Necessary condition of linear distributed parameter systems with exact controllability

In this paper, we studied the necessary condition of distributed parameter system with exact controllability. Let Φ0t be the control mapping. We introduce new a class of control operators that is called the I-class control which satisfy R(Φ0t)∩R(T(t)) is closed set for t>0. If the system is exactly controllable in finite time τ, then the semigroup T(t) must have closed range. In particular, if the generator of T(t) has compact resolvent, its spectra distribute in a strip parallel to imaginary axis. As an application we assert that the systems associated with the immediately norm-continuous semigroups are never exact controllable for zero-class or I-class controls.