On Attainable Set and Controllability for Abstract Control Problem with Unbounded Input Operator

For linear evolution control system described by $\dot{x}=Ax(t)+Bu(t),x(0)=x_0$ ($A$ generates a strongly continuous semigroup ${S(t)}_{t\ge 0}$ in a Banach space $X$; $B$ is a linear unbounded operator), the attainable set $K(t)$ is studied. Conditions of the independence of $t$ for its closure $\overline{K(t)}$ are established. Controllability conditions for some classes of evolution systems are obtained.