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ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws

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Abstract

A family of time-varying hyperbolic systems of balance laws is considered. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach to obtain a time-varying strict Lyapunov function. It allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of input-to-state stability (ISS) type. Two examples illustrate the results.

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Authors and Affiliations

  1. Department of Automatic Control, Gipsa-lab, 961 rue de la Houille Blanche, BP 46, 38402, Grenoble Cedex, France

    Christophe Prieur

  2. Team INRIA DISCO, L2S, CNRS-Supelec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France

    Frédéric Mazenc

Authors
  1. Christophe Prieur
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  2. Frédéric Mazenc
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Correspondence to Christophe Prieur.

Additional information

Work supported by HYCON2 Network of Excellence “Highly-Complex and Networked Control Systems”, grant agreement 257462.

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Prieur, C., Mazenc, F. ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math. Control Signals Syst. 24, 111–134 (2012). https://doi.org/10.1007/s00498-012-0074-2

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  • DOI: https://doi.org/10.1007/s00498-012-0074-2

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