Abstract
A principal technical result of this paper is that the one-dimensional heat equation with boundary control is exactly null-controllable with control restricted to an arbitrary set $\E\subset[0,T]$ of positive measure. A general abstract argument is presented to show that, in contrast to previous results, this implies the bang-bang property for time-optimal controls---i.e., such a control can take only extreme values of (the hull of) the constraint set---without imposing any condition regarding the target state.
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