not adjacent, uj satisfies the Cauchy problem ∂xx uj = λ2 uj , uj (ak ) = 0, and ∂x uj (ak ) = 0, where ak is an end of ej , then uj = 0; hence, by the first equation of (2.26), vj = 0 in L2 (0, j ), which contradicts the fact that y = (u, v, θ ) is an eigenvector of A. We conclude that iR ⊂ ρ(A).   Lemma 2.4 Let A be the operator given by (2.15), then condition (1.10) holds. Proof Suppose that (1.10) is not true, then there exists a sequence (βn ) of real numbers, with βn −→ ∞ (without loss of generality, we suppose that βn > 0), and a sequence of vectors (yn ) = (un , v n , θ n ) in D(A) with yn H = 1, such that (iβn − A)yn  −→ 0.

(2.27)

Our goal is to prove that this condition yields the contradiction yn H −→ 0 as n −→ 0. The proof is divided into two steps. In the first, we prove, as in the proof of Theorem 2 in [1], that ∂x uj,n , θj,n , and vj,n converge to 0 in L2 (0, j ) for j in J te . In the second, we prove the same properties for j in J e .

2 Exponential Stability

25

First Step Because Re((iβn − A)yn , yn H ) =

"

j ∈J te

 2 κj ∂x θj,n  , we obtain

  ∂x θj,n  −→ 0.

(2.28)

Now, writing condition (2.27) term by term, we obtain, for all j in {1, . . . , N }, iβn uj,n − vj,n −→ 0,

in H 1 (0, j ),

(2.29)

iβn vj,n − ∂xx uj,n + γj ∂x θj,n −→ 0,

in L2 (0, j ),

(2.30)

iβn θj,n + γj ∂x vj,n − κj ∂xx θj,n −→ 0,

in L2 (0, j ).

(2.31)

Let j ∈ J te . Multiplying (2.29) by γj and adding the result up to (2.31), we obtain iβn (γj ∂x uj,n + θj,n ) − κj ∂xx θj,n −→ 0 in L2 (0, j ).

(2.32)

γj ∂x uj,n + θj,n , which is bounded in βn 2 L (0, j ), and integrating by parts, we obtain

Taking L2 -inner product of (2.32) with

' &  2 κj   i γj ∂x uj,n + θj,n + ∂x θj,n , γj ∂xx uj,n + ∂x θj,n βn %x=j κj ∂x θj,n (γj ∂x uj,n + θj,n ) %% −→ 0. − % % βn

(2.33)

x=0

   ∂xx uj,n    is bounded, we can conclude that the product By (2.28) and the fact that  βn  in the aforementioned expression tends to 0. Recall the Gagliardo–Nirenberg inequality (1.13) of Theorem 1.33: there are two positive constants C1 and C2 such that, for any w in H 1 (0, j ), wL∞ ≤ C1 ∂x w1/2 w1/2 + C2 w . ∂x θj,n ∂x uj,n θj,n With this inequality applied to w = √ , w = √ and √ yield that the βn βn βn boundary terms in (2.33) converge to 0. Then, γj ∂x uj,n + θj,n −→ 0 in L2 (0, j ).

(2.34)

It follows from (2.34) and (2.28) that θj,n is uniformly bounded in H 1 (0, j ). Then, by the compactness of the embedding of H 1 (0, j ) into L2 (0, j ), there is

26

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

a subsequence of θj,n , still denoted θj,n , which is a Cauchy sequence in L2 (0, j ). Then, with (2.28) used again, θj,n is a Cauchy sequence in H 1 (0, j ). Let θ j be its limit. Then, from (2.29) and (2.34), it follows that uj,n converges to 0 in H 1 (0, j ) and θj = 0. Dividing (2.30) by βn , simplifying by taking into account that ∂x θj,n −→ 0 in L2 (0, j ), multiplying the result by vj,n , and integrating by parts, we obtain %  2  vj,n ∂x uj,n %x=j 1  % i vj,n  + ∂x vj,n , ∂x uj,n − −→ 0. % βn βn x=0

(2.35)

As for (2.33), we prove that the second and third terms in (2.35) converge to 0, so vj,n −→ 0 in L2 (0, j ). Second Step We will prove that ∂x uj,n and vj,n converge to 0 in L2 (0, j ) for j in J e. Let j in {1, . . . , N }. Equations (2.29) and (2.30) can be rewritten, respectively, as iβn uj,n − vj,n = fj,n −→ 0

in H 1 (0, j ),

(2.36)

iβn vj,n − ∂xx uj,n = gj,n −→ 0

in L2 (0, j ).

(2.37)

Substituting (2.36) into (2.37), we obtain ∂xx uj,n + βn2 uj,n = −gn − iβn fj,n .

(2.38)

From (2.36), we deduce immediately that  2  2 βn2 uj,n  − vj,n  −→ 0.

(2.39)

Next, let hj be a function in C 1 ([0, j ], C). We want to compute the real part of the inner product of (2.38) with hj ∂x uj,n in L2 (0, j ). It follows from integration by parts that  1  Re ∂xx uj,n , hj ∂x uj,n = 2



%x= % % %∂x uj,n (x)%2 hj (x)%% j x=0



j

− 0

 % %  %∂x uj,n (x)%2 ∂x hj (x)dx ,

 %x=j   %2 % 1 % βn2 %uj,n (x)% hj (x)% Re βn2 uj,n , hj ∂x uj,n = x=0 2

2 Exponential Stability

27



j

− 0

 %2 % βn2 %uj,n (x)% ∂x hj (x)dx ,

and 

%x=    iβn fj,n + gj,n , hj ∂x uj,n = iβn fj,n (x)hj (x)uj,n (x)%x=0j + ∂x fj,n , iβn uj,n hj     (2.40) + fj,n , iβn uj,n ∂x hj + gj,n , hj ∂x uj,n .

            By  taking  intoaccountthat gj,n , ∂x fj,n , and fj,n converge to 0 and that ∂x uj,n  and iβn uj,n  are bounded, we deduce that the products in the second member of (2.40) converge to zero. Hence, the real part of the inner product of (2.38) by hj ∂x uj,n gives %x=j %x=j %2 %2 1 2 %% 1 % % % βn uj,n (x)% hj (x)% + %∂x uj,n (x)% hj (x)% x=0 x=0 2 2

j   % % % % 1 %∂x uj,n (x)%2 + β 2 %uj,n (x)%2 ∂x hj (x)dx − n 2 0 %x=  + Re iβn fj,n (x)hj (x)uj,n (x) %x=0j −→ 0.

(2.41)

For j in J te , with hj (x) = j − x, and by taking into account that uj,n −→ 0 in and βn uj,n = −i(fj,n + vj,n ) −→ 0 in L2 (0, j ), we obtain

H 1 (0, j )

%2 1 % %2   1 % − βn2 %uj,n (0)% − %∂x uj,n (0)% − Re iβn fj,n (0)uj,n (0) −→ 0. 2 2

(2.42)

Now, because %2 % %2   1 % −Re iβn fj,n (0)uj,n (0) ≤ βn2 %uj,n (0)% + %fj,n (0)% , 4 we can deduce %2 1 % %2 %2   % 1 % − βn2 %uj,n (0)% − %∂x uj,n (0)% − Re iβn fj,n (0)uj,n (0) − %fj,n (0)% 2 2 %2 1 % %2 1 2 %% ≤ − βn uj,n (0)% − %∂x uj,n (0)% ≤ 0. (2.43) 4 2 The continuity condition (2.12) and the Dirichlet condition (2.13) of f n imply that fj,n (0) −→ 0.

28

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

Then, by (2.42), the first member of the inequalities in (2.43) converges to 0; hence, the second member converges to 0 and then %2 % %2 % βn2 %uj,n (0)% → 0 and %∂x uj,n (0)% → 0,

(2.44)

which implies, with the use of (2.42),   Re iβn fj,n (0)uj,n (0) → 0.

(2.45)

By the same manner, if we choose hj (x) = x, we deduce that %2 % %2 % βn2 %uj,n (j )% → 0, %∂x uj,n (j )% → 0,

(2.46)

  Re iβn fj,n (j )uj,n (j ) → 0.

(2.47)

and

Now, let j in J e , and let ak be an end of ej in Vint . By the fact that un is continuous at the inner nodes and using (2.44) and (2.45) or (2.46) and (2.47), we deduce that %2 %   βn2 %uj,n (ak )% → 0 and Re iβn fj,n (ak )uj,n (ak ) → 0. On the other hand, because un and θ n satisfy the balance condition in (2.16) and θ n converges to 0 at every inner node, then (2.44) or (2.46) implies % % %∂x uj,n (ak )%2 → 0. Hence, from (2.41) with hj (x) = j − x or hj (x) = x, we obtain

0

j

% % %  % %∂x uj,n (x)%2 + β 2 %uj,n (x)%2 dx −→ 0, n

which concludes that yn H −→ 0. This behavior contradicts the hypothesis that yn has the unit norm.   We can now state the main result of this chapter. Theorem 2.5 The C0 -semigroup S(t), generated by the operator A, is exponentially stable. Proof The proof is a direct consequence of Lemmas 2.3 and 2.4.

 

Remark 2.6 Consider the case where the thermoelastic edges are governed by Cattaneo’s law.

2 Exponential Stability

29

The linear system on the thermoelastic edge ej will be uj,tt − αj uj,xx + βj θj,x = 0, in (0, j ) × (0, ∞), θj,t + γj qj,x + δj uj,tx = 0, in (0, j ) × (0, ∞), τj qj,t + qj + κj θj,x = 0, in (0, j ) × (0, ∞), where qj = qj (x, t) is the heat flux at time t of the edge ej and αj , βj , γj , δj , τj , and κj are positive constants, with initial conditions uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x), θj (x, 0) = θj0 (x), qj (x, 0) = qj0 (x). Every elastic edge satisfies the following equation: uj,tt − αj uj,xx = 0

in (0, j ) × (0, +∞),

where αj is a positive constant, with initial conditions uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x). In addition, suppose that the whole system will satisfy the following boundary conditions: uj (ak , t) = 0,

j ∈ Jk , ak ∈ Vext ,

θj (ak , t) = 0,

j ∈ Jkte , ak ∈ Vext ,

uj (ak , t) = ul (ak , t) qj (ak , t) = 0  j ∈Jkte

dkj

j, l ∈ Jk , ak ∈ Vint ,

j ∈ Jkte , ak ∈ Vint ,

  δj  dkj αj ∂x uj (ak , t) = 0, ak ∈ Vint . αj ∂x uj (ak , t) − βj θj (ak , t) + βj e j ∈Jk

Then, we can also prove that solutions of the system decay exponentially.

2.2 Second Case Now, we consider the second case where the graph is a tree of elastic edges that ends by thermoelastic edges at leaves (Fig. 2.2). We can prove that the system is exponentially stable. Theorem 2.7 The C0 -semigroup, generated by the operator A, is exponentially stable.

30

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

Proof With the frequency domain condition, we only need to justify (1.9) and (1.10). The proofs are similar to those of Lemmas 2.3 and 2.4, respectively, with some modifications. Suppose that (1.9) is not true; as in the proof of Lemma 2.3, there are a real number β, β = 0, and a vector y = (u, v, θ ) ∈ D(A), y = 0, such that Ay = iβy. If ej is thermoelastic, then the functions uj , ∂x uj , and θj are zero in L2 (0, j ) and at the inner nodes; thus, (u, v) satisfies 

j = 1, . . . , N, vj = λuj , j = 1, . . . , N. ∂xx uj = λ2 uj ,

(2.48)

Now, if ej is an elastic edge attached to a thermoelastic one, then it satisfies the Cauchy problem ∂xx uj = λ2 uj , uj (ak ) = 0, and ∂x uj (ak ) = 0, where ak is an end of ej , and then uj = 0 in H 1 (0, j ) and vj = 0 in L2 (0, j ); moreover, uj and ∂x uj vanish at the ends of ej . We iterate this procedure from the leaves to the root, so that we obtain u = 0 in H01 (G), v = 0 in L2 (G), and θ = 0 in V (G), which contradict the fact that y = 0. For property (1.10), the difference from the proof of Lemma 2.4 is that, in the

j  % % %  % %∂x uj,n (x)%2 + %vj,n (x)%2 dx second step of such proof, we prove first that 0

converges to zero when ej is an elastic edge attached to a thermoelastic one, and then, by iteration using the same arguments, we prove that

j  % % % %  %∂x uj,n (x)%2 + %vj,n (x)%2 dx converges to zero for every elastic edge, which 0

contradicts the fact that yn  = 1.

 

3 Comments 3.1 Comment 1 We can consider a more general case (as in the next chapter): we assume that G contains at least one thermoelastic string, that Vext = ∅, that every maximal subgraph of elastic edges is a tree, the leaves of which thermoelastic edges are attached, and that every maximal subgraph of thermoelastic edges is not a circuit. Then, we can prove that the associated semigroup is exponentially stable.

3.2 Comment 2 In [37], G is a star-shaped network of interconnected elastic and thermoelastic rods (Fig. 2.3). The edges (rods) ej , j = 1, . . . , N occupy the intervals (0, j ), j > 0,

3 Comments

31

Fig. 2.3 Star-shaped thermoelastic network

respectively. The common node is identified to x = 0. The edges ej , j = 1, . . . , N1 (0 < N1 < N) are thermoelastic, and the other edges are all purely elastic. The thermoelastic–elastic network system under consideration is ⎧ ⎪ uj,tt (x, t) − uj,xx (x, t) + αj θj,x (x, t) = 0, x ∈ (0, j ), j = 1, 2, . . . , N1 , t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ θ ⎪ j,t (x, t) − θj,xx (x, t) + βj uj,tx (x, t) = 0, x ∈ (0, j ), j = 1, 2, . . . , N1 , t > 0, ⎪ ⎪ ⎪ ⎪ uj,tt (x, t) − uj,xx (x, t) = 0, x ∈ (0, j ), j = N1 + 1, N1 + 2, . . . , N, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ uj (j , t) = 0, j = 1, 2, . . . , N, t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ uj (0, t) = uk (0, t), ∀ j, k = 1, 2, . . . , N, t > 0, θk (k , t) = 0, k = 1, 2, . . . , N1 , t > 0, ⎪ ⎪ ⎪ θj (0, t) = θk (0, t), ∀ j, k = 1, 2, . . . , N1 , t > 0, ⎪ ⎪ ⎪ ⎪ N1 N1 α N " " ⎪ " j ⎪ ⎪ uj,x (0, t) = αj θj (0, t), ⎪ βj θj,x (0, t) = 0, t > 0, ⎪ ⎪ j =1 j =1 j =1 ⎪ ⎪ ⎪ ⎪ ⎪ θ k (t = 0) = θk0 , k = 1, 2, . . . , N1 , ⎪ ⎪ ⎪ ⎩ u (t = 0) = u0 , u (t = 0) = u1 , j = N + 1, N + 2, . . . , N, j

 where

j,t

j

j

1

N  N  N1 , u1j , θk0 k=1 u0j j =N1 +1 j =N1 +1

1

(2.49)

 is the given initial state.

The natural energy of this system is as follows: 1 2 N

E(t) =



j =1 0

j

N1  αj j 2 1 u2j,t + u2j,x dx + θ dx. 2 βj 0 j j =1

The space state is H=

H01 (G) × L2 (G) ×

#N 1  k=1

equipped with the inner product

$ 2

L (0, k ) ,

32

2 Exponential Stability of a Network of Elastic and Thermoelastic Materials

y, y ˜ H=

N 



∂x fj (x)∂x f˜j (x)dx +

0

j =1

+

j



j

gj (x)g˜j (x)dx

0

N1



k

k=1 0

$ αk ˜ hk (x)hk (x)dx , βk

˜ The space H is a Hilbert space. where y = (f , g, h) and y˜ = (f˜, g, ˜ h). In this Hilbert space, the authors defined the operator A : D(A) ⊆ H → H, by ⎞ 0 I 0 ⎟ ⎜ 0 −αIN ×N1 ∂x ⎠ , A = ⎝ ∂xx 0 −βINT ×N1 ∂x ∂xx ⎛

(2.50)

where α = diag(α1 , α2 , . . . , αN1 ), β = diag(β1 , β2 , . . . , βN1 ), IN ×N1 +T * IN1 ,0 , and IN1 is the N1 -unit matrix, with domain

=

⎧ ⎡ ⎤ N ⎨  D(A) = (u, v, θ ) ∈ ⎣H01 ∩ H 2 (0, j )⎦ × H01 (G) ⎩ j =1

×

N1  j =1

⎫ ⎬

H 2 (0, j ), satisfying (2.51) , ⎭

where ⎧ N N1 " " ⎪ ⎪ ∂ u (0) = αj ∂x θj (0), ⎪ x j ⎪ ⎪ j =1 j =1 ⎪ ⎪ ⎨ θj (j ) = 0, j = 1, 2 . . . , N1 , ⎪ θ j (0) = θk (0), j, k = 1, 2, . . . , N1 , ⎪ ⎪ ⎪ N1 ⎪ " αj ⎪ ⎪ ⎩ βj ∂x θj (0) = 0.

(2.51)

j =1

Then, the system (2.49) may be rewritten as the first-order evolution equation on H, ⎧ ⎨ d y = Ay, dt ⎩ y(0) = y 0 , where y = (u, ut , θ ) and y 0 = (u0 , u1 , θ 0 ). The authors proved the following results:

3 Comments

33

Theorem 2.8 Operator A generates a C0 -semigroup of contractions on H. Moreover, the energy of the system (2.49) decays to zero as t → ∞ if and only if one of the following two conditions is fulfilled: (1) N − N1 = 1 and / Q, i, j = N1 + 1, N1 + 2, . . . , N, i = j . (2) N − N1 ≥ 2 and i /j ∈ Theorem 2.9 The energy of system (2.49) decays to zero exponentially if and only if N − N1 ≤ 1, that is, if no more than one purely elastic undamped rod is involved in the network. To examine more closely the case N − N1 > 1, we need the following definition: Definition 2.10 ([18, 85]) Real numbers 1 , 2 , . . . , m are said to verify the conditions (S), if 1 , 2 , . . . , m are linearly independent over the field Q of rational numbers, and the ratios i /j are algebraic numbers for i, j = 1, 2, . . . , m. Theorem 2.11 When N − N1 > 1, one has lim inf tE(t) > 0. t→∞

(2.52)

Thus, we cannot expect a decay rate which is beyond first-order polynomial. Furthermore, if N1 +1 , N1 +2 , . . . , N (N − N1 > 1) satisfy the condition (S), then for any ε > 0, there always exists a constant Cε > 0 such that the energy of network (2.49) satisfies 1

E(t) ≤ Cε t − 1+ε (u0 , u1 , θ 0 )2D(A) , ∀ t ≥ 0 for all (u0 , u1 , θ 0 ) ∈ D(A). Thus, this decay rate is nearly sharp in the sense of (2.52).

Chapter 3

Exponential Stability of a Network of Beams

In this chapter, we consider the asymptotic behavior of a network of Euler–Bernoulli beams, some of them are thermoelastic (sensible to a thermal effect), while the others are elastic (without temperature change). Such study is motivated by the need for engineers to eliminate vibrations in some dynamical structures consisting of elastic beams, coupled in the form of chain or graph such as pipelines, bridges, and some cable networks. There are other complicated examples in the automotive industry. Our main question is whether dissipation over each thermoelastic edge is enough to produce an exponential decay rate of the whole system. As in [1, 88], where we considered a network of strings, we prove by using the frequency domain method that the answer is positive. The main difficulty in the present case is to “estimate” some boundary terms of higher order. To overcome it, we improve the method used in [88] by special multiplier techniques as in [5]. Many authors have studied similar networks, for example, Mercier and Regnier in [68, 69] studied the controllability and the spectrum of a network of Euler– Bernoulli beams. Dekoninck and Nicaise studied in [21] the exact controllability of a network of beams with boundary dampings. Ammari in [3] has proved the polynomial stability of a star-shaped network of elastic Euler–Bernoulli beams. In [100], the authors proved the exponential stability of a star-shaped network of beams with controls applied at external ends. In [35] the authors have proved the asymptotic stability of a star-shaped network of Timoshenko Beams. Ammari et al. considered in [12] a chain of Euler–Bernoulli beams and strings; they established some results of polynomial stability of such system. For the transmission problem between elastic and thermoelastic materials see, for example, [62, 81] for strings, and [84] for beams. Suppose that the equilibrium position of our network of elastic and thermoelastic beams coincides with the graph G of N edges, e1 , . . . , eN , and p vertices, a1 , . . . , ap . We assume that G contains at least one thermoelastic beam, that Vext = ∅, that every maximal subgraph of elastic edges is a tree, the leaves of which © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Ammari, F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-86351-7_3

35

36

3 Exponential Stability of a Network of Beams

Fig. 3.1 Elastic–thermoelastic graph

Fig. 3.2 In Fig. 3.1, there are three maximal subgraphs of elastic edges, each of them is a tree

Fig. 3.3 Some thermoelastic edges attached to every leaf of the trees of Fig. 3.2

thermoelastic edges are attached, and that every maximal subgraph of thermoelastic edges is not a circuit (Figs. 3.1, 3.2, and 3.3). We denote the displacement in the beam ej by uj = uj (x, t) and the variation of temperature between the actual state and a reference temperature in a thermoelastic beam ej by θj = θj (x, t), in position x at time t.

3 Exponential Stability of a Network of Beams

37

Assume that: • Every thermoelastic beam ej satisfies the following equations: uj,tt + uj,xxxx − γj θj,xx = 0

in (0, j ) × (0, ∞),

(3.1)

θj,t − θj,xx + γj uj,xxt = 0

in (0, j ) × (0, ∞),

(3.2)

where γj is a positive constant, with initial conditions uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x), θj (x, 0) = θj0 (x).

(3.3)

• Every elastic beam ej satisfies the following equation: uj,tt + uj,xxxx = 0

in (0, j ) × (0, ∞),

(3.4)

with initial conditions uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x).

(3.5)

Denote by Jkte the set of indices of thermoelastic beams adjacent to ak and Jke the  be the set of external nodes of maximal set of elastic beams adjacent to ak . Let Vext subgraph of thermoelastic beams. Then, the boundary conditions on the graph G are described as follows: uj (ak , t) = 0,

j ∈ Jk , ak ∈ Vext ,

(3.6)

θj (ak , t) = 0,

 j ∈ Jk , ak ∈ Vext ,

(3.7)

uj,xx (ak , t) = 0,

j ∈ Jk , ak ∈ Vext ,

(3.8)

uj (ak , t) = ul (ak , t),

j, l ∈ Jk , ak ∈ Vint ,

(3.9)

θj (ak , t) = θl (ak , t),

j, l ∈

uj,xx (ak , t) = ul,xx (ak , t),  

Jkte ,

ak ∈ Vint ,

j, l ∈ Jk , ak ∈ Vint ,

dkj uj,x (ak , t) = 0, ak ∈ Vint ,

(3.10) (3.11) (3.12)

j ∈Jkte

   dkj uj,xxx (ak , t) − γj θj,x (ak , t) + dkj uj,xxx (ak , t) = 0, ak ∈ Vint , j ∈Jke

j ∈Jkte



(3.13)   dkj γj uj,xt (ak , t) − θj,x (ak , t) = 0, ak ∈ Vint .

(3.14)

j ∈Jkte

Note that (3.9) and (3.10) imply the continuity of u and θ, and conditions (3.11)– (3.14) are transmission conditions at inner nodes.

