This paper 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 ). It also proves the Lebeau-Robbiano spectral inequality when Ω is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.where D is a subset of Ω × (0, T ), andwhere J is a subset of ∂Ω × (0, T ). Such apriori estimates are called observability inequalities.In the case that D = ω × (0, T ) and J = Γ × (0, T ) with ω and Γ accordingly open and nonempty subsets of Ω and ∂Ω, both inequalities (1.2) and (1.3) (where ∂Ω is smooth) were essentially first established, via the Lebeau-Robbiano spectral inequalities in [28] (See also [29,34,16]). These two estimates were set up to the linear parabolic equations (where ∂Ω is of class C 2 ), based on the Carleman inequality provided in [18]. In the case when D = ω × (0, T ) and J = Γ × (0, T ) 1991 Mathematics Subject Classification. Primary: 35B37.

This paper presents a new observability estimate for parabolic equations in × (0, T ), where is a convex domain. The observation region is restricted over a product set of an open nonempty subset of and a subset of positive measure in (0, T ). This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.

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We prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the euclidean space with homogeneous Dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves. We consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time. As a consequence of this equivalence, using the well-known variational characterization of minimal norm controls, we establish necessary and sufficient conditions for the minimal time and the corresponding control.

This paper studies connections among observable sets, the observability inequality, the Hölder-type interpolation inequality and the spectral inequality for the heat equation in R n . We present the characteristic of observable sets for the heat equation. In more detail, we show that a measurable set in R n satisfies the observability inequality if and only if it is γ-thick at scale L for some γ > 0 and L > 0. We also build up the equivalence among the abovementioned three inequalities. More precisely, we obtain that if a measurable set in R n satisfies one of these inequalities, then it satisfies others. Finally, we get some weak observability inequalities and weak interpolation inequalities where observations are made over a ball.