Flows of vector fields and the Kalman Theorem

TL;DR

This paper presents two novel proofs of the Kalman Theorem using basic flow properties of vector fields, which can be extended to broader control theory contexts.

Contribution

It introduces alternative proofs of the Kalman Theorem based on flow properties, enabling generalizations to new controllability criteria for nonlinear systems.

Findings

Provided two new proofs of the Kalman Theorem

One proof has been generalized to derive new controllability criteria

Enhanced understanding of control system properties through flow-based methods

Abstract

We give two proofs of the Kalman Theorem, alternative to the most common ones, which infer such a classical result of Control Theory using just very basic facts on flows of vector fields. These proofs are apt to be generalised in diverse directions -- in fact one of them has been already generalised, yielding new criteria for local controllability of non-linear real analytic controlled systems.