Control System Radii and Robustness Under Approximation

The purpose of this paper is twofold. First, we provide a short review and summarize results on the robustness of controllability and stabilizability for finite dimensional control problems. We discuss the computation of system radii which provide a measure of robustness. Second, we consider systems which arise as finite difference and finite element approximations to control systems defined by partial differential equations. In particular, we derive controllability criteria for approximations of the controlled heat equation which are easy to check numerically. For a particular example we establish tight theoretical upper and lower bounds on the controllability radii for the finite difference and finite element models and compare these bounds with numerical results. Finally, we present numerical results on stabilizability radii which suggests that conditioning of the LQR control problem may be measured by this radii.