Observation from measurable sets for parabolic analytic evolutions and applications

Abstract

We find new quantitative estimates on the space-time analyticity of solutions to linear parabolic equations with time-independent coefficients and apply them to obtain observability inequalities for its solutions over measurable sets.1991 Mathematics Subject Classification. Primary: 35B37.Theorem 2. Assume that J ⊂ ∂Ω × (0, T ) is a measurable set with positive surface measure in ∂Ω × (0, T ). Then, there is N = N (Ω, J, T, δ) such that the inequality

Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn

“…Arbitrarily fix t > 0. By applying Corollary 3.1 (where f (·) = u(t, ·) and a = 2t), we see that the radius of analyticity of u(t, ·) (which is treated as a function of x) is independent of t. It is an analogy result for solutions of the heat equation in a bounded domain with an analytic boundary (see [2,17]). This property plays a very important role in the proof of the observability estimates from measurable sets when using the telescope series method developed in [2,17].…”

“…The inequality (1.5) has been widely studied. See [19,29,34] for the case where ω is open; [1,2,17] for the case when ω is measurable.…”

Quantitative unique continuation for the heat equation with Coulomb potentials

“…Later on, it has been extended in [25] to the case of bounded domain with a C 2 -smooth boundary (see also [2,23]). Using sharp analyticity estimates for solutions to general parabolic equations or systems with analytic coefficients, such a kind of quantitative estimate have been established in a series of recent works [1,6,7].…”

“…As addressed in the introduction, such a kind of quantitative estimate of unique continuation has been proved to be applicable in the subject of control theory in recent years (see e.g. [1,6,7,23,24,29,30,31,32,34]). In particular, it can be used to establish the null controllability from measurable subsets of positive measure, and to obtain the bang-bang property of time and norm optimal control problems for the heat equation with singular Coulomb potentials.…”

Observability Inequalities from Measurable Sets for Some Abstract Evolution Equations

“…When the target z 1 is replaced by a ball in X, the bang-bang property follows from Pontryagin's maximum principle and the unique continuation property of adjoint equations. With respect to studies on the bang-bang property, we refer the readers to [2,8,10,17,19,23,29,30,37] (where the target is allowed to be a single point in the state space) and [15,16,38] (where the target is a ball in the state space).…”