Abstract
Abstract This article presents two observability inequalities for the heat equation over Ω× (0,T). In the first one, the observation is from a subset of positive measure in Ω× (0,T), while in the second, the observation is from a subset of positive surface measure on ∂Ω× (0,T). We will provide some applications for the above-mentioned observability inequalities, the bang-bang property for the minimal time, time optimal and minimal norm control problems, and also establish new open problems related to observability inequalities and the aforementioned applications.
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