Optimal Control of a Class of Variational–Hemivariational Inequalities in Reflexive Banach Spaces
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The present paper represents a continuation of Migórski et al. (J Elast 127:151–178, 2017). There, the analysis of a new class of elliptic variational–hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force.
KeywordsVariational–hemivariational inequality Clarke subdifferential Weak convergence Optimal pair Optimal control Elastic rod Contact problem
Mathematics Subject Classification47J20 47J22 49J53 74M15
- 16.Martins, J.A.C., Marques, M.D.P.M. (eds.): Contact Mechanics. Kluwer, Dordrecht (2002)Google Scholar
- 18.Matei, A., Micu, S.: Boundary optimal control for a frictional contact problem with normal compliance. Appl. Math. Optim. 1–23 (2017)Google Scholar
- 19.Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)Google Scholar