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On exact Observability for Compactly perturbed infinite dimension system

Nisrine Charaf, Faouzi Triki

TL;DR

The paper addresses exact observability for infinite-dimensional systems when the generator is compactly perturbed by a self-adjoint operator K. It proves an asymptotic relation ẇμ_n = μ_n(1 + f(μ_n)) with lim_{μ→∞} f(μ) = 0 between the perturbed and unperturbed eigenvalues and establishes sufficient conditions, including a growth condition on f and compactness of the commutator AK − KA, under which exact observability is preserved for the perturbed system. A weak necessary condition on the perturbed eigenfunctions is discussed, highlighting low-frequency behavior. The analysis combines min-max spectral characterizations, resolvent estimates, and Riesz-projection techniques to transfer observability from the unperturbed to the perturbed system, with implications for stability of inverse problems in infinite dimensions.

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.

On exact Observability for Compactly perturbed infinite dimension system

TL;DR

The paper addresses exact observability for infinite-dimensional systems when the generator is compactly perturbed by a self-adjoint operator K. It proves an asymptotic relation ẇμ_n = μ_n(1 + f(μ_n)) with lim_{μ→∞} f(μ) = 0 between the perturbed and unperturbed eigenvalues and establishes sufficient conditions, including a growth condition on f and compactness of the commutator AK − KA, under which exact observability is preserved for the perturbed system. A weak necessary condition on the perturbed eigenfunctions is discussed, highlighting low-frequency behavior. The analysis combines min-max spectral characterizations, resolvent estimates, and Riesz-projection techniques to transfer observability from the unperturbed to the perturbed system, with implications for stability of inverse problems in infinite dimensions.

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.
Paper Structure (5 sections, 8 theorems, 82 equations)

This paper contains 5 sections, 8 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Characterization of exact Observability
  3. Main results
  4. Proof of Theorem \ref{['eigenvalues']}
  5. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 2.1

Let $A$ be a self-adjoint, positive with compact resolvent operator, and let $C$ be an admissible operator for the system observability1. Assume the following gap condition holds for some constant $\gamma >0$. Then, the system observability1 is exactly observable if and only if there exists $\delta$$>$ 0 such that for all $k$$\in$$\mathbb N^{*}$

Theorems & Definitions (17)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • proof
  • Proposition 4.1
  • proof
  • ...and 7 more