Optimal and approximate control of ﬁnite-difference approximation schemes for the 1D wave equation - Enrique Zuazua (Rendiconti di Matematica e delle Sue Applicazioni, 2004)
We address the problem of control of numerical approximation schemes for the 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. Recently, it has been shown that, in the context of exact control, i.e., when the control is required to drive the solution to a final target exactly, due to high frequency spurious numerical solutions, convergent numerical schemes may lead to unstable approximations of the control. In other words, the classical convergence property of numerical schemes does not guarantee a stable and convergent approximation of controls. In this article we address the same problem in the context of optimal and approximate control in which the final requirement of achieving the target exactly is relaxed. We prove that, for those relaxed control problems, convergence (as the mesh-size tends to zero) holds. In particular, in the context of approximate control we show that, if the final condition is relaxed so that the final state is required to reach and ε-neighborhood of the final target with ε > 0, then the controls of numerical schemes (the so-called ε-controls) converge to the ε-controls of the wave equation. We also show that this result fails to be true in several space dimensions. Although convergence is proved in the context of these relaxed control problems, the fact that instabilities occur at the level of exact control have to be considered as a serious warning in the sense that instabilities may ultimately arise if the control requirement is reinforced to exactly achieve the final target, i.e., as ε is taken smaller and smaller.