In this paper we characterize a broad class of semilinear surjective operatorsgiven by the following formula ( ) H

As we have announced in the title of this work, we show that a broad class of linear evolution equations are exactly controllable. This class is represented by the following infinite dimensional linear control system:where Z, U are Hilbert spaces, the control function u belong to L 2 (0, t 1 ; U ), t 1 > 0, B ∈ L(U, Z), A generates a strongly continuous semigroup operator T (t) according to [5]. We give necessary and sufficient condition for the exact controllability of this system and apply this results to a linear controlled damped wave equation.