Convergence Results for Optimal Control Problems Governed by Elliptic Quasivariational Inequalities. (17th August 2020)

• Record Type: Journal Article
• Authors: Sofonea, Mircea; Tarzia, Domingo A.
• Is Part Of: Numerical functional analysis and optimization. Volume 41:Number 11(2020)
• Page Start: 1326
• Page End: 1351
• Publication Date: 2020-08-17
• Subjects: Convergence results -- frictional contact -- heat transfer -- optimal control -- optimal pair -- quasivariational inequality -- unilateral constraint; 47J20 -- 49J27 -- 49J40 -- 49K20 -- 74M15 -- 74M10; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2020.1772288
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: We consider an optimal control problem Q governed by an elliptic quasivariational inequality with unilateral constraints. We associate to Q a new optimal control problem Q ˜, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of Problem Q ˜ to a solution of Problem Q . The proofs are based on convergence results for elliptic quasivariational inequalities, obtained by using arguments of compactness, lower semicontinuity, monotonicity, penalty and various estimates. Finally, we illustrate the use of the abstract convergence results in the study of optimal control associated with two boundary value problems. The first one describes the equilibrium of an elastic body in frictional contact with an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance and unilateral constraint, associated to a version of Coulomb's law of dry friction. The second one describes a stationary heat transfer problem with unilateral constraints. For the two problems we prove existence, uniqueness and convergence results together with the corresponding physical interpretation.