Control of one and two homonuclear spins

Abstract

We consider the problem of steering control for the systems of one spin 1/2 particle and two interacting homonuclear spin 1/2 particles in an electro-magnetic field. The describing models 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 describing 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.