  One main goal of quantum control is to steer a quantum system toward an
expected state or dynamics. For measurement-induced quantum control,
measurements serve as the only control, which is like the cases in quantum Zeno
and anti-zeno effects. In this paper, this scenario will be investigated in a
general $N$-level quantum system. It is proved that, when the intial and
expected states are both pure, the control space could be reduced to an
effective subspace spanned by these two states only. This result will greatly
simplify the measurement-induced control strategy of an $N$-level quantum
system.