38

3 Exponential Stability of a Network of Beams

We define the total energy of the system by 1 2 N

E(t) =



j

j =1 0

%  j % %2 % % %  %uj,t %2 + %uj,xx %2 dx + 1 %θj % dx, 2 te 0 j ∈J

where J te is the set of thermoelastic edges; similarly, we will denote by J e the set of elastic edges. We prove that E(t) ≤ ME(0)e−wt

(3.15)

for some positive numbers M and w. That is, the energy decays exponentially to 0. This chapter is organized as follows. Section 1 gathers the functional setting, existence, and uniqueness of solutions. In Sect. 2, we show the exponential stability of the semigroup generated by the system. In Sect. 3, we briefly look at other cases of boundary conditions.

1 Functional Spaces, Existence, and Uniqueness of Solutions In this section, we consider the well-posedness of the system (3.1)–(3.14). We first rewrite it in a first-order evolution equation. To start, let us introduce the following spaces: L2 (G) =

N 

L2 (0, j ); H k (G) =

j =1

V (G) =

N 

N 

H k (0, j ), k = 2, 4;

j =1

Vj where Vj = L2 (0, j ) if j ∈ J te and Vj = {0} if j ∈ J e ;

j =1

V (G) = k

N 

Vjk where Vjk = H k (0, j ) if j ∈ J te and

j =1

Vjk = {0} if j ∈ J e , k = 1, 2 and F (G) = {f = (f1 , . . . , fN ) ∈ H 2 (G) satisfying (3.16)–(3.18)}, fj (ak ) = fl (ak ), j, l ∈ Jk , ak ∈ Vint ,

(3.16)

1 Functional Spaces, Existence, and Uniqueness of Solutions



39

fj (ak ) = 0, j ∈ Jk , ak ∈ Vext , dkj ∂x fj (ak ) = 0, ak ∈ Vint .

(3.17) (3.18)

j ∈Jk

Let the space H = F (G) × L2 (G) × V (G). Lemma 3.1 The map H × H → C, (y, y) ˜ → y, y ˜ H defined by y, y ˜ H=

N 



j

0

j =1



j

+

∂x2 fj (x)∂x2 f˜j (x)dx +



j

gj (x)g˜j (x)dx

0

 ˜ hj (x)hj (x)dx ,

(3.19)

0

˜ is an inner product on H. Moreover, where y = (f , g, h) and y˜ = (f˜, g, ˜ h) (H, ., .H ) is a Hilbert space. Proof It suffices to prove that for every f ∈ F (G) the assumption N 

 j =1

j 0

 ∂x2 fj (x)∂x2 fj (x)dx

=0

(3.20)

implies f = 0. Let f in F (G) satisfying (3.20). We use a matrix method [98]. First of all, we recall some definitions and notations already presented in the preliminary chapter. The incidence matrix D = (dkj )p×N and the adjacency matrix E = (eik )p×p are defined by,

dkj

⎧ ⎨ 1 if πj (j ) = ak = −1 if πj (0) = ak ⎩ 0 otherwise

 and eik =

1 if ai and ak are adjacent 0 otherwise.

For two matrices A = (aj k ) and B = (bj k ) of the same size, A ∗ B = (aj k bj k ), and for any function Q : R −→ R, we define the matrix Q(A) = (qik )p×p by  qik =

Q(aik ) if eik = 1, 0 otherwise.

40

3 Exponential Stability of a Network of Beams

In particular, we write A(r) = Q(A) if Q(x) = x r . Furthermore, the matrix L = (ik )p×p , is defined by  ik =

s(i,k) if eik = 1, 0 otherwise,

where s(i, k) = s(k, i) is the index of the edge connecting ai and ak . Now, we introduce the following definition used in [98] (see also [1]): To the function f is associated the matrix function F defined by F : [0, 1] −→ Cp×p , x −→ F (x) = (fj k (x))p×p , with 3  4 1 + dj s(j,k) − xdj s(j,k) . fj k (x) = ej k fs(j,k) s(j,k) 2 The continuity condition of f at the interior nodes and Dirichlet conditions at the external nodes can be⎛expressed in this manner: ⎞ ϕ1 ⎜ ⎟ There exists ϕ = ⎝ ... ⎠ ∈ Cp such that ϕk = 0 when ak is an external node, ϕp and F (0) = (ϕeT ) ∗ Eint ,

(3.21)

⎛ ⎞ 1 ⎜ .. ⎟ where e = ⎝ . ⎠ ∈ Rn and Eint = (ej k )p×p is the matrix defined by 1 ej k

 =

1 if aj and ak are adjacent and |Jk | > 1, 0 otherwise.

Recall that the matrix L(−1) = (Lik )p×p is defined as follows: 5 Lik =

1 s(i,k)

if eik = 1,

0

otherwise,

then the condition (3.18) applied to f is expressed as follows: (L(−1) ∗ F  (0) ∗ Eint )e = 0.

(3.22)

1 Functional Spaces, Existence, and Uniqueness of Solutions

41

Finally, we have F (1 − x) = F (x)T .

(3.23)

From (3.20), we deduce the following matrix-differential equation: F (x) = xL ∗ C + D,

(3.24)

where C = L(−1) ∗ F  (0) and D = F (0). By taking x = 1 in (3.24) and using (3.23), we obtain F (1) = D T = L ∗ C + D which implies that D T − D = L ∗ C, then, using (3.22), we get 

 L(−1) ∗ (D T − D) ∗ Eint e = 0.

(3.25)

We recall the following elementary rules for a matrix M ∈ Cp×p (see [22]): (M ∗ D T )e = Mϕ, (M ∗ D)e = diag(Me)ϕ.

(3.26)

Then, (3.25) implies that Kϕ = 0, where K = L(−1) ∗ Eint − diag



  L(−1) ∗ Eint e .

T ∗E Because Vext = ∅, the matrix obtained from K ∗ Eint int by removing rows and columns that are zero is invertible [1]. By taking into account that ϕj = 0 when aj is an exterior vertex, we deduce that ϕ = 0. Then D = 0 and C = 0. Hence, f = 0.  

Now, define the linear operator A : D(A) ⊆ H → H, ⎛

⎞ 0 A01 0 ⎜ ⎟ A = ⎝ −A41 0 A2γ ⎠ , 0 −A2γ A21

(3.27)

42

3 Exponential Stability of a Network of Beams

where Akγ = diag(γ1 ∂xk , . . . , γN ∂xk ), k ∈ N, and ∂x0 = I, and whose domain is given by    D(A) = (u, v, θ ) ∈ F (G) ∩ H 4 (G) × F (G) × V 2 (G) satisfying (3.28)–(3.33) below} ∂x2 uj (ak ) = ∂x2 ul (ak ), ∂x2 uj (ak ) = 0, θj (ak ) = θl (ak ), θj (ak ) = 0, 

j, l ∈ Jk , ak ∈ Vint , j ∈ Jk , ak ∈ Vext ,

(3.29)

j, l ∈

(3.30)

Jkte ,

ak ∈ Vint ,

 j ∈ Jk , ak ∈ Vext ,

  dkj ∂x3 uj (ak ) − γj ∂x θj (ak ) = 0, ak ∈ Vint ,

j ∈Jk



(3.28)

  dkj γj ∂x vj (ak ) − ∂x θj (ak ) = 0, ak ∈ Vint ,

(3.31) (3.32) (3.33)

j ∈Jk

with γj = 0 if j ∈ J e . For y = (u, v, θ ) ∈ D(A), the components of Ay are ⎧ j = 1, . . . , N, ⎨ vj , j = 1, . . . , N, −∂x4 uj + γj ∂x2 θj , ⎩ j = 1, . . . , N. −γj ∂x2 vj + ∂x2 θj , So that, the initial boundary value problem (3.1)–(3.14) can be written as an evolutionary equation in H ⎧ ⎨ d y = Ay, dt ⎩ y(0) = y 0 , where y = (u, ut , θ ) and y 0 = (u0 , u1 , θ 0 ). Lemma 3.2 Let H and A be defined as before. Then (i) The operator A is dissipative. (ii) 1 ∈ ρ(A): the resolvent set of A, and (I − A)−1 is compact.

(3.34)

1 Functional Spaces, Existence, and Uniqueness of Solutions

43

Proof (i) Let y = (u, v, w) ∈ D(A). Using the definition of the inner product (3.19), we have Re(Ay, yH ) = Re

N 

 j =1



j

+ 0

j

0

∂x2 vj ∂x2 uj dx

j

+ 0

(−∂x4 uj + γj ∂x2 θj )vj dx

 (−γj ∂x2 vj + ∂x2 θj )θj dx .

Performing some integrations by parts, ⎛ N  %x=j %x=j  % % ∂x2 uj ∂x vj % + (−∂x3 uj + γj ∂x θj )vj % Re(Ay, yH ) = Re ⎝ x=0

j =1

x=0

 %x= %x= 2  . −γj θj ∂x vj %x=0j + θj ∂x θj %x=0j − ∂x θj 

(3.35)

Using boundary conditions, it yields ⎛ ⎡ n    ⎣ vj (ak )(−∂x3 uj (ak ) + γj ∂x θj (ak )) Re(Ay, yH ) = Re ⎝ k=1

j ∈Jk

⎤⎞    θj (ak )(−γj ∂x vj (ak ) + ∂x θj (ak )) ⎦⎠ + j ∈Jkte N    ∂x θj 2 − j =1

=−

N    ∂x θj 2 ≤ 0,

(3.36)

j =1

and this proves the dissipativeness of the operator A in H. Next, we shall prove that 1 ∈ ρ(A). Let z = (f , g, h) ∈ H, we look for y = (u, v, θ ) ∈ D(A) such that (I − A)y = z

(3.37)

i.e., uj − vj = fj , j = 1, . . . , N,

(3.38)

vj + ∂x4 uj − γj ∂x2 θj = gj , j = 1, . . . , N,

(3.39)

θj − ∂x2 θj + γj ∂x2 vj = hj , j = 1, . . . , N.

(3.40)

44

3 Exponential Stability of a Network of Beams

Substituting (3.38) into (3.39) and (3.40), we obtain uj + ∂x4 uj − γj ∂x2 θj = gj + fj , j = 1, . . . , N,

(3.41)

θj − ∂x2 θj + γj ∂x2 uj = hj + γj ∂x2 fj , j = 1, . . . , N.

(3.42)

For the sequel, we need the following space:   F = (w, ϕ) ∈ H 2 (G) × V 1 (G) satisfying (3.43) below ⎧ wj (ak ) = wl (ak ), j, l ∈ Jk , ak ∈ Vint , ⎪ ⎪ ⎪ ⎪ ⎪ j, l ∈ Jkte , ak ∈ Vint , ϕ (a ) = ϕl (ak ), ⎪ ⎨ j k j ∈ Jk , ak ∈ Vext , wj (ak ) = 0,  , ⎪ (a ) = 0, j ∈ Jk , ak ∈ Vext ϕ ⎪ j k ⎪ " ⎪ ⎪ ⎪ dkj ∂x wj (ak ) = 0, ak ∈ Vint . ⎩

(3.43)

j ∈Jk

Equipped with the inner product   ˜ ϕ) ˜ (w, ϕ), (w,

F

=

N



j

j =1 0

∂x2 wj ∂x2 w˜j dx

+

N



j

∂x ϕj ∂x ϕ˜j dx,

j =1 0

F is a Hilbert space. For (w, ϕ) in F. Taking the inner product, in L2 (0, j ), of (3.41) by wj and (3.42) by ϕj , j = 1, . . . , N, and integrating by parts, we obtain, respectively

j 0



j

uj wj dx +

%x=j % ∂x2 uj ∂x2 wj dx + (∂x3 uj − γj ∂x θj )wj % x=0

0

%x=j % − ∂x2 uj ∂x wj % + γj x=0

j



j

∂x θj ∂x wj dx =

0

(gj + fj )wj dx

0

and

0

j

%x= θj ϕj dx + (−∂x θj + γj ∂x uj )ϕj %x=0j +

−γj 0

j



j

∂x uj ∂x ϕj dx = 0



j

∂x θj ∂x ϕj dx 0

(hj + γj ∂x2 fj )ϕj dx

for j = 1, . . . , N. Now summing up over j ∈ {1, . . . , N }, we find by taking into account the boundary conditions (3.16)–(3.18), (3.28)–(3.33), and (3.43), a((u, θ ), (w, ϕ)) = g(w, ϕ),

(3.44)

1 Functional Spaces, Existence, and Uniqueness of Solutions

45

where a((u, θ), (w, ϕ)) =

N j  j =1 0

+

uj wj dx +

N j  j =1 0

N j  j =1 0

∂x θj ∂x ϕj dx + γj

∂x2 uj ∂x2 wj dx + #

j 0

N j  j =1 0

θj ϕj dx $

(∂x θj ∂x wj − ∂x uj ∂x ϕj )dx

and g(w, ϕ) =

N



j

(gj + fj )wj dx +

j =1 0

+

 

N

 j =1 0

j

(hj + γj ∂x2 fj )ϕj dx

γj ∂x fj (ak ))ϕj (ak ).

ak ∈Vint j ∈Jk

a is a continuous sesquilinear form on F × F and g is a continuous anti-linear form on F. Moreover, for every (w, ϕ) ∈ F % %  2 % %   %a((u, θ ), (w, ϕ))% ≥ (w, ϕ) . F

It follows, using the Lax–Milgram’s lemma, that (3.44) has a unique solution (u, θ ) in F. N 6 If we consider respectively (w, ϕ) in {0} × D(0, j ) and (w, ϕ) in N 6 j =1

j =1

D(0, j ) × {0} then, (u, θ ) belongs to the space H 4 (G) × H 2 (G) and satisfies uj + ∂x4 uj − γj ∂x2 θj = gj + fj , j = 1, . . . , N, θj − ∂x2 θj + γj ∂x2 uj = hj + γj ∂x2 fj , j = 1, . . . , N.

By some integrations by parts, we deduce that the solution (u, θ ) satisfies the conditions ⎧ 2 2 j, l ∈ Jk , ak ∈ Vint , ⎪ ⎪ ∂x uj (ak ) = ∂x ul (ak ), ⎪ 2 ⎪ u (a ) = 0, j ∈ J , ak ∈ Vext , ∂ ⎪ j k k x ⎨ "   dkj ∂x3 uj (ak ) − γj ∂x θj (ak ) = 0, ak ∈ Vint , ⎪ ∈Jk ⎪   " ⎪ j" ⎪ ⎪ dkj −γj ∂x uj (ak ) + ∂x θj (ak ) = dkj γj ∂x fj (ak ), ak ∈ Vint . ⎩ j ∈Jk

j ∈Jk

46

3 Exponential Stability of a Network of Beams

Returning back to the Lax–Milgram’s lemma, y = (u, v, θ ) verifies y2H ≤ c z2H , where c is a positive constant independent of y. All of that prove that y = (u, v, θ ) ∈ D(A) and (I − A)−1 ∈ L(H), that is, 1 ∈ ρ(A). Moreover, one can deduce, by the Sobolev embedding theorem, that (I − A)−1 is a compact operator. The proof is complete.   It follows from the Lumer–Phillips theorem (Theorem 1.15): Corollary 3.3 The operator A is the infinitesimal generator of a C0 -semigroup of contraction (S(t))t≥0 on the Hilbert space H. Hence, for any y 0 ∈ H, the Cauchy problem (3.34) has a unique solution y ∈ C([0, +∞), H). Furthermore, if y 0 ∈ D(A), then y ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H).

2 Exponential Decay The aim of this section is to prove the exponential decay rate of the energy E(t) of the whole system, i.e., there exist two positive constants M and w verifying E(t) ≤ ME(0)e−wt

(3.45)

or equivalently, there are two positive constants M and w such that S(t) ≤ Me−wt , ∀t ≥ 0. To prove such property we use Theorem 1.25. We will first verify the condition (1.9) in Theorem 1.25. Lemma 3.4 Let A be the operator given by (3.27), then the assumption (1.9) holds, that is iR ⊂ ρ(A). Proof Suppose that (1.9) is not true. Then, there is a real number β ∈ R such that λ := iβ is in σ (A). Since (I − A)−1 is compact, λ must be an eigenvalue of A, then there is a vector y = (u, v, θ ) ∈ D(A), y = 0 such that Ay = iβy. Already, we have Re(Ay, y) = Re(λy, y) = 0,

2 Exponential Decay

47

which leads to −

N    ∂x θj 2 = 0. j =1

Then, using the continuity condition of θ at inner nodes, the Dirichlet condition of  , and the condition that every maximal subgraph of thermoelastic edges is θ in Vext not a circuit, we deduce that θj = 0 for j = 1, . . . , N. Thus, (u, v) satisfies ⎧ j = 1, . . . , N, ⎨ vj = λuj , j = 1, . . . , N, −∂x4 uj = λ2 uj , ⎩ 2 j = 1, . . . , N. −γj ∂x vj = 0,

(3.46)

If λ = 0 and ej is thermoelastic, then by the third and first equations in (3.46), ∂x2 uj = 0 and vj = 0 in L2 (0, j ). Furthermore, ∂xk uj (ak ) for k = 0, . . . , 3 and if ak is an end of ej . If λ = 0 and ej is the only elastic edge attached to a thermoelastic edge at the end ak , then uj satisfies the Cauchy problem: ∂x4 uj + λ2 uj = 0 and ∂xk uj (ak ) = 0, k = 0, . . . , 3. So that ∂xk uj , k = 0, . . . , 3 are zero in L2 (0, j ) and at both ends of ej . We iterate this procedure in each maximal subgraph of elastic edges of G; we then conclude that uj = 0 in H 2 (0, j ) and vj = 0 in L2 (0, j ). Now, suppose that λ = 0. Then, ⎧ j = 1, . . . , N, ⎨ vj = 0, j = 1, . . . , N, −∂x4 uj = 0, ⎩ j = 1, . . . , N. −γj ∂x2 vj = 0,

(3.47)

Multiplying the second equation in the above system by uj and then summing over j, we obtain, by taking into account conditions (3.18) and (3.32) N     2 2 ∂x uj  = 0 j =1

which implies that u = 0 in H 2 (G).

 

Next we prove that S(t)t≥0 satisfies (1.10) in Theorem 1.25. Lemma 3.5 Let A be the operator given by (3.27), then condition (1.10) holds for the semigroup (S(t))t≥0 .

48

3 Exponential Stability of a Network of Beams

Proof Suppose that (1.10) is not true, then there exists a sequence (βn ) of real numbers, with βn −→ ∞ (βn > 0 without loss of generality) and a sequence of vectors (yn ) = (un , v n , θ n ) in D(A) with yn H = 1, such that (iβn − A)yn H −→ 0. Writing this condition term by term we get, iβn uj,n − vj,n = fj,n −→ 0,

in H 2 (0, j ),

(3.48)

iβn vj,n + ∂x4 uj,n − γj ∂x2 θj,n = gj,n −→ 0,

in L2 (0, j ),

(3.49)

iβn θj,n − ∂x2 θj,n

+ γj ∂x2 vj,n

= hj,n −→ 0,

2

in L (0, j ),

(3.50)

for j in {1, . . . , N }. Step 1 Since Re((iβn − A)yn , yn H ) =

N   " ∂x θj,n  , we obtain

j =1

  ∂x θj,n  −→ 0, for j = 1, . . . , N which implies θj,n −→ 0, in L2 (0, j ) for j = 1, . . . , N, by (3.30), (3.31) and the fact that every maximal subgraph of thermoelastic edges is not a circuit. 2 te Step 2 We will prove that ∂x2 uj,n −→ 0 and vj,n  2−→0 in L (0, j ) for j in J . ∂ v  Let j in J te . From (3.48) we deduce that  xβnj,n  is bounded. Then it is easy  2  ∂ θ  to deduce, respectively, from (3.50) and (3.49), the boundedness of  xβnj,n  and  4   ∂x uj,n   βn .

Using (3.48) again to replace ∂x2 vj,n by βn ∂x2 uj,n in (3.50), multiplying the new ∂2u

equation by xβnj,n , we obtain, using the fact that θj,n −→ 0 and that ∂x2 uj,n is bounded in L2 (0, j ),  2  1  2   γj ∂x2 uj,n  − ∂x θj,n , γj ∂x2 uj,n −→ 0 in L2 (0, j ). βn Integrating by parts, we obtain,  2 γ %x=j  γj  j   % ∂x θj,n , ∂x3 uj,n −→ 0. ∂x θj,n ∂x2 uj,n % + γj ∂x2 uj,n  − x=0 βn βn

(3.51)

2 Exponential Decay

49

We want prove that the second and third terms in the left hand side of the above expression tend to 0 as n tends to ∞. To do this we will apply Theorem 1.33 more than one time. Inequality (1.14), applied to w = ∂x2 uj,n gives    2   ∂ 4 u 1/2  1/2 ∂ uj,n   x j,n   2  x ≤ C3  .  ∂x uj,n  + C4 1/2  βn  βn

 3  ∂ uj,n  x

1/2

βn

It follows that

 3  ∂ uj,n  x

(3.52)

is bounded and then the third term in (3.51) tends to zero,

1/2

βn

since ∂x θj,n −→ 0 in L2 (0, j ). Inequality (1.13) applied, respectively, to w = ∂x2 uj,n and w = ∂x θj,n gives, respectively,   2 ∂ uj,n 

∞

x

1/4

βn

   2   1/2  ∂ 3 u 1/2 ∂ uj,n   2   x j,n  x ≤ C1 ∂x uj,n   1/2  + C2 1/4  βn  βn

and   ∂x θj,n 

∞

1/2

βn which imply that

 1/2    2θ ∂x θj,n   1/2  ∂ j,n   x ≤ C1 ∂x θj,n    + C2 1/2  βn  βn

 2  ∂ uj,n  x

1/4

βn

∞

is bounded and

∂x θj,n ∞ 1/2

βn

tends to zero. Hence, the

second term in (3.51) converges to zero. Thus, (3.51) is reduced to 2    2 ∂x uj,n  −→ 0.

(3.53)

Going back to (3.48) and (3.50), we get, respectively, ∂x2 vj,n −→ 0 in L2 (0, j ) βn and ∂x2 θj,n −→ 0 in L2 (0, j ). βn Dividing (3.49) by βn , multiplying the result by vj,n and integrating by parts, we obtain %x=j   2 1 3 1  3 % i vj,n  + ∂x uj,n , ∂x vj,n −→ 0. ∂x uj,n vj,n % − x=0 βn βn

(3.54)

50

3 Exponential Stability of a Network of Beams

Rewriting again (1.13) with w = ∂x3 uj,n and w = vj,n , respectively, and (1.14) 3/4 1/4 with w = ∂x vj,n , dividing the three inequalities, respectively, by βn , βn , and 1/2

βn , we deduce, using (3.52) and the boundedness of the third term in (3.54) converge to zero. It follows that

∂x4 uj,n βn ,

that the second and

vj,n −→ 0 in L2 (0, j ). Step 3 We will prove that ∂x2 uj,n −→ 0 and vj,n −→ 0 in L2 (0, j ) for j in J e . Let j in {1, . . . , N }. Combining (3.48) and (3.49), we obtain − βn2 uj,n + ∂x4 uj,n − γj ∂x2 θj,n = gj,n + iβn fj,n .

