Small superconducting circuits containing Josephson junction have received a great deal of attention as promising candidates for scalable quantum bits. In this presentation, we will consider a current-biased dc SQUID in presence of an applied driving magnetic flux. The behavior of this Josephson device is analogous to a quantum particle in a 1−D anharmonic potential manipulated by a time-dependent external control field, i.e, driven multilevel quantum systems. The problem of finding the required time-dependent control field that yields the desired quantum evolution is formulated in the framework of optimal control theory. We first describe the sufficient conditions for complete controllability of N-level quantum systems subject to a single control field. Then, we calculate the optimal control that will steer the system from its initial state to a desired final state at specified time using optimal control theory. If an upper and lower bounds are imposed for the external control, the optimal solution is of bang-bang type and switches from the upper to the lower values of the control bounds. We will also present an analysis of the sensitivity of the optimal solution to external perturbations and fluctuations of the internal Hamiltonian. Since, it is of great practical importance to know how changes in the system parameters affect the optimal solution.

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