  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 which we
identify as the global optimum in the limit of infinite control 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. We generalize our findings to characterize the topologies
of paths that can be achieved by coherent control under the assumption of
finite power.