(3.55)

Let q be a function in C 2 ([0, j ], C) independent of n such that ∂x2 q = 0. Taking the inner product in L2 (0, j ) of (3.55) with q∂x uj,n , and integrating by parts, we obtain %x=j %x=j %2 %2 % 1 % 1 %% % % − βn2 %uj,n (x)% q(x)% − %∂x2 uj,n (x)% q(x)%% x=0 2 2 x=0  %x=j   %x=  % − Re iβn fj,n (x)q(x)uj,n (x)%x=0j + Re ∂x3 uj,n (x)q(x)∂x uj,n (x)% x=0

%x=  1 j 2 % %2 − γj Re ∂x θj,n (x)q(x)∂x uj,n (x)%x=0j + βn %uj,n % ∂x qdx 2 0

j

j

j % % 3 % 2 %2 + gj,n q∂x uj,n dx − iβn uj,n ∂x (qfj,n )dx %∂x uj,n % ∂x qdx = 2 0 0 0 

j %x=j  % . (3.56) − ∂x θj,n ∂x (q∂x uj,n )dx + Re ∂x2 uj,n (x)∂x uj,n (x)∂x q(x)% 

x=0

0

It is easy to verify that the first three terms in the right hand side of the above equation converge to zero. In the edge ej , the last term in (3.56) converges to zero  case of thermoelastic    ∂x2 uj,n    1/4  β since  and ∂ u  converge to zero by (1.13). Thus, equation  x j,n n  βn1/4  ∞ ∞ (3.56) is reduced to %x=j %x=j %2 %2 % 1 %% 1 % % % − %∂x2 uj,n (x)% q(x)%% − βn2 %uj,n (x)% q(x)% x=0 2 2 x=0  %x=j   %x=  % − Re iβn fj,n (x)q(x)uj,n (x)%x=0j + Re ∂x3 uj,n (x)q(x)∂x uj,n (x)% x=0

2 Exponential Decay

51

 %x=  1 j 2 % %2 − γj Re ∂x θj,n (x)q(x)∂x uj,n (x)%x=0j + βn %uj,n % ∂x qdx 2 0

j % % 3 % 2 %2 + %∂x uj,n % ∂x qdx −→ 0. 2 0

(3.57)

In particular, if ak is an end of ej then by taking q(x) = x or q(x) = j − x we have using results of step 1 and step 2, %2   %2 1 %% 1 % % − βn2 %uj,n (ak )% − %∂x2 uj,n (ak )% + Re ∂x3 uj,n (ak )∂x uj,n (ak ) 2 2     − Re iβn fj,n (ak )uj,n (ak ) − γj Re ∂x θj,n (ak )∂x uj,n (ak ) −→ 0.

(3.58)

Now, we show that for every interior end ak of a thermoelastic edge ej , ∂ 3 u (ak ) 1/2 , ∂x2 uj,n (ak ), βn ∂x uj,n (ak ), x j,n 1/2 βn

j

and βn uj,n (ak ) tend to zero.

Let j be in {1, . . . , N} and take the inner product of (3.55) with yields

j

1/2 3/2 βn e−βn (j −x) uj,n (x)dx

−

1/2 βn

0

= +

γj 1/2 βn

∂x θj,n (x)e



−1 1/2 βn



1 j



j

1/2

e−βn

0

1/2

e−βn

(j −x) ,

(j −x) 4 ∂x uj,n (x)dx

1/2  gj,n + iβn fj,n e−βn (j −x) dx

0

%x=j − γj %

−βn (j −x) % 1/2

1 1/2 βn

x=0

j

1/2

∂x θj,n e−βn

(j −x)

dx.

0

(3.59) The right hand side of (3.59) converges to zero since ∂x θj,n converges to zero and 1/2

e−βn (j −x) is bounded. Performing integration by parts to the left hand side, we obtain

j

1/2

βn e−βn 3/2

(j −x)

uj,n (x)dx −

0

=−

1 1/2

βn

1/2

−βn

1/2

∂x3 uj,n (x)e−βn

1/2

∂x uj,n (x)e−βn

1 1/2 βn

%x=j (j −x) % % x=0

0

j

1/2

e−βn

(j −x) 4 ∂x uj,n (x)dx 1/2

+ ∂x2 uj,n (x)e−βn

%x=j (j −x) % % x=0

%x=j %x=j 1/2 % + βn uj,n (x)e−βn (j −x )% . %

(j −x) %

x=0

x=0

52

3 Exponential Stability of a Network of Beams

It follows that −

1 3 1/2 ∂ u ( ) + ∂x2 uj,n (j ) − βn ∂x uj,n (j ) + βn uj,n (j ) 1/2 x j,n j βn

−→ 0. 1/2

We have used the fact that ∀α ∈ R, ∀k ∈ {0, 1, 2, 3}, βnα ∂xk uj,n (0)e−βn j tend to 0. 1 −βn1/2 x 1 −βn1/2 (j −x) With the same manner, by considering 1/2 e instead of 1/2 e , βn

we obtain that

βn

1 3 1/2 ∂ u (0) + ∂x2 uj,n (0) + βn ∂x uj,n (0) + βn uj,n (0) 1/2 x j,n βn

−→ 0.

In conclusion, for every inner node ak , 1 1/2 d ∂ 3 u (a ) + ε∂x2 uj,n (ak ) + βn dkj ∂x uj,n (ak ) + εβn uj,n (ak ) 1/2 kj x j,n k βn

−→ 0. (3.60)

with ε ∈ {−1, 1}. Summing over j ∈ Jk , this leads to ∂x2 uj,n (ak ) + βn uj,n (ak ) −→ 0.    ∂x θj,n   We have used the conditions (3.18) and (3.32), the fact that   1/2  βn

converges to

∞

zero, and the continuity of uj,n and ∂x2 uj,n at ak . Going back to (3.60), we get 1 1/2 d ∂ 3 u (a ) + βn dkj ∂x uj,n (ak ) 1/2 kj x j,n k βn

j

= λn −→ 0.

(3.61)

We fix j in J te and ak as an interior end of ej . Then, we have the following three inequalities: $ ∂x3 uj,n (ak ) 1/2 β ∂ u (a ) x j,n k n 1/2 βn   j 1/2 1/2 = Re (λn − βn ∂x uj,n (ak ))βn ∂x uj,n (ak ) %2 a % %2 % 1 %% j %%2 ≤ −βn %∂x uj,n (ak )% + βn %∂x uj,n (ak )% + %λn % , 2 2a

  Re ∂x3 uj,n (ak )∂x uj,n (ak ) = Re

#

%2 % %2   b % 1 %∂x θj,n (ak )% % % Re ∂x θj,n (ak )∂x uj,n (ak ) ≤ βn ∂x uj,n (ak ) + 2 2b βn

2 Exponential Decay

53

and %2 %2   c % 1 %% −Re iβn fj,n (ak )uj,n (ak ) ≤ βn2 %uj,n (ak )% + fj,n (ak )% 2 2c for any positive numbers a, b, and c. Going back to (3.58), %2   1% %2 1 % % % − βn2 %uj,n (ak )% + Re ∂x3 uj,n (ak )∂x uj,n (ak ) − %∂x2 uj,n (ak )% 2 2     − Re ∂x θj,n (ak )∂x uj,n (ak ) − Re iβn fj,n (ak )uj,n (ak ) %2 % %2 1 %% 1 %∂x θj,n (ak )% 1 %% j %%2 % f − (a ) − − %λn % j,n k 2a 2c 2b βn     %2 %2 %2 1 %% % % c b 1 a % βn2 %uj,n (ak )% + −1 + + βn %∂x uj,n (ak )% − %∂x2 uj,n (ak )% ≤ − + 2 2 2 2 2 ≤0

for a = b = c =

1 . Using that 4

%2 % %2 1 %% j %%2 1 %% 1 %∂x θj,n (ak )% % − %λn % − fj,n (ak ) − −→ 0 2a 2c 2d βn we deduce %2 % βn2 %uj,n (ak )% −→ 0,

% %2 %2 % % 2 % %∂x uj,n (ak )% and βn %∂x uj,n (ak )% −→ 0.

Moreover, all the expressions       Re ∂x3 uj,n (ak )∂x uj,n (ak ) , Re ∂x θj,n (ak )∂x uj,n (ak ) and Re iβn fj,n (ak )uj,n (ak )

converge to 0. Now, let an elastic edge ej be attached to a thermoelastic one at an internal node ak . Take q = x or q = j − x in (3.56). By using the continuity conditions of uj,n and ∂x2 uj,n at ak and the damping conditions (3.18) and (3.32), we have  

j % %2 %x=j  %2 % % % % 2 −→ 0. %∂x uj,n (x)% + 3βn2 %uj,n (x)% dx − 2Re ∂x2 uj,n (x)∂x uj,n (x)% 0

x=0

(3.62)

54

3 Exponential Stability of a Network of Beams

Similarly, taking q = 1 in (3.56) we obtain %2 %2 1 %% 1 % % − βn2 %uj,n (as )% − %∂x2 uj,n (as )% 2 2     + Re ∂x3 uj,n (as )∂x uj,n (as ) − Re iβn fj,n (as )uj,n (as ) −→ 0,

(3.63)

where as is the second end of ej . As for a thermoelastic edge we prove that all the expressions in the left hand side of (3.63) converge to zero, then

j

0

%  %2 %2 % % 2 % 2% % dx −→ 0. %∂x uj,n (x)% + 3βn uj,n (x)

(3.64)

We iterate such procedure in each maximal subgraph of elastic edges of G to obtain (3.64) for all j in J e . In summary, we have yn H −→ 0. This result contradicts the hypothesis that yn has the unit norm.   We can now state the main result of this chapter. Theorem 3.6 The C0 -semigroup S(t), generated by the operator A, is exponentially stable. More precisely, the energy of the whole system (3.1)–(3.14) decays exponentially to zero. Proof The proof is a direct consequence of Lemmas 3.4 and 3.5.

3 Comment If we replace the continuity condition of θ at inner nodes, θj (ak ) = θl (ak ) j, l ∈ J te (ak ), ak ∈ Vint and the condition  j ∈J te (a

dkj (γj uxt (ak , t) − θj,x (ak , t)) = 0, ak ∈ Vint k)

by the following: θj (ak ) = 0 j ∈ J te (ak ), ak ∈ Vint

 

3 Comment

55

and Kirchhoff’s law, 

dkj θj,x (ak , t) = 0, ak ∈ Vint

j ∈J te (ak )

then we obtain the same results. Furthermore, if we consider the following boundary conditions: uj (ak , t) = 0, uj,x (ak , t) = 0, θj (ak , t) = 0,

j ∈ Jk , ak ∈

uj (ak , t) = ul (ak , t)

j, l ∈ Jk , ak ∈ Vint ,

θj (ak , t) = θl (ak , t)

j, l ∈ Jkte , ak ∈ Vint ,

uj,x (ak , t) = ul,x (ak , t)  j ∈Jkte



j, l ∈ Jk , ak ∈ Vint ,

  dkj uj,xx (ak , t) − γj θj (ak , t) = 0, ak ∈ Vint ,    dkj uj,xxx (ak , t) − γj θj,x (ak , t) + dkj uj,xxx (ak , t) = 0, ak ∈ Vint , j ∈Jke

j ∈Jkte



j ∈ Jk , ak ∈ Vext ,  Vext ,

dkj θj,x (ak , t) = 0, ak ∈ Vint ,

j ∈Jkte

we can prove that the energy of the system decays exponentially to zero.

Chapter 4

Stability of a Tree-Shaped Network of Strings and Beams

In past decades, the dynamic behavior of networks of flexible structures has been studied by some authors, see, for instance, [7, 18, 49] and the references therein. The importance of these studies lies in the need for engineering to eliminate vibrations in such composite structures. In this chapter, we consider a model (S) of a tree-shaped network of N elastic materials, constituting strings and Euler–Bernoulli beams. Suppose that the equilibrium position of the tree of elastic strings and beams coincides with the tree T of N edges, e1 , . . . , eN and p = N + 1 vertices, a1 , . . . , ap ; and recall that I = {1, . . . , p} and J = {1, . . . , N } denote, respectively, the set of indices of the vertices and the set of indices of edges. We suppose that a1 is the root of T , and that a1 and a2 are ends of e1 . Our model is then described as follows: Every string ej satisfies the following equation: uj,tt − uj,xx = 0 in (0, j ) × (0, ∞),

(4.1)

and every beam ej satisfies the following equation: uj,tt + uj,xxxx = 0 in (0, j ) × (0, ∞),

(4.2)

where uj = uj (x, t) is the function describing the displacement of the string or beam ej . The initial conditions are uj (x, 0) = u0j (x), uj,t (x, 0) = u1j (x).

(4.3)

Denote by Jks (resp. Jkb ), the set of indices strings (resp. beams) adjacent to ak and by s and V b , respectively, the set of external nodes of strings and those of beams, Vext ext

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Ammari, F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-86351-7_4

57

58

4 Stability of a Tree-Shaped Network of Strings and Beams

different from a1 . Then, the transmission conditions at the inner nodes are ⎧ uj (ak , t) = ul (ak , t), j, l ∈ Jk , ak ∈ Vint , ⎪ ⎪ ⎪ ⎪ ⎪ (a , t) = u (a , t), j, l ∈ Jkb , ak ∈ Vint , u j,xx k l,xx k ⎪ ⎨ " dkj uj,x (ak , t) = 0, ak ∈ Vint , b ⎪ j ∈J ⎪ "k " ⎪ ⎪ ⎪ dkj uj,xxx (ak , t) − dkj uj,x (ak , t) = 0, ak ∈ Vint , ⎪ ⎩

(4.4)

j ∈Jks

j ∈Jkb

and the boundary conditions are 

ujk (ak , t) = 0, ak ∈ Vext , b , ujk ,xx (ak , t) = 0, ak ∈ Vext

(4.5)

where jk is the index of the unique edge adjacent to ak ∈ Vext . For a classical solution u of (S), the energy is defined as the sum of the energy of its components, that is, 1 E(t) = 2 N



j =1 0

j

 j % % % % %uj,t (x, t)%2 dx + 1 %uj,x (x, t)%2 dx 2 s 0

%2 1  j %% uj,xx (x, t)% dx, + 2 b 0

j ∈J

j ∈J

where J s and J b are the respective index sets of strings and beams in the tree. Differentiate formally the energy function with respect to time t, we get dE (t) = 0, dt and the system is conservative. Stability of such models of networks of strings or of beams has been proved before, by applying a control at an external node or by forcing the damping conditions at inner nodes. In [4], the authors prove the polynomial stability of a star-shaped network of strings when a feedback is applied at the common node, and in [10] and [94], the authors prove a similar result for a tree of strings when the feedback is applied at an external node. In [3], we considered a network of beams. See also [100] for exponential stability of a star-shaped network of beams and [35] for asymptotic stability of a star-shaped network of Timoshenko beams. In [88] and [89] we added thermoelastic edges to the network of elastic materials to obtain an exponential stability result. For strings-beams networks, see [5] where the authors considered a star-shaped network of beams and a string, with controls applied at all the exterior nodes. They proved a result of exponential stability. Some results of polynomial stability have

4 Stability of a Tree-Shaped Network of Strings and Beams

59

Fig. 4.1 First tree

Fig. 4.2 Second tree

proved before, for coupled string-beam systems [11] (see [7] for general setting) and for chains of alternated beams and strings [12], when feedbacks are applied at inner nodes. The case of a 2 − d coupled system of a wave equation and a plate equation has been studied by Ammari and Nicaise in [6]. They proved a result of exponential stability under some geometric conditions. In this chapter, we study a more general case of networks, in fact, it is the model (S) presented at the beginning, stabilized by applying feedbacks at all leaves (the ∗ =V root remains free): Figs. 4.1 and 4.2. For this, let V ∗ = V − {a1 }, Vext ext − {a1 } and let δ in {0, 1} with δ = 1 if e1 is a string and δ = 0 if e1 is a beam. Then instead of (4.5) we take u1 (a1 , t) = 0, (1 − δ)u1,xx (a1 , t) = 0, ujk ,x (ak , t) = −dkj ujk ,t (ak , t),

s ak ∈ Vext ,

ujk ,xxx (ak , t) = dkj ujk ,t (ak , t), ujk ,x (ak , t) = 0,

b ak ∈ Vext ,

where some derivative boundary feedbacks are applied at external nodes (except the root).

60

4 Stability of a Tree-Shaped Network of Strings and Beams

Formally, we have  % % dE %uj ,t (ak , t)%2 ≤ 0. (t) = − k dt ∗ ak ∈Vext

So the system is dissipative. We prove different decay results of the energy of the system depending on the position of beams relative to strings. Precisely, (S) is exponentially stable if there is no beam following a string from the root to leaves and polynomially stable if not. Moreover, we give an example corresponding to the last case which is not exponentially stable. The method that we use to show exponential or polynomial stability is based on the resolvent approach. From now, for simplicity, we use sometimes the following two notations: un = o(vn ) and un = O(vn ) for two sequences of complex numbers un and vn where vn is non-zero for n large enough; they mean uvnn −→ 0 as n −→ ∞ and uvnn is bounded, respectively The chapter is organized as follows: In Sect. 1, we reformulate the system (S) as an evolution equation in a Hilbert space and prove that it is associated with a C0 -semigroup of contractions, and in Sect. 2, by using frequency domain method, we first prove, under some conditions, that the system (S) is exponentially stable, then we give a result of polynomial stability.

1 Abstract Setting First, Recall that J s and J b are the respective sets of indices of strings and beams in the tree. Then, for a function f on T , we set f s = (fj )j ∈J s and f b = (fj )j ∈J b , and we rewrite f as f = (f s , f b ). The aim of this section is to rewrite the system (S) as an evolution equation in an appropriate Hilbert space. We then prove the existence and uniqueness of solutions of the problem using semigroup theory. Let us consider ⎫ ⎧ ⎬ ⎨   H 1 (0, j ) × H 2 (0, j ) | f satisfies (4.6) , V = f = (f s , f b ) ∈ ⎭ ⎩ s b j ∈J

j ∈J

where ⎧ f1 (a1 ) = 0, ⎪ ⎪ ⎪ ⎪ ⎨ fj (ak ) = fl (ak ), j, l ∈ Jk , ak ∈ Vint , b , ∂x fjk (ak ) = 0, ak ∈ Vext ⎪ " ⎪ ⎪ dkj ∂x fj (ak ) = 0, ak ∈ Vint . ⎪ ⎩ j ∈J b (ak )

(4.6)

1 Abstract Setting

61

Note that we can rewrite the last two equations in one, as follows: 

dkj ∂x fj (ak ) = 0, ak ∈ V ∗ .

j ∈Jkb

Define the energy space of (S) by H=V ×

N 

L2 (0, j )

j =1

endowed by the inner product y, y ˜ H :=



N       ∂x fj , ∂x f˜j + ∂x2 fj , ∂x2 f˜j + gj , g˜ j ,

j ∈J s

j =1

j ∈J b

˜ Then, H is a Hilbert space. where y = (f , g), and y˜ = (f˜, g), Now define the operator A in H by D (A ) =

⎧ ⎫ 6 2 6 4 ⎨ y = (us , ub , vs , vb ) ∈ V × V | us ∈ H (0, j ), ub ∈ H (0, j ) ⎬ j ∈J s

⎩

j ∈J b

and y satisfies (4.7)

⎭

,

where ⎧ s , ∂x ujk (ak ) = −dkj vjk (ak ), ak ∈ Vext ⎪ ⎪ ⎪ 2 ⎪ (1 − δ)∂x u1 (a1 ) = 0, ⎪ ⎪ ⎨ 2 ∂x uj (ak ) = ∂x2 ul (ak ), j, l ∈ Jkb , ak ∈ Vint , b , ⎪ ak ∈ Vext ∂x3 ujk (ak ) = dkj vjk (ak ), ⎪ ⎪ " " ⎪ 3 ⎪ dkj ∂x uj (ak ) − dkj ∂x uj (ak ) = 0, ak ∈ Vint , ⎪ ⎩ j ∈Jks

j ∈Jkb

and ⎛

⎞ ⎛ ⎞ us vs ⎜ ub ⎟ ⎜ vb ⎟ ⎟ ⎜ ⎟ A⎜ ⎝ vs ⎠ = ⎝ ∂ 2 us ⎠ , (us , ub , vs , vb ) ∈ D(A). x vb −∂x4 ub

(4.7)

62

4 Stability of a Tree-Shaped Network of Strings and Beams

Then, putting y = (u, ut ), we write the system (S) into the following first-order evolution equation: 

dy dt

= Ay, y(0) = y 0 ,

(4.8)

on the energy space H, where y0 = (u0 , u1 ). We have the following result: Lemma 4.1 The operator A is the infinitesimal generator of a C0 -semigroup of contractions (T (t))t≥0 . Proof By Lumer–Phillips Theorem (Theorem 1.15), it suffices to show that A is m-dissipative. First, for any y ∈ D(A) we have ⎛ Re(Ay, yH ) = Re ⎝



0

j ∈J s

+



j ∈J b

j

(∂x vj ∂x uj dx + ∂x2 uj vj )dx ⎞

j

0

(∂x2 vj ∂x2 uj dx − ∂x4 uj vj )dx ⎠ .

Using integration by parts, we obtain using, boundary and transmission conditions (4.6) and (4.7), Re(Ay, yH ) = −

 % % %vj (ak )%2 ≤ 0. k ∗ ak ∈Vext

Then the operator A is dissipative. We show now that every positive real number λ belongs to ρ(A). Let z = (f , g) ∈ H, we look for y = (u, v) ∈ D(A) such that (λ − A)y = z, i.e., λuj − vj = fj , j = 1, . . . , N,

(4.9)

λvj − ∂x2 uj = gj , j ∈ J s ,

(4.10)

λvj + ∂x4 uj = gj , j ∈ J b .

(4.11)

1 Abstract Setting

63

Then, λ2 uj − ∂x2 uj = gj + λfj , j ∈ J s ,

(4.12)

λ2 uj + ∂x4 uj = gj + λfj , j ∈ J b .

(4.13)

Let w in V . Multiplying the first equation by ws and the second equation by wb , and integrating by parts, we get, respectively,



j

λ2

j

uj wj dx +

0

0

% ∂x uj ∂x wj dx − ∂x uj wj %0j =



j

(gj + λfj )wj dx,

0

for j in J s and



j

λ2 0

j

uj wj dx + 0



j

=

%j %j % % ∂x2 uj ∂x2 wj dx + ∂x3 uj wj % − ∂x2 uj ∂x wj % 0

0

(gj + λfj )wj dx,

0

for j in J b . Then, summing the two obtained equations, the left hand side will be λ2

N



j

j =1 0

+

uj wj dx +

j ∈J s

⎛

j

∂x uj ∂x wj dx +

0



j ∈J b

j 0

∂x2 uj ∂x2 wj dx ⎞

 ⎜   ⎟ dkj wj (ak )∂x uj (ak ) + dkj wj (ak )∂x3 uj (ak )⎠ ⎝− ak ∈Vint

+





j ∈Jks

 

vjk (ak )wjk (ak ) −

ak ∈Vext

j ∈Jkb

dkj ∂x2 uj (ak )∂x wj (ak ).

ak ∈Vint j ∈J b k

We find, by taking into account (4.6) and (4.7) a(u, w) = F (w),

(4.14)

where a(u, w) = λ2

N



j

j =1 0

+λ



ak ∈Vext

uj wj dx +



j ∈J s

ujk (ak )wjk (ak )

j 0

∂x uj ∂x wj dx +



j ∈J b

j 0

∂x2 uj ∂x2 wj dx

64

4 Stability of a Tree-Shaped Network of Strings and Beams

and F (w) =

N



j



(gj + λfj )wj dx + λ

j =1 0

fjk (ak )wjk (ak ).

ak ∈Vext

a is a continuous sesquilinear form on V × V , and F is a continuous anti-linear form on V . Moreover, there exists C > 0 such that, for every w ∈ V ,   % % %a(w, w)% ≥ C w 2 , V where N   2 u = V



j =1 0

j



% %2 %uj % dx + j ∈J s

j



% % %∂x uj %2 dx +

0

j ∈J b

0

j

% % % 2 %2 %∂x uj % dx.

