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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.

Observability Inequalities for Parabolic Equations over Measurable Sets and Some Applications Related to the Bang-Bang Property for Control Problems