A necessary and sufficient condition for the controllability of linear systems in Hilbert spaces and applications

As we have announced in the title of this work, we show that a broad class of linear evolution equations is exactly controllable. This class is represented by the following infinite-dimensional linear control system: $$\dot{z} = \mathcal{A}z + \mathcal{B}u(t),\quad t > 0,\ z \in Z, u(t) \in U,$$where Z, U are Hilbert spaces, the control function u belong to L²(0, t1 ;U), t1 > 0, $\mathcal{B}\in L(U,Z)$ and $\mathcal{A}$ generates a strongly continuous semigroup operator T(t) according to Pazy. We give a necessary and sufficient condition for the exact controllability of this system and apply this result to a linear controlled damped wave equation.