Nonsmooth Lur’e Dynamical Systems in Hilbert Spaces

Abstract

International audienceIn this paper, we study the well-posedness and stability analysis of set-valued Lur’e dynamical systems in infinite-dimensional Hilbert spaces. The existence and unique- ness results are established under the so-called passivity condition. Our approach uses a regularization procedure for the term involving the maximal monotone operator. The Lya- punov stability as well as the invariance properties are considered in detail. In addition, we give some sufficient conditions ensuring the robust stability of the system in finite- dimensional spaces. The theoretical developments are illustrated by means of two examples dealing with nonregular electrical circuits and an other one in partial differential equations. Our methodology is based on tools from set-valued and variational analysis

On a Class of Lur’e Dynamical Systems with State-Dependent Set-Valued Feedback

“…Proof From Assumption 2, there exists κ ∈ R such that (κI, B, C, D) is passive by using Lemma 2. By using change of variables, without loss of generality, we can suppose that P ≡ I, the identity matrix (see, e.g., [4,14]). Let us use the following implicit scheme to approximate (8).…”

“…where F t n i+1 ,x n i := −κI + B(N −1 K(t n i+1 ,x n i ) + D) −1 C is a maximal monotone operator (see, e.g., [14,4,5]). Then we can compute x n i+1 uniquely as follows…”

“…In [3], the authors studied the time-dependent case K(t, x) ≡ K(t) with motivation for considering the viability control problem and the output regulation problem, particularly in power converters. To the best of our knowledge, it can be considered as the first work which considers non-static set-valued feedbacks with non-zero D (see, e.g., [4,5,6,10,11,12,13,14] for static cases). (A5) It holds that rge(D) ⊆ rge(C) and K : [0, ∞) ⇒ R m has closed and convex values for each t ≥ 0.…”

Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions

“…Proof. We use an implicit scheme to approximate problem (1). In details, let T = T − t 0 and for each given positive integer n, we set h n = T /n and t n i = t 0 + ih for 0 ≤ i ≤ n. We can construct the sequence (x n i ) 0≤i≤n with x n 0 = x 0 as follows:…”

“…There are also various works for the case of time-dependence, i.e., A t,x ≡ A t (see [13,15,17,26,32] and the reference therein). Among important contributions are sweeping processes [3,16,18,22,23,24,25,30,31], Skorohod problem [29], hysteresis operators [14] and recently Lur'e dynamical systems [1,7,8,9,10,19,27,28]. In particular, when A t,x ≡ N C(t) , the normal cone of a moving closed convex set, one obtains the sweeping processes   ẋ (t) ∈ f (t, x(t)) − N C(t) (x(t)), a.e.…”