Attainable set for rank 3 step 2 free Carnot group with positive controls

TL;DR

This paper characterizes the attainable set for a rank 3, step 2 free Carnot group with positive controls, using Pontryagin's maximum principle to analyze extremal trajectories and boundary structure.

Contribution

It provides a detailed description of the boundary of the attainable set for this control system, including bounds on control switchings and a parametrization of boundary faces.

Findings

Bounded the number of switchings in optimal controls.

Parametrized the boundary faces of the attainable set.

Analyzed extremal trajectories for bang-bang, singular, and mixed controls.

Abstract

We find the attainable set for a control system on the free Carnot group of rank $3$ and step $2$ with positive controls. This kind of control systems is connected with the theory of free Lie semigroups; with some estimates for probabilities of inequalities for independent random variables; with the nilpotent approximation of robotic control systems and with contour recovering without cusps in image processing. We investigate the boundary of the attainable set with the help of the Pontryagin maximum principle for the time-optimal control problem. We study extremal trajectories that correspond to bang-bang, singular and mixed controls. We obtain upper bounds for the number of switchings for optimal controls. This implies a parametrization of the boundary faces of the attainable set.