By the Lax–Milgram’s lemma, problem solution u in V . It is 6 2 (4.14) has6a unique easy to verify that: u belongs to H (0, j ) × H 4 (0, j ), v = λu − f ∈ V , j ∈J s

j ∈J b

us and ub satisfies, respectively, (4.10) and (4.11), and the conditions ⎧ s , ∂x ujk (ak ) = −dkj vjk (ak ), ak ∈ Vext ⎪ ⎪ ⎪ 2 ⎪ (1 − δ)∂x u1 (1 ) = 0, ⎪ ⎪ ⎨ 2 ∂x uj (ak ) = ∂x2 ul (ak ), j, l = Jkb , ak ∈ Vint , b , ⎪ ak ∈ Vext ∂x3 ujk (ak ) = dkj vjk (ak ), ⎪ ⎪ " " ⎪ 3 ⎪ dkj ∂x uj (ak ) − dkj ∂x uj (ak ) = 0, ak ∈ Vint . ⎪ ⎩ j ∈Jks

j ∈Jkb

Furthermore, y2H ≤ c z2H , where c is a positive constant independent of y. In conclusion, y = (u, v) ∈ D(A) and (λ − A)−1 ∈ L(H), that is, λ ∈ ρ(A).   Corollary 4.2 For an initial datum y 0 ∈ H, there exists a unique solution y ∈ C([0, +∞), H) of problem (4.8). Moreover, if y 0 ∈ D(A), then y ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞), H). Remark 4.3 Note that, by the Sobolev embedding theorem, we deduce that the canonical injection D(A) → H is compact, then the operator A has compact resolvent (Proposition 1.5). Hence, the spectrum of A consists of all isolated eigenvalues, i.e., σ (A) = σp (A).

2 Asymptotic Behavior

65

2 Asymptotic Behavior The aim of this section is to show that the system (S) is asymptotically stable. Moreover, we will prove that the solution is exponentially stable if there is no beam following a string, from the root to the leaves, as in the first tree (Fig. 4.1), and polynomially stable if at least a beam follows a string, as in the second tree (Fig. 4.2). Finally, the lack of exponential stability is proved on an example. For the sequel, we need some definitions and notations. We say that a string verifies the NFB property if it is not followed by any beam, and such string is called a NFB-string. A beam satisfies the NFS property if it is not followed by any string and is not attached simultaneously to a string and a beam that is followed by a string; it is called a NFS-beam. The NFB-strings form the first layer of strings; it may be empty, and then we remove it. The first layer of beams is formed of NFS-beams in the new tree T1 . The second layer of strings is composed of NFB-strings of the tree T2 obtained from T1 by removing the NFS-beams, etc. Let r be the number of layers of beams in T not containing e1 . Note that r ≥ 1. Returning back to the tree in Fig. 4.2; the NFB-strings of T are grouped into two connected components {e12 }, and {e14 , e18 , e16 , e20 }. Then the first layer of beams is the union of its three connected components {e4 , e9 , e5 , e10 }, {e11 }, and {e13 , e17 , e15 , e19 }. For a subgraph G of T , we denote by Vext (G) the set of the endpoints of G, except the outer node (the nearest node of G to the root), and by Vint (G) the set of the inner nodes, except the outer node. Finally, if ak is a node of G, then we denote by Jk (G) the index set of edges belonging to G and attached to ak .

2.1 Asymptotic Stability In this section, we prove condition (1.9) in Theorem 1.25, then the system (S) is asymptotically stable. Theorem 4.4 The semigroup (T (t))t≥0 , generated by the operator A is asymptotically stable. Proof Using Corollary 1.22 it suffices to show that iR ⊂ ρ(A); otherwise, by taking into account Remark 4.3, there is a real number β, such that λ := iβ is an eigenvalue of A. Let y = (u, v) the corresponding eigenvector. We have ⎧ for j in {1, . . . , N }, ⎨ vj = λuj 2 ∂ u = λvj for j in J s , ⎩ x 4j −∂x uj = λvj for j in J b .

(4.15)

66

4 Stability of a Tree-Shaped Network of Strings and Beams

If λ = 0, multiplying the second and the third equations of (4.15) by uj and summing, we obtain, using (4.6) and (4.7),      2 2 ∂x uj 2 + ∂x uj  = 0 j ∈J s

j ∈J b

which implies that uj is constant on ej for every j ∈ J e and ∂x uj is constant on ej for every j ∈ J b . b , ∂ u = 0, then u is constant on e . Moreover, using the fourth For ak ∈ Vext x jk jk jk condition in (4.6) we deduce that uj is constant on ej for every j ∈ J b . Hence, u = 0, by the first and the second condition in (4.6). Using again (4.15), with λ = 0, we deduce that v = 0. Thus y = 0. In the sequel, we suppose that λ = 0. Taking the real part of the inner product of λy − Ay = 0 with y in H, we obtain Re(Ay, yH ) = −

 % % %vj (ak )%2 = 0. k ∗ ak ∈Vext

∗ and then u (a ) = 0 for a ∈ V ∗ , which holds Thus vjk (ak ) = 0 for ak ∈ Vext jk k k ext b and ∂ u (a ) = 0 for a ∈ V s . for k = 1; ∂x3 ujk (ak ) = 0 for ak ∈ Vext x jk k k ext Then, u is zero on every edge attached to a leaf, and by iteration, on every maximal subgraph of strings not followed by beams. Now let G be a maximal subgraph of beams not followed by strings. We want to prove that u is zero on G.

First Case G = T . For each j in {1, . . . , N}, substituting the first equation of (4.15) into the third, we obtain, ∂x4 uj + λ2 uj = 0.

(4.16)

Again, as in the previous chapter, we use a matrix method as follows (see page 40 for any explanation on the notations used): To a function f on T is associated the matrix function F defined by F : [0, 1] −→ Cp×p , x −→ F (x) = (fj k (x))p×p , with 3 fj k (x) = ej k fs(j,k) s(j,k)



1 + dj s(j,k) − xdj s(j,k) 2

4 .

The system (4.16) is then rewritten as L(−4) ∗ U  + λ2 U = 0.

(4.17)

2 Asymptotic Behavior

67

Integrating equation (4.17), we obtain 7 7 U (x) = A1 ∗ cos( βLx) + A2 ∗ sin( βLx) 7 7 + B1 ∗ cosh( βLx) + B2 ∗ sinh( βLx),

(4.18)

where without loss of generality we have supposed that β > 0 and with A1 , A2 , B1 , B2 ∈ Cp×p . Then, 7  7 7 L(−1) ∗ U  = β −A1 ∗ sin( βLx) + A2 ∗ cos( βLx)  7 7 +B1 ∗ sinh( βx) + B2 ∗ cosh( βLx) , (4.19)  7 7 L(−2) ∗ U  = β −A1 ∗ cos( βLx) − A2 ∗ sin( βLx)  7 7 +B1 ∗ cosh( βx) + B2 ∗ sinh( βLx) , (4.20)  7 7 L(−3) ∗ U  = β 3/2 A1 ∗ sin( βLx) − A2 ∗ cos( βLx)  7 7 +B1 ∗ sinh( βx) + B2 ∗ cosh( βLx) . (4.21) The function U satisfies also, U (1 − x) = U (x)T .

(4.22)

The boundary and transmission conditions can be expressed as follows: ⎛ ⎞ ϕ1 ⎜ .. ⎟ For the continuity condition of u at the inner nodes, there exists ϕ = ⎝ . ⎠ ∈ Cp ϕp such that U (0) = (ϕeT ) ∗ E,

(4.23)

⎛ ⎞ 1 ⎜ .. ⎟ where e = ⎝ . ⎠ ∈ Rp . 1 Since u is zero at all external nodes, then ϕk = 0 when ak is an external node. The continuity condition of ∂x2 u at the interior nodes and the fact that ∂x2 u is zero at the root can be expressed in this manner,

68

4 Stability of a Tree-Shaped Network of Strings and Beams

⎞ ψ1 ⎟ ⎜ there exists ψ = ⎝ ... ⎠ ∈ Cp such that ψ1 = 0, and ⎛

ψp L(−2) ∗ U  (0) = (ψeT ) ∗ E.

(4.24)

The fourth condition of (4.6) and the fifth of (4.7) applied to u are expressed, respectively, as follows: (L(−1) ∗ U  (0) ∗ E ∗ )e = 0,

(4.25)

(L(−3) ∗ U  (0) ∗ E ∗ )e = 0,

(4.26)

and

where E ∗ is obtained from E by annulling the first line. Substituting (4.23)–(4.24) in (4.19)–(4.21), and (4.25)-(4.26) in (4.18)–(4.20) leads to 1 (U (0) − L(−2) ∗ U  (0)) = 2 1 B1 = (U (0) + L(−2) ∗ U  (0)) = 2

A1 =

1 ((ϕ − ψ)eT ) ∗ E, 2 1 ((ϕ + ψ)eT ) ∗ E, 2

(4.27) (4.28)

and (A2 ∗ E ∗ )e = 0,

(4.29)

∗

(4.30)

(B2 ∗ E )e = 0.

By taking x = 1 in (4.18) and (4.20) and using (4.22), we get, by combining the two obtained equations, 7 7 sinh(−1) ( βL) ∗ (B1T − B1 cosh( βL)) = B2 .

(4.31)

Multiplying, in the Hadamard product, the above equation by E ∗ , we get, using (4.30)   7 7 sinh(−1) ( βL) ∗ (B1T − B1 cosh( βL)) ∗ E ∗ e = 0.

(4.32)

We recall the following elementary rules for a matrix M ∈ Cp×p (see [22]): (M ∗ B1T )e = M(ϕ + ψ), (M ∗ B1 )e = diag(Me)(ϕ + ψ).

(4.33)

2 Asymptotic Behavior

69

Then, (4.32) implies J (ϕ + ψ) = 0, where   7 7 7 J = sinh(−1) ( βL) ∗ E ∗ − diag sinh(−1) ( βL) ∗ cosh( βL) ∗ E ∗ e . The matrix, obtained from J ∗ E ∗T ∗ E ∗ by removing rows and columns that are zero, is a strictly diagonally dominant matrix. Since ϕ1 = ψ1 = 0, this implies that the vector ϕ + ψ, and hence, the matrix B1 , is zero. Returning to (4.31), we deduce that B2 = 0. For j = 1, . . . , N, the expression of uj is then   7 7 j j j j uj (x) = a1 cos( βx) + a2 sin( βx), a1 , a2 ∈ C , which easily implies, using the transmissions conditions and the fact that u and ∂x u vanish at the leaves, that u = 0. Second Case G = T . Let a  be the nearest node of G to a1 . Then, a  is an end of at least one string. For simplicity of notations, we will suppose, in this part, that a  = a1 and G = T but with boundary conditions at a1 : ⎧ ⎪ ⎨ uj (a1 ) = ul (a1 ) j, l ∈ J1 , ∂x2 uj (a1 ) = ∂x2 ul (a1 ) j, l ∈ J1 , " ⎪ ⎩ j ∈J b d1j ∂x uj (a1 ) = 0. 1

⎛

⎞ ψ1 ⎜ ⎟ ∗ , As for the first case, there is ψ = ⎝ ... ⎠ in Cp with ϕk = 0 when ak ∈ Vext ψp such that U (0) = (ϕeT ) ∗ E and L(−2) ∗ U  (0) = (ψeT ) ∗ E.

70

4 Stability of a Tree-Shaped Network of Strings and Beams

The third and fourth conditions of (4.6) and the fifth of (4.7) applied to u are expressed as follows: (L(−1) ∗ U  (0) ∗ E)e = 0 and (L(−3) ∗ U  (0) ∗ E ∗ )e = 0. As in the first case, we obtain (4.27) to (4.31). Moreover, (ϕk + ψk )k=2,...,p will be the trivial solution of a homogeneous linear system whose matrix is invertible. Then, B1 ∗ E ∗ and B2 ∗ E ∗T ∗ E ∗ are zero. For j ∈ {1, . . . , N } − J1 , the expression of uj is then   7 7 j j j j uj (x) = a1 cos( βx) + a2 sin( βx), a1 , a2 ∈ C , which easily implies, by using the transmissions conditions and the fact that u and ∂x u vanish at the leaves, uj = 0. Then, we can suppose that G is a start of beams ej , j ∈ J1 . Without loss of generality, we identify ej with (0, j ) by taking πj (0) = a1 . In such case, we have the following system: ⎧ ⎪ u ( ) = 0, j ∈ J1 , ⎪ ⎪ j j ⎪ ⎨ ∂x uj (j ) = 0 and ∂x3 uj (j ) = 0, j ∈ J1 , uj (0) = ul (0) and ∂x2 uj (0) = ∂x2 ul (0), j, l ∈ J1 , ⎪ " ⎪ ⎪ ⎪ ⎩ ∂x uj (0) = 0. j

taking into account that uj (x) is of the form   7 7 7 7 j j j j j j j j uj (x) = a1 cos( βx) + a2 sin( βx) + b1 cosh( βx) + b2 sinh( βx) a1 , a2 , b1 , b2 ∈ C ,

This implies ⎧ √ √ √ √ j j j j ⎪ a1 cos( βj ) + a2 sin( βj ) + b1 cosh( βj ) + b2 sinh( βj ) = 0, j ∈ J1 , ⎪ ⎪ √ √ ⎪ j j ⎪ ⎪ −a sin( βj ) + a2 cos( βj ) = 0, j ∈ J1 , ⎪ ⎨ j 1 √ √ j b1 sinh( βj ) + b2 cosh( βj ) = 0, j ∈ J1 , j ⎪ ⎪ b1 = b1l , j, l ∈ J1 , ⎪ ⎪ " ⎪ j j ⎪ ⎪ ⎩ a2 + b2 = 0. j

2 Asymptotic Behavior

71

The discriminant of the above system is =



⎛ ⎝

j



⎞   7 7 7 cosh( βk ) sin( βj ) + sinh( βj ) ⎠ ,

k =j

which is different from zero. We conclude that uj = 0, j ∈ J1 . That is, u is null on G. By iteration, and using transmission conditions, we conclude that u is zero on T . The above discussion is sufficient to conclude that y = 0, which contradicts the fact that y = 0.  

2.2 Exponential Stability In this section, we suppose that there are no beam following a string (from the root to leaves), that is to say on every tree branch there is no beam between a string and a leaf (Fig. 4.1). We prove that the solution of the whole system (S) is exponentially stable. Note that in this case, all strings are NFB and there is at most only and only maximal subgraph of beams (which necessarily contains e1 ). Theorem 4.5 If there are no beam following a string, then the system (S) is exponentially stable. Proof It suffices to prove that (1.10) holds. Suppose the conclusion is false. Then, there exists a sequence (βn ) of real numbers, without loss of generality, with βn −→ +∞, and a sequence of vectors (yn ) = (un , v n ) in D(A) with yn H = 1, such that (iβn I − A)yn H −→ 0, which is equivalent to iβn uj,n − vj,n = fj,n −→ 0,

in H 1 (0, j ), j in J s ,

(4.34)

iβn uj,n − vj,n = fj,n −→ 0,

in H 2 (0, j ), j in J b ,

(4.35)

iβn vj,n − ∂x2 uj,n = gj,n −→ 0,

in L2 (0, j ), j in J s ,

(4.36)

iβn vj,n + ∂x4 uj,n = gj,n −→ 0,

in L2 (0, j ), j in J b .

(4.37)

Then, − βn2 uj,n − ∂x2 uj,n = gj,n + iβn fj,n , j in J s ,

(4.38)

−βn2 uj,n + ∂x4 uj,n = gj,n + iβn fj,n , j in J b ,

(4.39)

72

4 Stability of a Tree-Shaped Network of Strings and Beams

and     vj,n 2 − β 2 uj,n 2 −→ 0, j = 1, . . . , N. n

(4.40)

First, since Re((iβn − A)yn , yn H ) =

 % %vj

k ,n

%2 (ak )% ,

∗ ak ∈Vext

we obtain % %vj

k ,n

% ∗ (ak )% −→ 0, for ak ∈ Vext .

(4.41)

Next, we decompose the rest of the proof into two steps. Step 1 We will prove that for every j in J s     ∂x uj,n  , vj,n  −→ 0.

(4.42)

s . Then, using We start by strings attached to external nodes. To do this let ak ∈ Vext (4.34), (4.41), and damping conditions at external nodes, we have

βn ujk ,n (ak ) −→ 0 and ∂x ujk ,n (ak ) −→ 0.

(4.43)

Taking the real part of the inner product of (4.38) by qjk ∂x ujk ,n with qjk = x or qjk = (jk − x) yields % %j %2 %j k %2 % k 1 2 %% 1 %% % % % βn ujk ,n (ak ) % + ∂x ujk ,n (x) qjk (x)%% 2 2 0 0

j   % % % k % 1 %∂x uj ,n (x)%2 + β 2 %uj ,n (x)%2 ∂x qj (x)dx − n k k k 2 0 %j  + Re iβn fjk ,n (x)qjk (x)ujk ,n (x) %0 k −→ 0.

(4.44)

Using (4.43) in (4.44), qjk (x) = x or qjk (x) = jk − x, leads to 1 − 2



j k 0

% %∂x uj

k ,n

%2 %2  % (x)% + βn2 %ujk ,n (x)% dx → 0.

Moreover, if we take qjk = 1, we obtain that %2 1 % %2   1 % − βn2 %ujk ,n (ak  )% − %∂x ujk ,n (ak  )% − Re iβn fjk ,n (ak  )ujk ,n (ak  ) −→ 0, 2 2

2 Asymptotic Behavior

73

where ak  is the second end of ejk different from ak , and as in [88] it follows that, βn ujk ,n (ak  ) −→ 0 and ∂x ujk ,n (ak  ) −→ 0.

(4.45)

Now, let ak be an external node of the tree T  obtained from T by removing all the strings attached to leaves. Using (4.45), the fifth condition in (4.7) and the continuity of un at ak , we deduce that ak satisfies (4.43). Then, by iterations, we have that for every j in J s (even for j = 1) −

1 2

0

j

% % %  % %∂x uj,n (x)%2 + β 2 %uj,n (x)%2 dx −→ 0, n

(4.46)

and βn uj,n (ak  ) −→ 0 and ∂x uj,n (ak  ) −→ 0,

(4.47)

where ak  is an end of ej . In particular, using (4.40), we get     ∂x uj,n  , vj,n  −→ 0. If there is no beam in the tree, we conclude that yn H → 0 which contradicts the fact that yn H = 1 and the proof is then complete. Step 2 Now, we suppose that there is at least one beam. We denote by G the only maximal subgraph of beams of T and we will show that for every beam ej in G,       2 ∂x uj,n  , vj,n  −→ 0. We start by beams ended by external nodes of G (i.e., those attached to stings). Let ak ∈ Vext (G), then βn ujk ,n (ak ) −→ 0 and ∂x3 ujk ,n (ak ) −→ 0.

(4.48)

Indeed, if ak is an external node of the initial tree T , then (4.48) is due to (4.35) and (4.41), and damping conditions at external nodes, if ak is attached to some strings, then it is due to (4.47) by taking into account the continuity condition of un and the fifth condition in (4.7) at inner nodes. Let j be in J b and qj be a function in C 2 ([0, j ], C) such that ∂x2 qj = 0. We want to calculate the real part of the inner product of (4.39) with qj ∂x uj,n .

74

4 Stability of a Tree-Shaped Network of Strings and Beams

Straightforward calculations give %j     %2 1 % % −βn2 uj,n , qj ∂x uj,n +Re ∂x4 uj,n , qj ∂x uj,n = − βn2 %uj,n (x)% qj (x)% 0 2  

j %  % % 1 2 %j + βn2 %uj,n % ∂x qj dx + Re ∂x3 uj,n (x)qj (x)∂x uj,n (x)% 0 2 0 %

%2 %2 % j 1 %% 3 j %% 2 % % − %∂x2 uj,n (x)% qj (x)%% + %∂x uj,n % ∂x qj dx 2 2 0 0  % j  % , −Re ∂x2 uj,n (x)∂x uj,n ∂x qj (x)%

Re

0

and   Re gj,n + iβn fj,n , qj ∂x uj,n = Re 

− Re iβn 0

j





j

gj,n ∂x uj,n qj dx



0

% ∂x (qj fj,n )uj,n dx + Re iβn fj,n (x)qj (x)uj,n (x)%0j .

Since gj,n , fj,n and ∂x (qj fj,n ) converge to 0 in L2 (0, j ) and iβn uj,n and ∂x uj,n are bounded in L2 (0, j ), the first and the second terms of the right member of the previous equality converge to 0. It follows  % j % j  %2 1 % % % − βn2 %uj,n (x)% qj (x)% + Re ∂x3 uj,n (x)qj (x)∂x uj,n (x)% 0 0 2 % %2 %j 1 %% % − %∂x2 uj,n (x)% qj (x)%% 2 0  %j   % % − Re iβn fj,n (x)qj (x)uj,n (x) %0j − Re ∂x2 uj,n (x)∂x uj,n (x)∂x qj (x)% 0

+

1 2



j 0

% %2 3 βn2 %uj,n % ∂x qj dx + 2



j 0

% %2 % 2 % %∂x uj,n % ∂x qj dx −→ 0.

(4.49)

In particular, if ak is in Vext (G), then with j = jk and qjk (x) = x or qjk (x) = jk −x (4.49) becomes, using that ∂x ujk ,n (ak ) = 0, %2   1 j k % %2 jk %% 2 % 2   − βn2 %ujk ,n % dx %∂x ujk ,n (ak )% + Re dkjk ∂x ujk ,n (ak )∂x ujk ,n (ak ) + 2 2 0

j % %2 k % 3 % + (4.50) %∂x2 ujk ,n % dx → 0, 2 0 where ak  is the end of ejk different from ak .

2 Asymptotic Behavior

75

Multiplying (4.39) by obtain

1 1/2 βn

1/2

e−βn

(jk −x)

or by

1 1/2 βn

1/2

e−βn

x,

then, as in [88], we

1 d ∂ 3 u (a ) + ε∂x2 ujk ,n (ak ) + εβn ujk ,n (ak ) 1/2 kjk x jk ,n k βn

−→ 0,

with ε ∈ {−1, 1}. Since ∂x3 ujk ,n (ak ) and βn ujk ,n (ak ) tend to 0, we deduce that ∂x2 ujk ,n (ak ) −→ 0.

