A counterexample to exact boundary control for a parabolic obstacle problem

We give a smooth one-dimensional obstacle and a smooth admissible initial state for which no endpoint controls can steer the parabolic obstacle problem exactly to its stationary energy minimizer in finite time. The obstacle has two interior maxima of equal height. The stationary solution is constant between them, while the obstacle prevents the evolving state from falling below that height at either end of this interval. A positive sine test function then gives a strictly positive lower bound on the terminal $L^2$ error for every finite time. This answers the universal boundary-control question in AIM Problem 398 negatively.