Spectral analysis and stabilization of a chain of

Denis Mercier

Description

Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings Kaıs Ammari ?, Denis Mercier †, Virginie Regnier † and Julie Valein ‡ Abstract. We consider N Euler-Bernoulli beams and N strings alternatively connected to one another and forming a particular network which is a chain begin- ning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We prove that the energy of the so- lution of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions on the length of the strings and beams. On another hand we prove a polynomial decay result of the energy of the second sys- tem, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a frequency domain method and combines a con- tradiction argument with the multiplier technique to carry out a special analysis for the resolvent. 2010 Mathematics Subject Classification. 35L05, 35M10, 35R02, 47A10, 93D15, 93D20. Key words and phrases. Network, wave equation, Euler-Bernoulli beam equation, spec- trum, resolvent method, feedback stabilization.

• feedback law


• kaıs ammari


• explicit decay


• decay rate


• ing conservative


• euler-bernoulli beams


• interesting stability results


• system

Sujets

• maths

Informations

• Publié par: pefav
• Nombre de lectures: 16
• Langue: English

Extrait

Spectral analysis and stabilization of a chain of

serially connected

∗
KaısAmmari,

Euler-Bernoulli beams and

strings

†
Denis Mercier,
and
‡
Julie Valein

†
Virginie R´gnier

Abstract.We considerNEuler-Bernoulli beams andNstrings alternatively
connected to one another and forming a particular network which is a chain
beginning with a string.We study two stabilization problems on the same network and
the spectrum of the corresponding conservative system:the characteristic equation
as well as its asymptotic behavior are given.We prove that the energy of the
solution of the first dissipative system tends to zero when the time tends to infinity
under some irrationality assumptions on the length of the strings and beams.On
another hand we prove a polynomial decay result of the energy of the second
system, independently of the length of the strings and beams, for all regular initial
data. Ourtechnique is based on a frequency domain method and combines a
contradiction argument with the multiplier technique to carry out a special analysis
for the resolvent.

2010 Mathematics Subject Classification.35L05, 35M10, 35R02, 47A10, 93D15, 93D20.
Key words and phrases.Network, wave equation, Euler-Bernoulli beam equation,
spectrum, resolvent method, feedback stabilization.
∗
D´partement de Math´matiques, Facult´ des Sciences de Monastir, 5019 Monastir, Tunisie, e-mail:
[e-mail],
†
Univ Lille Nord de France, F-59000 Lille, France, UVHC, LAMAV, FR CNRS 2956, F-59313
Valenciennes, France, email:[e-mail], [e-mail]
‡
Institut Elie Cartan Nancy (IECN), Nancy-Universit´ & INRIA (Project-Team CORIDA), B.P.
70239, F-54506 Vandoeuvre-l`s-Nancy Cedex France, email:[e-mail]

1

1

Introduction

In this paper we study two feedback stabilization problems for a string-beam network.
In the following, only chains will be considered as mathematically described in Section
5 of [25] (see also [26] and Figure 1).

•

0

k1

string

•

l10

k2

beam

•

k3

•

l20l30
string

k4

beam

•

k5

k6

•

•

l40l50l6
string beam

Figure 1:A chain with 2N= 6 edges

Following Ammari/Jellouli/Mehrenberger ([10]), we study a linear system modelling the
vibrations of a chain of alternated Euler-Bernoulli beams and strings but withNbeams
andNstrings (instead of one string-one beam in [10]).For each edgekj(representing
a string ifjis odd and a beam ifjis even) of the chain, the scalar functionuj(x, t) for
x∈(0, lj) andt >0 contains the information on the vertical displacement of the string
ifjis odd and of the beam ifjis even (1≤j≤2N), wherelj>0 is the length of the
edgekj.

More precisely we consider the evolution problems (P1) and (P2) described by the
fol

2

lowing systems of 2Nequations :

2 2
u−∂ u)
(∂t2j−1x2j−1(t, x) = 0, x∈(0, l2j−1), t∈(0,∞), j= 1, ..., N,
2 4
u u)(t, x) = 0, x∈(
(∂t2j+∂x2j0, l2j), t∈(0,∞), j= 1, ..., N,
u1(t,0) = 0, u2N(t, l2N) = 0, t∈(0,∞),

2 2
∂ u(t u
x2j,0) =∂x2j(t, l2j) = 0, t∈(0,∞), j= 1, ..., N,
(P1)
uj(t, lj) =uj+1(t,0), t∈(0,∞), j= 1, ...,2N−1,
3
∂ u) +∂ u
x2j(t,0x2j−1(t, l2j−1) =−∂tu2j−1(t,l2j−1), t∈(0,∞), j= 1, ..., N,
3
∂ u(t, l) +∂ u= 1, ..., N,
x2j2j x2j+1(t,0) =∂tu2j(t,l2j), t∈(0,∞), j

0 1
u(0, x) =u(x)u, ∂(0, x) =u(x) ), j= 1, ...,2N,
j jt jj, x∈(0, lj

and
2 2
∂ uu
(t2j−1−∂x2j−1)(t, x) = 0, x∈(0, l2j−1), t∈(0,∞), j= 1, ..., N,
2 4
∂ u)(t, x) = 0, x
(∂tu2j+x2j∈(0, l2j), t∈(0,∞), j= 1, ..., N,
2
u(t, u(t, l) =
10) = 0, u2N(t, l2N) = 0, ∂x2N2N0, t∈(0,∞),
22
u(t,
∂ u2j(t,0) =∂tx 2j0), t∈(0,∞), j= 1, ..., N,
x

22
(P)
2∂ u2j(t, l2j) =−∂u2j(t,l2j), t∈(0,∞), j= 1, ..., N−1,
xtx
uj(t, lj) =uj+1(t,0), t∈(0,∞), j= 1, ...,2N−1,
3
u(t,(0,∞), j= 1, ..., N,
∂x2j0) +∂xu2j−1(t, l2j−1) =−∂tu2j−1(t,l2j−1), t∈
3
∂xu2j(t, l2j) +∂xu2j+1(t,0) =∂tu2j+1(t,0), t∈(0,∞), j= 1, ..., N−1,

0 1
u(0, x) =u(x), ∂u( (x), x∈(0
j jt j0, x) =uj, lj), j= 1, ...,2N.

Models of the transient behavior of some or all of the state variables describing the
motion of flexible structures have been of great interest in recent years, for details about
physical motivation for the models, see [13], [19], [21] and the references therein.
Transmission problems on networks can be viewed as special cases of interaction
problems or problems on multistructures.They have been studied since the 1980ies for
example by J.P. Roth, J.v.Below, S. Nicaise, F. Ali Mehmeti.They use the
terminology of networks which had been fixed in earlier contributions of Lumer and Gramsch.
For an outline of recent developpements see [1].Mathematical analysis of transmission
partial differential equations is also detailed in [21].

Moreover, the control, observation and stabilization problems of networks have been the
object of intensive research (see [19, 21, 35] and the references therein).These works use
results from several domains:non-harmonic Fourier series, Diophantine approximations,
graph theory, wave propagation techniques.

3