Geometry of Optimal Control for Control-Affine Systems

TL;DR

This paper explores the geometric structure of control-affine systems in low dimensions, computing invariants and analyzing geodesic trajectories to understand their complex behaviors.

Contribution

It provides a detailed geometric analysis of control-affine systems with metric structures in 2D and 3D, including invariants and geodesic solutions.

Findings

Computed local isometric invariants for systems with 2 and 3 states.

Derived geodesic trajectories for homogeneous control-affine systems.

Revealed rich and varied behaviors even in low-dimensional cases.

Abstract

Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagin's maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied.