Control of hyperbolic and parabolic equations on networks and singular limits

We study the controllability properties of transport equations and of parabolic equations with vanishing diffusivity posed on a tree-shaped network. Using a control localized on the exterior nodes, we obtain a null-controllability result for both systems. The hyperbolic proof relies on the method of characteristics; while the parabolic one on duality arguments and Carleman inequalities. In particular, we estimate the cost of the null-controllability of advection-diffusion equations with diffusivity ε > 0 and study its asymptotic behavior when ε → 0+. More specifically, we show that the cost of null-controllability decays exponentially for a time sufficiently large and explodes for short times. The core of the proof consists in proving an observability estimate keeping track of the viscosity parameter by relying on a suitable Carleman inequality.