(4.51)

Hence, (4.50) can be rewritten as   1 j k % %2 2 Re dkjk ∂x ujk ,n (ak  )∂x ujk ,n (ak  ) + βn2 %ujk ,n % dx 2 0

j % % k % 3 %2 + %∂x2 ujk ,n % dx −→ 0. 2 0

(4.52)

Now, we rewrite (4.49), with qj = 1 and j = jk , %2   1% %2 1 % % % − βn2 %ujk ,n (ak  )% + Re ∂x3 ujk ,n (ak  )∂x ujk ,n (ak  ) − %∂x2 ujk ,n (ak  )% 2 2   − Re iβn fjk ,n (ak  )ujk ,n (ak  ) → 0. (4.53) For j ∈ Jk  (G), multiplying (4.39) by

1 1/2 βn

1/2

e−βn

x

or by

1 1/2 βn

1/2

e−βn

(j −x) ,

we get

1 1/2 d  ∂ 3 u (a  ) + ε∂x2 uj,n (ak  ) + βn dk  j ∂x uj,n (ak  ) + εβn uj,n (ak  ) 1/2 k j x j,n k βn

−→ 0, (4.54)

with ε ∈ {−1, 1}. In the case of ak  = a1 , summing over j ∈ Jk  (G), then by taking into account the continuity condition of un and ∂x2 un , the fourth condition in (4.6), and the fifth in (4.7) at ak  we get ∂x2 ujk ,n (ak  ) + βn ujk ,n (ak  ) −→ 0. It yields from (4.54), 1 3 1/2 ∂ u (a  ) + βn ∂x ujk ,n (ak  ) 1/2 x jk ,n k βn

:= αjk ,n −→ 0,

(4.55)

76

4 Stability of a Tree-Shaped Network of Strings and Beams

which holds also if ak  = a1 (due to (4.54) and the boundary conditions at a1 ). Hence, for any positive real number a, we have  3    ∂ uj ,n (a  ) 1/2 Re ∂x3 ujk ,n (ak  )∂x ujk ,n (ak  ) = Re x k1/2 k βn ∂x ujk ,n (ak  )  βn  1/2 1/2 = Re (αjk ,n − βn ∂x ujk ,n (ak  ))βn ∂x ujk ,n (ak  ) %2 %2 % % ≤ −βn %∂x ujk ,n (ak  )% + a2 βn %∂x ujk ,n (ak  )% % % 2 1 % αjk ,n % . + 2a (4.56) Moreover, for any real positive number b we have %2 %2  b %  1 %% fjk ,n (ak  )% . −Re iβn fjk ,n (ak  )ujk ,n (ak  ) ≤ βn2 %ujk ,n (ak  )% + 2 2b

(4.57)

Taking j = jk and combining (4.56), (4.57), and (4.53), we obtain the following framing for jk : %2   1% %2 1 % % % − βn2 %ujk ,n (ak  )% + Re ∂x3 ujk ,n (ak  )∂x ujk ,n (ak  ) − %∂x2 ujk ,n (ak  )% 2 2 %2 %2   1 %% 1 %% fjk ,n (ak  )% − αjk ,n % − Re iβn fjk ,n (ak  )ujk ,n (ak  ) − 2b 2a %2 %2 1 %% 2 1 b 2 %% % ≤ (− + )βn ujk ,n (ak  )% − %∂x ujk ,n (ak  )% 2 2 2 %2 % a + (−1 + )βn %∂x ujk ,n (ak  )% ≤ 0, 2 with a = b = 12 . This implies, using that fjk ,n (ak  ) −→ 0 and following properties: βn ujk ,n (ak  ) −→ 0, ∂x2 ujk ,n (ak  ) −→ 0,

7

1 2a

% % %αj,n %2 −→ 0 the

βn ∂x ujk ,n (ak  ) −→ 0,

1 and √ ∂x3 ujk ,n (ak  ) −→ 0. βn

(4.58)

  We have used (4.55) for the last property. Moreover, Re ∂x2 ujk ,n (ak  )∂x ujk ,n (ak  ) tends to 0 as n goes to infinity, then (4.52) leads to 1 2

0

j k

%

%

2 βn2 %ujk ,n % dx

3 + 2

0

j k

% %2 % 2 % %∂x ujk ,n % dx −→ 0.

Now, let ak be an external node of the graph G  obtained from G by removing all the edges ended by external nodes in G. Using (4.58), the fifth condition in (4.7),

2 Asymptotic Behavior

77

and the continuity of un at ak , we deduce that βn ujk ,n (ak ) −→ 0, ∂x2 ujk ,n (ak ) −→ 0,

7 βn ∂x ujk ,n (ak ) −→ 0,

1 and √ ∂x3 ujk ,n (ak ) −→ 0. βn Then by iterations, we have that for every beam ej (even if j = 1) 1 2



j 0

%2 % 3 βn2 %uj,n (x)% dx + 2



j 0

% %2 % 2 % %∂x uj,n (x)% dx −→ 0.

(4.59)

Finally, using (4.59), it yields       2 ∂x uj,n  , βn uj,n  −→ 0. In conclusion, yn H converge to 0, which contradicts the hypothesis that yn H = 1.  

2.3 Polynomial Stability In this section we suppose that there is at least a beam following a string (Fig. 4.2). We will prove that the solution of the whole system (S) is polynomially stable and non-exponentially stable at least in a simple case of two components. Recall that r is the number of layers of beams in T not containing e1 . Then we have the following result: Theorem 4.6 If at least one beam follows a string from the root to a leaf, then the C0 -semigroup (T (t)t≥0 is polynomially stable. More precisely: There are C > 0 such that    tA  e y0 

H

≤

C t 1/r

y0 D(A)

for every y0 ∈ D(A). Proof We have proved that iR ⊂ ρ(A) (proof of Theorem 4.4) then, in view of Theorem 1.26, it suffices to prove that (1.12) holds for α = r. Suppose the conclusion is false. Then there exists a sequence (βn ) of real numbers, without loss of generality, with βn −→ +∞, and a sequence of vectors (yn ) = (un , v n ) in D(A) with yn H = 1, such that  r  β (iβn I − A)yn  −→ 0, n H

78

4 Stability of a Tree-Shaped Network of Strings and Beams

which is equivalent to βnr (iβn uj,n − vj,n ) = fj,n −→ 0, in H 1 (0, j ), j in J s ,

(4.60)

βnr (iβn uj,n − vj,n ) = fj,n −→ 0, in H 2 (0, j ), j in J b ,

(4.61)

= gj,n −→ 0,

in L (0, j ), j in J ,

(4.62)

βnr (iβn vj,n + ∂x4 uj,n ) = gj,n −→ 0,

in L2 (0, j ), j in J b .

(4.63)

βnr (iβn vj,n

− ∂x2 uj,n )

2

s

Then, − βnr (βn2 uj,n + ∂x2 uj,n ) = gj,n + iβn fj,n , j in J s ,

(4.64)

βnr (−βn2 uj,n + ∂x4 uj,n ) = gj,n + iβn fj,n , j in J b ,

(4.65)

and  2  2 βnr vj,n  − βn2+r uj,n  −→ 0, j = 1, . . . , N.   Since Re( βnr (iβn − A)yn , yn H ) =

" ∗ ak ∈Vext

(4.66)

%2 % βnr %vjk ,n (ak )% , we obtain

r % % ∗ . βn2 %vjk ,n (ak )% −→ 0, for ak ∈ Vext

We will prove that yn H converge to zero, which contradicts the fact that yn H = 1. The rest of the proof will be decomposed into four steps, in which we need several times the real part of the inner product of (4.64) and (4.65) with qj ∂x uj,n respectively, where qj = 1 or qj = x or qj = j − x. They give %j %j %2 % %2 % 1 2+r %% 1 % uj,n (x)% qj (x)%% + βnr %∂x uj,n (x)% qj (x)%% βn 2 2 0 0

j   % % % % 1 %∂x uj,n (x)%2 + β 2 %uj,n (x)%2 ∂x qj (x)dx − βnr n 2 0  % + Re iβn fj,n (x)qj (x)uj,n (x) % j −→ 0, 0

in the first case (that is j ∈ J s ) and  %j % j  %2 1 2+r %% % % r 3 % uj,n (x) qj (x)% + Re βn ∂x uj,n (x)qj (x)∂x uj,n (x)% − βn 0 0 2 % %2 % j 1 %% % − βnr %∂x2 uj,n (x)% qj (x)%% 2 0

(4.67)

2 Asymptotic Behavior

 − Re 1 + βnr 2

79

%j 

% βnr ∂x2 uj,n (x)∂x uj,n ∂x qj (x)%



j

0

%

%

2 βn2 %uj,n % ∂x qj dx

0

3 + βnr 2

 % − Re iβn fj,n (x)qj (x)uj,n (x) %0j



j 0

% %2 % 2 % %∂x uj,n % ∂x qj dx −→ 0,

(4.68)

in the second case (that is j ∈ J b ). Step 1 We will prove that for every edge ej of the first layer of strings we have 1+ 2r

βn

  r  uj,n  , βn2 ∂x uj,n  −→ 0.

(4.69)

Let G be a maximal subgraph of NFB-strings of T (i.e., a maximal subgraph in the first layer of T ). s . By using (4.60) and damping conditions at external nodes, we have Let ak ∈ Vext 1+ 2r

βn

r

ujk ,n (ak ) −→ 0 and βn2 ∂x ujk ,n (ak ) −→ 0.

(4.70)

Then, from (4.70) and (4.67), with j = jk and qjk (x) = x or qjk (x) = jk − x, we have that (4.69) holds for j = jk . Moreover, as in the previous theorem, (4.69) holds for any edge ej of G, and if ak is an end of such edge then 1+ 2r

βn

r

uj,n (ak ) −→ 0 and βn2 ∂x uj,n (ak ) −→ 0.

(4.71)

In fact, (4.69) and (4.67) hold for every string ej in the first layer of strings. Step 2 Now, we assume that we have removed the first layer of strings. Let G be a subgraph of beams, maximal for the property that every edge is NFS (G is a maximal subgraph in the second layer of T ). Note that there is no string attached to a node in Vext (G) ∪ Vint (G). Let ak ∈ Vext (G). If ak is an external node of T , then by (4.60) and damping conditions at external nodes, we get 1+ 2r

βn

r

ujk ,n (ak ) −→ 0, βn2 ∂x3 ujk ,n (ak ) −→ 0 and ∂x ujk ,n (ak ) = 0,

which holds if ak is not an external node of T , due to the continuity condition of un , the fourth condition in (4.6), and the fifth in (4.7) at ak and (4.71). Again, as in the previous proof, using (4.68) in some iterations, we have that for every beam ej of G, not attached to a string, 1+ 2r

βn

 r     uj,n  , βn2  ∂x2 uj,n  −→ 0,

(4.72)

80

4 Stability of a Tree-Shaped Network of Strings and Beams

and if ak is a node of G, not belonging to a string, then for every j ∈ Jk (G), 1+ 2r

βn

r

uj,n (ak ) −→ 0, βn2

− 12 3 ∂x uj,n (ak )

r

−→ 0, βn2

+ 12

∂x uj,n (ak ) → 0,

r 2

and βn ∂x2 uj,n (ak ) −→ 0.

(4.73)

Moreover, we will prove that (4.73) holds even if ak is an end of a string. Note that, in this case, ak must be the outer node of G and that every beam attached to ak lies in G. Taking qj = 1 in (4.68), with j in Jkb , we obtain %2   1 % %2 % 1 % % − βn2+r %uj,n (ak )% + Re βnr ∂x3 uj,n (ak )∂x uj,n (ak ) − βnr %∂x2 uj,n (ak )% → 0. 2 2 (4.74) Multiplying (4.65) by r− 12

βn

1 −βn1/2 x βn e

or by

1 −βn1/2 (j −x) βn e

r+ 12

dkj ∂x3 uj,n (ak )+εβnr ∂x2 uj,n (ak )+βn

leads to

dkj ∂x uj,n (ak )+εβn1+r uj,n (ak ) −→ 0. (4.75)

Summing over j in Jkb we get, using the fifth condition in (4.7) and fourth in (4.6), r− 12

βn



  dkj ∂x uj,n (ak ) + Rε βnr ∂x2 uj,n (ak ) + βn1+r uj,n (ak ) −→ 0,

j ∈Jks

where R is the cardinal of Jkb . Returning back to (4.75), r− 12

βn

dkj ∂x3 uj,n (ak )−

1 r− 12  r+ 1 βn dki ∂x ui,n (ak )+βn 2 dkj ∂x uj,n (ak ) := γj,n −→ 0. R s i∈Jk

(4.76) 1

Multiplying now (4.76) by βn2 dkj ∂x uj,n (ak ) at the left, and summing the real parts, we obtain    % % %∂x uj,n (ak )%2 Re ∂x uj,n (ak )∂x3 uj,n (ak ) + βnr+1 βnr j ∈Jkb

   1 2 = Re γj,n βn dkj ∂x uj,n (ak ) , j ∈Jkb

j ∈Jkb

(4.77)

2 Asymptotic Behavior

81

due to condition fifth in (4.7). We deduce that βnr



  Re ∂x uj,n (ak )∂x3 uj,n (ak )

j ∈Jkb

≤ −βnr+1

% % % % %∂x uj,n (ak )%2 + 1 β r+1 %∂x uj,n (ak )%2 n 2 b b

j ∈Jk

j ∈Jk

%2 1  %% γj,n % . + r 2βn b

(4.78)

j ∈Jk

Therefore, we can deduce from (4.77), after summing over j in Jkb , that (4.73) holds for all j in Jkb . Moreover, by taking qj = x or qj = j − x in (4.68), we obtain that (4.72) holds for every j in Jkb . Thus, (4.72) and (4.73) are verified for every beam ej of the second layer of T . Step 3 Let G be a maximal subgraph of NFB-strings in the new tree obtained by removing the last layer of beams. For every node ak in Vext (G) we have 1+ 2r

βn

r−1

ujk ,n (ak ) −→ 0 and βn 2 ∂x ujk ,n (ak ) −→ 0,

due to the transmission conditions at ak and (4.73). As in Step 1, if ej is a string of G then 1+ r−1 2

βn

  r−1   uj,n  , βn 2 ∂x uj,n  −→ 0

(4.79)

and 1+ r−1 2

βn

r−1

uj,n (ak  ) −→ 0 and βn 2 ∂x uj,n (ak  ) −→ 0,

where ak  is an end of ej . If e1 ∈ / G. Suppose that we remove the last layer of strings and let G  be a maximal subgraph of NFS-beams in the new tree. Then ∀ak ∈ Vext (G  ) 1+ r−1 2

βn

r−1

r

ujk ,n (ak ) −→ 0, βn 2 ∂x3 ujk ,n (ak ) −→ 0 and βn2

+ 12

∂x ujk ,n (ak ) −→ 0.

82

4 Stability of a Tree-Shaped Network of Strings and Beams

As in Step 2 , we have by iteration, that for every beam ej of G  (even if j = 1), 1+ r−1 2

βn

 r−1     uj,n  , βn 2  ∂x2 uj,n  −→ 0,

(4.80)

and for every node ak of ej , 1+ r−1 2

βn

r−1

uj,n (ak ) −→ 0, βn 2

− 12 3 ∂x uj,n (ak )

r−1

−→ 0, βn 2

+ 12

∂x uj,n (ak ) → 0

r−1

and βn 2 ∂x2 uj,n (ak ) −→ 0. Step 4 We iterate such procedure in T − {e1 } (from leaves to the root). If e1 is a string, we have, using the transmission and damping conditions at a2 , 1+ r−r 2

βn

r−r

u1,n (a2 ) −→ 0 and βn 2 ∂x u1,n (a2 ) −→ 0,

then, by taking j = 1 in (4.67), we get 1+ r−r 2

βn

 r−r     u1,n  , βn 2  ∂x2 u1,n  −→ 0.

(4.81)

If e1 is a beam, then 1+ r−r 2

βn

r−r

u1,n (a2 ) −→ 0, βn 2

− 12 3 ∂x u1,n (a2 )

r−r

−→ 0 and βn 2

+ 12

∂x u1,n (a2 ) → 0. (4.82)

Moreover, r−r

βn 2 ∂x2 u1,n (a2 ) −→ 0. Using continuity conditions of un and ∂x2 un , and the fourth condition in (4.6) and fifth in (4.7) at a2 , (4.68) leads to 1+ r−r 2

βn

 r−r     u1,n  , βn 2  ∂x2 u1,n  −→ 0.

(4.83)

In conclusion yn  converge to 0, which contradicts the hypothesis that yn H = 1.  

2 Asymptotic Behavior

83

2.4 Lack of Exponential Stability Now, we consider a reduced system composed of one string e1 and one beam e2 such that 1 = 2 = π and with control is applied on the beam. Precisely, we consider the system ⎧ ⎪ u1,tt − u1,xx = 0 in (0, π ) × (0, ∞), ⎪ ⎪ ⎪ ⎪ ⎨ u2,tt + u2,xxxx = 0 in (0, π ) × (0, ∞), (S0 ) : u1 (0, t) = u2 (0, t), u2,x (0, t) = 0, u2,xxx (0, t) = u1,x (0, t), ⎪ ⎪ ⎪ u1 (π, t) = 0, u2,xxx (π, t) = u2,t (π, t), u2,x (π, t) = 0, ⎪ ⎪ ⎩ u (x, 0) = u0 (x), u (x, 0) = u1 (x), j = 1, 2. j j,t j j In view of Theorem 4.6, the system (S0 ) is polynomial stable, and we will prove that it is not exponentially stable. Note that if the control is applied on the string instead of the beam, then the system is exponentially stable (by Theorem 4.5). Theorem 4.7 The system (S0 ) is not exponentially stable in the energy space H. Proof We prove that the corresponding semigroup (T (t))t≥0 is not exponentially stable. √ √ For n ∈ N, such that n is integer and even let βn = n2 + 2 n + n1 and fn = (0, 0, − sin βn x, 0), then βn → +∞ and fn is in H and is bounded. Let yn = (u1,n , u2,n , v1,n , v2,n ) ∈ D(A) such that (A − iβn )yn = fn . We will prove that yn → +∞. We have βn2 u1,n + ∂x2 u1,n = sin βn x, −βn2 u2,n + ∂x4 u2,n = 0.

(4.84) (4.85)

Then u1,n and u2,n are of the form x + c2 ) cos(βn x), 2βn 7 7 7 7 = d1 sin( βn x) + d2 cos( βn x) + d3 sinh( βn x) + d4 cosh( βn x).

u1,n = c1 sin(βn x) + (− u2,n

The transmission and boundary conditions are rewritten as follows: 7

d2 + d4 = c2 , βn (d1 + d3 ) = 0,

3/2

βn (−d1 + d3 ) = −

(4.86) (4.87)

1 + βn c1 , 2βn

(4.88)

84

4 Stability of a Tree-Shaped Network of Strings and Beams

and c1 sin(βn π ) + (− 7

7

π + c2 ) cos(βn π ) = 0, 2βn

7

7

(4.89)

d1 cos( βn π ) − d2 sin( βn π ) + d3 cosh( βn π ) + d4 sinh( βn π ) = 0, (4.90) and 7 7 7 7 3/2 βn (−d1 cos( βn π ) + d2 sin( βn π ) + d3 cosh( βn π ) + d4 sinh( βn π )) 7 7 7 = iβn (d1 sin( βn π ) + d2 cos( βn π ) + d3 sinh( βn π ) 7 +d4 cosh( βn π )). (4.91) Summing (4.90) and (4.91), we obtain 7 7 7 2 βn (d3 cosh( βn π ) + d4 sinh( βn π )) 7 7 7 = i(d1 sin( βn π ) + d2 cos( βn π ) + d3 sinh( βn π ) 7 +d4 cosh( βn π )).

(4.92)

Substituting (4.86) and (4.87) into (4.90) leads to d4 =

√ √ √ cosh( βn π ) − cos( βn π ) sin( βn π ) d1 + c2 . √ √ √ √ sinh( βn π ) + sin( βn π ) sinh( βn π ) + sin( βn π )

(4.93)

Now, by substituting (4.86)–(4.88) and (4.93) into (4.92), we get 7 7 7 7 βn h( βn π ) − 2βn tan(βn π ) sin( βn π ) sinh( βn π )   7 7 7 7 +i 1 − cosh( βn π ) cos( βn π ) + βn tan(βn π )h( βn π )   7 7 7 7 = 2 βn sin( βn π ) sinh( βn π ) − ih( βn π )   π 1 , (4.94) × − 2 tan(βn π ) + 2βn 2βn

2d1

with 7 7 7 7 7 h( βn π ) = cosh( βn π ) sin( βn π ) + sinh( βn π ) cos( βn π ). 3/2

We want to prove that βn d1 is equivalent to

√

2

n π2 as n goes to infinity.

2 Asymptotic Behavior

85

√ √ Since βn = n2 + 2 n + n1 , then βn = n +

√1 n

= n(1 + o( √1n )) and

7 π π 1 sin( βn π ) = sin( √ ) = √ + o( √ ), n n n 7 π cos( βn π ) = cos( √ ) = 1 + o(1), n π 1 π tan(βn π ) = tan( ) = + o( ), n n n 7 1 enπ cosh( βn π ) = cosh((n + √ )π ) = (1 + o(1)), 2 n 7 enπ (1 + o(1)), sinh( βn π ) = 2 then 7 enπ h( βn π ) = (1 + o(1)), 2 7 7 7 enπ o(1), βn sinh( βn π ) sin( βn π ) tan(βn π ) = 2 7 7 7 √ enπ (1 + o(1)). βn sinh( βn π ) sin( βn π ) = π n 2 Hence, (4.94) implies   7 enπ 1 π 1 enπ √ 2d1 βn n (2π + o(1)) − 2 O( ) + (1 + o(1)) = 2 2 n 2βn 2βn that leads to √ 3/2 2βn d1 ∼ π 2 n.

(4.95)

when n tends to infinity. Now taking the real part of the inner product of (4.84) with (π − x)∂x u1,n , we get % %2   % 1 %     %− % − π |βn c2 |2 = − 1 β 2 u1,n 2 + ∂x u1,n 2 + β c n 1 n % 2β % 2 2 n  π  + Re sin(βn x)(π − x)∂x u1,n dx .

π − 2

0

 2  2 Then, by taking into account (4.87-4.88) and (4.95), βn2 u1,n  + ∂x u1,n  must be not bounded. In conclusion yn is not bounded.  

86

4 Stability of a Tree-Shaped Network of Strings and Beams

Remark 4.8 In view of Theorem 1.26, a slight modification in the proof of the previous theorem allows us to deduce that the semigroup associated to system (S0 ) cannot be polynomially stable of order α for every α > 2. Moreover, such result persists if we consider two components of any length. Remark 4.9 We believe that this last result of non-exponential stability remains true in the general case, when at least a beam following a string (Fig. 4.2).

