There is the concept of controllability of a quantum system, i.e. do the given set of controls permit you to create any state or unitary? Usually this is computed by looking at the Lie Algebra of the system, and can be quite messy; you need to take the individual Hamiltonian terms that you can control, and calculate all their commutators to arbitrary orders. If you can take linear combinations of those and make any arbitrary Hamiltonian, then your full Hilbert space is controllable; you can make any unitary you want, and any quantum state is said to be reachable from any other. See here for an example.

One important point to emphasise, however, is that this tells you nothing about efficiency, i.e. how long it takes you to reach a given state (as a function of the system size). There is an explicit construction that you can make based on the above decomposition in terms of commutators, but the time required scales exponentially in the order of the commutator required. Numerical techniques of control theory are methods that attempt to find the required control fields (as a function of time) in a more efficient manner, but (to my knowledge) rarely give you any guarantees. So, if you have fixed $\Theta$ and bounded $c_k(t)$, the controllability concept may not be sufficient.

There is the concept of controllability of a quantum system, i.e. do the given set of controls permit you to create any state or unitary? Usually this is computed by looking at the Lie Algebra of the system, and can be quite messy; you need to take the individual Hamiltonian terms that you can control, and calculate all their commutators to arbitrary orders. If you can take linear combinations of those and make any arbitrary Hamiltonian, then your full Hilbert space is controllable; you can make any unitary you want, and any quantum state is said to be reachable from any other. See Complete controllability of quantum systems (PRA 2001) for an example.

One important point to emphasise, however, is that this tells you nothing about efficiency, i.e. how long it takes you to reach a given state (as a function of the system size). There is an explicit construction that you can make based on the above decomposition in terms of commutators, but the time required scales exponentially in the order of the commutator required. Numerical techniques of control theory are methods that attempt to find the required control fields (as a function of time) in a more efficient manner, but (to my knowledge) rarely give you any guarantees. So, if you have fixed $\Theta$ and bounded $c_k(t)$, the controllability concept may not be sufficient.