Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Stability of random-switching systems of differential equations

Authors: C. Zhu, G. Yin and Q. S. Song
Journal: Quart. Appl. Math. 67 (2009), 201-220
MSC (2000): Primary 60J27, 60J75, 93D05, 93D20, 93E15
Published electronically: February 18, 2009
MathSciNet review: 2514632
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Abstract | References | Similar Articles | Additional Information

Abstract: This work is devoted to the stability of random-switching systems of differential equations. After presenting the formulation of random-switching systems, the notion of stability is recalled, and sufficient conditions in terms of the Liapunov function are presented. Then easily verifiable conditions for stability and instability of systems arising in approximation are established. Using a logarithm transformation, necessary and sufficient conditions are derived for systems that are linear in the continuous state component. Several examples are provided as demonstrations. Among other things, a somewhat different behavior from the well-known Hartman-Grobman theorem is observed.

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Additional Information

C. Zhu
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201
Email: zhu@uwm.edu

G. Yin
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: gyin@math.wayne.edu

Q. S. Song
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089
Email: qingshus@usc.edu

DOI: https://doi.org/10.1090/S0033-569X-09-01092-8
Keywords: Random switching, hybrid system, stability, necessary condition, sufficient condition
Received by editor(s): September 6, 2007
Published electronically: February 18, 2009
Additional Notes: Research of the first author was supported in part by the National Science Foundation under DMS-0304928
Research of the second author was supported in part by the National Science Foundation under DMS-0603287, and in part by the National Security Agency under MSPF-068-029
Research of the third author was supported in part by the U.S. Army Research Office MURI grant W911NF-06-1-0094 at the University of Southern California
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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