Existence conditions for stabilizing and antistabilizing solutions to the nonautonomous matrix Riccati differential equation

In this paper several necessary and sufficient conditions for existence of stabilizing and antistabilizing solutions to the nonautonomous matrix Riccati differential equation (RDE) are presented. The conditions are reduced to existence of a solution to the corresponding Riccati type matrix inequality, or to existence of exponential dichotomy for the associated Hamiltonian linear differential system, or to convergence of the Newton type iterative algorithm for construction of the stabilizing or antistabilizing solution. Existence of extremal solutions to the nonautonoms RDE is a well-known fact (history of the problem and a list of references may be found in fundamental papers [1], [2]). In the present paper it is shown that, in the class of all bounded solutions to the nonautonomous RDE, the stabilizing solution is the maximal solution and the antistabilizing solution is the minimal one (for the autonomous RDE these properties of extremal solutions were reported in [3], [4]). Asymptotic properties of the extremal solutions to the nonautonomous RDE are investigated in the paper and the sets of the associated attracting solutions are indicated. The stabilizing or antistabilizing solution to the RDE turns out to be very useful in many optimal control design and identification problems [5 — 10]. We shall apply the obtained results to the linear nonstationary control system optimal stabilization problem under a quadratic performance criterion of arbitrary form. This problem is of a great interest in the theory of optimization and invariance of linear control systems