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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:
kais.ammari@fsm.rnu.tn,
†
Univ Lille Nord de France, F-59000 Lille, France, UVHC, LAMAV, FR CNRS 2956, F-59313
Valenciennes, France, email:denis.mercier@univ-valenciennes.fr, virginie.regnier@univ-valenciennes.fr
‡
Institut Elie Cartan Nancy (IECN), Nancy-Universit´ & INRIA (Project-Team CORIDA), B.P.
70239, F-54506 Vandoeuvre-l`s-Nancy Cedex France, email:julie.valein@iecn.u-nancy.fr
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