## Some properties of Lyapunov function sets

HTML articles powered by AMS MathViewer

- by Emmanuel Moulay PDF
- Proc. Amer. Math. Soc.
**138**(2010), 4067-4073 Request permission

## Abstract:

The purpose of this paper is to study some geometrical and topological properties of Lyapunov function sets. These functions are very useful in control theory to solve stability problems. We focus our attention on the set of Lyapunov functions associated with continuous and discontinuous nonlinear systems.## References

- Jean-Pierre Aubin and Arrigo Cellina,
*Differential inclusions*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR**755330**, DOI 10.1007/978-3-642-69512-4 - F. H. Clarke, Yu. S. Ledyaev, and R. J. Stern,
*Asymptotic stability and smooth Lyapunov functions*, J. Differential Equations**149**(1998), no. 1, 69–114. MR**1643670**, DOI 10.1006/jdeq.1998.3476 - Nir Cohen and Izchak Lewkowicz,
*Convex invertible cones and the Lyapunov equation*, Linear Algebra Appl.**250**(1997), 105–131. MR**1420573**, DOI 10.1016/0024-3795(95)00424-6 - Klaus Deimling,
*Multivalued differential equations*, De Gruyter Series in Nonlinear Analysis and Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1992. MR**1189795**, DOI 10.1515/9783110874228 - A. F. Filippov,
*Differential equations with discontinuous righthand sides*, Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. MR**1028776**, DOI 10.1007/978-94-015-7793-9 - Wolfgang Hahn,
*Stability of motion*, Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York, 1967. Translated from the German manuscript by Arne P. Baartz. MR**0223668** - J. Kurzweil,
*On the inversion of Lyapunov’s second theorem on stability of motion*, Amer. Math. Soc. Transl. (2)**24**(1963), 19–77. - Thomas J. Laffey and Helena Šmigoc,
*Common solution to the Lyapunov equation for $2\times 2$ complex matrices*, Linear Algebra Appl.**420**(2007), no. 2-3, 609–624. MR**2278236**, DOI 10.1016/j.laa.2006.08.028 - Y. Li and X. Chen,
*Stability on multi-robot formation with dynamic interaction topologies*, IEEE International Conference on Intelligent Robots and Systems (Alberta, Canada), 2005, pp. 1325–1330. - S. Mehdi Madani-Esfahani, Mehrez Hached, and Stanisław H. Żak,
*Estimation of sliding mode domains of uncertain variable structure systems with bounded controllers*, IEEE Trans. Automat. Control**35**(1990), no. 4, 446–449. MR**1047998**, DOI 10.1109/9.52300 - Oliver Mason and Robert Shorten,
*The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem*, Electron. J. Linear Algebra**12**(2004/05), 42–63. MR**2139459** - A. N. Michel, N. R. Sarabudla, and R. K. Miller,
*Stability analysis of complex dynamical systems: some computational methods*, Circuits Systems Signal Process.**1**(1982), no. 2, 171–202. MR**736885**, DOI 10.1007/BF01600051 - C. Zuily and H. Queffelec,
*Eléments d’analyse pour l’agrégation*, Masson, Paris, 1996.

## Additional Information

**Emmanuel Moulay**- Affiliation: Xlim (UMR-CNRS 6172), Département Signal Image Communications, Université de Poitiers, Bvd. Marie et Pierre Curie, B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France
- Email: emmanuel.moulay@univ-poitiers.fr
- Received by editor(s): February 4, 2008
- Received by editor(s) in revised form: November 23, 2009
- Published electronically: June 18, 2010
- Additional Notes: The author was supported in part by the CNRS
- Communicated by: Peter A. Clarkson
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**138**(2010), 4067-4073 - MSC (2010): Primary 93D20, 34D20, 37B25; Secondary 52A05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10298-8
- MathSciNet review: 2679627