On exact Observability for Compactly perturbed infinite dimension system

TL;DR

This paper investigates conditions under which the exact observability of an infinite-dimensional system with a self-adjoint generator is preserved after a compact self-adjoint perturbation, using spectral analysis techniques.

Contribution

It provides new sufficient conditions ensuring the preservation of exact observability under compact perturbations of infinite-dimensional systems.

Findings

Derived spectral estimates for perturbed operators

Established conditions for observability preservation

Enhanced understanding of spectral effects on system observability

Abstract

In this paper, we study the observability of compactly perturbed infinite dimensional systems. Assuming that a given infinite-dimensional system with self-adjoint generator is exactly observable we derive sufficient conditions on a compact self adjoint perturbation to guarantee that the perturbed system stays exactly observable. The analysis is based on a careful asymptotic estimation of the spectral elements of the perturbed unbounded operator in terms of the compact perturbation. These intermediate results are of importance themselves.