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Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity. (English) Zbl 1176.93017

Summary: This article deals with the approximation of the boundary control of the linear one-dimensional wave equation. It is known that the high frequency spurious oscillations introduced in the classical methods of finite difference and finite element lead to nonuniform controllability properties (see [J. A. Infante and E. Zuazua, \(M2AN\) Math. Model. Numer. Anal. 33, 407–438 (1999; Zbl 0947.65101)]. A space-discrete scheme with an added numerical vanishing viscous term is presented and analyzed. The extra numerical damping filters out the high numerical frequencies and ensures the convergence of the sequence of discrete controls to a control of the continuous conservative wave equation when the mesh size tends to zero.

MSC:

93B05 Controllability
93B40 Computational methods in systems theory (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L05 Wave equation

Citations:

Zbl 0947.65101
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