Controllability of a wave equation with a boundary obstacle

On the controllability property of a homogeneous linear string of length one submitted to a lower time dependant obstacle at the extremity x = 1, described by the function {ψ(t)} 0 ≤ t ≤ T. The Dirichlet control acts on the other extremity x = 0. The string is modelled by the wave equation yⁿ - y_zz=0 in (t,x) ∈ (0,T) x (0,1) while the obstacle is represented by the Signorinis conditions y(t,1) ≥ ψ(t), y_z(t,1) ≥ 0, y_z(t,1) (y(t,1) - ψ(t)) = 0 in (0,T).

When introduce a penalised system in y_{∈y transforming the Signorinis condition into the simpler one} y_∈x(t,1) = ∈^-1[y_∈(t,1) - ψ(t)], ∈ being a small positive parameter. We construct explicitly a family if controls of the penalised problem, (u_∈)_∈>0, which are uniformly bounded tithe respect to ∈: in H¹(0,T) and we pass to the limit obtaining control for the initial equation. Thus, we prove that for any T>2 and initial data (y⁰,y¹) from a subset of H¹(0,T) x L²(0,1), the initial system is null controllable with controls in H¹(0,T).