State-dependent implicit sweeping process in the framework of quasistatic evolution quasi-variational inequalities

Samir Adly (1) , Tahar Haddad (2) , Le Ba Khiet (3)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 UMSBJ - Université Mohammed Seddik Benyahia [Jijel] | University of Jijel
3 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]

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.

Mots clés: Quasistatic frictional contact problems; Unilateral constraints; Evolution variational inequalities; Moreau’s sweeping process

Domaines: Optimisation et contrôle [math.OC]

Samir Adly, Tahar Haddad, Le Ba Khiet. State-dependent implicit sweeping process in the framework of quasistatic evolution quasi-variational inequalities. Journal of Optimization Theory and Applications, 2019, 182 (2), pp.473-493. ⟨10.1007/s10957-018-1427-x⟩. ⟨hal-02386012⟩