# Morphisms of tautological control systems

## TL;DR

This paper explores the relationships and conditions under which morphisms between tautological control systems preserve reachability, with applications to second-order control systems, advancing the understanding of system mappings in control theory.

## Contribution

It introduces conditions for the existence of morphisms between tautological control systems and relates these to reachability and lifting properties, with specific results for second-order systems.

## Key findings

- Sufficient conditions for reachability transfer between systems.
- Establishment of a correspondence between lifting control systems and morphisms.
- Application of results to second-order control systems.

## Abstract

In this paper, we investigate morphisms of tautological control systems. Given a tautological control system $\mathfrak{H}$ on the manifold N and a mapping $\Phi: M \to N$, we study existence of tautological control system $\mathfrak{G}$ on the manifold $M$ such that there exists a trajectory-preserving morphism $(\Phi, \Phi^ #)$ from $\mathfrak{G}$ to $\mathfrak{H}$. Sufficient conditions are given such that reachability of $\mathfrak{H}$ implies the reachability of $\mathfrak{G}$. Correspondence between the notion of lifting ordinary control systems and morphisms of tautological control systems are examined. We give an application of the above results to the class of second-order type control systems, where the special structure of second-order type leads to additional results.