We consider the optimal control problem of transferring population between states of a
quantum system where the coupling proceeds only via intermediate states that are subject
to decay. We pose the question whether it is generally possible to carry out this
transfer. For a single intermediate decaying state, we recover the Stimulated Raman
Adiabatic Passage (STIRAP) process for which we present analytic solutions in the finite
time case. The solutions yield perfect state transfer only in the limit of infinite
time. We also present analytic solutions for the case of transfer that has to proceed
via two consecutive intermediate decaying states. We show that in this case, for finite
power the optimal control does not approach perfect state transfer even in the infinite
time limit. Our four-level results agree with those of Khaneja et al. J. Magnet. Reson.
162, 311 (2003) derived in a different way. We generalize our findings to characterize
the topologies of paths that can be achieved by coherent control.