A Bang-Bang Principle of Time Optimal Internal Controls of the Heat Equation

In this paper, we study a time optimal internal control problem governed by the heat equation in $\Omega\times [0,\infty)$. In the problem, the target set $S$ is nonempty in $L^2(\Omega)$, the control set $U$ is closed, bounded and nonempty in $L^2(\Omega)$ and control functions are taken from the set $\uad=\{u(\cdot, t): [0,\infty)\ra L^2(\Omega) {measurable}; u(\cdot, t)\in U, {a.e. in t} \}$. We first establish a certain null controllability for the heat equation in $\Omega\times [0,T]$, with controls restricted to a product set of an open nonempty subset in $\Omega$ and a subset of positive measure in the interval $[0,T]$. Based on this, we prove that each optimal control $u^*(\cdot, t)$ of the problem satisfies necessarily the bang-bang property: $u^*(\cdot, t)\in \p U$ for almost all $t\in [0, T^*]$, where $\p U$ denotes the boundary of the set $U$ and $T^*$ is the optimal time. We also obtain the uniqueness of the optimal control when the target set $S$ is convex and the control set $U$ is a closed ball.