Optimal control of a spin system with weighted energy cost functional

Optimal control of quantum mechanical system with weighted energy cost function has been derived by representing the unitary operator in terms of the projection operators of the Hamiltonian of the control system. The admissible Hilbert space of controllers of the system is expressed as the direct sum of the Hilbert spaces corresponding to the weights of the controllers of the quantum mechanical system. The optimal control which steers the initial state to a target state, minimizing the weighted energy, is formulated in terms of the controllability operator of the system. As an example, the weighted optimal control problem of the time evolution of a quantum spin system subjected to an external field with the minimum energy function is formulated in terms of the quantum spin up and spin down states of the Pauli two-level system.