3 Comments 3.1 Comment 1 As in [37], the Authors take in [36] a transmission problem coupling heat and wave equations on a star-shaped network. For a simple model (Fig. 4.3) one can see [107] and [106] where some results of controllability and polynomial stability are obtained. Now, let us describe briefly the star-shaped network G (Fig. 4.4) considered in [36]. The edges (segments) ej , j = 1, . . . , N occupy the intervals (0, j ), j > 0, respectively. The common node is identified to x = 0. The heat equations arise on the intervals (0, k ), k = 1, 2, . . . , N1 (0 < N1 < N) in the network with state θk , respectively; the wave equations hold on the intervals (0, j ), k = N1 + 1, N1 + Fig. 4.3 Coupled heat-wave

Fig. 4.4 Star-shaped network: heat equations (red), wave equations (black)

3 Comments

87

2, . . . , N in the network with state (uj , uj,t ). The authors choose the following heat-wave system on star-shaped network. ⎧ θk,t (x, t) − θk,xx (x, t) = 0, x ∈ (0, j ), k = 1, 2, · · · , N1 , t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ uj,tt (x, t) − uj,xx (x, t) = 0, x ∈ (0, j ), j = N1 + 1, · · · , · · · , N, t > 0, ⎪ ⎪ ⎪ ⎪ θk (k , t) = uj (j , t), ∀ j, k = 1, 2, · · · , N, t > 0, ⎪ ⎪ ⎪ ⎪ uj (j , t) = 0, j = 1, 2, · · · , N, t > 0, ⎪ ⎪ ⎪ ⎪ uj (0, t) = uk (0, t), ∀ j, k = 1, 2, · · · , N, t > 0, ⎪ ⎨ θk (k , t) = 0, k = 1, 2, · · · , N1 , t > 0, ⎪ ⎪ θ ⎪ k (0, t) = uj (0, t), ∀ k = 1, 2, · · · , N1 , j = N1 + 1, N1 + 2, · · · , N, t > 0, ⎪ ⎪ ⎪ N1 N " " ⎪ ⎪ ⎪ u (0, t) + θk,x (0, t) = 0, t > 0, j,x ⎪ ⎪ ⎪ j =N1 +1 k=1 ⎪ ⎪ ⎪ 0 k ⎪ ⎪ θ (t = 0) = θk , k = 1, 2, · · · , N1 , ⎪ ⎩ uj (t = 0) = u0j , uj,t (t = 0) = u1j , j = N1 + 1, N1 + 2, · · · , N, (4.96)    N  0 N1  0 N 1 is the given initial state. , uj where θk k=1 , uj j =N1 +1

j =N1 +1

The energy of this system is defined as follows: 1 1 E(t) = 2

N



k

k=1 0

N j   1  |uj,x |2 + |uj,t |2 dx. |θk | dx + 2 0 2

j =N1 1

It satisfies 1  dE(t) =− dt

N



k

|θk,t |2 dx ≤ 0

k=1 0

and therefore the energy is decreasing. Then, the authors wrote the system as an abstract Cauchy problem in some appropriate Hilbert space H: ⎧ ⎨ d y = Ay, dt ⎩ y(0) = y 0 , where y = (u, ut , θ ) and y 0 = (u0 , u1 , θ 0 ) ∈ H is given.

88

4 Stability of a Tree-Shaped Network of Strings and Beams

The authors proved the following results. Theorem 4.10 The energy of the system (4.96) decays to zero as t → ∞ if and only if one of the following two conditions is fulfilled: (1) N − N1 = 1, (2) N − N1 ≥ 2 and i /j ∈ / Q, i, j = N1 + 1, N1 + 2, · · · , N, i = j . Theorem 4.11 The energy of system (4.96) does not decay exponentially as t → ∞, as soon as the network involves at least one wave equation. The authors examinate more deeply the case N − N1 > 1, in particular, they give the following estimate for N − N1 = 1. Theorem 4.12 Case N − N1 = 1: The heat-wave star contains only one edge described by a wave equation. The energy of the corresponding system decays polynomially as follows: There exists a positive constant C such that E(t) ≤ Ct −4 (u0 , u1 , θ 0 )2D(A) , ∀ t ≥ 0 for every (u0 , u1 , θ 0 ) ∈ D(A). This decay rate is sharp.

3.2 Comment 2 Some of the following are recent works related to the classical and fractional heat equation on graphs: [63, 67, 72, 90, 99].

Chapter 5

Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction on a Tree

In this chapter, we study the stability of a model of fluid propagating in a 1-d network, under some feedbacks applied at exterior nodes, with the presence of point mass at inner nodes, see Fig. 5.1. At rest, the network coincides with the tree T = (E, V), where E = {e1 , . . . , eN } is the set of edges and V = {a1 , . . . , aN +1 } is the set of nodes (vertices). We assume that a1 is the root of T , which will be denoted by R, that e1 is the edge containing R, and that a2 is its vertex different from R. More precisely, we consider the following initial and boundary value problem: ⎧ yj,tt − yj,xx = 0 in (0, j ) × (0, ∞), j ∈ J := {1, . . . , N}, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ " ⎪  ⎪ k ∈ Iint , ⎪ ⎪ j ∈J dkj yj,x (ak , t) = sk (t), ⎪ ⎪ k ⎪ ⎪  ⎪ k ∈ Iint , ⎪ ⎨ sk (t) + sk (t) = −yt (ak , t), yj (ak , t) = yl (ak , t), j, l ∈ Jk , k ∈ Iint , ⎪ ⎪ y1 (a1 , t) = 0, ⎪ ⎪ ⎪ ∗ , ⎪ k ∈ Iext ⎪ ⎪ dkjk yjk ,x (ak , t) = −yt (ak , t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ sk (0) = sk0 , sk (0) = sk1 , k ∈ Iint , ⎪ ⎪ ⎩ yj (x, 0) = yj0 (x), yj,t (x, 0) = yj1 (x), x ∈ (0, j ), j ∈ J,

(5.1)

where yj : [0, j ] × (0, ∞) −→ R, j ∈ J , represents the velocity potential of the fluid on the edge ej and sk : (0, ∞) −→ R, k ∈ Iint , denotes the movement of the point mass occupant the node ak . These functions allow us to identify the network with its rest graph. This simplified model of fluid–structure interaction draws on the work of Ervedoza and Vanninathan [24]; they considered a fluid occupying a domain in two dimensions and a solid immersed in it and proved some results of controllability of such system, see also [92] for more details. Then, we consider a corresponding one-dimensional model (where the dimension of space is reduced to 1), with many © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Ammari, F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-86351-7_5

89

90

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

Fig. 5.1 A tree-shaped network

mass points (which model the structure). We refer to [17] where the authors gave some mathematical models of vibrations of fluid–solid structures corresponding to some physical situations, as the tube bundles vibrating inside a moving fluid in a nuclear reactor. The problem of fluid–structure interaction in one dimension has been studied by several authors. In [95], the authors studied the asymptotic behavior of a one-dimensional model of mass point moving in a fluid. They considered the same system in [96], but with a finite number of mass points floating in the fluid. Recently Tucsnak et al. [59] studied the controllability of a similar system. In [97], the authors modeled a biological problem (an intracranial saccular aneurysm), into a coupled fluid–structure interaction problem, in one dimension, consisting of a wave equation with a dynamical condition at one end. Note that the pointwise (or boundary) stabilization on the wave equation has been treated during the last few years, see, for example, [9] for one string and [4, 7, 10] for some networks of strings. The main result of this chapter asserts that, under some conditions, the energy of the solutions of the dissipative system decays exponentially to zero when the time tends to infinity. The method is based on a frequency domain method and a special analysis for the resolvent. If (y, s) = ((yj )j ∈J , (sk )k∈Iint ) is a solution of (5.1), we define the energy of (y, s) at instant t by E(t) =

1 2 j ∈J

0

j

%   % % % %  % %yj,t (x, t)%2 + %yj,x (x, t)%2 dx+ 1 %s  (t)%2 + |sk (t)|2 . k 2 k∈Iint

5

Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

91

Simple formal calculations show that a sufficiently smooth solution of (5.1) satisfies the energy estimate  t% % %yj ,t (ak , s)%2 ds, k

E(0) − E(t) =

∗ k∈Iext

∀t ≥ 0.

(5.2)

0

In particular, (5.2) implies that E(t) ≤ E(0),

∀t ≥ 0.

Define ⎫ ⎧ ⎬ ⎨  H 1 (0, j ), 1 (R) = 0, j (ak ) = l (ak ), j, l ∈ Jk , k ∈ Iint , V = ∈ ⎭ ⎩ j ∈J

and the natural well-posedness space for (5.1) is

H =V ×



⎛ L2 (0, j ) × ⎝

j ∈J



⎞2 C⎠

k∈Iint

endowed with the inner product   Z, Z˜

H

=

 

j ∈J

+

j

∂x fj (x)∂x f˜j (x)dx +

0



j

 gj (x)g˜j (x)dx

0

   ck c˜k + dk d˜k ,

k∈Iint

˜ where Z = (f , g, c, d) and Z˜ = (f˜, g, ˜ c, ˜ d). Then, we can as a first-order differential equation, by   rewrite the system (5.1)   putting z(t) = y(t), y (t), s(t), s (t) :   z (t) = Az(t), z(0) = z0 = y 0 , y 1 , s 0 , s 1 , where A(y, v, p, q) = (v, ∂x2 y, q, −p − v M ), ∀(y, v, p, q ∈ D(A),

92

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

with v M = (v(ak ), k ∈ Iint ), and ⎧ ⎪ ⎨

⎡

D(A) = (y, v, p, q) ∈ ⎣

⎪ ⎩

 j ∈J

⎤

⎛

H 2 (0, j ) ∩ V ⎦ × V × ⎝



⎞2 C⎠ satisfying (5.3)

k∈Iint

⎫ ⎪ ⎬ ⎪ ⎭

,

where 5"

j ∈Jk dkj ∂x yj (ak ) = qk , ∀k ∈ Iint , ∗ . dkj ∂x yjk (ak ) = −vjk (ak ), ∀k ∈ Iext

(5.3)

To simplify the notations, sometimes, we will write y(ak ) instead of yj (ak ) for y in V . The outline of this work is the following. In Sect. 1, we prove the existence and uniqueness of solutions for system (5.1). Section 2 is devoted to proving the exponential stability of the associated semigroup. Finally, in Sect. 3, we prove the lack of exponential stability of a graph containing a circuit and a tree with two uncontrolled exterior nodes (Figs. 5.2 and 5.3). The Sect. 4 is devoted to the study of the case of a chain with non-equal mass points (Fig. 5.4). Fig. 5.2 Circuit

Fig. 5.3 Star

1 Well-Posedness

93

Fig. 5.4 A chain of strings

1 Well-Posedness Lemma 5.1 The operator A generates a C0 -semigroup of contraction (S(t))t≥0 on H. Proof It is clear that the operator A is dissipative, and moreover, for every z = (y, v, p, q) ∈ D(A), Re (Az, zH ) = −

 % % %vj (ak )%2 ≤ 0. k

(5.4)

∗ k∈Iext

Now, we prove that every positive real number λ belongs to ρ(A), the resolvent set of A. For this, let Z = (f , g, c, d) ∈ H , and we solve the equation (λ − A)z = Z

(5.5)

⎧ λyj − vj = fj , j ∈ J, ⎪ ⎪ ⎨ λvj − ∂x2 yj = gj , j ∈ J, ⎪ λp − qk = ck , k ∈ Iint , ⎪ ⎩ k λqk + pk + v(ak ) = dk , k ∈ Iint .

(5.6)

with z = (y, v, p, q) in D(A). We rewrite (5.5) explicitly

We eliminate (v, q) in (5.6) to get λ2 yj − ∂x2 yj = gj + λfj , j ∈ J, (λ2 + 1)pk + v(ak ) = dk + λck , k ∈ Iint .

(5.7) (5.8)

Let w in V . Multiplying (5.7) by wj in L2 (0, j ) and summing over j ∈ J, 



yj wj dx +

0

j ∈J

+

j

λ2 

∗ k∈Iext

w(ak )v(ak ) =

j

  ∂x yj ∂x wj dx − w(ak )qk

0



j ∈J

0

k∈Iint j

(gj + λfj )wj dx.

(5.9)

94

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

Multiplying (5.8) by rk ∈ C and summing over k ∈ Iint , we get 

(λ2 + 1)

pk r k +

k∈Iint





v(ak )rk =

k∈Iint

(dk + λck )rk .

(5.10)

k∈Iint

Summing (5.9) and (5.10), we get  j ∈J

+

     λ2 yj , wj + ∂x yj , ∂x wj + (λ2 + 1) pk r k  

v(ak )rk − w(ak )qk

k∈Iint

+



w(ak )v(ak ) =

∗ k∈Iext

k∈Iint



  gj + λfj , wj + (dk + λck )rk

 j ∈J

k∈Iint

to obtain a((y, p), (w, r)) = F (w, r), where a((y, p), (w, r)) = +λ

 

 j ∈J

     λ2 yj , wj + ∂x yj , ∂x wj + (λ2 + 1) pk r k 

y(ak )rk − w(ak )pk + λ



k∈Iint

w(ak )y(ak )

∗ k∈Iext

k∈Iint

and f (w, r) =

   gj + λfj , wj + (dk + λck )rk − w(ak )ck

 j ∈J

+



k∈Iint

f (ak )rk +



k∈Iint

w(ak )f (ak ).

∗ k∈Iext

k∈Iint

6 a is a continuous sesquilinear form on V × C and F is a continuous anti-linear k∈Iint 6 C. Moreover, form on V × k∈Iint

a((w, r), (w, r)) =

   2   2  |rk |2 λ2 wj  + ∂x wj  + (λ2 + 1) j ∈J

⎛

−2iλI m ⎝

 k∈Iint

⎞ w(ak )rk ⎠ + λ

k∈Iint

 % % %w(ak )%2 . ∗ k∈Iext

2 Exponential Stability

95

We have ⎡ ⎤     2   %  % 2 %a((w, r), (w, r))% ≥ ⎣ |rk |2 ⎦ , λ2 wj  + ∂x wj  + λ2 j ∈J

k∈Iint

that is, a is coercive. The conclusion results immediately from the Lax–Milgram lemma.   According to Theorem 1.15, we have the following results:   Proposition 5.2 Suppose that y 0 , y 1 , s 0 , s 1 ∈ H. Then, the problem (5.1) admits a unique solution (y, y  , s, s  ) ∈ C([0, +∞); H ).   If y 0 , y 1 , s 0 , s 1 ∈ D(A), then (y, y  , s, s  ) ∈ C([0, +∞), D(A)) ∩ C 1 ([0, +∞); H ). Moreover, (y, s) satisfies the energy estimate (5.2).

2 Exponential Stability It is clear that if T contains an edge ej , not attached to a leaf, with length j ∈ π N, then i is eigenvalue of A with eigenvector z = (y, v, p, q) such that yj = i sin x, vj = − sin x, pk = 1, and qk = i when ak is the nearest end of ej to the root R, and all the other components of z are null. In the following, the tree T is said to be a Pi-tree if the length of every edge not attached to a leaf is different from mπ for every m in N∗ . Then, we have the following result. Lemma 5.3 The spectrum of A contains no point on the imaginary axis if and only if T is a Pi-tree. Proof By the Sobolev embedding theorem, we can deduce that (I − A)−1 is a compact operator. Then, the spectrum of A only consists of eigenvalues. We will show that the equation Az = iβz with z = (y, v, p, q) ∈ D(A) and β ∈ R has only trivial solution.

(5.11)

96

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

By taking the inner product of (5.11) with z ∈ H and using that Re (Az, zH ) = −

 % % %v(ak )%2 , ∗ k∈Iext

we obtain that ∗ . v(ak ) = 0 for k ∈ Iext

(5.12)

Furthermore, by the second condition in (5.3), we also have ∗ . ∂x y(ak ) = 0 for k ∈ Iext

(5.13)

Now, Eq. (5.11) can be rewritten explicitly as vj = iβyj , j ∈ J,

(5.14)

∂x2 yj = iβvj , j ∈ J,

(5.15)

qk = iβpk , k ∈ Iint ,

(5.16)

−pk − v(ak ) = iβqκ , k ∈ Iint .

(5.17)

If β = 0, then v = 0, q = 0, and p = 0. Multiplying the second equation in the above system by yj and then summing over j, we obtain   ∂x yj 2 = 0, j ∈J

which implies, using the continuity condition of y at inner nodes and its Dirichlet condition at R, that y = 0. Next, we suppose that β = 0. From (5.12) and (5.14), and we get ∗ . yjk (ak ) = 0 for k ∈ Iext

Using (5.14)–(5.17), we have β 2 yj + ∂x2 yj = 0, j ∈ J, (β 2 − 1)pk = v(ak ), k ∈ Iint . The function yj , j ∈ J , is then of the form yj = α1 sin(βx) + α2 cos(βx).

(5.18) (5.19)

2 Exponential Stability

97

∗ , then Using (5.12), (5.13), and (5.18), we obtain that yjk = 0, for every k in Iext ∗ using (5.14), vjk = 0, for every k in Iext . For the sequel of the proof, we consider two cases.

First Case: β 2 = 1 From (5.19), (5.14), and the Dirichlet condition at R, we deduce that yj (0) = yj (j ) = 0 for j ∈ J. Since T is a Pi-tree, we conclude that yj = 0 for j ∈ J. Return back to the balance conditions and (5.16), one can deduce that q = p = 0 and hence z = 0. Second Case: β 2 = 1 Let ak be the second end of an edge ej attached to a leaf. Since yj and vj are zero, and using (5.19) and (5.16), we deduce that pk = 0 and qk = 0. Next, let el be an internal edge (i.e., not containing a leaf) attached at a node ak to an edge ej ended by a leaf. Using again the balance condition and the continuity of y at ak , we obtain that yl (ak ) = 0 and ∂x yl (ak ) = 0. Then, by (5.18) , yl = 0. We iterate such procedure from leaves to root to conclude that z = 0.   The main result of this chapter concerns the precise asymptotic behavior of the solution of (5.1). Our technique is based on a frequency domain method and a special analysis for the resolvent. Recall that the system (5.1) is said to be exponentially stable if there exist two  constants M, ω > 0, such that for all y 0 , y 1 , s 0 , s 1 ∈ H,  2   E(t) ≤ Me−ωt  y 0 , y 1 , s 0 , s 1  , ∀t ≥ 0. H

Then, our main result is the following: Theorem 5.4 The system defined by Eq. (5.1) is exponentially stable if and only if T is a Pi-tree. Proof The system defined by Eq. (5.1) is exponentially stable if and only if the C0 -semigroup of contraction (S(t))t≥0 , generated by A, is exponentially stable. By Theorem 1.25, it suffices to show that the operator A satisfies the following two conditions:     (5.20) ρ(A) ⊃ {iβ | β ∈ R} ≡ iR and lim sup (iβ − A)−1  < ∞. β∈R, |β|→∞

By Lemma 5.3, the first condition in (5.20) is satisfied if and only if T is a Pi-tree. Then, we suppose that T is a Pi-tree, and by contradiction, we suppose that the second condition in (5.20) is false. Then, there exist a sequence of real numbers βn → ∞ (βn > 0 without loss of generality) and a sequence of vector zn = (y n , v n , p n , q n ) ∈ D(A) with zn H = 1 such that (iβn I − A)zn H −→ 0 as n −→ ∞,

(5.21)

98

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

i.e., iβn yj,n − vj,n ≡ fj,n −→ 0 iβn vj,n − ∂x2 yj,n ≡ gj,n −→ 0

in H 1 (0, j ),

(5.22)

in L2 (0, j ),

(5.23)

iβn pk,n − qk,n ≡ hk,n −→ 0 in C,

(5.24)

iβn qk,n + pk,n + v n (ak ) ≡ rk,n −→ 0 in C,

(5.25)

for every j in J and every k in Iint . Our goal is to derive from (5.21) that zn H converge to zero, thus, a contradiction. The proof is divided into three steps. First Step Recall that for every j in J,        1  vj  ≤ C1 ∂x vj 1/2 vj 1/2 + C2 vj  , ∞ 1/2 1/2 βn βn

1/2 βn

for some positives constants C1 and C2 (we have used the Gagliardo–Nirenberg v  inequality (1.14)). This implies, using (5.22), that j,n1/2 ∞ is bounded. βn

Then, for every k in Iint , by dividing (5.25) by βn , we deduce that qk,n converge to zero, and then βn pk,n converge to zero in view of (5.24). In particular, pk,n converge to zero. Second Step Notice that from (5.4), we have  % % % % %v (ak )%2 . (iβn I − A)zn H ≥ %Re iβn I − A)zn , zn H % = n

(5.26)

∗ k∈Iext

Then, by (5.21), % % %v (ak )% −→ 0, ∀k ∈ I ∗ . n ext This further leads to % % % % ∗ %βn y n (ak )% −→ 0, ∀k ∈ Iext

(5.27)

due to (5.22) and the trace theorem. We also have dyjk ∗ (ak ) −→ 0, ∀k ∈ Iext . dx

(5.28)

2 Exponential Stability

99

Third Step Substitute (5.22) into (5.23) to get − βn2 yj,n − ∂x2 yj,n = gj,n + iβn fj,n , j ∈ J.

(5.29)

Next, we take the inner product of (5.29) with ∂x yj,n b in L2 (0, j ) for b ∈ C 1 [0, j ], we get j j %2 %2 1 2 %% 1 %% βn yj,n (x)% b(x) + ∂x yj,n (x)% b(x) 0 0 2 2 *  +j + Re iβn fj,n (x)yj,n (x)b(x) 0

%2 % %2  1 j %% βn yj,n % + %∂x yj,n % ∂x b(x)dx −→ 0. − 2 0

Using (5.27) and (5.28), this leads to 1 2

0

j k

% %βn yj

k ,n

%2 % % + %∂x yj

k ,n

%2  % dx −→ 0

∗ , by taking b = x or b =  − x. It follows that for every k in Iext jk

%2 1 % %2   1 2 %% βn yjk ,n (as )% + %∂x yjk ,n (as )% + Re iβn fjk ,n (as )yjk ,n (as ) −→ 0, 2 2

(5.30)

where as is the end of ejk , different from ak . We will show that all the terms in the first members of (5.30) converge to zero. To do that, we use the following inequality: %2 % %2   1 % Re iβn fjk ,n (as )yjk ,n (as ) ≥ − βn2 %yjk ,n (as )% − %fjk ,n (as )% 4 to obtain %2 1 % %2 %2   % 1 2 %% βn yjk ,n (as )% + %∂x yjk ,n (as )% + Re iβn fjk ,n (as )yjk ,n (as ) + %fjk ,n (as )% 2 2 %2 1 % %2 1 2 %% ≥ βn yjk ,n (as )% + %∂x yjk ,n (as )% ≥ 0. (5.31) 4 2 In addition, we have fjk ,n (as ) −→ 0. Then, (5.31) combined with (5.30) implies that βn yjk ,n (as ) −→ 0, and ∂x yjk ,n (as ) −→ 0.

100

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

Furthermore, we also have   Re iβn fjk ,n (as )yjk ,n (as ) −→ 0. We then conclude by iteration, as in [88], that for every j in J,

j

% % % %  %βn yj,n %2 + %∂x yj,n %2 dx −→ 0.

0

Finally, in view of (5.22), we also get   vj,n  −→ 0, for j ∈ J.   In conclusion, pk,n and qk,n converge to zero for every k in Iint , and ∂x yj,n  and   vj,n  converge to zero for every j in J, which implies that zn H −→ 0 clearly contradicts (5.21).  

3 Two Examples of Non-exponential Stability In this section, we consider two particular cases. In the first case (Fig. 5.2), there is a circuit in the graph. We prove that, even with much more controls, the exponential stability fails. In the second case (Fig. 5.3), we eliminate a control of a leaf. We prove that the new system is also not exponentially stable.

