# Global Stability of a Class of Difference Equations on Solvable Lie Algebras

## TL;DR

This paper investigates the global stability of a class of difference equations derived from sampled-data flows on matrix Lie groups, establishing conditions for stability based on linearization and algebraic invariants.

## Contribution

It introduces a novel stability analysis framework for difference equations on solvable Lie algebras, leveraging properties of the algebraic structure and linearization.

## Key findings

- Global asymptotic stability under certain conditions
- Semiglobal exponential stability for nilpotent Lie algebras
- Stability determined by Jacobian linearization at the origin

## Abstract

Motivated by the ubiquitous sampled-data setup in applied control, we examine the stability of a class of difference equations that arises by sampling a right- or left-invariant flow on a matrix Lie group. The map defining such a difference equation has three key properties that facilitate our analysis: 1) its power series expansion enjoys a type of strong convergence; 2) the origin is an equilibrium; 3) the algebraic ideals enumerated in the lower central series of the Lie algebra are dynamically invariant. We show that certain global stability properties are implied by stability of the Jacobian linearization of dynamics at the origin. In particular global asymptotic stability. If the Lie algebra is nilpotent, then the origin enjoys semiglobal exponential stability.