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Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems

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Abstract

We propose a class of locally Lipschitz functions with piecewise structure for use as Lyapunov functions for hybrid dynamical systems. Subject to some regularity of the dynamics, we show that Lyapunov inequalities can be checked only on a dense set and thus we avoid checking them at points of nondifferentiability of the Lyapunov function. Connections to other classes of locally Lipschitz or piecewise regular functions are also discussed, and applications to hybrid dynamical systems are included.

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Notes

  1. In [12], interval-splitting sets with the additional property (11) are used to construct locally Lipschitz functions for which the steepest descent/subdifferential flow generated by the Clarke subdifferential has the pathological behavior of generating strictly increasing orbits.

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Authors and Affiliations

  1. LAAS-CNRS, Université de Toulouse, Toulouse, France

    Matteo Della Rossa, Aneel Tanwani & Luca Zaccarian

  2. Department of Industrial Engineering, University of Trento, Trento, Italy

    Luca Zaccarian

  3. Department of Statistics and Mathematics, Loyola University, Chicago, IL, USA

    Rafal Goebel

Authors
  1. Matteo Della Rossa
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  2. Rafal Goebel
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  3. Aneel Tanwani
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  4. Luca Zaccarian
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Correspondence to Matteo Della Rossa.

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Work supported in part by ANR via grant HANDY (number ANR-18-CE40-0010).

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Della Rossa, M., Goebel, R., Tanwani, A. et al. Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems. Math. Control Signals Syst. 33, 123–149 (2021). https://doi.org/10.1007/s00498-020-00273-9

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