Singularities can arise in the external field obtained by tracking control of quantum mechanical systems. Whether or not the trajectory is disturbed by the presence of a singular point is shown to mainly depend on the average momentum along the trajectory at the moment of passing the singular point. If the singularity occurs on a turning point, the tracking will be quite unstable since the direction taken by the trajectory is very sensitive to field errors. The theoretical analysis of these situations yields detailed conclusions about the impact of field singularities in quantum tracking control. A rank index is defined to characterize nontrivial singularities, and the rank is shown to play an important role in determining the tracking quality while passing over a singular turning point where the field has a unique solution. A special class of nontrivial singularities is identified by the ability to remove the singularily under a proper limiting process. These insights into the nature and influence of singularities in tracking control of quantum systems are beneficial for developing numerical schemes and for designing controls.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)
- Physical and Theoretical Chemistry