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A uniformly controllable and implicit scheme for the 1-D wave equation

Arnaud Münch (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using...

An Ingham type proof for a two-grid observability theorem

Michel Mehrenberger, Paola Loreti (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1 D -wave equation for a time T > 2 2 ; this time, if the observation is made in ( - T / 2 , T / 2 ) , is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I 338 (2004) 413–418]. Our proof follows an Ingham type approach.

An Ingham type proof for a two-grid observability theorem

Paola Loreti, Michel Mehrenberger (2007)

ESAIM: Control, Optimisation and Calculus of Variations

Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time T > 2 2 ; this time, if the observation is made in ( - T / 2 , T / 2 ) , is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I338 (2004) 413–418]. Our proof follows an Ingham type approach.

Analytic controllability of the wave equation over a cylinder

Brice Allibert (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We analyze the controllability of the wave equation on a cylinder when the control acts on the boundary, that does not satisfy the classical geometric control condition. We obtain precise estimates on the analyticity of reachable functions. As the control time increases, the degree of analyticity that is required for a function to be reachable decreases as an inverse power of time. We conclude that any analytic function can be reached if that control time is large enough. In the C∞ class, a...

Arbitrary high-order finite element schemes and high-order mass lumping

Sébastien Jund, Stéphanie Salmon (2007)

International Journal of Applied Mathematics and Computer Science

Computers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elementsof order k and show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta...

Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass

C. Castro (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a hybrid, one-dimensional, linear system consisting in two flexible strings connected by a point mass. It is known that this system presents two interesting features. First, it is well posed in an asymmetric space in which solutions have one more degree of regularity to one side of the point mass. Second, that the spectral gap vanishes asymptotically. We prove that the first property is a consequence of the second one. We also consider a system in which the point mass is replaced...

Asymptotic behaviour of the scattering phase for non-trapping obstacles

Veselin Petkov, Georgi Popov (1982)

Annales de l'institut Fourier

Let S ( λ ) be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle 𝒪 ⊂ R n , n ≥ 3 with Dirichlet or Neumann boundary conditions on ∂ 𝒪 . The function s ( λ ) , called scattering phase, is determined from the equality e - 2 π i s ( λ ) = det S ( λ ) . We show that s ( λ ) has an asymptotic expansion s ( λ ) ∼ ∑ j = 0 ∞ c j λ n - j as λ → + ∞ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

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