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A uniformly controllable and implicit scheme for the 1-D wave equation. (English) Zbl 1130.93016
Summary: 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 \(\Delta t\). We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order \(h^2\) and \(\Delta t\). Using a discrete version of Ingham’s inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in \(L^2(0,T)\) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal \(L^2\)-norm control. The results are illustrated with several numerical experiments.

MSC:
93B05 Controllability
35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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