Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 293930, 11 pages
http://dx.doi.org/10.1155/2015/293930
Research Article

Stabilizing Solution for a Discrete-Time Modified Algebraic Riccati Equation in Infinite Dimensions

“Constantin Brâncuși” University of Tirgu-Jiu, B-dul Republicii No. 1, 210152 Târgu Jiu, Romania

Received 12 March 2015; Revised 19 May 2015; Accepted 26 May 2015

Academic Editor: Driss Boutat

Copyright © 2015 Viorica Mariela Ungureanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We provide necessary and sufficient conditions for the existence of stabilizing solutions for a class of modified algebraic discrete-time Riccati equations (MAREs) defined on ordered Banach spaces of sequences of linear and bounded operators. These MAREs arise in the study of linear quadratic (LQ) optimal control problems for infinite-dimensional discrete-time linear systems (DTLSs) affected simultaneously by multiplicative white noise (MN) and Markovian jumps (MJs). Unlike most of the previous works, where the detectability and observability notions are key tools for studying the global solvability of MAREs, in this paper the conditions of existence of mean-square stabilizing solutions are given directly in terms of system parameters. The methods we have used are based on the spectral theory of positive operators and the properties of trace class and compact operators. Our results generalise similar ones obtained for finite-dimensional MAREs associated with stochastic DTLSs without MJs. Also they complete and extend (in the autonomous case) former investigations concerning the existence of certain global solutions (as minimal, maximal, and stabilizing solutions) for generalized discrete-time Riccati type equations defined on infinite-dimensional ordered Banach spaces.