  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.
