Trajectory-constrained optimal local time-continuous waveform controls for state transitions in $N$-level quantum systems

Based on a parametrization of pure quantum states we explicitly construct a sequence of (at most) $4N-5$ local time-continuous waveform controls to achieve a specified state transition for $N$-level quantum systems when sufficient controls of the Hamiltonian are available. The control magnitudes are further optimized in terms of a time-energy performance, which is a generalization of the time performance index. Trajectory-constrained optimal local time-continuous waveform controls, including both local sine-waveforms and $n^{\rm th}$-order-polynomial waveform controls are obtained in terms of time-energy performance. It is demonstrated that constrained optimal local $n^{\rm th}$-order-polynomial waveform controls approach constrained optimal bang-bang controls when $n\rightarrow\infty$.