Evolution equations governed by the sweeping process

Castaing, Charles; Hā, Trúóng Xuân Dúc; Valadier, Michel

doi:10.1007/bf01027688

Cited by 80 publications

(77 citation statements)

References 14 publications

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“…Generalizations of the sweeping process have been the object of many studies, see e.g. [5, 19, 32-34, 36, 41, 67, 68, 72] and more references in [18,35,42]. Recently it was also shown that quite similar formalisms apply to nonsmooth electrical networks as well as some problems of absolute stability [10,23].…”

Section: Introductionmentioning

confidence: 97%

Higher order Moreau’s sweeping process: mathematical formulation and numerical simulation

2006

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In this paper we present an extension of Moreau's sweeping process for higher order systems. The dynamical framework is carefully introduced, qualitative, dissipativity, stability, existence, regularity and uniqueness results are given. The time-discretization of these nonsmooth systems with a timestepping algorithm is also presented. This differential inclusion can be seen as a mathematical formulation of complementarity dynamical systems with arbitrary dimension and arbitrary relative degree between the complementaryslackness variables. Applications of such high-order sweeping processes can be found in dynamic optimization under state constraints and electrical circuits with ideal diodes.

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Section: Introductionmentioning

confidence: 97%

Higher order Moreau’s sweeping process: mathematical formulation and numerical simulation

2006

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“…In this subsection we study the family of set-valued maximal monotone operators used as feedback control laws for the system (13). First, some results about the existence and (in some cases) uniqueness of solutions are presented.…”

Section: Set-valued Controllermentioning

confidence: 99%

Set-Valued Sliding-Mode Control of Uncertain Linear Systems: Continuous and Discrete-Time Analysis

Miranda-Villatoro¹,

Brogliato²,

Castaños³

2018

SIAM J. Control Optim.

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In this paper we study the closed-loop dynamics of linear time-invariant systems with feedback control laws that are described by set-valued maximal monotone maps. The class of systems considered in this work is subject to both, unknown exogenous disturbances and parameter uncertainty. It is shown how the design of conventional sliding mode controllers can be achieved using maximal monotone operators (which include the set-valued signum function). Two cases are analyzed: continuous-time and discrete-time controllers. In both cases well-posedness together with stability results are presented. In discrete time we show how the implicit scheme proposed for the selection of control actions makes sense resulting in the chattering effect being almost suppressed even with uncertainty in the system.

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“…It is well known (see for example [6]) that N S (x) the normal cone of a closed convex set S at x ∈ H can be defined in terms projection operator P roj S (·) as follows N S (x) = {ξ ∈ H, there exists r > 0 such that x ∈ P roj S (x + rξ)}.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Let S be a nonempty closed convex subset of H and x be a point in S. The convex normal cone of S at x is defined by (see for instance [1,6])…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Here N C(t,u(t)) (u(t)) denotes the convex normal cone to C(t, u(t)) at u(t). Such problems with delay and convex compact valued upper semicontinuous perturbations have been studied [5,6,8] in the case when C does not depend on the state u ∈ H. Recently, in [4], the authors have obtained an existence result for (1.1) (i.e C depends on the time t ∈ [0, T ] and state u ∈ H) via an implicit discretization technique based on the fixed point theorem. The main purpose of this paper is to give a new proof of the the existence of solutions for (1.1).…”

Section: Introductionmentioning

confidence: 99%

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