Finite time stabilization of the waves on an infinite network
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 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 converges when .
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 . 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 where is the solution of the considered wave system and is a constant. The free case, , 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 . This is the subject of [2] and this paper. In [2], we have proved that when 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 as , for large t. The decay rate is given by with is the common length of the finite edges and . 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 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 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 restricted to a some Hilbert space is constant after 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 and N strictly positive numbers. We consider a network Γ with N finite edges with lengths respectively. We attach to these edges an infinite edge identified to the semi-infinite interval . We consider that the network has a single vertex identified to 0 and that the edge is identified to the interval , for . This configuration is described by Fig. 1.
We deal with the wave equation 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 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 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 and and one infinite edge . This type of network is called a Y-junction. We assume that the numbers and are rationally independent. Thus . 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 with and as . 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 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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