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Lyapunov Stability of Differential Inclusions Involving Prox-Regular Sets via Maximal Monotone Operators

  • Samir Adly
  • Abderrahim HantouteEmail author
  • Bao Tran Nguyen
Article
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Abstract

In this paper, we study the existence and the stability in the sense of Lyapunov of differential inclusions governed by the normal cone to a given prox-regular set, which is subject to a Lipschitzian perturbation. We prove that such apparently more general non-smooth dynamics can be indeed remodeled into the classical theory of differential inclusions, involving maximal monotone operators. This result is new in the literature. It permits to make use of the rich and abundant achievements in the class of monotone operators to study different stability aspects, and to give new proofs for the existence, the continuity, and the differentiability of solutions. This going back and forth between these two models of differential inclusions is made possible thanks to a viability result for maximal monotone operators. Applications will concern Luenberger-like observers associated with these differential inclusions.

Keywords

Differential inclusions Prox-regular sets Maximal monotone operators Lyapunov functions a-Lyapunov pairs Invariant sets Observer designs 

Mathematics Subject Classification

34A60 34D05 37B25 49J15 47H05 49J52 93B05 

Notes

Acknowledgements

The authors wish to thank the referees for providing valuable comments, which allowed to improve the manuscript. The research of the second and the third authors was supported by Conicyt grants: Fondecyt no. 1151003, Conicyt-Redes no. 150040, Mathamsud 17-MATH-06, and Conicyt-Pcha/Doctorado Nacional/2014-63140104.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.XLIM UMR-CNRS 7252Université de LimogesLimogesFrance
  2. 2.Center for Mathematical Modeling (CMM)Universidad de ChileSantiagoChile
  3. 3.Universidad de O’HigginsRancaguaChile

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