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

TL;DR

This paper develops a generalized method for observability estimates in Banach spaces, combining uncertainty principles and dissipation estimates, with applications to semigroups from Lévy processes and subordination techniques.

Contribution

It introduces an iterative approach for sharp observability estimates that handle a broad class of growth and decay rates, extending previous results.

Findings

Derived explicit asymptotic observation constants.

Extended dissipation estimates to subordinated semigroups.

Applied results to semigroups associated with Lévy 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 L{\'e}vy processes.