Some properties of Lyapunov function sets

Moulay, Emmanuel

doi:10.1090/S0002-9939-2010-10298-8

Some properties of Lyapunov function sets
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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.