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Article: Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps
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TitlePositive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps
 
AuthorsLi, ZY2
Zhou, B2
Lam, J1
Wang, Y2
 
KeywordsIteration
Jump System
Lyapunov Equation
Markov Process
Mean Square Stability
Positive Operator
Stochastic System
 
Issue Date2011
 
PublisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/amc
 
CitationApplied Mathematics And Computation, 2011, v. 217 n. 21, p. 8179-8195 [How to Cite?]
DOI: http://dx.doi.org/10.1016/j.amc.2011.01.031
 
AbstractThis paper studies the iterative solutions of Lyapunov matrix equations associated with Itô stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms. © 2011 Elsevier Inc. All rights reserved.
 
ISSN0096-3003
2013 Impact Factor: 1.600
 
DOIhttp://dx.doi.org/10.1016/j.amc.2011.01.031
 
ISI Accession Number IDWOS:000290622200004
Funding AgencyGrant Number
National Natural Science Foundation of China60904007
61074111
10771044
China Postdoctoral Science Foundation20100480059
Foundation for Innovative Research Group of the National Natural Science Foundation of China601021002
Development Program for Outstanding Young Teachers at Harbin Institute of TechnologyHITQNJS.2009.054
Heilongjiang Postdoctoral Foundation of ChinaLRB10-194
HKU CRCG 201007176243
GRE HKU7138/10E
Funding Information:

This work is supported in part by the National Natural Science Foundation of China under grant numbers 60904007, 61074111 and 10771044, the China Postdoctoral Science Foundation under grant number 20100480059, the Foundation for Innovative Research Group of the National Natural Science Foundation of China under grant 601021002, the Development Program for Outstanding Young Teachers at Harbin Institute of Technology under grant number HITQNJS.2009.054, the Heilongjiang Postdoctoral Foundation of China under Grant No. LRB10-194, HKU CRCG 201007176243, and by GRE HKU 7138/10E.

 
ReferencesReferences in Scopus
 
DC FieldValue
dc.contributor.authorLi, ZY
 
dc.contributor.authorZhou, B
 
dc.contributor.authorLam, J
 
dc.contributor.authorWang, Y
 
dc.date.accessioned2012-08-08T08:45:24Z
 
dc.date.available2012-08-08T08:45:24Z
 
dc.date.issued2011
 
dc.description.abstractThis paper studies the iterative solutions of Lyapunov matrix equations associated with Itô stochastic systems having Markovian jump parameters. For the discrete-time case, when the associated stochastic system is mean square stable, two iterative algorithms with one in direct form and the other one in implicit form are established. The convergence of the implicit iteration is proved by the properties of some positive operators associated with the stochastic system. For the continuous-time case, a transformation is first performed so that it is transformed into an equivalent discrete-time Lyapunov equation. Then the iterative solution can be obtained by applying the iterative algorithm developed for discrete-time Lyapunov equation. Similar to the discrete-time case, an implicit iteration is also proposed for the continuous case. For both discrete-time and continuous-time Lyapunov equations, the convergence rates of the established algorithms are analyzed and compared. Numerical examples are worked out to validate the effectiveness of the proposed algorithms. © 2011 Elsevier Inc. All rights reserved.
 
dc.description.naturelink_to_subscribed_fulltext
 
dc.identifier.citationApplied Mathematics And Computation, 2011, v. 217 n. 21, p. 8179-8195 [How to Cite?]
DOI: http://dx.doi.org/10.1016/j.amc.2011.01.031
 
dc.identifier.citeulike8960256
 
dc.identifier.doihttp://dx.doi.org/10.1016/j.amc.2011.01.031
 
dc.identifier.epage8195
 
dc.identifier.hkuros208784
 
dc.identifier.isiWOS:000290622200004
Funding AgencyGrant Number
National Natural Science Foundation of China60904007
61074111
10771044
China Postdoctoral Science Foundation20100480059
Foundation for Innovative Research Group of the National Natural Science Foundation of China601021002
Development Program for Outstanding Young Teachers at Harbin Institute of TechnologyHITQNJS.2009.054
Heilongjiang Postdoctoral Foundation of ChinaLRB10-194
HKU CRCG 201007176243
GRE HKU7138/10E
Funding Information:

This work is supported in part by the National Natural Science Foundation of China under grant numbers 60904007, 61074111 and 10771044, the China Postdoctoral Science Foundation under grant number 20100480059, the Foundation for Innovative Research Group of the National Natural Science Foundation of China under grant 601021002, the Development Program for Outstanding Young Teachers at Harbin Institute of Technology under grant number HITQNJS.2009.054, the Heilongjiang Postdoctoral Foundation of China under Grant No. LRB10-194, HKU CRCG 201007176243, and by GRE HKU 7138/10E.

 
dc.identifier.issn0096-3003
2013 Impact Factor: 1.600
 
dc.identifier.issue21
 
dc.identifier.scopuseid_2-s2.0-79956076188
 
dc.identifier.spage8179
 
dc.identifier.urihttp://hdl.handle.net/10722/157115
 
dc.identifier.volume217
 
dc.languageeng
 
dc.publisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/amc
 
dc.publisher.placeUnited States
 
dc.relation.ispartofApplied Mathematics and Computation
 
dc.relation.referencesReferences in Scopus
 
dc.subjectIteration
 
dc.subjectJump System
 
dc.subjectLyapunov Equation
 
dc.subjectMarkov Process
 
dc.subjectMean Square Stability
 
dc.subjectPositive Operator
 
dc.subjectStochastic System
 
dc.titlePositive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps
 
dc.typeArticle
 
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Author Affiliations
  1. The University of Hong Kong
  2. Harbin Institute of Technology