Optimal control for the infinity obstacle problem

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 p -obstacle problem to the minimal value of our optimal control problem for the ∞ \infty -obstacle problem, as p → ∞ p\to \infty .