Controllability of One Spin and Two Interacting Spins

Abstract.

 We consider the problem of steering control for the systems of one spin ½ particle and two interacting spin ½ particles in an electro-magnetic field. The models described are bilinear systems whose state varies on the Lie group of special unitary matrices of dimensions two and four, respectively. By performing decompositions of Lie groups, taking into account the described equations at hand, we derive control laws to steer the state of the system to any desired final configuration. Explicit formulas are given for the parameters involved in the control algorithms. Moreover, the proposed algorithms allow for arbitrary bounds on the magnitude of the controls and for some flexibility in the specification of the final time which must be greater than a given value but otherwise arbitrary. The controllability of the two models is also analyzed.