Time-optimal decompositions in SU(2)

TL;DR

This paper investigates the problem of minimizing the total time for decomposing elements in the SU(2) group into products of exponentials of two generators, providing a complete solution for time-optimal decompositions.

Contribution

It provides a complete characterization of time-optimal decompositions in SU(2), a problem previously unresolved.

Findings

Derived explicit formulas for optimal decompositions

Identified the structure of minimal-time decompositions

Solved the time-optimal control problem for SU(2)

Abstract

A connected Lie group G is generated by its two 1-parametric subgroups exp(tX), exp(tY) if and only if the Lie algebra of G is generated by {X, Y}. We consider decompositions of elements of G into a product of such exponentials with times t > 0 and study the problem of minimizing the total time of the decompositions for a fixed element of G. We solve this problem for the group SU(2) and describe the structure of the time-optimal decompositions.