Joint BCAM-UPV/EHU Analysis and PDE seminar: Cost of null controllability for parabolic equations with vanishing viscosity

Abstract
The transport-diffusion equation with vanishing diffusivity describes the dynamics
of physical and biological phenomena in which the transport dynamics dominates the
diffusive dynamics. Since these systems are of parabolic nature, it is well-known that
they are null controllable. However, there are many open questions on the asymptotic
behaviour of the cost of null contrallability when the diffusion parameter vanishes.

In this talk we study the transport-diffusion equation with Neumann, Robin and mixed
boundary conditions. The main results concern the behaviour of the cost of the null
controllability when the diffusivity vanishes and the control acts in the interior. First, we
prove that if we almost have Dirichlet boundary conditions in the part of the boundary
in which the flux of the transport enters, the cost of the controllability decays for a time
T sufficiently large. Next, we show some examples of Neumann and mixed boundary
conditions in which for any time T > 0 the cost explodes exponentially. Finally, we study
the cost of the problem with Neumann boundary conditions when the control is localized
in the whole domain.


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