Outer invariance entropy for discrete-time linear systems on Lie groups

Fritz Colonius,Alexandre J Santana,João A.N Cossich

doi:10.1051/cocv/2021054

Abstract

We introduce discrete-time linear control systems on connected Lie groups and present an upper bound for the outer invariance entropy of admissible pairs (K, Q). If the stable subgroup of the uncontrolled system is closed and K has positive measure for a left invariant Haar measure, the upper bound coincides with the outer invariance entropy.

Highlights

In 2009, Colonius and Kawan [6] introduced the theory of invariance entropy for control systems
Invariance entropy for continuous-time linear systems on Lie groups has been analyzed by Ayala et al [2], cf. the references therein for the theory of continuous-time linear control systems on Lie groups
The present paper introduces linear control systems on connected Lie groups and starts the investigation of invariance entropy

Summary

Introduction

In 2009, Colonius and Kawan [6] introduced the theory of invariance entropy for control systems. The present paper introduces linear control systems on connected Lie groups and starts the investigation of invariance entropy. The upper bound coincides with the outer invariance entropy, if the stable subgroup of the uncontrolled system (cf the definition after formula (2.2)) is closed and K has positive measure for a left invariant Haar measure μ on G This bears some similarity to the characterization of outer invariance entropy in the continuous-time case, cf Da Silva [7].

Discrete-time linear control systems on Lie groups

Linear systems induced on homogeneous spaces

Outer invariance entropy

Closedness of the stable subgroup