Global solutions of a class of discrete‐time backward nonlinear equations on ordered Banach spaces with applications to Riccati equations of stochastic control

Abstract

SUMMARYIn this paper, we consider the problem of existence of certain global solutions for general discrete‐time backward nonlinear equations, defined on infinite dimensional ordered Banach spaces. This class of nonlinear equations includes as special cases many of the discrete‐time Riccati equations arising both in deterministic and stochastic optimal control problems. On the basis of a linear matrix inequalities approach, we give necessary and sufficient conditions for the existence of maximal, stabilizing, and minimal solutions of the considered discrete‐time backward nonlinear equations. As an application, we discuss the existence of stabilizing solutions for discrete‐time Riccati equations of stochastic control and filtering on Hilbert spaces. The tools provided by this paper show that a wide class of nonlinear equations can be treated in a uniform manner. Copyright © 2012 John Wiley & Sons, Ltd.