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We discuss a control problem for the Lamé system which naturally leads to the following uniqueness problem: Given a bounded domain of , are there non-trivial solutions of the evolution Lamé system with homogeneous Dirichlet boundary conditions for which the first two components vanish? We show that such solutions do not exist when the domain is Lipschitz. However, in two space dimensions one can build easily polygonal domains in which there are eigenvibrations with the first component being identically...
We analyze the problem of switching controls for control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other in a systematic way so that, in each instant of time, only one actuator is active. We first address a finite-dimensional model and show that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. To do this we introduce a new variational principle building...
We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability...
We address three null controllability problems related to the heat equation. First we show that the heat equation with a rapidly oscillating density is uniformly null controllable as the period of the density tends to zero. We also prove that the same result holds for the finite-difference semi-discretization in space of the constant coefficient heat equation as the step size tends to zero. Finally, we prove that the null controllability of the constant coefficient heat equation can be obtained...
We analyze the problem of boundary observability of the finite-difference space semidiscretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the meshsize)
boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [9]. Our method uses previously known uniform observability inequalities for low
frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm...
We propose a finite difference semi-discrete scheme for the
approximation of the boundary exact controllability problem of
the 1-D beam equation modelling the transversal vibrations
of a beam with fixed ends.
First of all we show that, due to the high frequency spurious
oscillations, the uniform (with respect to the mesh-size)
controllability property of the semi-discrete model fails in the
natural functional setting.
We then prove that there are two ways of restoring the uniform
controllability...
This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012.
In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set , with Dirichlet boundary conditions. The observation is done on a subset of Lebesgue measure , where is fixed. We denote by the...
We consider the wave and Schrödinger equations on a bounded open connected subset of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset of during a time interval with . It is well known that, if the pair satisfies the Geometric Control Condition ( being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be...
We analyze the controllability of the motion of a fluid by means of the
action of a vibrating shell coupled at the boundary of the fluid. The
model considered is linear. We study its approximate controllability,
i.e. whether the fluid may reach a dense set of final configurations at
a given time. We show that this problem can be reduced to a unique
continuation question for the Stokes system. We prove that this unique
continuation property holds generically among analytic domains and
therefore,...
We consider space semi-discretizations of the 1- wave equation in a bounded
interval with homogeneous Dirichlet boundary conditions. We analyze the problem
of boundary observability, , the problem of whether the total energy of
solutions can be estimated uniformly in terms of the energy concentrated on the
boundary as the net-spacing → 0. We prove that, due to the spurious modes
that the numerical scheme introduces at high frequencies, there is no such a
uniform bound. We prove however a uniform...
We consider a dynamical one-dimensional nonlinear von Kármán model for beams depending on a parameter and study its asymptotic behavior for large, as . Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponentially uniformly with respect to the parameter . In order for this to be true the damping mechanism has to have the appropriate scale with respect to . In the limit as we obtain damped Berger–Timoshenko beam models...
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