Minimum Time Optimal Synthesis for a Control System on SU(2)

TL;DR

This paper derives explicit solutions for minimum time control of a quantum system modeled on SU(2), providing a comprehensive method to determine optimal trajectories and control strategies for desired state evolutions.

Contribution

It explicitly solves the time optimal control problem on SU(2), describing all optimal trajectories and a simple method for minimum time control for any final state.

Findings

Optimal trajectories can be characterized in the complex unit disk.

A circular separatrix divides different classes of optimal trajectories.

A critical trajectory intersects all optimal paths, guiding control design.

Abstract

For the time optimal control on an invariant system on SU(2), with two independent controls and a bound on the norm of the control, the extremals of the maximum principle are explicit functions of time and the resulting differential equations can be explicitly integrated. We use this fact here to perform the optimal synthesis for these systems, i.e., find all optimal trajectories. As a consequence, we describe a simple method to find the minimum time control for every desired final condition. Although the Lie group SU(2) is three dimensional, optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. Our results are of interest to find the minimum time needed to achieve a given evolution of a two level quantum system.