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Stabilization of the wave equation on d networks

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Niveau: Supérieur, Doctorat, Bac+8
Stabilization of the wave equation on 1-d networks Julie Valein?, Enrique Zuazua †‡ August 1, 2008 Abstract In this paper we study the stabilization of the wave equation on general 1-d networks. For that, we transfer known observability results in the context of control of conservative systems (see [14]) into a weighted observability estimate for the dissipative one. Then we use an interpolation inequality similar to the one proved in [7] to obtain the explicit decay estimate of the energy for smooth initial data. The obtained decay rate depends on the geometric and topological properties of the network. We give also some examples of particular networks in which our results apply yielding different decay rates. Keywords 0.1 stabilization, wave equation, network 93D15, 93B07, 35L05 1 Introduction and main results In this paper, we consider a planar network of elastic strings that undergoes small perpendicular vibrations. Recently, the control, observation and stabilization problems of these networks have been the object of intensive research (see [14, 16] and the references therein). Here we are interested in the problem of stabilization of the network by means of a damping term located on one single exterior node. The aim of this paper is to develop a systematic method to address this issue and to give a general result allowing to transform an observability result for the corresponding conservative system into a stabilization one for the damped one.

  • conservative systems

  • neumann boundary

  • well known

  • results apply

  • observability results

  • ∂?1 ∂x

  • corresponding conservative system

  • ?v ?

  • observability inequalities


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Language English

Stabilization of the

wave equation on 1-d networks

∗ †‡
Julie Valein, Enrique Zuazua

August 1, 2008

Abstract
In this paper we study the stabilization of the wave equation on general 1-d networks.For
that, we transfer known observability results in the context of control of conservative systems
(see [14]) into a weighted observability estimate for the dissipative one.Then we use an
interpolation inequality similar to the one proved in [7] to obtain the explicit decay estimate
of the energy for smooth initial data.The obtained decay rate depends on the geometric and
topological properties of the network.We give also some examples of particular networks in
which our results apply yielding different decay rates.

Keywords 0.1stabilization, wave equation, network

1

93D15, 93B07, 35L05

Introduction and main results

In this paper, we consider a planar network of elastic strings that undergoes small perpendicular
vibrations. Recently,the control, observation and stabilization problems of these networks have
been the object of intensive research (see [14, 16] and the references therein).
Here we are interested in the problem of stabilization of the network by means of a damping
term located on one single exterior node.The aim of this paper is to develop a systematic method
to address this issue and to give a general result allowing to transform an observability result for
the corresponding conservative system into a stabilization one for the damped one.
Before going on, let us recall some definitions and notations about1−dnetworks used in the
paper. Werefer to [1, 19, 24] for more details.
n
A1−dnetworkRis a connected set ofR,n≥1, defined by

N
[
R=ej
j=1


Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et
Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 France, julie.valein@univ-valenciennes.fr

IMDEA-Matemáticas & Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de
Madrid, 28049 Madrid, Spain, enrique.zuazua@uam.es.

The work of the second author is supported by the Grant MTM2005-00714 of the Spanish MEC, the DOMINO
Project CIT-370200-2005-10 in the PROFIT program of the MEC (Spain) and the SIMUMAT project of the CAM
(Spain). Thiswork started while the first author was visiting the Universidad Autónoma de Madrid during six
months partially supported by a grant ’bourse régionale de mobilité à l’internationale’ of the ’Région Nord-Pas de
Calais’ (France).

1

whereejis a curve that we identify with the interval(0, lj), lj>0,and such that fork6=j, ej∩ek
is either empty or a common end called a vertex or a node (hereejstands for the closure ofej).
For a functionu:R −→R,we setuj=u|ejthe restriction ofuto the edgeej.
We denote byE={ej; 1≤j≤N}the set of edges ofRand byVthe set of vertices ofR.For
a fixed vertexv,let
Ev={j∈ {1, ..., N};v∈ej}
be the set of edges havingvas vertex.If card(Ev) = 1, vis an exterior node, while if card(Ev)≥2,
vWe denote byis an interior one.Vextthe set of exterior nodes and byVintthe set of interior
ones. Forv∈ Vext, the single element ofEvis denoted byjv.
We now fix a partition ofVext:Vext=D ∪ {v1}.In this way, we distinguish the conservative
exterior nodes,D, in which we impose Dirichlet homogeneous boundary condition, and the one in
which the damping term is effective,v1. Tosimplify the notation, we will assume thatv1is located
at the end0of the edgee1.
Letuj=uj(t, x) :R×[0, lj]→Rbe the function describing the transversal displacement in
timetof the stringejof lengthljus denote by. LetLthe sum of the lengths of all edges of the
network, the total length of the network.
We assume that the displacementsujsatisfy the following system
2 2
∂ uj∂ uj
2(x, t)−2(x, t) = 00< x < lj, t >0,∀j∈ {1, ..., N},
∂t ∂x
uj(v, t) =ul(v, t)∀j, l∈ Ev, v∈ Vint, t>0,
X
∂uj

(v, t) = 0∀v∈ Vint, t>0,
∂nj
(1)
j∈Ev
ujv(v, t) = 0∀v∈ D, t>0,
∂u1∂u1
(0, t) =(0, t)∀t >0,
∂x ∂t

(0)∂u(1)
u(t= 0) =u ,(t= 0) =u ,
∂t
where∂uj/∂nj(v, .)stands for the outward normal (space) derivative ofujat the vertexv. We
denote byuthe vectoru= (uj)j=1,...,N.
The above system has been considered by several authors in some particular situations.We
refer, for instance, to [6], [2], [3, 4], [5] and [25], where explicit decay rates are obtained for networks
with some special structures.We also refer to [20] where the problem is considered in the presence
of delay terms in the feedback law.
The object of this paper is not to give an additional result in a particular case, but rather
to develop a systematic method allowing to address the issue in a general context.We do this
transfering known observability results for the corresponding conservative system into stabilization
results for the dissipative one.This provides a new proof for the existing results mentioned above
and allows getting new ones.
As mentioned above, in this paper, we consider the case where the dissipation is located on an
external node of the network, but the method can be adapted to treat the case where the damping
term is located in several nodes, both exterior and interior ones.
In order to study system (1) we need a proper functional setting.We define the Hilbert spaces
2 2
L(R) ={u:R →R;uj∈L(0, lj),∀j= 1,∙ ∙ ∙, N},

and

N
Y
1
V:={φ∈H(0, lj) :φj(v) =φk(v)∀j, k∈ Ev,∀v∈ Vint;φjv(v) = 0∀v∈ D},
j=1

2