Wavelet filtering for exact controllability of the wave equation

In this paper we propose and analyze a wavelet-based numerical method for exact controllability of the one-dimensional wave equation, concentrating on the particular case of Dirichlet controls. We use the Hilbert uniqueness method (HUM) and prove this result in the context of the finite element space semidiscretization with spline wavelet basis functions. The algorithm for the numerical construction of the controls is based on a conjugate gradient method proposed by Glowinski [J. Comput. Phys., 103 (1992), pp. 189-221] and a two-grid filtering technique in the wavelet basis. We show that wavelet filtering yields numerically exact controllability in the optimal time of the continuous problem and the convergence of the numerical controls using a multilevel discretization filtering technique. Classical arguments then allow proving the uniform boundedness of the numerical controls and passing to the limit as the mesh size tends to zero. We prove that the condition number of the discretized HUM operator. with wavelet filtering is uniformly bounded with respect to the mesh width and prove convergence of the numerical controls obtained with the two-grid filtering technique. Numerical experiments show that wavelet filtering yields numerical controls with the same control time T as in the continuous case.