Advantages for controls imposed in a proper subset

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

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