We study about the approximate controllability of an abstract semilinear control system. By introducing a new sufficient condition, we proved the approximate controllability of the system.

By the intuitive idea of approximate an infinite dimensional manifold by finite dimensional manifolds, concepts of projected subsystems and project limit controllability are proposed.

In this paper, the approximate perturbation range of the state matrix of a linear time-invariant system is given, keeping the system controllability invariant.