A complex-analytic approach to the problem of uniform controllability of a transport equation in the vanishing viscosity limit

We revisit a result by Coron and Guerrero stating that the one-dimensional transport-diffusion equationut+Mux−εuxx=0in (0,T)×(0,L), controlled by the left Dirichlet boundary value is zero-controllable at a bounded cost as ε→0+ε→0+, when T>4.3L/MT>4.3L/M if M>0M>0 and when T>57.2L/|M|T>57.2L/|M| if M<0M<0. By a completely different method, relying on complex analysis, we prove that this still holds when T>4.2L/MT>4.2L/M if M>0M>0 and when T>6.1L/|M|T>6.1L/|M| if M<0M<0.