Well-Posedness and Output Regulation for Implicit Time-Varying Evolution Variational Inequalities

Abstract

A class of evolution variational inequalities (EVIs), which comprises ordinary differential equations (ODEs) coupled with variational inequalities (VIs) associated with time-varying set-valued mappings, is proposed in this paper. We first study the conditions for existence and uniqueness of solutions. The central idea behind the proof is to rewrite the system dynamics as a differential inclusion which can be decomposed into a single-valued Lipschitz map, and a time-dependent maximal monotone operator. Regularity assumptions on the set-valued mapping determine the regularity of the resulting solutions. Complementarity systems with time-dependence are studied as a particular case. We then use this result to study the problem of designing state feedback control laws for output regulation in systems described by EVIs. The derivation of control laws for output regulation is based on the use of internal model principle, and two cases are treated: First, a static feedback control law is derived when full state feedback is available; In the second case, only the error to be regulated is assumed to be available for measurement and a dynamic compensator is designed. As applications, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. Another application is seen in designing control inputs for regulation in power converters.

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

“…The paper studies the well-posedness and asymptotic behaviour for a class of Lur'e dynamical systems where the set-valued feedback depends not only on the time but also on the state. Let us emphasis that the obtained solutions are strong, comparing with the weak solutions acquired in [3]. The main tool is a new implicit discretization scheme, which is an advantage for implementation in numerical simulations.…”

“…Let us show that we can use our result to answer this question. Indeed, we can rewrite (S) as follows y(t) = Cx(t) + Dλ(t), λ(t) ∈ −N K(t) (ȳ(t) + (C − C)x(t)) = −NK (t,x(t)) (ȳ(t)), t ≥ 0; Therefore we can not apply the result in [3] but can use our result to deduce the existence of solutions for the associated dynamical system. Indeed, we can see that (A, B, B, D,K) satisfies all assumptions of Theorem 1 whereK(t, x) = [f 1 (t) − εx 1 , +∞) × [f 2 (t) − εx 2 , +∞).…”

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

“…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.…”

Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators

“…In some case, a kind of "distance" between such operators A(t) is used to express and evaluate continuity properties of the corresponding map t → A(t). Thus, bounded Hausdorff (or Attouch-Wets)-distances, or Brezis-Haraux and Fitzpatrick functions have been used; see the recent papers of Attouch et al [2] and Tanwani et al [30].…”

“…The computation and the application of Vladimirov's pseudo distance to specific examples may not always be easy and it may demand more work; this, however, may well be a common feature, as, for instance, Brezis-Haraux function and Vladimirov pseudo distance are not completely unrelated. With our techniques and/or results, a number of subsidiary classes of problems may be solved and then applied in different studies and fields; these include second-order evolution inclusions, relaxation and viscosity in control theory, variational inequalities, Skorohod problems and sweeping processes, asymptotic theory, etc; cf., say, [26,12,3,29,30,4,5,15,2].…”