A Weak Notion of Symmetry for Dynamical Systems

TL;DR

This paper introduces the concept of weak invariance in dynamical systems, a flexible symmetry notion that generalizes classical invariance and enables new decomposition methods for control and observation.

Contribution

It proposes weak invariance as a middle ground between classical symmetry and partial symmetry, establishing its properties and implications for system decomposition.

Findings

Weak invariance implies the associated diffeomorphism is an automorphism.

A vector field is weakly invariant iff its flow is weakly invariant.

Weakly invariant systems admit a cascade decomposition with group affine dynamics.

Abstract

Many nonlinear dynamical systems exhibit symmetry, affording substantial benefits for control design, observer architecture, and data-driven control. While the classical notion of group invariance enables a cascade decomposition of the system into highly structured subsystems, it demands very rigid structure in the original system. Conversely, much more general notions (e.g., partial symmetry) have been shown to be sufficient for obtaining less-structured decompositions. In this work, we propose a middle ground termed "weak invariance", studying diffeomorphisms (resp., vector fields) that are group invariant up to a diffeomorphism of (resp., vector field on) the symmetry group. Remarkably, we prove that weak invariance implies that this diffeomorphism of (resp., vector field on) the symmetry group must be an automorphism (resp., group linear). Additionally, we demonstrate that a vector field is weakly invariant if and only if its flow is weakly invariant, where the associated group linear vector field generates the associated automorphisms. Finally, we show that weakly invariant systems admit a cascade decomposition in which the dynamics are group affine along the orbits. Weak invariance thus generalizes both classical invariance and the important class of group affine dynamical systems on Lie groups, laying a foundation for new methods of symmetry-informed control and observer design.