# On Weak Observability For Evolution Systems with Skew-Adjoint Generators

## TL;DR

This paper investigates the weak observability and stability of inverse problems for evolution equations generated by skew-adjoint operators, introducing new resolvent inequalities and Fourier methods.

## Contribution

It provides new resolvent inequalities and analytical techniques for assessing weak observability in evolution systems with skew-adjoint generators.

## Key findings

- Established conditions for well-posedness of the inverse problem
- Derived a new resolvent inequality for skew-adjoint operators
- Linked stability estimates to weak observability inequalities

## Abstract

In the paper we consider the linear inverse problem that consists in recovering the initial state in a first order evolution equation generated by a skew-adjoint operator. We studied the well-posedness of the inversion in terms of the observation operator and the spectra of the skew-adjoint generator. The stability estimate of the inversion can also be seen as a weak observability inequality. The proof of the main results is based on a new resolvent inequality and Fourier transform techniques which are of interest themselves.