Quantum conditions on the control of dynamics of a system coupled to an environment are obtained. Specifically, consider a system initially in a system subspace H(0) of dimensionality M(0), which evolves to populate system subspaces H(1), H(2) of dimensionalities M(1), M(2). Then, there always exists an initial state in H(0) that does not evolve into H(2) if M(0)>dM(2), where 2<or=d<or=(M(0)+M(1)+M(2))(2) is the number of operators in the Kraus representation. Note, significantly, that the maximum d can be far smaller than the dimension of the bath. If this condition is not satisfied, then dynamics from H(0) that avoids H(2) can only be attained physically under stringent conditions. An example from molecular dynamics and spectroscopy, i.e., donor to acceptor energy transfer, is provided.