Reduction of state dependent sweeping process to unconstrained differential inclusion

Haddad, Tahar; Kecis, Ilyas; Thibault, Lionel

doi:10.1007/s10898-014-0220-0

Cited by 18 publications

(16 citation statements)

References 21 publications

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“…3.3 for more details; this model has been largely developed by Cornet [21]. For the existence of solutions with time-varying convex/nonconvex sets C(t) we refer the reader to [9,[14][15][16][25][26][27][28][32][33][34]37,58]. Applications to the crowd motion modeling have been realized in [36].…”

Section: Introductionmentioning

confidence: 99%

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

2014

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International audienceIn this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $C(t)$, supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $C(t)$. This class of problems subsumes as a particular case, the evolution variational inequalities. Assuming that the moving subset $C(t)$ has a continuous variation for every $t \in [0, T ]$ with $C (0)$ bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model, to the planning procedure in mathematical economy, and to nonregular electrical circuits containing nonsmooth electronic devices like diodes. The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau’s seminal work

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Section: Introductionmentioning

confidence: 99%

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

2014

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“…we get the first equality in (20). The other two equalities in (20) The main idea behind the previous existence theorems, Theorems 4.5 and 4.6, as well as the forthcoming results on Lyapunov stability in the next section, is that differential inclusion (1) is in some sense equivalent to a differential inclusion governed by a (Lipschitz continuous perturbation of a) maximal monotone operator.…”

Section: The Existence Resultsmentioning

confidence: 88%

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

Adly

Hantoute

Nguyen

2018

J Optim Theory Appl

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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.

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“…The extended Moreau sweeping process, as involved in electrical circuit (see, e.g., [1,7]) and in crowd motion (see, e.g., [20]), can be stated as the (measure) differential inclusion (ESP ) du ∈ −N C(t); u(t) − Φ(t, u(t)) and u(0) = u 0 ∈ C(0), where N (•; •) denotes a normal cone. The uniform r-prox-regularity of all the sets C(t) is known to be the general condition under which (ESP ) admits a (unique) solution with bounded variation (see, e.g., [1,7,15,17]). Concrete problems are considered in [1,7,27] where the sets C(t) are in the form either C(t) = {x ∈ H : g 1 (t, x) ≤ 0, .…”

Section: Introductionmentioning

confidence: 99%

Preservation of Prox-Regularity of Sets with Applications to Constrained Optimization

Adly¹,

Nacry²,

Thibault³

2016

SIAM J. Optim.

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International audienceIn this paper, we first provide counterexamples showing that sublevels of prox-regular functions and levels of differentiable mappings with Lipschitz derivatives may fail to be prox-regular. Then, we prove the uniform prox-regularity of such sets under usual verifiable qualification conditions. The preservation of uniform prox-regularity of intersection and inverse image under usual qualification conditions is also established. Applications to constrained optimization problems are given

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Reduction of state dependent sweeping process to unconstrained differential inclusion

Cited by 18 publications

References 21 publications

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

Convex sweeping process in the framework of measure differential inclusions and evolution variational inequalities

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

Preservation of Prox-Regularity of Sets with Applications to Constrained Optimization

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