  We investigate the time-optimal solution of the selective control of two
uncoupled spin 1/2 particles. Using the Pontryagin Maximum Principle, we derive
the global time-optimal pulses for two spins with different offsets. We show
that the Pontryagin Hamiltonian can be written as a one-dimensional effective
Hamiltonian. The optimal fields can be expressed analytically in terms of
elliptic integrals. The time-optimal control problem is solved for the
selective inversion and excitation processes. A bifurcation in the structure of
the control fields occurs for a specific offset threshold. In particular, we
show that for small offsets, the optimal solution is the concatenation of
regular and singular extremals.
