Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Review on computational methods for Lyapunov functions

Pages: 2291 - 2331, Volume 20, Issue 8, October 2015      doi:10.3934/dcdsb.2015.20.2291

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Peter Giesl - Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom (email)
Sigurdur Hafstein - School of Science and Engineering, Reykjavik University, Menntavegi 1, IS-101 Reykjavik, Iceland (email)

Abstract: Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them.
    Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ different methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function.

Keywords:  Lyapunov function, stability, basin of attraction, dynamical system, contraction metric, converse theorem, numerical method.
Mathematics Subject Classification:  Primary: 37M99, 34D20; Secondary: 34D05, 37C75, 34D45.

Received: August 2014;      Revised: January 2015;      Available Online: August 2015.