Controllability of a wave equation with a boundary obstacle

Date: Fri, Nov 13 2009

Hour: 12:30

Location: Bizkaia Technology Park, Building 500 E-48160 DERIO - Basque Country- Spain

Speakers: Sorin Micu

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 - yzz=0 in (t,x) ∈ (0,T) x (0,1) while the obstacle is represented by the Signorinis conditions y(t,1) ≥ ψ(t), yz(t,1) ≥ 0, yz(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).

Confirmed speakers:

Sorin Micu