CONVERGENCE OF A SEMIDISCRETE TWO-GRID ALGORITHM FOR THE CONTROLLABILITY OF THE 1-d WAVE EQUATION.

The problem of exact controllability of elastic strings has been extensively studied during the last years. We consider the problem of computing numerically the boundary control for a finite-dimensional system obtained by discretizing in space the 1 - d wave equation. More precisely, we analyze whether the controls of numerical approximation schemes converge to the control of the continuous wave equation as the mesh size tends to zero. It is by now well known that, due to high-frequency spurious oscillations, numerical instabilities occur and may lead to the failure of convergence of some apparently natural numerical algorithms. In other words, the classical convergence property of numerical schemes does not guarantee a stable and convergent approximation of controls. Several remedies have been proposed in the literature to compensate for this fact: Tychonoff regularization, Fourier filtering, and mixed finite elements. In this paper we prove that the two-grid method proposed by Glowinski in [J. Comput. Phys., 103 (1992), pp. 189-221] to numerically approximate the control of the wave equation converges in 1 - d. We prove this result in the context of the finite-element space semidiscretization. Our method of proof relies essentially on the particular properties of the Fourier representation of the initial data of the coarse mesh when projected into the fine one. The explicit representation formula of the solutions shows that the high-frequency components are modulated by some weights that diminish the effect of these spurious components. This fact, combined with discrete multipliers techniques, allows us to prove uniform observability inequalities. Classical arguments then allow proving the uniform boundedness of the controls and passing to the limit as the mesh size tends to zero. In this way we prove the convergence of the controls of the finite-element semidiscrete approximation of the 1 - d wave equation with a boundary control on one extreme. It is important to underline that the controls obtained by this two-grid algorithm are not exact in the sense that they only guarantee the controllability of the projection of the solutions into the coarse mesh. [ABSTRACT FROM AUTHOR]