Geometrical aspects of exact boundary controllability for the wave equation - a numerical study

Abstract

Abstract. This essentially numerical study, sets out to investigate various geometrical properties of exact boundary controllability o f t h e w ave equation when the control is applied on a part of the boundary. R elationships between the geometry of the domain, the geometry of the controlled boundary, the time needed to control and the energy of the control are dealt with. A new norm of the control and an energetic cost factor are introduced. These quantities enable a detailed appraisal of the numerical solutions obtained and the detection of trapped rays. Contents

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

“…The first section briefly recalls some controllability results for the wave equation (1). In the second section, the fully discrete model is introduced and the main results of existence, characterization and convergence of the discrete control are presented.…”

“…We introduce a finite difference approximation in space and time of the wave equation (1). Let us consider J ∈ N, h = 1/(J + 1) and a uniform grid of the space interval (0, 1) given by 0 = x 0 < x 1 < ... < x J < x J+1 = 1, with x j = jh, j = 0, ..., J + 1.…”

“…Moreover, the limit v is the HUM control of the continuous system (1). If the convergence in (37) is strong, then the above convergences are strong too.…”

“…We compare the result obtained from the usual finite difference scheme (FDS), i.e. (S 0,0 h,∆t ), ∆t < h, the scheme (S θ,α h,∆t ) for (θ, α, ν) ∈ C (MDS) and the bi-grid method (BI-GRID) introduced in [9] (see also [1]). When the initial conditions (y 0 , y 1 ) are both continuous, the usual centered scheme (FDS) provides good numerical results, and convergence of the discrete control is observed (see for instance [5] for numerical simulations in 2-D).…”

“…In [9], these authors described an algorithm based on the HUM method and on the usual centered finite difference approximation of the wave system (1). This approximation is of the type:…”