Controllability and observabiliy of an artificial advection-diffusion problem

In this paper we study the controllability of an artificial advection-diffusion system through the boundary. Suitable Carleman estimates give us the observability on the adjoint system in the one dimensional case. We also study some basic properties of our problem such as backward uniqueness and we get an intuitive result on the control cost for vanishing viscosity. Introduction Artificial advection-diffusion problem In the present paper we deal with some advection-diffusion problem with small viscosity truncated in one space direction. Our interest for the linear advection diffusion equation comes from the Navier-Stokes equation, but it arises also in other fields as, for example, meteorology. For a given viscosity ε > 0, the incompressible Navier-Stokes equation can be written as { ft + (f.∇)f − ε∆f +∇p = 0 div(f) = 0 where f is the velocity vector field, p the pressure, ∇ the gradient and ∆ the usual Laplacian. Considering the flow around a body, we have that f is almost constant far away from the body and equal to a (see [10]). Linearizing the equation, we get the following equation for the vorticity ut + a.∇u− ε∆u = 0. In the sequel, we assume for simplicity that a is the nth unit vector of the canonical basis of R. When one computes the solution of this problem, one can only solve numerically this problem on a bounded domain. A good way to approximate the solution on the whole space may be given by the use of artificial boundary conditions (see [6]). For any T > 0, we hence consider the following control problem on Ω = R × (−L, 0) (L some positive constant) (Sv)    ut + ∂nu− ε∆u = 0 ε(ut + ∂νu) = v ε(ut + ∂νu) + u = 0 u(0, .) = u0 in (0, T )× Ω, on (0, T )× Γ0, on (0, T )× Γ1, in Ω, where Γ0 := R n−1 × {0} and Γ1 := R n−1 × {−L} forms a partition of the boundary ∂Ω. Here we have denoted ∂n the partial derivative with respect to xn and ∂ν the normal derivative. We are interested in the so-called null controllability of this system for givenu0, find v such that the solution of(Sv) satisfiesu(T ) ≡ 0. Using classical duality arguments, we will be interested on the observability of the adjoint system through Γ0. 1