Vj. Mizel et Ti. Seidman, AN ABSTRACT BANG-BANG PRINCIPLE AND TIME-OPTIMAL BOUNDARY CONTROL OF THE HEAT-EQUATION, SIAM journal on control and optimization, 35(4), 1997, pp. 1204-1216
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 epsilon subset of [0, T] of po
sitive measure, A general abstract argument is presented to show that,
in contrast to previous results, this implies the bang-bang property
ibr time-optimal controls - i.e., such a control can take only extreme
values of (the hull of) the constraint set - without imposing any con
dition regarding the target state.