Optimal Control of Two-level Quantum System Withweighted Energy Cost Functional

We have derived in this paper optimal control of quantum mechanical system with weighted energy cost function 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 state of the quantum mechanical system from 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 quantum spin of Pauli two-level system subjected to an external field with the minimum energy function is also illustrated and formulated in terms of the quantum spin up and spin down states of the Pauli two-level system.