Singular optimal control for a transport-diffusion equation

In this paper we consider a transport-diffusion equation with coefficient of diffusion epsilon > 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 epsilon SE arrow 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 epsilon SE arrow 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 epsilon SE arrow 0(+), and moreover decays exponentially as epsilon SE arrow 0(+) as soon as all trajectories cross the boundary.