3.1 A Circuit (Fig. 5.2) In this part, we suppose that T contains a circuit (Fig. 5.2), with 1 = 2 = 3 = 1 and with feedbacks at each inner node. Then, the second equation in (5.1) will be  j ∈Jk

dkj yj,x (ak , t) = s1 (t) − y t (ak , t), ak ∈ Iint ,

and we can rewrite the system (5.1) in the Hilbert space H as z (t) = Az(t). The associated state space is     3 2 2 2 H = V × L (0, 1) × L (0, 4 ) × C3

3 Two Examples of Non-exponential Stability

101

with   3 V =  ∈ H 1 (0, 1) × H 1 (0, 4 ), 1 (1) = 0, 1 (0) = 2 (0) = 3 (0) , 2 (1) = 4 (0), 3 (1) = 4 (4 )} . The system (5.1) can be rewritten in the Hilbert space H as   z (t) = Az(t), z(0) = y 0 , y 1 , s 0 , s 1 , where A is the operator defined on H by A(y, v, p, q) = (v, ∂x2 y, q, −p − v M ), ∀(y, v, p, q) ∈ D(A), with v M = (v(a2 ), v(a3 ), v(a4 )) and 3  4  3  2 H 2 (0, 1) × H 2 (0, 4 ) ∩ V × V × C3 D(A) = (y, v, p, q) ∈ satisfying (5.32) , where ⎧ " 3 ⎪ ⎨ − j =1 ∂x yj (0) = q2 − v(a2 ), ∂x y2 (1) − ∂x y4 (0) = q3 − v(a3 ), ⎪ ⎩ ∂ y (1) + ∂ y ( ) = q − v(a ). x 3 x 4 4 4 4

(5.32)

The operator A generates a C0 -semigroup of contraction (S(t))t≥0 satisfying the first result of asymptotic behavior. Theorem 5.5 (S(t))t≥0 is asymptotically stable if and only if 4 is irrational and not in π Z. Proof First, if 4 is in π Z, then i is an eigenvalue of A (as in the case of a tree), and if 4 = ab with a and b integer, then ibπ is an eigenvalue of A. Now, we suppose that 4 ∈ / Q and 4 ∈ / π Z. We only need to prove that iR ⊂ ρ(A). For this, we will prove that the equation Az = iβz with z = (y, v, p, q) ∈ D(A) and β ∈ R has only trivial solution.

(5.33)

102

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

The real part of the inner product of (5.33) with z ∈ H is Re (Az, zH ) = −

 % % % % % % % % %v(ak )%2 = −(%v(a2 )%2 + %v(a3 )%2 + %v(a4 )%2 );

(5.34)

k∈Iint

then v(ak ) = 0 for every k ∈ 2, 3, 4. Now Eq. (5.33) can be rewritten as follows: vj = iβyj , j ∈ J,

(5.35)

∂x2 yj = iβvj , j ∈ J,

(5.36)

qk = iβpk , k ∈ 2, 3, 4,

(5.37)

−pk − v(ak ) = iβqκ , k ∈ 2, 3, 4.

(5.38)

If β = 0, then, as in for the initial example, we show that y = 0. Next, we suppose that β = 0. We have y(ak ) = 0 for every k ∈ 2, 3, 4

(5.39)

in view of (5.34)–(5.36), and β 2 yj + ∂x2 yj = 0, j ∈ J, (β − 1)pk = 0, k ∈ 2, 3, 4. 2

(5.40) (5.41)

By using (5.35)–(5.38), the condition (5.39) implies that the solution of (5.40) is of the form yj = αj sin(βx), where αj ∈ R. As in the case of a tree, we consider two cases: β 2 = 1 and β 2 = 1. First Case: β 2 = 1 We have, using again (5.36), α1 = α2 = α3 = 0, and since 4 ∈ / π Z, α4 = 0. Return back to the balance condition and (5.37), at inner nodes, one can deduce that q = p = 0 and hence z = 0. Second Case: β 2 = 1 We have pk = qk = 0 for every k in Iint . Now, if β ∈ / π Z, then (5.38) gives that α1 = α2 = α3 = 0, and the balance condition at a3 implies that α4 = 0. If β ∈ π Z, then (5.38) gives α4 = 0, since 4 ∈ / Q. Using again the balance condition, respectively, at a3 , a4 , and a2 , we obtain that α2 = α3 = α1 = 0. Then, z = 0.   Theorem 5.6 The semigroup (S(t))t≥0 is not exponentially stable, even if 4 is irrational and not in π Z.

3 Two Examples of Non-exponential Stability

103

Proof To prove that (S(t))t≥0 is not exponentially stable, we consider the sequence fn of vectors of H defined by fn = (0, g n , 0, 0), where g n = (0, − sin βn x, 0, 0) and βn is a sequence of real numbers satisfying βn −→ +∞ and that will be defined later. We then prove that the sequence zn = (y n , v n , p n , q n ) of elements of D(A) such that (iβn − A)zn = fn is not bounded. The sequences y n and q n should satisfy ⎧ 2 2 ⎪ ⎨ βn yj,n + ∂x yj,n = 0, for j = 1, 3, 4, 2 βn y2,n + ∂x2 y2,n = sin βn x, ⎪ ⎩ q = − βn2 y (a ) = −β c y (a ), for k = 2, 3, 4, k,n n n n k β 2 −1 n k n

n with cn = β 2β−1 . Then, for j = 1, 3, 4, there exist two complex numbers aj and bj n (depending of n) such that



yj,n = aj sin(βn x) + bj cos(βn x), ∂x yj,n = −βn bj sin(βn x) + βn aj cos(βn x),

and there exist two complex numbers a2 and b2 (depending of n) such that 5

y2,n = a2 sin(βn x) + (− 2βx n + b2 ) cos(βn x), ∂x y2,n = ( x2 − βn b2 ) sin(βn x) + (− 2β1n + βn a2 ) cos(βn x).

The boundary and transmission conditions are expressed as follows: ⎧ a1 sin(βn ) + b1 cos(βn ) = 0, ⎪ ⎪ ⎪ ⎪ b1 = b2 = b3 , ⎪ ⎪ ⎪ ⎪ − 1 + βn a1 + βn a2 + βn a3 = (i + cn )βn b1 , ⎪ ⎪ 2βn ⎪ ⎪ ⎨ a2 sin(βn ) + (− 2β1n + b2 ) cos(βn ) = b4 ,   1 1 ⎪ ⎪ βn a4 − ( 2 − βn b2 ) sin(βn ) + (− 2βn + βn a2 ) cos(βn ) = (i + cn )βn b4 , ⎪ ⎪ ⎪ ⎪ a3 sin(βn ) + b3 cos(βn ) = a4 sin(βn 4 ) + b4 cos(βn 4 ), ⎪ ⎪ ⎪ ⎪ ⎪ −β b sin(βn ) + βn a3 cos(βn ) − βn b4 sin(βn 4 ) + βn a4 cos(βn 4 ) = ⎪ ⎩ n 3 −(i + cn )βn (a3 sin(βn ) + b3 cos(βn )). (5.42) Our goal is to prove that βn a3 converge to infinity. A straightforward calculation leads to (F A + GB)βn a3 = BH + F C,

(5.43)

104

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

where A = sin βn + ((i + cn ) sin βn + cos βn ) sin βn 4 + sin βn cos(βn 4 ),   2 cos βn 2 + 3(i + cn ) cos βn − (2 − cn − 2icn ) sin βn sin(βn 4 ) B = − cos βn + sin βn + (2 cos βn − (i + cn ) sin βn ) cos(βn 4 ),   1 i + cn i + cn (1 + cos βn sin(βn 4 ) ) sin βn − C= 2 βn 2 1 sin βn + ( − cos βn ) cos(βn 4 ), 2 βn and F = − sin βn + (i + cn ) cos βn − (2 cos βn + (i + cn ) sin βn ) sin(βn 4 )   2 cos βn 2 + 3(i + cn ) cos βn − (2 − cn − 2icn ) sin βn cos(βn 4 ), + sin βn G = cos βn + (i + cn ) sin βn + sin βn sin(βn 4 ) − ((i + cn ) sin βn + cos βn ) cos(βn 4 ),     i + cn 1 i + cn 1 sin βn (1 + cos βn − cos βn sin(βn 4 ) − ) sin βn − H = 2 βn 2 βn 2 × cos(βn 4 ). Now, by using the Asymptotic Dirichlet theorem [86], there exists (Pn , Qn ) ∈ N2 Pn such that Q converge to 4 , Pn and Qn tend to infinity as n goes to infinity and for n every n in N, |Qn 4 − Pn | < Take βn = 2π Qn + integer n ≥ n0 , 0 < λn := −

2π 1/4 ; Qn

1 . Qn

then, there exists a positive integer n0 such that for every

2π 2π 4 2π 2π 4 π + 1/4 < βn 4 − 2π Pn < μn := + 1/4 < Qn Q 2 n Qn Qn

and sin(λn ) < sin(βn 4 ) < sin(μn ), cos(μn ) < cos(βn 4 ) < cos(λn ).

3 Two Examples of Non-exponential Stability

105

Moreover, sin(βn 4 ) and cos(βn 4 ) satisfy the following asymptotic approximations: # # $ $ 1 1 2π 4 sin(βn 4 ) = 1/4 + o , cos(βn 4 ) = 1 + o . 1/4 1/4 Qn Qn Qn We also have # sin(βn ) = sin

$

2π 1/4

Qn # 1/4

Qn cot(βn ) = 2π

2π

=

1/4

#

1+o

Qn

# +o $$

1 1/4

1 1/4

Qn

#

$ , cos(βn ) = 1 + o

$

1 1/4

, and

Qn

.

Qn

It follows that 2π(2 + 4 )

#

1

$

1 C = − + o(1), 2 # $$ # $ # 1/4 1 Qn 1 i 2π F = , G=o , H = − + o(1). 1 + 4i 1/4 + o 1/4 1/4 2π 2 Qn Qn Qn A=

1/4 Qn

+o

1/4 Qn

,

B = 1 + 4 + o(1),

Returning back to (5.43), we could write 1/4

i Qn (2 + 4 + o(1))βn a3 = − (1 + 4 ) + o(1) − 2 4π

# 1 + 4i

2π 1/4

Qn

# +o

1 1/4

$$ .

Qn

Hence, βn a3 ∼ −

1 1/4 Qn , 4π(2 + 4 )

  which implies that y3,n  converges to infinity as n goes to infinity and that consequently zq n is not bounded.   Remark 5.7 A small change in the proof leads to the conclusion that a polynomial stability cannot be better than t12 in the case of this special circuit (by using a frequency domain characterization of polynomial stability of a C0 -semigroup of contraction due to Borichev and Tomilov and given in Theorem 1.26). Precisely, we prove that the system is not t1α -polynomially stable for every α in (2, ∞).

106

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

3.2 A Star with Two Fixed Endpoints (Fig. 5.3) In this example, we have taken 1 = 2 = 1, and the two exterior ends of e1 and e2 are supposed to be fixed. More precisely, we consider the following system: ⎧ yj,tt − yj,xx = 0 in (0, 1) × (0, ∞), j ∈ {1, 2}, ⎪ ⎪ ⎪ ⎪ ⎪ y3,tt − y3,xx = 0 in (0, 3 ) × (0, ∞), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ " ⎪ ⎪ yj,x (0, t) = s  (t), s  (t) + s(t) = −yt (0, t), ⎪ ⎪ ⎪ ⎨ j =1 yj (0, t) = yl (0, t), j, l ∈ {1, 2, 3}, ⎪ ⎪ ⎪ y2 (1, t) = y3 (3 , t) = 0, y1,x (1, t) = −yt (1, t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s(0) = s0 , s  (0) = s1 , ⎪ ⎪ ⎪ ⎪ yj (x, 0) = yj0 (x), yj,t (x, 0) = yj1 (x), x ∈ (0, 1), j ∈ {1, 2}, ⎪ ⎪ ⎩ y3 (x, 0) = y30 (x), y3,t (x, 0) = y31 (x), x ∈ (0, 3 ).

(5.44)

We can rewrite the system (5.44) in the Hilbert space H as z (t) = Az(t), where H =V ×

  2 L2 (0, 1) × L2 (0, 3 ) × C2

with   2 V =  ∈ H 1 (0, 1) × H 1 (0, 3 ), 2 (1) = 3 (3 ) = 0, j (0) = l (0), j, l ∈ {1, 2, 3}



and       A y, v, p, q = v, ∂x2 y, q, −p − v(0) , ∀ y, v, p, q ∈ D(A), with   4  3 2 2 2 D(A) = y, v, p, q ∈ H (0, 1) × H (0, 3 ) ∩ V × V × C2 ; 3  j =1

⎫ ⎬

∂x yj (0) = q, and ∂x y1 (1) = −v1 (1) . ⎭

3 Two Examples of Non-exponential Stability

107

The operator A generates a C0 -semigroup of contraction (S(t))t≥0 , and we have the following result. Theorem 5.8 The semigroup (S(t))t≥0 is asymptotically stable if and only if 3 is irrational and not in π Z. Even if 3 is irrational and not in π Z, the semigroup (S(t))t≥0 is not exponentially stable. Proof As in the case of a circuit, we consider the sequence fn of vectors of H defined by fn = (0, g n , 0, 0), where g n = (0, − sin βn x, 0) and βn is a sequence of real numbers satisfying βn −→ +∞ and that will be defined later, such that the sequence zn = (y n , v n , pn , qn ) of elements of D(A) satisfying (iβn − A)zn = fn is not bounded. The sequences y n and qn should satisfy ⎧ 2 2 ⎪ ⎨ βn yj,n + ∂x yj,n = 0, for j = 1, 3, 2 βn y2,n + ∂x2 y2,n = sin βn x, ⎪ ⎩ q = − βn2 y (0) = −β c y (0), n

βn2 −1 n

n n n

n with cn = β 2β−1 . Then, for j = 1, 3, there exist two complex numbers aj and bj n (depending of n) such that



yj,n = aj sin(βn x) + bj cos(βn x), ∂x yj,n = −βn bj sin(βn x) + βn aj cos(βn x),

and there exist two complex numbers a2 and b2 (depending of n) such that 5

yn2 = a2 sin(βn x) + (− 2βx n + b2 ) cos(βn x), ∂x y2,n = ( x2 − βn b2 ) sin(βn x) + (− 2β1n + βn a2 ) cos(βn x).

The boundary and transmission conditions are expressed as follows: ⎧ a2 sin(βn ) + (− 2β1n + b2 ) cos(βn ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ a3 sin(βn 3 ) + b1 cos(βn 3 ) = 0, b1 = b2 = b3 , ⎪ 1 ⎪ ⎪ − ⎪ 2βn + βn a1 + βn a2 + βn a3 = −βn cn b1 , ⎪ ⎩ −βn b1 sin(βn ) + βn a1 cos(βn ) = −βn a1 sin(βn ) − iβn b1 cos(βn ).

(5.45)

From the first, the second, and the sixth equations in (5.45), we deduce  a2 =

 1 − b2 cot(βn ), 2βn

a3 = −b3 cot(βn 3 ), and a1 = −ib1 .

(5.46)

108

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

Using (5.46) in the fifth equation of (5.45), taking into account that b1 = b2 = b3 , to get βn b1 [−cn + i + cot(βn ) + cot(βn 3 )] = −

1 1 + cot(βn ). 2βn 2

(5.47)

Pn As in the previous case, there exists (Pn , Qn ) ∈ N2 such that Q converge to 3 , n Pn and Qn tend to infinity as n goes to infinity and for every n in N,

|Qn 4 − Pn | < We take again βn = 2π Qn + 1/4

Qn cot(βn 3 ) = 2π 3

# 1+o

#

2π 1/4 , Qn

1

1 . Qn

and then we get (as in the first case)

$$

1/4

Qn

1/4

Qn , and cot(βn ) = 2π

# 1+o

#

1 1/4

$$ .

Qn

It follows from (5.47) that 1/4

βn b1 (1 + 3 )

1/4

Qn Qn ∼ 2π 3 4π 3

as n goes to infinity. Hence, using the second equality in (5.46), we obtain 1 1/4 Qn 4π(1 + 3 )   as n goes to infinity, which implies that y3,n  converges to infinity as n goes to infinity and that consequently zn is not bounded.   βn a3 ∼ −

4 A Chain with Non-equal Mass Points In this section, we consider a particular network which is a chain of N edges (N ≥ 2) and p = N + 1 vertices such that the (N − 1) interior vertices aj are point masses with mass mj . But the masses mj are not necessarily equal (Fig. 5.4). Precisely, we consider the following system:

4 A Chain with Non-equal Mass Points

109

⎧ yj,tt − yj,xx = 0 in (0, j ) × (0, ∞), j ∈ {1, . . . , N }, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ yj,x (0, t) − yj −1,x (j −1 , t) = sj (t), j ∈ {2, . . . , N }, ⎪ ⎪ ⎪ ⎪ mj s  (t) + sj (t) = −yt (0, t), ⎪ j ∈ {2, . . . , N }, ⎪ j ⎪ ⎨ yj −1 (j −1 , t) = yj (0, t), j ∈ {2, . . . , N }, ⎪ y1,x (0, t) = yt (0, t), ⎪ ⎪ ⎪ ⎪ y N (N , t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0  1 ⎪ ⎪ j ∈ {2, . . . , N }, ⎪ sj (0) = sj , sj (0) = sj , ⎪ ⎩ 0 yj (x, 0) = yj (x), yj,t (x, 0) = yj1 (x), x ∈ (0, j ), j ∈ {1, . . . , N }.

(5.48)

Note that the feedback is applied at the vertex a1 . We give a necessary and sufficient condition for the exponential stability of system (5.48). The general case of a tree with distinct masses at inner nodes is complicated for the moment, because the calculations are based on some recurrence relations, something we could not do for a general tree. To start, we quickly redefine the associated state space H and the operator A as follows: H =V ×

N 

⎛ L2 (0, j ) × ⎝

j =1

N −1 

⎞2 C⎠

j =1

with ⎧ ⎫ N ⎨ ⎬  V = ∈ H 1 (0, j ), N (N ) = 0, j −1 (j −1 ) = j (0), j = 2, . . . , N ⎩ ⎭ j =1

and       A y, v, p, q = y, ∂x2 y, q, −m−1 (p + v M ) , ∀ y, v, p, q ∈ D(A) 1 with −m−1 (p + v M ) = (− mj pj −

D(A) =

⎧ ⎪ ⎨ ⎪ ⎩



⎡

y, v, p, q ∈ ⎣

1 mj



⎤



k∈Iint

⎫ ⎬ ⎭

⎛

H 2 (0, j ) ∩ V ⎦ × V × ⎝

j ∈J

satisfying (5.49)

v(j ))j ∈{2,...,N −1} and

,

⎞2 C⎠

110

5 Feedback Stabilization of a Simplified Model of Fluid–Structure Interaction. . .

where 

∂x yj (0) − ∂x y(j −1 ) = qj , ∂xy1 (0) = v1 (0).

j ∈ {2, . . . , N },

(5.49)

Then, the operator A generates a C0 -semigroup of contraction (S(t))t≥0 . Moreover, σ (A) = σp (A). For every mass point m, we denote by i1 (m), . . . , ikm (m) the indices of the interior nodes with masses equal to m and ordered as follows: i1 (m) < i2 (m) < . . . < ikm (m). For r = r(m) ∈ {1, . . . , km }, we define the scalars #s−1 



m,r(m),s =

ir =j0 H = < (u, v, (u,



j ∈J

j

  vj (x) v˜ j (x) + ∂x uj (x) ∂x u(x) ˜ dx.

(6.11)

0

System (6.3)–(6.7) can be rewritten as the first-order evolution equation # $ ⎧ # $ ⎪ ∂ u u ⎪ ⎨ = A ∂u , ∂u ∂t ∂t ∂t ⎪ ⎪ ⎩ u(0) = u0 , ∂u = u1 ∂t

(6.12)

120

6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping

where the operator A : D(A) ⊂ H → H is defined by # $   v u := A , ∂x (∂x u + β ∗ ∂x v) v with β := (βj )j ∈J and β ∗ ∂x v := (βj ∂x vj )j ∈J , and D(A) :=

⎧ ⎨ ⎩

(u, v) ∈ H, v ∈ V , (∂x u + β ∗ ∂x v)

∈

 j ∈J



⎫ ⎬

H 1 (0, j ) : (u, v) satisfies (6.13) , ⎭

  dkj ∂x uj (ak ) + βj (ak )∂x vj (ak ) = 0,

t > 0, ak ∈ Vint .

(6.13)

j ∈Jk

Lemma 6.1 The operator A is dissipative, 0 ∈ ρ(A) : the resolvent set of A. Proof For (u, v) ∈ D(A), we have   Re( A(u, v), (u, v) H )

  j ∂x vj ∂x uj dx + = Re j ∈J

0

j

 ∂x (∂x uj + βj ∂x vj )vj dx .

0

Performing integration by parts and using transmission and boundary conditions, a straightforward calculation leads to    Re( A(u, v), (u, v) H ) = − j ∈J



j

% %2 βj (x) %∂x vj (x)% dx ≤ 0

0

which proves the dissipativeness of the operator A in H. Next, using Lax–Milgram’s lemma, we prove that 0 ∈ ρ(A). For this, let (f , g) ∈ H and we look for (u, v) ∈ D(A) such that A(u, v) = (f , g)

1 Well-Posedness of the System

121

which can be written as vj = fj , j ∈ J,

(6.14)

∂x (∂x uj + βj ∂x vj ) = gj , j ∈ J.

(6.15)

v is completely determined by (6.14). Let w ∈ V ; multiplying (6.15) by wj , then summing over j ∈ J , we obtain, using transmission and boundary conditions, 

j ∈J

j

   ∂x uj + βj ∂x vj ∂x wj dx = −

0



j ∈J

j

(6.16)

gj wj dx. 0

Replacing vj in the last equality by (6.14), we get ϕ(u, w) = ψ(w),

(6.17)

where ϕ(u, w) =



j

∂x uj ∂x wj 0

j ∈J

and ψ(w) = −

 

j

gj wj dx +

0

j ∈J



j

βj ∂x fj ∂x wj dx . 0

The function ϕ is a continuous sesquilinear form on V × V and ψ is a continuous anti-linear form on V ; here V is equipped with the inner product  

f,g =



j ∈I

j

∂x fj ∂x gj .

0

Since ϕ is coercive on V , by the Lax–Milgram lemma, Eq. (6.17) has a unique solution u ∈ V . Then taking w ∈ D(0, j ) in (6.17) and integrating by parts, we j ∈J  deduce that (∂x u + β ∗ ∂x v) ∈ H 1 (0, j ) and (u, v) satisfies (6.14) and (6.15). j ∈J

Moreover (u, v) satisfies (6.13). Returning back to the Lax–Milgram lemma, (u, v) verifies      (u, v) ≤  (f , g)   . H H

In conclusion (u, v) ∈ A and A−1 ∈ L(H), which assert that 0 ∈ ρ(A).

 

122

6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping

By the Lumer–Phillip’s theorem (see [79, 92]), we have the following proposition. Proposition 6.2 The operator A generates a C0 -semigroup of contraction (Sd (t))t≥0 on the Hilbert space H. 0 1  for an initial datum (u , u ) ∈ H, there exists a unique solution  Hence, ∂u u, ∂t ∈ C([0, +∞), H) to problem (6.12). Moreover, if (u0 , u1 ) ∈ D(A), then  u,

∂u ∂t

 ∈ C([0, +∞), D(A)). ∂u

Furthermore, the solution (u, ∂t ) of (6.3)–(6.7) with initial datum in D(A) satisfies (6.9). Therefore the energy is decreasing.

2 Asymptotic Behavior In order to analyze the asymptotic behavior of system (6.3)–(6.7), we shall use Theorems 1.25 and 1.26 which respectively characterize exponential and polynomial stability of a C0 -semigroup of contraction: Lemma 6.3 (Asymptotic Stability) The operator A verifies (1.9) and then the associated semigroup (S(t))t≥0 is asymptotically stable on H. Proof Since 0 ∈ ρ(A) we only need here to prove that (iwI − A) is a one-to-one correspondence in the energy space H for all w ∈ R∗ . The proof will be done in two steps: In the first step we will prove the injective property of (iwI − A) and in the second step we will prove the surjective property of the same operator. • Suppose that there exists w ∈ R∗ such that Ker(iwI − A) = {0}. So λ = iw is an eigenvalue of A, and then let (u, v) be an eigenvector of D(A) associated to λ. For every j in J we have vj = iwuj ,

(6.18)

∂x (∂x uj + βj ∂x vj ) = iwvj .

