Detectability via observability in a nonuniform framework: dual relationship with controllability and stabilizability

TL;DR

This paper introduces a new nonuniform observability concept for finite-dimensional nonautonomous linear control systems, establishing its duality with controllability and its implications for exponential detectability.

Contribution

It proposes the nonuniform complete observability property, generalizing existing notions, and links it to controllability and stability in nonautonomous systems.

Findings

Nonuniform complete observability is more general than uniform observability.

A dual relationship between observability and controllability is established.

Nonuniform complete observability guarantees nonuniform exponential detectability.

Abstract

In this paper we propose a new observability property for nonautonomous linear control systems in finite dimension: the nonuniform complete observability, which is more general than the uniform complete observability. A dual relationship is established between this new notion of observability and the recently defined nonuniform complete controllability property, with the aim of obtaining the main result, which proves that nonuniform complete observability guarantees the nonuniform exponential detectability, a concept related to exponential stability in this framework.