Observability Inequalities and Measurable Sets

Observation from measurable sets for parabolic analytic evolutions and applications

“…Lemmas 3 and 5 imply in a similar way to the reasonings in [3,Theorem 11] the following result. Lemma 6.…”

“…We explain the technical details for parabolic higher order equations with constant coefficients and for second order systems with time independent analytic coefficients. We believe that this set of examples will make it clear to the experts that the combination of the methods in [35], [2], [32] with others here imply analog results to those in [3] for parabolic evolutions associated to possibly non self-adjoint higher order elliptic equations or second order systems with time independent analytic coefficients over analytic domains: existence of bounded null-controls acting over measurable sets and the uniqueness and bang-bang property of certain optimal controls.…”

“…Mixing up ideas developed in [35], [2] and [32], it was shown in [3] that the heat equation over bounded domains Ω in R n can be null controlled at all times T > 0 with interior and bounded controls acting over space-time measurable sets D ⊂ Ω×(0, T ) with positive Lebesgue measure, when Ω is a Lipchitz polyhedron or a C 1 domain in R n . [3] also established the boundary null-controllability with bounded controls over measurable sets J ⊂ ∂Ω × (0, T ) with positive surface measure.…”

“…In this work we explain the techniques necessary to apply the same methods in [3] in order to obtain the interior and boundary null controllability of some higher order or non self-adjoint parabolic evolutions with time-independent analytic coefficients over analytic domains Ω of R n and with bounded controls acting over measurable sets with positive measure. We also show the null-controllability with controls acting over possibly different measurable regions over each component of the Dirichlet data of higher order parabolic equations or over each component of the solution to second order parabolic systems; both at the interior and at the boundary.…”

“…The method we use relies on the telescoping series method -built with ideas borrowed from [24] and first used in [32] -and on local observability inequalities for analytic functions over measurable sets: the Lemma 1 as in [2,3] and a new extension of Lemma 1, the Lemma 2 below. We use Lemma 2 in the proof of Theorem 8.…”