Minimal-time trajectories of a linear control system on a homogeneous space of the 2D Lie group

TL;DR

This paper solves a minimal-time control problem for a linear system on a 2D Lie group homogeneous space, providing explicit solutions and revealing points with multiple optimal trajectories.

Contribution

It explicitly characterizes optimal trajectories and minimal times for a linear control system on a cylinder modeled as a 2D Lie group homogeneous space.

Findings

Existence of optimal trajectories between any two states.

Explicit calculation method for minimal time.

Presence of points with two distinct minimal-time trajectories.

Abstract

Through the Pontryagin maximum principle, we solve a minimal-time problem for a linear control system on a cylinder, considered as a homogeneous space of the solvable Lie group of dimension two. The main result explicitly shows the existence of an optimal trajectory connecting every couple of arbitrary states on the manifold. It also gives a way to calculate the corresponding minimal time. Finally, the system admits points with two distinct minimal-time trajectories connecting them.