Attainable subspaces and the bang-bang property of time optimal controls for heat equations

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In this paper, we study two subjects on internally controlled heat equations with time varying potentials: the attainable subspaces and the bang-bang property for some time optimal control problems. We present some equivalent characterizations on the attainable subspaces, and provide a sufficient conditions to ensure the bang-bang property. Both the above-mentioned characterizations and the sufficient condition are closely related to some function spaces consisting of some solutions to the adjoint equations. It seems for us that the existing ways to derive the bang-bang property for heat equations with time-invariant potentials (see, for instance, [4],[7],[16],[26]) do not work for the case where the potentials are time-varying. We provide another way to approach it in the current paper.