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

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