Minimum Time Optimal Synthesis for a Control System on SU(2)
(2014)cite arxiv:1407.7491.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.
Description
Minimum Time Optimal Synthesis for a Control System on SU(2)