(6.19)

We have     Re A(u, v), (u, v) H = − j ∈J

Then βj ∂x vj = 0 a.e. on (0, j ).

0

j

% %2 βj %∂x vj % dx = 0.

2 Asymptotic Behavior

123

Let ej be a K-V edge. According to (6.18) and the fact that βj ∂x vj = 0 a.e. on (0, j ), we have ∂x uj = 0 a.e. on j . Using (6.19), we deduce that vj = 0 on ωj . Returning back to (6.18), we conclude that uj = 0 on j . Putting y = ∂x uj + βj ∂x vj = (1 + iwβj )∂x uj , we have y ∈ H 2 (0, j ) and ∂x y = −w 2 uj . Hence y satisfies the Cauchy problem ∂x2 y +

w2 y = 0, y(z0 ) = 0, ∂x y(z0 ) = 0 1 + iwβj

for some z0 in j . Then y is zero on (0, j ) and hence ∂x uj and uj are zero on (0, j ). Moreover uj and ∂x uj + βj ∂x vj vanish at 0 and at j . If ej is a purely elastic edge attached to a K-V edge at one of its ends, denoted by xj , then uj (xj ) = 0, ∂x uj (xj ) = 0. Again, by the same way we can deduce that ∂x uj and uj are zero in L2 (0, j ) and at both ends of ej . We iterate such procedure on every maximal subgraph of purely elastic edges of G (from leaves to the root), to obtain finally that (u, v) = 0 in D(A), which is in contradiction with the choice of (u, v). • Now given (f , g) ∈ H, we solve the equation (iwI − A)(u, v) = (f , g) or equivalently, 5

v = iwu − f w 2 u + ∂x2 u + iw ∂x (β ∗ ∂x u) = (∂x β ∗ ∂x f ) − iwf − g.

(6.20)

Let us define the operator Au = −∂x2 u − iw ∂x (β ∗ ∂x u),

∀ u ∈ V.

It is easy to show that A is an isomorphism from V onto V  (where V  is the dual space of V obtained by means of the inner product in H ). Then the second line of (6.20) can be written as follows:   u − w 2 A−1 u = A−1 g + iwf − ∂x (β ∗ ∂x f ) .

(6.21)

If u ∈ Ker(I − w 2 A−1 ), then w 2 u − Au = 0. It follows that w 2 u + ∂x2 u + iw∂x (β ∗ ∂x u) = 0.

(6.22)

124

6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping

Multiplying (6.22) by u and integrating over T , then by Green’s formula we obtain w2



j ∈J

− iw

j

|uj (x)|2 dx −

0

j ∈J



j ∈J



j

j

|∂x uj (x)|2 dx

0

βj (x) |∂x uj (x)|2 dx = 0.

0

This shows that 

j ∈J

j

βj (x) |∂x uj |2 dx = 0,

0

which imply that β ∗ ∂x u = 0 in G. Inserting this last equation into (6.22), we get w2 u + ∂x2 u = 0,

in G.

According to the first step, we have that Ker(I − w2 A−1 ) = {0}. On the other hand thanks to the compact embeddings V → H and H → V  we see that A−1 is a compact operator in V . Now thanks to Fredholm’s alternative, the operator (I −w 2 A−1 ) is bijective in V , hence Eq. (6.21) has a unique solution in V , which yields that the operator (iwI − A) is surjective in the energy space H. The proof is thus complete.   Before stating the main result, we define a property (P) on β as follows: ∀j ∈ J, βj , βj ∈ L∞ (0, j ) and

∀ak ∈ Vint ,



dkj βj (ak ) ≤ 0.

(P)

j ∈Jk

Theorem 6.4 Suppose that the function β satisfies property (P), then (i) If β is continuous at every inner node of T , then (Sd (t))t≥0 is exponentially stable on H. (ii) If β is not continuous at least at an inner node of T , then (Sd (t))t≥0 is polynomially stable on H, in particular there exists C > 0 such that for all t > 0 we have    At 0 1  e (u , u )

H

≤

 C  0 1  (u , u ) , ∀ (u0 , u1 ) ∈ D(A).   D ( A) t2

2 Asymptotic Behavior

125

Proof According to Theorems 1.25, 1.26, and Lemma 6.3, it suffices to prove that for γ = 0, when β is continuous at every inner node, or γ = 1/2, when β is not continuous at an inner node, there exists r > 0 such that   |w|γ (iwI − A)(u, v)H ≥ r. inf (u,v)H ,w∈R

(6.23)

Suppose that (6.23) fails. Then there exists a sequence of real numbers wn , with wn → ∞ (without loss of generality,  suppose that wn > 0) and a sequence of vectors (un , v n ) in D(A) with (un , v n )H = 1 such that  γ  wn (iwn I − A)(un , v n )H → 0.

(6.24)

  We shall prove that (un , v n )H = o(1), which contradict the hypotheses on (un , v n ). Writing (6.24) in terms of its components, we get for every j ∈ J, γ

wn (iwn uj,n − vj,n ) =: fj,n = o(1) in H 1 (0, j ),

(6.25)

wn (iwn vj,n − ∂x (∂x uj,n + βj ∂x vj,n )) =: gj,n = o(1) in L2 (0, j ).

(6.26)

γ

Note that γ

wn



j ∈J

j

% %2 βj (x) %∂x vj (x)% dx

0

 γ   = Re wn (iwn I − Ad )(un , v n ), (un , v n ) H = o(1).

Hence, for every j ∈ J   γ  1  2  β ∂ v wn2   j x j,n 

L2 (0,j )

= o(1).

(6.27)

Then from (6.25), we get that    12   wn βj wn ∂x uj,n   γ 2

L2 (0,j )

= o(1).

(6.28) −γ

Define Tj,n = (∂x uj,n + aj ∂x vj,n ) and multiplying (6.26) by wn qTj,n where q is any real function in H 2 (0, j ), we get

j

Re 0

iwn vα,n ¯ qTj,n dx − Re

j 0

∂x Tj,n qTj,n dx = o(1).

(6.29)

126

6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping

Using (6.25) we have

j

iwn vα,n ¯ qTj,n dx

Re 0



j

= −Re



−γ

j

vj,n q(∂x vj,n + wn ∂x fj,n )dx + Re

0

iwn vj,n qβj ∂x vj,n dx

% %2 j % %2 1 1 j q(x) %vj,n (x)% + ∂x q %vj,n % dx 0 2 2 0

j −I m qβj wn vj,n ∂x vj,n dx + o(1).

0

=−

(6.30)

0

On the other hand, integrating the second term in (6.29) by parts yields

j

Re

∂x Tj,n qTj,n dx =

0

% %2 1 q(x) %Tj,n (x)% 2

j 0

−

1 2



j

% %2 ∂x q %Tj,n % dx.

(6.31)

0

Hence, by substituting (6.30) and (6.31) into (6.29), we obtain 1 2

0

j

% % %2 %2 1 j ∂x q %vj,n % dx + ∂x q %Tj,n % dx − I m qβj wn vj,n ∂x vj,n dx 2 0 0   % % 2 j  % %2 j 1  % % % % = o(1). (6.32) q(x) vj,n (x) − + q(x) Tj,n (x) 0 0 2

j

Lemma 6.5 The following property holds:

j

Im

qβj wn vj,n ∂x vj,n dx = o(1).

(6.33)

0 γ

1

Proof Since wn2 βj2 ∂x vj,n → 0 in L2 (0, j ) and q ∈ L∞ (0, j ), it suffices to prove that    1− γ2  12  wn βj vj,n  = O(1). (6.34)  L2 (0,j )

1−2γ

For this, taking the inner product of (6.26) by iwn 2−γ wn

 2  12  β vj,n   j  2

=

L (0,j )

1−γ −iwn



βj vj,n leads to

j

∂x Tj,n βj vj,n dx 0

1−2γ − iwn



j

gj,n βj vj,n dx. 0

(6.35)

2 Asymptotic Behavior

127

Since βj ∈ L∞ (0, j ) and gj,n → 0 in L2 (0, j ) we can deduce the inequality 1−2γ



j

− Re(iwn

gj,n βj vj,n dx) ≤

0

 2 1  1 2−γ  2  β v + o(1). wn  j j,n   4 L2 (ωj )

(6.36)

On the other hand, we have 

1−γ − Re iwn



j

∂x Tj,n βj vα,n ¯ dx

0

 1−γ = −Re iwn Tj,n (x)βj (x)vj,n (x) 3

1−γ + Re iwn

j 0

  4   βj ∂x uj,n vj,n + βj βj ∂x vj,n vj,n + βj ∂x uj,n ∂x vj,n dx .

j 0

(6.37) Using (6.27) and (6.28) we have 

1−γ Re iwn



j

= o(1).

βj ∂x uj,n ∂x vj,n dx

(6.38)

0

Using again (6.27) and the fact that βj ∈ L∞ (0, j ), we conclude that 

1−γ Re iwn

j 0

βj βj ∂x vj,n vj,n dx

 2 1  1 2−γ  2  β wn  v +o(1). j j,n   4 L2 (0,j )

 ≤

(6.39)

Now by (6.26), we obtain after integrating by parts that 3

1−γ Re iwn 0

j

βj ∂x uj,n vj,n dx

3

−γ = Re wn 0

α¯

βj (∂x vj,n

% %2 1  −γ  wn βj (x) %vj,n (x)% = 2

4

−γ + wn ∂x fj,n )vj,n dx j 0

1 −γ − wn 2

0

j

4

% %2 βj %vj,n % dx + o(1).

Furthermore, using that βj ∈ L∞ (0, j ) and that vj,n is bounded, we deduce 3

1−γ Re iwn

j 0

βj ∂x uj,n vj,n dx

4 ≤

% %2 1  −γ  wn βj (x) %vj,n (x)% 2

j 0

+ O(1). (6.40)

128

6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping

Combining (6.38)–(6.40) with (6.37), we get 1−γ − Re(iwn



j

 1−γ ∂x Tj,n βj vj,n dx) ≤ −Re iwn Tj,n (x)βj (x)vj,n (x)

0

j 0

% %2 j 1  −γ + wn βj (x) %vj,n (x)% 0 2  2 1  1 2−γ  2  + wn  + O(1). βj vj,n  2 4 L (0,j ) (6.41)

Thus, substituting (6.36) and (6.41) into (6.35) leads to 1 2−γ wn 2

 2  12  β vj,n   j  2

 1−γ ≤ −Re iwn Tj,n (x)βj (x)vj,n (x)

L (0,j )

+

% %2 1  −γ  wn βj (x) %vj,n (x)% 2

j 0

j 0

+ O(1). (6.42)

Summing over j ∈ J,  j ∈J

 2  1  2 2 wn βj vj,n   2

≤ −2

L (0,j )

⎛



Re ⎝iwn

1−γ

v n (ak )

ak ∈Vint −γ

+ wn



⎞ dkj βjk (ak )Tjk ,n (ak )⎠

j ∈Jk

 % %  %v (ak )%2 dkj βj k (ak ) + O(1). n

ak ∈Vint

j ∈Jk

(6.43) We have used the continuity condition of v n and the compatibility condition (6.7) at inner nodes and the Dirichlet condition of u and v at external nodes. Note that from property (P) we have  % %  %v (ak )%2 dkj βj (ak ) ≤ 0. n ak ∈Vint

(6.44)

j ∈Jk

then to conclude, it suffices to estimate ⎛ ⎞   1−γ Re ⎝iwn v n (ak ) dkj βjk (ak )Tjk ,n (ak )⎠ . ak ∈Vint

j ∈Jk

Case (i), corresponding to γ = 0: Here β is continuous in all inner nodes. It follows   " " 1−γ that ak ∈Vint Re iwn v n (ak ) j ∈Jk dkj βjk (ak )Tjk ,n (ak ) = 0

2 Asymptotic Behavior

129

Then, (6.43) and (6.44) yield  2  1  2 2 wn βj vj,n   2 L (0,j )

= O(1)

for every j ∈ J, and the proof of Lemma 6.5 is complete for case (i). Case (ii), corresponding to γ = 12 : Recall that here the function β is not continuous at some internal nodes. We want estimate the first term in the right hand 1−γ side of (6.42). To do this it suffices to estimate Re(iwn Tj,n (x)βj (xj )vj,n (x)) at an inner node x = xj when βj (xj ) = 0. For simplicity and without loss of generality we suppose that xj is the end of ej identified to 0 via πj . Since βj is continuous on [0, j ], there exists a positive number kj < j such that βj (x) = 0 on [0, j ]. We first prove  2 wn vj,n L2 (0,k ) = o(1).

(6.45)

j

Using Gagliardo–Nirenberg inequality (1.14), (6.27), (6.28) and the boundedness of vj,n ,   vj,n 

L∞ (0,kj )

 1  1 ≤ C1 vj,n L2 2 (0,k ) ∂x vj,n L2 2 (0,k ) j j   +C2 vj,n  2 = O(1), L (0,kj )

 −3  wn 8 Tj,n L∞ (0,k

j)

 1  14 2  w ≤ C1  T  n j,n  2

L (0,kj )

 1  −1   2 w T  n j,n  2

L (0,kj )

 −3  +C2 wn 8 Tj,n L2 (0,k ) = o(1). j

− 12

It follows that iwn

as

1  2 * +k Tj,n (x)vj,n (x) 0j = o(1) and then wn2 vj,n L2 (0,k ) = o(1). j

−1 Then, we multiply (6.26) by iwn 2 vj,n and we repeat exactly 1  2 before, using (6.26) and wn2 vj,n  = o(1), we obtain (6.45).

the same strategy

1

We are now ready to estimate −Re(iwn2 Tj,n (0)vj,n (0)). Applying Gagliardo–Nirenberg inequality (1.14) to w = vj,n we obtain, using (6.27),  3 1 1   4 2   w wn2 vj,n L∞ (0,k ) ≤ C1  v j,n n   2 j

L (0,kj )

 1 1  4 2 wn ∂x vj,n    2

  3   4  ≤ o(1) + wn vj,n   o(1).

L (0,kj )

1   + C2 wn2 vj,n L2 (0,k ) j

130

6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping

Using again the Gagliardo–Nirenberg inequality (1.14) with w = Tj,n ,   Tj,n 

L∞ (0,kj )

 1  14 2  w ≤ C1  T  n j,n  2

L (0,kj )

 1  − 14 2   wn ∂x Tj,n  + C2 Tj,n  2   L (0,kj )

1  2  14  ≤ o(1) wn ∂x Tj,n   2

+ o(1)

L (0,kj )

   34   w ≤ o(1) + o(1)  v  n j,n 

.

L2 (0,kj )

Here, we have used (6.26)–(6.28). Then 1 1     |Re(iwn2 Tj,n (0)vj,n (0))| ≤ wn2 vj,n L∞ (0, ) Tj,n L∞ (0, j

≤

j)

2 1 32  wn vj,n L2 (0,k ) + o(1) j 4

(6.46)

and 3 1 4kj 1     − Re iwn2 Tj,n (x)vj,n (x) ≤ 2wn2 vj,n L∞ (0, ) Tj,n L∞ (0, j

0

≤

2 1 32  wn vj,n L2 (0,k ) + o(1). j 2

j)

(6.47)

Multiplying (6.26) by ivj,n in L2 (0, kj ) and integrating by parts, we obtain 3  1 * 1 2 +k wn2 vj,n L2 (0,k ) = −iwn2 Tj,n (x)vj,n (x) 0j + iwn2 j



kj

Tj,n ∂x vj,n dx + o(1).

0

(6.48) Using (6.27) and (6.28), the second term on the left hand side of (6.48) converges to zero. We conclude, using (6.47) that 3  2 wn2 vj,n L2 (0,k ) = o(1). j

Return back to (6.46) which yields 1

−Re(iwn2 Tj,n (0)vj,n (0)) = o(1).

2 Asymptotic Behavior

131

We obtain the same result if we suppose that xj is the end of ej,n identified to j via πj , that is 1

−Re(iwn2 Tj,n (j )vj,n (j )) = o(1), and we then conclude that the first term on the right hand side of (6.43) converges to zero. Then, again, using (6.44), we obtain that  j ∈J

 2  12   wn βj wn vj,n   2 1 2

= O(1),

L (0,j )

then  2 3  1  2  β wn2  v  j j,n  2

= O(1)

L (0,j )

for every j ∈ I, and the proof of Lemma 6.5 is complete for case (ii).

 

Return back to the proof of Theorem 6.4. Substituting (6.33) in (6.32) leads to 1 2



% % %2 %2 1 j ∂x q %vj,n % dx + ∂x q %Tj,n % dx 2 0 0   %2 % %2  j % 1 − q(x) %vj,n (x)% + %Tj,n (x)% = o(1) 0 2 j

(6.49)

for every j ∈ J. Let j ∈ J such that ej is a K-V string. First, note that from (6.34), we deduce that  2  12  β vj,n   j  2

= o(1).

L (0,j )

Then, we take q(x) = 1 2



j

0

x 0

βj (s)ds in (6.49) to obtain

% %2 1 βj %Tj,n % dx − 2



j

βj (s)ds

 % % % %  %vj,n (j )%2 + %Tj,n (j )%2 = o(1).

0

(6.50) Since

1 j 2 0

% %2

 βj %Tj,n % dx = o(1) and 0 j βj (s)ds > 0, then (6.50) implies % % % % %Tj,n (j )%2 + %vj,n (j )%2 = o(1).

(6.51)

132

6 Stability of a Graph of Strings with Local Kelvin–Voigt Damping

Therefore (6.49) can be rewritten as

% % %2 %2 1 α¯ ∂x q %vj,n % dx + ∂x q %Tj,n % dx 2 0 0  % %2 % %2  1 + q(0) %vj,n (0)% + q(0) %Tj,n (0)% = o(1). (6.52) 2     Taking q = x + 1 in (6.52) implies that vj,n L2 (0, ) = o(1) and Tj,n L2 (0, ) = j j     o(1). Moreover, ∂x uj,n  2 = Tj,n − βj vj,n  2 = o(1). Also we have 1 2



j

L (0,j )

L (0,j )

vj,n (0) = o(1) and Tj,n (0) = o(1).

(6.53)

Finally, notice that (6.51) signifies that vj,n (j ) = o(1) and Tj,n (j ) = o(1).

(6.54)

To conclude, it suffices to prove that   vj,n 

L2 (0,j )

    = o(1) and ∂x uj,n L2 (0, ) = Tj,n L2 (0, ) = o(1) j

j

(6.55)

for every j ∈ I such that ej is purely elastic. To do this, starting by a string ej attached at one end to only K-V strings. Using continuity condition of v and the compatibility condition at inner nodes, implies that eα¯ satisfies (6.53) or (6.54). Moreover, by taking q = 1 in (6.49), we conclude that ej satisfies (6.53) and (6.54). Then using again (6.49) with q = x, we deduce that (6.55) is satisfied by ej . We iterate such procedure on each maximally connected subgraphof purely elastic strings (from leaves to the root).   Thus (un , v n )H = o(1), which contradicts the hypothesis (un , v n )H = 1.   Remark 6.6 (1) If for every j ∈ J , βj is continuous on [0, j ] and not vanishes in such interval, then we do not need the property (P) in the Theorem 6.4. Indeed (P) is used only to estimate 

1−γ −Re iwn



j

∂x Tj,n βj vj,n dx 0

in (6.35), according to

1− γ wn 2

   12  β vj,n   j 

L2 (0,j )

.

2 Asymptotic Behavior

133

This is equivalent to estimate 

1−γ −Re iwn



j

∂x Tj,n vj,n dx

0 1− γ2

according to wn



1−γ −Re iwn

  vj,n 

L2 (0,j )

: 

j

∂x Tj,n vj,n dx

 1−γ = −Re iwn Tj,n vj,n

0



1−γ



+Re iwn 

j 0



j

Tj,n ∂x vj,n dx 0

1−γ

= −Re iwn

Tj,n (x) vj,n (x)

j 0

+ o(1)

as in case (ii) (proof of Theorem 6.4) we prove without using (P) that  1−γ −Re iwn Tj,n (x) vj,n (x)

j 0

2−γ

≤

wn 4

  vj,n 2 2

L (0,j )

+ o(1).

(2) We find here the particular cases studied in [39, 53, 54, 56, 87]. Note that concerning the result of polynomial stability in [39, 87] the authors proved that the t12 decay rate of solution is optimal when the damping coefficient is a characteristic function.

Conclusion

The mean purpose of this book is to study the stability of infinite-dimensional mathematical models of elastic networks with taking into account different forms of dissipation. The considered networks consist of finitely interconnected strings and beams or their combination. Such study is based on the semigroup approach to elaborate different kinds of stability (specially, exponential and polynomial stability) for the considered systems. When the dissipation is produced by a thermal effect, by coupling the wave (or the Petrowsky equation) with the heat equation on some components, we establish, under some geometric conditions of the network, that the whole system is exponentially stable. Note that the conditions imposed for a general graph of elastic– thermoelastic beams, to elaborate the exponential stability, can be applied to graphs of elastic–thermoelastic strings, to obtain similar results (Comment 1 of Chap. 2). Furthermore, same geometric conditions applied to a graph of strings with the Kelvin–Voigt damping produce exponential or polynomial decay of the associated semigroup, depending on the regularity of the damping coefficient function at inner nodes. In the case where external feedbacks are applied on the leafs of a tree of strings and beams, we proved that the associated semigroup (in an appropriate Hilbert space) is exponentially stable if there is no beam following a string (from leaves to the root); we proved the lack of exponential stability (on a simple example) when such condition fails , but the system is always polynomially stable. With the same type of feedback, applied on a tree modeling fluid–structure interaction, with the presence of point mass at inner nodes, we proved some results of stability depending on algebraic conditions on the length of some edges. The exponential stability fails in the case of a circuit attached to an edge or a star with two undamped external nodes. Some Remarks or Future Developments • In the case of a graph of elastic–thermoelastic of beams (or strings), what happens if the graph contains a maximal subgraph of thermoelastic components, which is © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Ammari, F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-86351-7

135

136

Conclusion

a circuit or a subgraph of purely elastic components with at least two endpoints in Vext ? Part of the answer is in [37], which briefly presented in Comment 2 of Chap. 2 where the authors consider a star-shaped network of elastic– thermoelastic rods. • As in Chap. 4, we can expect the stability of a graph of strings and beams, some of them are damped by thermal effect, in the two cases of Fourier’s law and Cattaneo’s law. One can extend the study to the nonlinear case as for a single string or a single beam. • The presence of circles in the networks considered in this book can be expected. What algebraic and geometric conditions on the network that guaranty the exponential or polynomial stability of the corresponding system? PDEs on cyclic graph can be found in some recent works, such as [65] where the authors studied a nonlinear fractional boundary value problem on a particular graph, namely, a circular ring with an attached edge. We also quote paper [16], where the authors studied the exponential stability of a system of transport equations with intermittent damping on a network of N circles intersecting at a single point O. See also [66] where the authors considered a Schrödinger equation on a tadpole graph. • Most models presented in this book can benefit, in future studies, from numerical simulation to consolidate the results obtained. See, for example, [37] for elastic– thermoelastic star-shaped network and [12] where the authors studied a nodal feedback stabilization of a star-shaped network of beams and a string. We can found, in some recent papers, numerical method to estimate solutions of EDPs on networks, such as [90? ].

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© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Ammari, F. Shel, Stability of Elastic Multi-Link Structures, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-030-86351-7

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