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

Abstract

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.

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

“…Passivity conditions are also used for the semi-global stability of FOSwP with proxregular sets S(t) is analysed in [522, Theorem 3.2, Proposition 3.5], using r from Definition A.2 as a parameter to determine the size of the basin of attraction (the stability of the convex case being recovered as r → +∞). This seems to be the first result of this type in the literature, a partial extension of [522,Theorem 3.2] is in [33,Theorem 6.1, Corollary 6.1]. The result in [522, Theorem 3.2] may be considered as the first version of absolute stability with a nonconvex set-valued feedback nonlinearity.…”

“…It is noteworthy that the input/output constraint P B = C T , P = P T ≻ 0 (see Section 3.4) which is implied by (D.1) when D + D T = 0, and its generalization ker(D + D T ) ⊂ ker(P B − C T ) used in Corollary 5.9 (see [139,Propositions 3.62,3.63] [151,155] for structural properties of dissipative systems), are passivity-like conditions stemming from the celebrated Kalman-Yakubovich-Popov (KYP) Lemma [139,155]. It follows that the results in [9,11,12,20,28,36,31,39,134,135,136,138,143,280,386,522,524], extended in [523,159] and in [143,Section 4], rely on a sort of passivity condition on the continuous-time system, not only for stability but also for well-posedness purposes. One consequence is that state jumps (according to the jump mappings in Section 2.4.4) are easily incorporated in the stability analysis because the storage function of Definition 2.2 is a good Lyapunov function candidate.…”

“…Possible future generalizations could be to replace convex functions ϕ(•) by prox-regular functions [492], as done for FOSwP in [204,238,530,143]. One important issue lies in the fact that in the non convex case, the biconjugate (ϕ ⋆ ) ⋆ may not equal ϕ, so that manipulations used to invert the set-valued part may no longer be valid (one path could be to use the results in [31]). Other extensions which have not been tackled, could be: consider a mixed LCP instead of an LCP in (2.22) (b), yielding the MLCS: [107] for preliminary results), time-varying LCS with A(t), B(t), C(t), D(t) (see Section 3.4 for a possible path), and control-related issues (optimal control has been tackled in [287,555,554], yielding MPEC problems, with applications in process control [484,96], trajectory tracking, etc).…”

“…This transformation has been used afterwards in [9,11,12,20,28,36,31,39,33,134,135,136,138,143,159,280,375,386,522,524], extended in [523,159] and in [143,Section 4] for the nonlinear case. Let the quadruple (A, B, C, D) of the LCS in (2.22) be passive (see Section D).…”

“…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.…”