Approximation of some control problems by finite-difference method

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Résumé : In this talk, we will explain some difficulties that may arise when one tries to approximate a control problem, notably the loss of uniform controllability (with respect to the mesh-size) that prevents from recovering a control for the continuous problem as a limit of controls for the discrete problem. As an example, we consider a finite-difference semi-discrete scheme for the approximation of internal controls of a one-dimensional wave (or fractional wave-like) equations). The continuous problem is controllable. However, the high frequency numerical spurious oscillations lead to a loss of the uniform controllability property. We will detail how to restore the uniform controllability property thanks to an appropriate filtering method on the initial condition. The proof is mainly based on the moment method.