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Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method

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In this article one discusses the controllability of a semi-discrete system obtained by discretizing in space the linear 1-D wave equation with a boundary control at one extremity. It is known that the semi-discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero (see [8]).

Here we introduce a new semi-discrete model based on a mixed finite element method with two different basis functions for the position and velocity. We show that the controls obtained with these semi-discrete systems can be chosen uniformly bounded in L2(0,T) and in such a way that they converge to the HUM control of the continuous wave equation, i.e. the minimal L2-norm control. We illustrate the mathematical results with several numerical experiments.

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Correspondence to Carlos Castro.

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Supported by Grant BFM 2002-03345 of MCYT (Spain) and the TMR projects of the EU ``Homogenization and Multiple Scales" and ``New materials, adaptive systems and their nonlinearities: modelling, control and numerical simulations".

Partially Supported by Grant BFM 2002-03345 of MCYT (Spain), Grant 17 of Egide-Brancusi Program and Grant 80/2005 of CNCSIS (Romania).

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Castro, C., Micu, S. Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102, 413–462 (2006).

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