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.
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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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Adly, S., Hantoute, A. & Nguyen, B.T. Lyapunov Stability of Differential Inclusions Involving Prox-Regular Sets via Maximal Monotone Operators. J Optim Theory Appl 182, 906–934 (2019). https://doi.org/10.1007/s10957-018-1446-7
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DOI: https://doi.org/10.1007/s10957-018-1446-7
Keywords
- Differential inclusions
- Prox-regular sets
- Maximal monotone operators
- Lyapunov functions
- a-Lyapunov pairs
- Invariant sets
- Observer designs