Local Nash Realizations

TL;DR

This paper studies the realization theory of Nash systems, a class of non-linear systems with semi-algebraic analytic functions, providing local minimality characterizations and isomorphism conditions.

Contribution

It offers a local characterization of minimality and isomorphism for Nash systems, advancing understanding of their realization theory.

Findings

Minimal Nash systems are characterized by observability and reachability.

Minimal Nash systems are isomorphic when they satisfy certain conditions.

Results are currently local but may be extendable globally.

Abstract

In this paper we investigate realization theory of a class of non-linear systems, called Nash systems. Nash systems are non-linear systems whose vector fields and readout maps are analytic semi-algebraic functions. In this paper we will present a characterization of minimality in terms of observability and reachability and show that minimal Nash systems are isomorphic. The results are local in nature, i.e. they hold only for small time intervals. The hope is that the presented results can be extended to hold globally.