Observability inequalities and measurable sets

Abstract

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.

Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

“…It was first observed in that the null controllability from measurable subsets implies the bang–bang property of time optimal controls. Then, by Corollary and the almost same argument in , one can obtain the following result.…”

“…Next, we shall make use of Propositions and , as well as Lemmas and , to establish an interpolation inequality from measurable sets for any solution u to Equation . For similar results, we refer the reader to .…”

“…However, our interest here is to study whether or not any norm optimal control of ( N P ) M is of the bang–bang type, and whether the norm optimal control is unique. By Corollary and the almost same argument in or , we have the following result:…”

“…For the subject of the observation being on a measurable subset of positive measure, the observability inequality has been studied in recent works [12][13][14][15]. It also has independent interest in control theory.…”

On a kind of time optimal control problem of the heat equation

“…For some other interesting work, we refer the reader to [22][23][24]. The approximate controllability of system (1.1) has been studied in much work (see, e.g., [14,[25][26][27][28]). It is clear that, for each ε > 0, we have y(T; y 0 , 0) ≤ ε when T is large enough.…”