Null boundary controllability of a circular elastic arch

We consider a circular arch of thickness ε and curvature r−1 whose elastic deformations are described by a 2×2 system of linear partial differential equation. The system of the type y′′ + Ay = 0,y = (y1, y3) involves the tangential y1 and normal y3 component of the arch displacement. We analyze in this work the null controllability of these two components by acting on the boundary through a Dirichlet and a Neumann control simultaneously. Using the multiplier technic we show that, for any ε > 0 fixed, the arch may be exactly controlled provided that the curvature be small enough. Then, we consider the numerical approximation of the controllability problem, using a C − C finite element method and analyze some experiments with respect to the curvature and the thickness of the arch. We also highlight and discuss numerically the loss of uniform controllability as the thickness ε goes to zero, due to apparition of an essential spectrum for the limit operator A.