Control and numerical approximation of the wave and heat equations

Abstract

In recent years important progress have been done in the context of numerical approximation of controllability problems for PDEs. It is by now well known that, often, numerical approximation schemes that are stable for solving initial-boundary value problems, develop instabilities when applied to controllability problems. This is due to the presence of spurious high frequency numerical solutions that the control mechanisms are not able to control uniformly as the mesh-size tends to zero. However, the theory is far from being complete. In this article we present some new results in this framework for the wave and the heat equations, which also raise a number of open questions and future directions of research. We first prove that a two-grid method, introduced by R. Glowinski, that is by now well-known to guarantee convergence for the 1 − d wave equation, also converges in the semilinear setting for globally Lipschitz nonlinearities. This result provides a further evidence of the robustness of the two-grid method. We then show that boundary controls for finite-difference space semi-discretizations of the heat equation converge when applied all along the boundary of the domain, a fact that does not hold for wave-like equations. This confirms that the strong irreversibility of the heat equation enhances the control properties of its numerical approximation schemes. This result fails when the control is restricted to some subsets of the boundary because of the lack of unique continuation of some high frequency eigenvectors of the underlying discrete eigenvalue problem.