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

• Published in: Journal of optimization theory and applications Vol. 182; no. 3; pp. 906 - 934
• Main Authors: Adly, Samir, Hantoute, Abderrahim, Nguyen, Bao Tran
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Engineering; Inclusions; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Operators (mathematics); Optimization; Stability; Theory of Computation; Viability; Maximal monotone operators; Lyapunov pairs; Prox-regular sets; Observer designs; 37B25; 34D05; Differential inclusions; Lyapunov functions; Invariant sets; 49J52; 93B05; 47H05; 34A60; 49J15
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-018-1446-7

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.