Solvability and stationarity for the optimal control of variational inequalities with point evaluations in the objective functional

Abstract

Motivated by applications in economics and engineering, we consider the optimal control of a variational inequality with point evaluations of the state variable in the objective. This problem class constitutes a specific mathematical program with complementarity constraints (MPCC). In our context, the problem is posed in an adequate function space and the variational inequality involves second order linear elliptic partial differential operators. The necessary functional analytic framework complicates the derivation of stationarity conditions whereas the non-convex and non-differentiable nature of the problem challenges the design of an efficient solution algorithm. In this paper, we present a penalization and smoothing technique to derive first order type conditions related to C-stationarity in the associated Sobolev space setting.

Numerical Analysis of a Family of Optimal Distributed Control Problems Governed by an Elliptic Variational Inequality

“…Several optimal control problems are governed by elliptic variational inequalities ( [1], [2], [3], [5], [6], [13], [14], [27], [31], [32], [39]) and there exists an abundant literature about continuous and numerical analysis of optimal control problems governed by elliptic variational equalities or inequalities ( [4], [10], [11], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [26], [29], [30], [35], [36], [40]) and by parabolic variational equalities or inequalities ( [7], [28]).…”

“…with M > 0 a given constant and u g is the corresponding solution of the elliptic variational inequality (1.3) associated to the control g. Several continuous optimal control problems are governed by elliptic variational inequalities, for example: the process of biological waste-water treatment; reorientation of a satellite by propellers; and economics: the problem of consumer regulation of a monopoly, etc. There exist an abundant literature for optimal control problems [4,42,50], for optimal control problems governed by elliptic variational equalities or inequalities [2,3,5,6,7,8,9,11,19,20,26,28,30,32,34,38,40,45,46,52,53,54], for numerical analysis of variational inequalities or optimal control problems [10,13,14,15,16,17,21,22,23,24,25,27,33,35,36,37,43,47,48,49,<...>…”