Finite time stabilization of the waves on an infinite network

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Abstract

In this paper the finite time stabilization of an infinite network of vibrating strings is studied. We consider a network with NN,N2 finite edges and one infinite edge. This network is subject to homogeneous Dirichlet conditions at the endpoints of the finite edges. At the internal vertex 0, we put the continuity condition and a balanced damping condition of the form j=0N(xuj(0,t)tuj(0,t))=0. By a spectral analysis technique, we prove that when the finite edges lengths are rationally proportional, the energy of the system becomes constant within a finite time Te. We also prove that for an equilateral network we have Te=2, where is the common edge length. This is a finite time stabilization result of the system to a state with constant energy. This finite time limit energy is given in terms of the initial data and computed for a generic network. The main idea is to calculate the resolvent and to prove that it is of finite exponential type on some subspace of the energy space.

Introduction

The stabilization of systems of differential equations on multistructured domains is of a great interest not only for the mathematical analysis but also in engineering and nanotechnology development. The study of the stabilization of a given system is related to the behaviour of its energy for large time t. The energy of a system described by some differential problem (ODE or PDE) is usually defined as a norm of the solution of that problem. When trying to stabilize some system, people look for the convergence of the energy E(t)=12u(t)X2 when t tends to infinity to some finite limit energy. For damped systems, the energy is a non increasing function of t and the amount of dissipated energy is related to the damping term included in the differential problem. When the energy is decreasing, a frequently asked question is about the nature of the decay (exponential, polynomial, logarithmic, etc) and when this question is answered another natural question is to look for the rate of the decay and to which limit the energy E(t) converges when t+.

A more interesting objective is to look for the case when a given system can be stabilized at a finite time. This means that the system reaches an equilibrium state and its energy becomes constant within a finite time T0. We did not find a wide mathematical literature about this topic despite its importance. In [1], the authors studied the finite time stabilization of wave equation on bounded networks of strings. It was proven therein that under some conditions on the damping terms and the velocities of the waves on the edges, the considered system can be stabilized in finite time. The condition for that is relating the damping coefficient at each internal node (or vertex) to the number of edges connected to it. The technique used is to look for some Riemann invariants and then to use them to solve explicitly some transport equations. In [3], [4] and [2], the exponential decay of the energy for the damped wave equation on some unbounded networks is considered. The damping is introduced as a condition at the vertex identified to 0. This condition is written as j=0Nxuj(0,t)=αtu0(0,t)=0,t0 where (u0,,uN) is the solution of the considered wave system and α0 is a constant. The free case, α=0, was the subject of [3] and [4]. The exponential decay of the local energy was proved in [3]. The decay rate and the limit energy were explicitly given when the finite edges of the network have the same length. The authors used a spectral approach. An analysis of the spectrum and some estimations on the resolvent of the generator of the wave semigroup were used to prove the results therein.

The same questions were asked in the damped case α>0. This is the subject of [2] and this paper. In [2], we have proved that when αN+1 where N is the number of finite edges, then the system of wave equations is exponentially stable. The energy of the solution decays to some limit energy E as |E(t)E|C.eω(α)t, for large t. The decay rate is given byω(α)={1Arcoth(r)ifα>N+11Argth(r)ifα<N+1 with >0 is the common length of the finite edges and r=1αN. The results are proved using a computation of the spectrum and the resolvent then a semigroup expansion that leads to the exponential decay via an application of the residue theorem.

The particular case α=N+1 can not be covered by the methods used in [2]. This is the aim of this paper. In this situation, the operator has only purely imaginary eigenvalues of finite multiplicities, when the edges lengths are rationally dependant. In this case, and using some explicit formulas for the resolvent R(λ) we prove that up to some closed subspaces of the energy space, the resolvent is an entire function of finite exponential type. By means of Paley-Wiener theorem, we prove that the wave semigroup U(t) restricted to a some Hilbert space is constant after t=2 and hence the system is finite time stabilized to a state with a constant energy that we may compute. We discuss also the case of rationally proportional lengths and we give a description of the spectrum. In this case, the finite time stabilization holds true. Finally we consider the irrational case. For a Y-junction network, we prove that there is no stabilization since the operator has a sequence of resonances asymptotically close to the imaginary axis.

The paper is organized as follows. In section 2, we set the problem and we prove its wellposedness in some adapted functional spaces then we define the energy of the solution and we state that it is decaying. In section 3, we compute the pure point spectrum and a family of associated eigenvectors. This is done in the rational case. In section 4, then we give explicit formulas for the resolvent then we prove that it is an entire function of finite exponential type on a closed subspace of the energy space. Then the main result of the paper is stated therein. Namely, we prove that a finite time stabilization of the wave semigroup occurs and the finite time limit energy is computed. In the final section, we discuss the irrational situation, namely for a Y-junction where the two finite edges have rational independent lengths.

Section snippets

The problem and its wellposedness

Let NN,N2 and 1,,N N strictly positive numbers. We consider a network Γ with N finite edges e1,,eN with lengths 1,,N respectively. We attach to these edges an infinite edge e0 identified to the semi-infinite interval [0,+[. We consider that the network has a single vertex identified to 0 and that the edge ej is identified to the interval [0,j], for j=1,,N. This configuration is described by Fig. 1.

We deal with the wave equation (t2x2)u(x,t)=0 on the network Γ. We apply the

Spectral study

In this section, we plan to compute the eigenvalues of H and a set of corresponding eigenvectors. We prove first that these eigenvalues are the roots of some holomorphic function in the complex domain. The factorization of that function is related to the arithmetic properties of the ratios jk of the edges lengths. The rational case will be considered than we treat the particular situation where all of the edges have the same length. In this case, all of the spectral information is explicitly

Stabilization of the wave semigroup

We study in this section the time evolution of the wave group U(t) by relating it to the properties of the resolvent of H. First we give some preliminary results in the general setting then we apply them to our situation. We consider the situation where the lengths of the finite edges are the same.

Case of rationally independent lengths

In this paragraph, we consider the case of network which has two finite edges with lengths 1 and 2 and one infinite edge e0=[0,+[. This type of network is called a Y-junction. We assume that the numbers 1 and 2 are rationally independent. Thus 12Q. In this case the operator H has no pure imaginary eigenvalues. However, we will prove that the equation (5) has at least a sequence of solutions (λn)n with (λn)<0 and |(λn)|+ as n. According to the lemma (18) below, that we employ from

Conclusion and comments

We have discussed the stabilization of the wave system on a unbounded network. We proved that for rationally dependent edge lengths, the energy becomes constant within a finite time given in terms of the edge lengths. In the irrational case, there is a sequence of complex eigenvalues asymptotically close to the imaginary axis in C and this indicates that no exponential or finite time stabilization can occur. The decay rate in this case will be discussed in a further paper.

A possible

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