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The advection-diffusion equation yεt − εyεxx + Myεx = 0, (x, t) ∈ (0, 1) × (0, T) is null controllable for any strictly positive values of the diffusion coefficient ε and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient ε goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to ε if T is greater than a minimal time TM, with TM in the interval \(\left[ {1,2\sqrt 3 } \right]/M\) for M > 0 and in the interval \(\left[ {2\sqrt 2 ,2\left( {1 + \sqrt 3 } \right)} \right]/|M|\) for M< 0. The exact value of TM is however unknown.

On the Controllability of an Advection-diffusion Equation with Respect to the Diffusion Parameter: Asymptotic Analysis and Numerical Simulations