Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain

Journal of Mathematical Analysis and Applications

2018

Self Cite

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 [5, 6], 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.

Section: Introductionsupporting

confidence: 78%

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

Journal of Mathematical Analysis and Applications

2018

Self Cite

“…Proposition 4.4 Let S ⊂ domA satisfy condition (6), and take x 0 ∈ S. If there is some ρ > 0 such that for any…”

Section: Strong and Weak Invariant Setsmentioning

confidence: 99%

Lyapunov Stability of Differential Inclusions with Lipschitz Cusco Perturbations of Maximal Monotone Operators

2019

Set-Valued Var. Anal

Self Cite

“…An extensive research has been done to solve this problem for different kinds of differential inclusions and equations ( [14,15]). Complete primal and dual characterizations are given in [6,7]. Proposition 3.3 Assume that A is a monotone operator, and let S be a closed subset of domA.…”

Section: Differential Inclusions Involving Maximal Monotone Operatorsmentioning

confidence: 99%

“…The concept of invariant sets will be the key tool to go back and forth between inclusions (1) and (2). Invariant sets with respect to differential inclusions governed by maximal monotone operators have been studied and characterized in [6,7]. Other references for invariant sets, also referred to as viable sets, and the related theory of Lyapunov stability are [9,13,21,26] among others.…”

Section: Introductionmentioning

confidence: 99%

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

2018

J Optim Theory Appl

Self Cite

In this paper, we study the existence and the stability in the sense of Lyapunov of solutions for differential inclusions governed by the normal cone to a prox-regular set and subject to a Lipschitzian perturbation. We prove that such, apparently, more general nonsmooth dynamics can be indeed remodelled into the classical theory of differential inclusions involving maximal monotone operators. This result is new in the literature and permits us to make use of the rich and abundant achievements in this class of monotone operators to derive the desired existence result and stability analysis, as well as the continuity and differentiability properties of the 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. As an application, we study a Luenberger-like observer, which is shown to converge exponentially to the actual state when the initial value of the state's estimation remains in a neighborhood of the initial value of the original system.