Optimal control and directional differentiability for elliptic quasi-variational inequalities

TL;DR

This paper investigates elliptic quasi-variational inequalities, establishing existence, differentiability, and optimal control results, thereby advancing the mathematical understanding and solution methods for these complex inequalities.

Contribution

It provides new existence theorems, extends differentiability results to infinite dimensions, and derives stationarity conditions for QVI-constrained optimal control problems.

Findings

Multiple existence theorems for solutions to elliptic QVIs.

Directional differentiability of the solution map in general data.

Derivation of stationarity conditions for QVI-constrained optimal control.

Abstract

We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general data and locally Hadamard differentiable obstacle mappings, thereby extending in particular the results of our previous work which provided the first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.