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

Abstract

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.

Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

“…Time-varying systems. Still dealing with systems possessing continuous solutions, let us cite the so-called Lyapunov pairs (V (t, x), W (x)) (whose definition is very close to dissipation inequality in (D.2) with null input) [ 300,301,21,34,35,38,32,31,30,556,341,376], which are used to prove stability of equilibria, but also existence of solutions and invariance of sets in the FOSwP. The proximal normal cone and the proximal subdifferential (see Sections A.…”

An approximate solution of a differential inclusion with maximal monotone operator

“…The differential inclusion has been studied by S. Adly, A. Hantoute, T. Nguyen, M. Thera and L. B. Khiet in [1][2][3], and by Y. Shang [4][5][6] in the study of fixed-time stability in the field of control engineering. We refer the interested reader to [7,8] for more details regarding related applications (mathematical questions tied to analysis dealing with porous media or biological systems, respectively).…”

“…In 2006, Mordukhovich [9] contemplated the accompanying differential inclusion ν (s) ∈ G(ν(s)) a.e s ∈ [0, b] where G is a Lipschitz multifunction. In 2019, he used in the work [10] the same method to study the following differential inclusion: We present in this work a method for obtaining the best approximate solution of (1). The existence of the solution of this inclusion is a classical result proved by Brezis in [11].…”

“…[12][13][14]). Our idea to prove the main result of this paper is to use this problem to find an approximate solution of our principle problem (1).…”

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

“…Typical examples of (4) involve the Fenchel subdifferential of proper, lower semicontinuous convex functions ([2]). System (4) has been extensively studied; namely, regarding existence, regularity and properties of the solutions [13], while Lyapunov stability of such systems have been initiated in [32]; see, also, [4,5,6] for recent contributions on the subject. Different criteria using the semi-group generated by the operator A can also be found in [31], where Lyapunov functions are characterized as viscosity-type solutions of Hamilton-Jacobi equations, and in [15], using implicit tangent cones associated to the invariant sets candidates.…”