Periodicity, Detectability and the Matrix Riccati Equation

Abstract

This paper discusses the periodic solution of matrix Riccati differential equations with periodic coefficients. Such equations arise in linear filtering and control and in many other applications. The principal result: the existence of a periodic solution is equivalent to detectability and stabilizability of certain coefficient pairs. This result generalizes the Kalman–Wonham–Kucera theorem for algebraic Riccati equations. Among the numerous preliminaries is a discussion, apparently new, of detectability for linear periodic control systems. Another important result, for a linear matrix differential equation, is the equivalence of a bounded solution, an exponentially stable solution and a periodic solution. Finally, the periodic solution is shown to be an equilibrium solution in the sense of Kalman.