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A solution to the quantum Zermelo problem for control Hamiltonians withgeneral energy resource bounds is provided. Interestingly, the energy resourceof the control Hamiltonian and the control time define a pair of conjugatevariables that minimize the energy-time uncertainty relation. The resultingcontrol protocol is applied to a single qubit as well as to a two-interactingqubit system represented by a Heisenberg spin dimer. For these low-dimensionalsystems, it is found that physically realizable control Hamiltonians exist onlyfor certain, quantized, energy resources.

Quantum Zermelo problem for general energy resource bounds