Null controllability of the 1-d heat equation as limit of the controllability of dissipative wave equations

Abstract

We consider the 1 - d wave equation epsilon u(tt) - u(tt) + u(t) = 0 With Dirichlet boundary conditions in a bounded interval. It is well known that for any epsilon > 0 this system is exactly controllable, with controls distributed on some open subinterval. It is also well known that the heat equation u(t) - u(xx) = 0 is null controllable. In this Note we prove that the null controls for the dissipative wave equations can be built so that they converge to a null control for the heat equation as epsilon --> 0. This shows the continuity of the null controls with respect to the singular perturbation under consideration