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En utilisant une méthode dépendante du temps, nous démontrons la complétude asymptotique
pour l'équation des ondes dans une classe d'espaces-temps stationnaires et
asymptotiquement plats. On introduit l'observable de vitesse asymptotique et on décrit
son spectre (sous des hypothèses plus faibles que pour la complétude asymptotique). Les
méthodes utilisées sont inspirées par celles de l'analyse du problème à deux corps en
mécanique quantique.
A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.
We consider the controllability and observation problem for a simple model describing the interaction between a fluid and a beam. For this model, microlocal propagation of singularities proves that the space of controlled functions is smaller that the energy space. We use spectral properties and an explicit construction of biorthogonal sequences to show that analytic functions can be controlled within finite time. We also give an estimate for this time, related to the amount of analyticity of the...
We study boundary control problems for the wave, heat, and Schrödinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. Exact controllability in L₂-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. Null controllability for the heat equation...
In this paper necessary and sufficient conditions of L∞-controllability and
approximate L∞-controllability are obtained for the control system
wtt = wxx − q2w,
w(0,t) = u(t),
x > 0, t ∈ (0,T), where
q ≥ 0, T > 0,
u ∈ L∞(0,T) is a control. This system is
considered in the Sobolev spaces.
In this paper necessary and sufficient conditions of L∞-controllability and approximate L∞-controllability are obtained for the control system wtt = wxx − q2w, w(0,t) = u(t), x > 0, t ∈ (0,T), where q ≥ 0, T > 0, u ∈ L∞(0,T) is a control. This system is considered in the Sobolev spaces.
In this paper necessary and sufficient conditions of L∞-controllability and
approximate L∞-controllability are obtained for the control system
wtt = wxx − q2w,
w(0,t) = u(t),
x > 0, t ∈ (0,T), where
q ≥ 0, T > 0,
u ∈ L∞(0,T) is a control. This system is
considered in the Sobolev spaces.
In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram’s...
In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram’s...
In this paper, we introduce a new method for feedback controller design for the complex distributed parameter networks governed by wave equations, which ensures the stability of the closed loop system. This method is based on the uniqueness theory of ordinary differential equations and cutting-edge approach in the graph theory, but it is not a simple extension. As a realization of this idea, we investigate a bush-type wave network. The well-posedness of the closed loop system is obtained via Lax-Milgram’s...
This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported.
The problem of modeling acoustic waves scattered by an object with Neumann boundary condition is considered. The boundary condition is taken into account by means of the fictitious domain method, yielding a first order in time mixed variational formulation for the problem. The resulting system is discretized with two families of mixed finite elements that are compatible with mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always correctly...
The problem of modeling acoustic waves scattered by an object with
Neumann boundary condition is considered. The boundary condition is
taken into account by means of the fictitious domain method, yielding
a first order in time mixed variational formulation for the
problem. The resulting system is discretized
with two families of mixed finite elements that are compatible with
mass lumping. We present numerical results illustrating that the Neumann boundary condition on the object is not always...
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