State-Dependent Implicit Sweeping Process in the Framework of Quasistatic Evolution Quasi-Variational Inequalities

This paper deals with the existence and uniqueness of solutions for a class of state-dependent sweeping processes with constrained velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty, we introduce a new notion called h...

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Bibliographic Details
Published inJournal of optimization theory and applications Vol. 182; no. 2; pp. 473 - 493
Main Authors Adly, Samir, Haddad, Tahar, Le, Ba Khiet
Format Journal Article
LanguageEnglish
Published New York Springer US 01.08.2019
Springer Nature B.V
Springer Verlag
Subjects
Applications of Mathematics
Calculus of Variations and Optimal Control; Optimization
Engineering
Evolution
Hilbert space
Inequalities
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Optimization and Control
Sweeping
Theory of Computation
Viscoelastic materials
Viscoelasticity
74M10
47J22
58E35
Quasistatic frictional contact problems
74G25
47J20
49J40
Unilateral constraints
Evolution variational inequalities
74M15
Moreau’s sweeping process
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Summary:This paper deals with the existence and uniqueness of solutions for a class of state-dependent sweeping processes with constrained velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty, we introduce a new notion called hypomonotonicity-like of the normal cone to the moving set, which is satisfied by many important cases. Combining this latter notion with an adapted Moreau’s catching-up algorithm and a Cauchy technique, we obtain the strong convergence of approximate solutions to the unique solution, which is a fundamental property. Using standard tools from convex analysis, we show the equivalence between this implicit state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities. As an application, we study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-018-1427-x