  We consider the optimal control problem in a two-qubit system with bounded
amplitude. Two cases are studied: quantum state preparation and entanglement
creation. Cost functions, fidelity and concurrence, are optimized over bang-off
controls for different values of the total duration, respectively. For quantum
state preparation problem, three critical time points are determined with high
precision, and optimal controls are obtained for different durations. A better
estimation of the quantum speed limit is obtained, so is the time-optimal
control. For entanglement creation problem, two critical time points are
determined, one of them is the minimal time to achieve maximal entanglement
(unit concurrence) starting from the product state. In addition, the optimal
control to reach the unit concurrence is found.
