Nonlinear Analysis 94 (2014) 84–99 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Boundary optimal control for quasistatic bilateral frictional contact problems Anca Capatina a,∗ , Claudia Timofte b a Institute of Mathematics ‘‘Simion Stoilow’’, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania b University of Bucharest, Faculty of Physics, Bucharest-Magurele, Romania article info Article history: Received 6 November 2012 Accepted 3 August 2013 Communicated by Enzo Mitidieri MSC: 49J40 74M10 74M15 Keywords: Boundary control Contact problems Variational inequalities Optimality conditions abstract We consider a mathematical model describing the quasistatic process of bilateral contact with friction between an elastic body and a rigid foundation. Our goal is to study a related optimal control problem which allows us to obtain a desired field of displacements on the contact boundary, by acting with a control on another part of the boundary of the body. Using penalization and regularization techniques, we derive the necessary conditions of optimality. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction The aim of this paper is to analyze a boundary optimal control of a quasistatic bilateral frictional contact problem. We consider an elastic body, which, under the influence of volume forces and surface tractions, is in bilateral contact with a rigid foundation. The friction is described by a nonlocal version of Coulomb’s law and we assume that the forces and tractions vary slowly with time and, therefore, the accelerations in the system are negligible. Also, we assume that there is no loss of contact between the body and the foundation. We are interested in finding the tractions acting on a given part of the boundary so that the resulting displacement on the contact boundary is close to a desired profile, while the norm of these surface forces remains small enough. The mathematical formulation of this problem is a state-control boundary optimal control problem where, in addition, the state is the solution of an implicit quasivariational inequality for which we do not have uniqueness results. Due to these difficulties, for obtaining the existence of at least one optimal control, we are forced to consider a family of penalized optimal control problems governed by an implicit variational inequality. We prove the existence of an optimal control for the penalized problem and the convergence of the sequence of penalized optimal controls to an optimal control for the initial problem. Since we do not have enough information about the dependence control-state, in order to obtain the necessary optimality conditions, we use some regularization techniques leading us to a control problem of a variational equality. Existence results and necessary optimality conditions for these regularized problems are established. The asymptotic analysis of these smoother problems provides that the sequence of optimal regularized controls converges to an initial optimal control. ∗ Corresponding author. Tel.: +40 212236364. E-mail addresses: ancacapatina@yahoo.fr, Anca.Capatina@imar.ro (A. Capatina). 0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.08.004