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Local Lyapunov Functions for Periodic and Finite-Time ODEs

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Recent Trends in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 35))

Abstract

Lyapunov functions for general systems are difficult to construct. However, for autonomous linear systems with exponentially stable equilibrium, there is a classical way to construct a global Lyapunov function by solving a matrix equation. Consequently, the same function is a local Lyapunov function for a nonlinear system.In this paper, we generalise these results to time-periodic and, in particular, finite-time systems with an exponentially attractive zero solution. We show the existence of local Lyapunov functions for nonlinear systems. For finite-time systems, we consider a generalised notion of a Lyapunov function, which is not necessarily continuously differentiable, but just locally Lipschitz continuous; the derivative is then replaced by the Dini derivative.

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Authors and Affiliations

  1. Department of Mathematics, University of Sussex, Falmer Campus, Brighton, BN1 9QH, UK

    Peter Giesl

  2. School of Science and Engineering, Reykjavik University, Menntavegi 1, 101, Reykjavik, Iceland

    Sigurdur Hafstein

Authors
  1. Peter Giesl
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  2. Sigurdur Hafstein
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Corresponding author

Correspondence to Peter Giesl .

Editor information

Editors and Affiliations

  1. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Andreas Johann

  2. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Hans-Peter Kruse

  3. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Florian Rupp

  4. Zentrum Mathematik, Technische Universität München, Garching bei München, Germany

    Stephan Schmitz

Additional information

Dedicated to Jürgen Scheurle on the occasion of his 60th birthday

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Giesl, P., Hafstein, S. (2013). Local Lyapunov Functions for Periodic and Finite-Time ODEs. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_7

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