Equivalent Hamiltonians for State to State Transfer in the Case of   Partial Quantum Control
  Given a fixed initial state, a desired Hamiltonian, and an amount of time, we provide a complete characterization of the set of Hamiltonians which perform the same action as the desired Hamiltonian on the state of interest. An example is provided where the desired Hamiltonian lies outside of the control Lie algebra, but implementable Hamiltonians exist within the characterization. 
Ian N. Hincks, David G. Cory, and Chandrasekhar Ramanathan
quant-ph