Final state observability in Banach spaces with applications to subordination and semigroups induced by Lévy processes

Dennis Gallaun; Jan Meichsner; Christian Seifert

doi:10.3934/eect.2023002

Article Contents

2023, Volume 12, Issue 4: 1102-1121. Doi: 10.3934/eect.2023002

This issue Previous Article Partial approximate controllability results for fractional order stochastic evolution equations using approximation method Next Article A new class of small initial data which may shift the critical power and lifespan estimates for the classical damped wave equations

Final state observability in Banach spaces with applications to subordination and semigroups induced by Lévy processes

1.
Technische Universität Hamburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, 21073 Hamburg, Germany
2.
FernUniversität in Hagen, Lehrgebiet Analysis, Fakultät Mathematik und Informatik, Universitätsstraße 47, 58084 Hagen, Germany
3.
Technische Universität Dresden, Arbeitsgruppe Astronomie, Lohrmann-Observatorium, August-Bebel-Straße 30, 01219 Dresden, Germany

^*Corresponding author: Christian Seifert
^*Corresponding author: Christian Seifert

Received: June 2022

Revised: January 2023

Early access: February 2023

Published: August 2023

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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.

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Mathematics Subject Classification: Primary: 47D06, 35Q93; Secondary: 47N70, 93D20, 93B05, 93B07.

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