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Article: Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps

TitlePositive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps
Authors
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
Citation
Applied Mathematics And Computation, 2011, v. 217 n. 21, p. 8179-8195 How to Cite?
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.
Persistent Identifierhttp://hdl.handle.net/10722/157115
ISSN
2014 Impact Factor: 1.551
2014 SCImago Journal Rankings: 0.958
ISI Accession Number ID
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.

References

 

DC FieldValueLanguage
dc.contributor.authorLi, ZYen_US
dc.contributor.authorZhou, Ben_US
dc.contributor.authorLam, Jen_US
dc.contributor.authorWang, Yen_US
dc.date.accessioned2012-08-08T08:45:24Z-
dc.date.available2012-08-08T08:45:24Z-
dc.date.issued2011en_US
dc.identifier.citationApplied Mathematics And Computation, 2011, v. 217 n. 21, p. 8179-8195en_US
dc.identifier.issn0096-3003en_US
dc.identifier.urihttp://hdl.handle.net/10722/157115-
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.en_US
dc.languageengen_US
dc.publisherElsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/amcen_US
dc.relation.ispartofApplied Mathematics and Computationen_US
dc.subjectIterationen_US
dc.subjectJump Systemen_US
dc.subjectLyapunov Equationen_US
dc.subjectMarkov Processen_US
dc.subjectMean Square Stabilityen_US
dc.subjectPositive Operatoren_US
dc.subjectStochastic Systemen_US
dc.titlePositive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumpsen_US
dc.typeArticleen_US
dc.identifier.emailLam, J:james.lam@hku.hken_US
dc.identifier.authorityLam, J=rp00133en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1016/j.amc.2011.01.031en_US
dc.identifier.scopuseid_2-s2.0-79956076188en_US
dc.identifier.hkuros208784-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-79956076188&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume217en_US
dc.identifier.issue21en_US
dc.identifier.spage8179en_US
dc.identifier.epage8195en_US
dc.identifier.isiWOS:000290622200004-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridLi, ZY=35240185200en_US
dc.identifier.scopusauthoridZhou, B=7401906664en_US
dc.identifier.scopusauthoridLam, J=7201973414en_US
dc.identifier.scopusauthoridWang, Y=35294070900en_US
dc.identifier.citeulike8960256-

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