On the Phase Portrait of the Matrix Riccati Equation Arising from the Periodic Control Problem

Abstract

A comprehensive description is provided of the properties of the matrix Riccati differential equation in which the coefficient matrices are T-periodic for some $T > 0$. The main results include: (1) a classification of all the periodic equilibria (T-periodic solutions); (2) necessary and sufficient conditions for convergence to the uniformly asymptotically stable (completely unstable) periodic equilibrium as $t \to \infty $$(t \to - \infty )$; (3) necessary and sufficient conditions for a solution to escape in finite forward or backward time; (4) a description of every almost periodic solution; (5) a proof that every solution which does not escape in finite time is asymptotically almost periodic and an explicit formula for the limiting almost periodic solution.