A necessary condition for dynamic equivalence

TL;DR

This paper establishes that two dynamically equivalent control systems are either statically equivalent or both ruled, with implications for understanding the structure and classification of control systems.

Contribution

It proves a necessary condition for dynamic equivalence, linking it to static equivalence or the ruled property, extending previous results on differentially flat systems.

Findings

Dynamically equivalent systems are either statically equivalent or both ruled.

Higher-dimensional systems in a dynamic equivalence must be ruled.

Differentially flat systems are a special case of ruled systems.

Abstract

If two control systems on manifolds of the same dimension are dynamic equivalent, we prove that either they are static equivalent --i.e. equivalent via a classical diffeomorphism-- or they are both ruled; for systems of different dimensions, the one of higher dimension must ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. Dynamic equivalence is also known as equivalence by endogenous dynamic feedback, or by a Lie-B\"acklund transformation when control systems are viewed as underdetermined systems of ordinary differential equations; it is very close to absolute equivalence for Pfaffian systems. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is assumed to be "trivial" (or linear controllable).