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

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) o...

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Bibliographic Details
Published in:Discrete Dynamics in Nature and Society Vol. 2015; no. 2015; pp. 870 - 880-085
Main Author: Ungureanu, Viorica Mariela
Format: Journal Article
Language:English
Published: Cairo, Egypt Hindawi Limiteds 01.01.2015
Hindawi Publishing Corporation
John Wiley & Sons, Inc
Wiley
Subjects:
Algebra
Banach space
Discrete-time systems
Linear quadratic
Linear systems
Markov processes
Mathematical analysis
Mathematical research
Operators
Riccati equation
ISSN:1026-0226, 1607-887X
Online Access:Get full text
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Summary: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.
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ISSN:1026-0226
1607-887X
DOI:10.1155/2015/293930