Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators

• Published in: Journal of mathematical analysis and applications Vol. 457; no. 2; pp. 1017 - 1037
• Main Authors: Adly, Samir, Hantoute, Abderrahim, Nguyen, Bao Tran
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.01.2018
• Subjects: Differential inclusions; Invariant sets; lsc Lyapunov pairs and functions; Lyapunov stability; Maximal monotone operators; Variational and nonsmooth analysis; Lyapunov stability; Variational and nonsmooth analysis; Differential inclusions; Maximal monotone operators; lsc Lyapunov pairs and functions; Invariant sets
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2017.04.059

We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [3,4], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.