Minimum Second Moment State For The Existence Of Average Optimal Stationary Policies In Linear Stochastic Systems

Abstract

This note considers the long-run aver- age cost control problem for a class of discrete-time stochastic systems. The stochastic system is assumed to be linear with respect to the state but the controls possess a general structure, possibly a nonlinear one. The main contribution of this paper is to show that the existence of a minimal second moment system state implies the existence of an optimal stationary policy for the long-run average cost problem. A numerical example illustrates the derived result. © 2010 AACC.373377Syrmos, V., Abdallah, C., Dorato, P., Grigoriadis, K., Static output feedback- A survey (1997) Automatica, 33, pp. 125-137Vargas, A.N., Do Val, J.B.R., On the existence of stationary optimal policies for the average-cost control problem of linear systems with abstract state-feedback (2008) Proc. 47th IEEE Conf. on Decision and Control, pp. 3682-3687. , Cancun, MexicoA bounded cost condition for the existence of average optimal stationary policies of linear stochastic systems (2009) Proc. European Control Conference, pp. 38-42A controllability condition for the existence of average optimal stationary policies of linear stochastic systems (2009) Proc. European Control Conference, pp. 32-37Approximation of the optimal average cost for a class of linear stochastic control systems (2010) American Control Conference, , (submitted)Hernández-Lerma, O., Lasserre, J.B., (1996) Discrete-Time Markov Control Processes: Basic Optimality Criteria, , Springer-Verlag, New YorkAubin, J.-P., Optima and equilibria: An introduction to nonlinear analysis (1998) Graduate Texts in Mathematics, 140. , Springer-Verlag, seriesAnderson, B.D.O., Moore, J.B., (1979) Optimal Filtering, , Prentice-Hall, Englewood Cliffs, N.JWu, M.Y., A note on stability of linear time-varying systems (1974) IEEE Trans. Automat. Control, 19, pp. 162-162Bertsekas, D.P., Shreve, S.E., (1978) Stochastic Optimal Control: The Discrete Time Case, , Academic PressSchal, M., Average optimality in dynamic programming with general state space (1993) Math. Oper. Res., 18, pp. 163-17