Continuity properties of solutions to H2 and H∞ Riccati equations

In most H2 and H∞ control problems solutions of the algebraic Riccati equation play a crucial role. Note that in general for continuous time systems we have to use quadratic matrix inequalities instead of Riccati equations. For details we refer to [5]. In particular we are interested in the stabilizing solution of these Riccati equations and quadratic matrix inequalities. However, if the system has zeros on the imaginary axis (continuous time) or on the unit circle (discrete time), we have to study semistabilizing solutions. These are solutions of the Riccati equation/quadratic matrix inequality associated to eigenvalues in the closed left-half plane (continuous time) or in the closed unit circle (discrete time). The standard way to obtain semi-stabilizing solutions is a cheap control argument where we perturb the system parameters to obtain a system without problems induced by for instance the zeros on the boundary of the stability domain. A natural question is then whether the semi-stabilizing solutions depend continuously on the system parameters. There are simple examples where the solution does not depend continuously on the system parameters (see e.g. [2]). On the other hand, [8] identifies a class of perturbations which guarantee a continuous behaviour. We would like to study this question in more detail. We will clearly identify what kind of perturbations can yield discontinuous behaviour and in the process show that for a very large class of systems discontinuities never occur. We will consider both continuous and discrete time systems. Notation in this paper is mostly standard. By M† we denote the Moore-Penrose inverse of M.