Final State Observability in Banach spaces with applications to Subordination and Semigroups induced by L{é}vy processes

Abstract

This paper generalizes the abstract method of proving an observability estimate by combining an uncertainty principle and a dissipation estimate. In these estimates we allow for a large class of growth/decay rates satisfying an integrability condition. In contrast to previous results, we use an iterative argument which enables us to give an asymptotically sharp estimate for the observation constant and which is explicit in the model parameters. We give two types of applications where the extension of the growth/decay rates naturally appear. By exploiting subordination techniques we show how the dissipation estimate of a semigroup transfers to subordinated semigroups. Furthermore, we apply our results to semigroups related to Lévy processes.

Final state observability estimates and cost-uniform approximate null-controllability for bi-continuous semigroups

“…Final state observability estimates have been studied in various contexts due to its relation to null-controllability, see e.g. [1,3,8,9,17,20,22,24,28] and references therein. Classically, the space Z is some L r -space with r ∈ [1, ∞] (when working in Hilbert spaces, one usually chooses r = 2), and then the final state observability estimate yields the form…”