Aspects of control theory on infinite-dimensional Lie groups and G-manifolds

TL;DR

This paper explores geometric control theory on potentially infinite-dimensional Lie groups and G-manifolds, focusing on differential equations, reachability, and the importance of regularity properties.

Contribution

It extends control theory to infinite-dimensional Lie groups and G-manifolds, analyzing differential equations and reachability within this broader geometric framework.

Findings

Existence and uniqueness of differential equations on G-manifolds with L^1 controls.

Characterization of closures of reachable sets in infinite-dimensional settings.

Role of regularity properties in control and reachability analysis.

Abstract

We develop aspects of geometric control theory on Lie groups G which may be infinite dimensional, and on smooth G-manifolds M modelled on locally convex spaces. As a tool, we discuss existence and uniqueness questions for differential equations on M given by time-dependent fundamental vector fields which are L^1 in time. We then discuss the closures of reachable sets in M for controls in the Lie algebra of G, or within a compact convex subset of the Lie algebra. Regularity properties of the Lie group G play an important role.