An accessibility condition for discrete-time linear systems on Lie groups

TL;DR

This paper characterizes the accessibility of discrete-time linear control systems on Lie groups using a novel derivative concept and provides criteria for accessibility and local controllability.

Contribution

It introduces a new approach to assess accessibility on Lie groups and establishes conditions for controllability based on infinitesimal automorphisms.

Findings

Maximal dimension of the subalgebra implies system accessibility

Provides simple criteria for accessibility in discrete-time systems

Establishes a sufficient condition for local controllability at the identity

Abstract

In this paper, we characterize the accessibility of discrete-time linear control systems on Lie groups. Using an exceptional notion of derivative, we construct a subalgebra $\mathfrak{h}$ based on the infinitesimal automorphism of the system such that if its dimension is maximal, the system is accessible. Our criteria provide simple conditions in a general context for the discrete-time case. Additionally, we prove a sufficient condition for local controllability at the identity using the infinitesimal automorphism, akin to the ad-rank condition in the continuous case.