A Time-Periodic Lyapunov Approach for Motion Planning of Controllable Driftless Systems on SU(n)

TL;DR

This paper introduces a Lyapunov-based control method for motion planning of controllable driftless systems on SU(n), ensuring global convergence to a periodic trajectory using a time-periodic reference and numerical control synthesis.

Contribution

It develops a novel control approach leveraging a periodic Lyapunov function and return method for SU(n) systems, enabling global asymptotic convergence.

Findings

Successfully applied to generate C--NOT quantum gate for n=4

Proved convergence using periodic LaSalle invariance principle

Demonstrated effectiveness through simulations

Abstract

For a right-invariant and controllable driftless system on SU(n), we consider a time-periodic reference trajectory along which the linearized control system generates su(n): such trajectories always exist and constitute the basic ingredient of Coron's Return Method. The open-loop controls that we propose, which rely on a left-invariant tracking error dynamics and on a fidelity-like Lyapunov function, are determined from a finite number of left-translations of the tracking error and they assure global asymptotic convergence towards the periodic reference trajectory. The role of these translations is to avoid being trapped in the critical region of this Lyapunov-like function. The convergence proof relies on a periodic version of LaSalle's invariance principle and the control values are determined by numerical integration of the dynamics of the system. Simulations illustrate the obtained controls for $n=4$ and the generation of the C--NOT quantum gate.