An important problem in coherent spectroscopy and quantum information science is to find limits on how close an open quantum dynamical system can be driven to a target state in the presence of dissipation and decoherence. What is the optimal excitation that achieves this objective? We describe these problems in the context of the design of multidimensional NMR experiments that maximize the efficiency of transfer of coherence between coupled spins in the presence of decoherence with the goal of optimizing the sensitivity of these experiments. We present some new mathematical techniques for computing limits on how much coherence or polarization can be transferred between coupled spins in multiple spin topologies.