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

The advection-diffusion equation y(t)(epsilon) - epsilon y(xx)(epsilon) + My(x)(epsilon) = 0, (x, t) (0, 1) x (0, T) is null controllable for any strictly positive values of the diffusion coefficient epsilon and of the controllability time T. We discuss here the behavior of the cost of control when the coefficient epsilon goes to zero, according to the values of T. It is actually known that this cost is uniformly bounded with respect to epsilon if T is greater than a minimal time T-M, with T-M in the interval [1,2]/M for M > 0 and in the interval [22,2(1+3)]/|M| for M< 0. The exact value of T-M is however unknown.We investigate in this work the determination of the minimal time T-M employing two distincts but complementary approaches. In a first one, we numerically estimate the cost of controllability, reformulated as the solution of a generalized eigenvalue problem for the underlying control operator, with respect to the parameter T and epsilon. This allows notably to exhibit the structure of initial data leading to large costs of control. At the practical level, this evaluation requires the non trivial and challenging approximation of null controls for the advection-diffusion equation. In the second approach, we perform an asymptotic analysis, with respect to the parameter epsilon, of the optimality system associated to the control of minimal L-2-norm. The matched asymptotic expansion method is used to describe the multiple boundary layers.