Exact controllability of semilinear evolution equation and applications

where Z, U are Hilbert spaces, A : D(A) ⊂ Z −→ Z is the infinitesimal generator of strongly continuous semigroup {T (t)}t≥0 inZ, B ∈ L(U,Z), the control function u belongs to L(0, τ ;U) and F : [0, τ ]× Z × U −→ Z is a suitable function. First, we give a necessary and sufficient condition for the exact controllability of the linear system z′ = Az + Bu(t); Second, under some conditions on F , we prove that the exact controllability of the linear system is preserved by the semilinear system, in this case the control u steering an initial state z0 to a final state z1 at time τ > 0 is given by the following formula: u(t) = B∗T ∗(τ − t)W−1 (I +K)−1(z1 − T (τ)z0), according to Theorem 3.1. Finally, these results can be applied to the controlled damped wave equation.