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In this paper, we study time optimal control problems for heat equations on$\Omega\times \mathbb{R}^+$. Two properties under consideration are theexistence and the bang-bang properties of time optimal controls. It is provedthat those two properties hold when controls are imposed on some proper subsetsof $\Omega$; while they do not stand when controls are active on the whole$\Omega$. Besides, a new property for eigenfunctions of $-\Delta$ withDirichlet boundary condition is revealed.

Advantages for controls imposed in a proper subset