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

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 [1,23]/M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {1,2\sqrt 3 } \right]/M$$\end{document} for M > 0 and in the interval [22,2(1+3)]/|M|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ {2\sqrt 2 ,2\left( {1 + \sqrt 3 } \right)} \right]/|M|$$\end{document} for M< 0. The exact value of TM is however unknown.