Singular Optimal Control for a Transport-Diffusion Equation

In this paper we consider a transport-diffusion equation with coefficient of diffusion ϵ > 0 small and coefficient of transport M(x, t). We study the asymptotic behavior of the cost of the null controllability of such a system when ϵ 0 + . If at least one trajectory associated to M(x, t) does not enter the control zone, we prove that this cost explodes exponentially as ϵ 0 + . On the other hand, as long as trajectories reach the control region and the controllability time is sufficiently large, we prove that the cost is bounded as ϵ 0 + , and moreover decays exponentially as ϵ 0 + as soon as all trajectories cross the boundary.