Well-Posedness of History/State-Dependent Implicit Sweeping Processes
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Well-Posedness of History/State-Dependent Implicit Sweeping Processes Shengda Zeng1,2 · Emilio Vilches3 Received: 25 January 2020 / Accepted: 24 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is devoted to the study of a new class of implicit state-dependent sweeping processes with history-dependent operators. Based on the methods of convex analysis, we prove the equivalence of the history/state dependent implicit sweeping process and a nonlinear differential equation, which, through a fixed point argument for historydependent operators, enables us to prove the existence, uniqueness, and continuous dependence of the solution in a very general framework. Moreover, we present some new convergence results with respect to perturbations in the data, including perturbations of the associated moving sets. Finally, the theoretical results are applied to prove the well-posedness of a history-dependent quasi-static contact problem. Keywords Implicit sweeping process · History-dependent operator · Frictional contact problem · Viscoelastic material · Unilateral constraints · Moreau’s sweeping process · Evolution variational inequality Mathematics Subject Classification 49J40 · 47J20 · 47J22 · 35L15 · 35L86 · 35L87 · 74HXX · 74M10
Communicated by Boris S. Mordukhovich.
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Emilio Vilches [email protected] Shengda Zeng [email protected]
1
Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, P.R. China
2
Jagiellonian University in Krakow, Chair of Optimization and Control, ul. Lojasiewicza 6, 30348 Krakow, Poland
3
Universidad de O’Higgins, Rancagua, Chile
123
Journal of Optimization Theory and Applications
1 Introduction The Moreau’s sweeping process is a first-order differential inclusion, involving the normal cone to a moving set depending on time. Roughly speaking, a point is swept by a moving closed set. The sweeping process was introduced and deeply studied by Moreau in a series of papers (see [1–3]) to model an elasto-plastic mechanical system. Since then, many other applications have been given, such as applications in switched electrical circuits [4], nonsmooth mechanics [5,6], crowd motion [7], hysteresis in elasto-plastic models [8], among others. Moreover, due to the development of new techniques to deal with differential inclusions involving normal cones, new variants of the sweeping process have been introduced. We can mention the state-dependent sweeping process, the second-order sweeping process, the implicit sweeping process [9–12], and some others variants. For more details, we refer to [13–17] and the references therein. The aim of this paper is to study the existence, uniqueness, and stability for a class of history/state-dependent implicit sweeping processes. The latter was introduced in [18] (for the special, case where the moving sets are state-independent) to model a history-dependent viscoelastic contact problem (see [18, Sect. 4]). Moreover, the
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