Optimal control for the infinity obstacle problem

TL;DR

This paper investigates an optimal control framework for the infinity obstacle problem, establishing existence of optimal controls and states, and analyzing the convergence from p-obstacle to infinity obstacle problems as p approaches infinity.

Contribution

It proves the existence of an optimal control that is also an optimal state for the infinity obstacle problem and demonstrates convergence from p-obstacle to infinity obstacle problems.

Findings

Existence of an optimal control and state for the infinity obstacle problem.

Convergence of minimal values from p-obstacle to infinity obstacle problem as p→∞.

Theoretical foundation for control problems involving the infinity obstacle.

Abstract

In this note, we show that a natural optimal control problem for the $\infty$-obstacle problem admits an optimal control which is also an optimal state. Moreover, we show the convergence of the minimal value of an optimal control problem for the $p$-obstacle problem to the minimal value of our optimal control problem for the $\infty$-obstacle problem, as $p\to\infty$.