Article: Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps
| Title | Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Authors | Li, ZY2 Zhou, B2 Lam, J1 Wang, Y2 | ||||||||||||||
| Keywords | Iteration Jump System Lyapunov Equation Markov Process Mean Square Stability Positive Operator Stochastic System | ||||||||||||||
| Issue Date | 2011 | ||||||||||||||
| Publisher | Elsevier 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?] DOI: http://dx.doi.org/10.1016/j.amc.2011.01.031 | ||||||||||||||
| Abstract | This 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. | ||||||||||||||
| ISSN | 0096-3003 2011 Impact Factor: 1.317 2011 SCImago Journal Rankings: 0.062 | ||||||||||||||
| DOI | http://dx.doi.org/10.1016/j.amc.2011.01.031 | ||||||||||||||
| ISI Accession Number ID | WOS:000290622200004
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 | References in Scopus |
| dc.contributor.author | Li, ZY | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| dc.contributor.author | Zhou, B | ||||||||||||||
| dc.contributor.author | Lam, J | ||||||||||||||
| dc.contributor.author | Wang, Y | ||||||||||||||
| dc.date.accessioned | 2012-08-08T08:45:24Z | ||||||||||||||
| dc.date.available | 2012-08-08T08:45:24Z | ||||||||||||||
| dc.date.issued | 2011 | ||||||||||||||
| dc.description.abstract | This 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.nature | Link_to_subscribed_fulltext | ||||||||||||||
| dc.identifier.citation | Applied 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.citeulike | 8960256 | ||||||||||||||
| dc.identifier.doi | http://dx.doi.org/10.1016/j.amc.2011.01.031 | ||||||||||||||
| dc.identifier.epage | 8195 | ||||||||||||||
| dc.identifier.hkuros | 208784 | ||||||||||||||
| dc.identifier.isi | WOS:000290622200004
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.issn | 0096-3003 2011 Impact Factor: 1.317 2011 SCImago Journal Rankings: 0.062 | ||||||||||||||
| dc.identifier.issue | 21 | ||||||||||||||
| dc.identifier.scopus | eid_2-s2.0-79956076188 | ||||||||||||||
| dc.identifier.spage | 8179 | ||||||||||||||
| dc.identifier.uri | http://hdl.handle.net/10722/157115 | ||||||||||||||
| dc.identifier.volume | 217 | ||||||||||||||
| dc.language | eng | ||||||||||||||
| dc.publisher | Elsevier Inc. The Journal's web site is located at http://www.elsevier.com/locate/amc | ||||||||||||||
| dc.publisher.place | United States | ||||||||||||||
| dc.relation.ispartof | Applied Mathematics and Computation | ||||||||||||||
| dc.relation.references | References in Scopus | ||||||||||||||
| dc.subject | Iteration | ||||||||||||||
| dc.subject | Jump System | ||||||||||||||
| dc.subject | Lyapunov Equation | ||||||||||||||
| dc.subject | Markov Process | ||||||||||||||
| dc.subject | Mean Square Stability | ||||||||||||||
| dc.subject | Positive Operator | ||||||||||||||
| dc.subject | Stochastic System | ||||||||||||||
| dc.title | Positive operator based iterative algorithms for solving Lyapunov equations for Itô stochastic systems with Markovian jumps | ||||||||||||||
| dc.type | Article |
Author Affiliations
- The University of Hong Kong
- Harbin Institute of Technology