# Stability of the solution set of quasi-variational inequalities and optimal control

## TL;DR

This paper investigates the stability of solutions and optimal control problems for a class of obstacle-type quasi-variational inequalities, focusing on how solutions respond to perturbations in the forcing term.

## Contribution

It introduces a novel approach to analyze solution stability of QVIs under monotone perturbations and establishes well-posedness of associated non-standard optimal control problems.

## Key findings

- Solution set stability under perturbations is characterized.
- Different assumptions are identified for decreasing and increasing perturbations.
- Optimal control problems are shown to be well-posed.

## Abstract

For a class of quasi-variational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to {monotone} perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.