Approximate controllability of semilinear impulsive strongly damped wave equation

Rothe’s fixed-point theorem is applied to prove the interior approximate controllability of a semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in the space Z1/2 = D((-Δ)1/2) × L2(Ω), where Ω is a bounded domain in Rn (n ≥ 1). Under some conditions we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state z0 to a neighborhood of the final state z1 at time τ > 0.