Optimal Control of Quasivariational Inequalities with Applications to Contact Mechanics

Abstract

This chapter deals with the optimal control of a class of elliptic quasivariational inequalities. We start with an existence and uniqueness result for such inequalities. Then we state an optimal control problem, list the assumptions on the data and prove the existence of optimal pairs. We proceed with a perturbed control problem for which we state and prove a convergence result, under general conditions. Further, we present a relevant particular case for which these conditions are satisfied and, therefore, our convergence result works. Finally, we illustrate the use of these abstract results in the study of a mathematical model which describes the equilibrium of an elastic body in frictional contact with an obstacle, the socalled foundation. The process is static and the contact is modeled with normal compliance and unilateral constraint, associated with the Coulomb's law of dry friction. We prove the existence, uniqueness, and convergence results together with the corresponding mechanical interpretation. We illustrate these results in the study of a one-dimensional example. Finally, we end this chapter with some concluding remarks.

Tykhonov Well-posedness of a Heat Transfer Problem with Unilateral Constraints

“…The results in [12,27,28,31,37] concern the analysis of hemivariational inequalities and are based on properties of the subdifferential in the sense of Clarke, defined for locally Lipschitz functions, which may be nonconvex. Results in the study of optimal control for variational and hemivariational inequalities have been discussed in several works, including [2,3,4,8,19,23,25,26,29,32,33,34,35,36] and, more recently, in [36,39].…”