Controllability of shadow reaction-diffusion systems

Abstract

We study the null controllability of linear shadow models for reaction-diffusion systems arising as singular limits when the diffusivity of some of the components is very high. This leads to a coupled system where one component solves a parabolic partial differential equation (PDE) and the other one an ordinary differential equation (ODE).We analyze these shadow systems from a controllability perspective and prove two types of results. First, by employing Carleman inequalities and ODE arguments, we prove that the null controllability of the shadow model holds. This result, together with the effectiveness of the controls to control the original dynamics, is illustrated by numerical simulations.We also obtain a uniform Carleman estimate for the reaction-diffusion equations which allows to obtain the null control for the shadow system as a limit when the diffusivity tends to infinity in one of the equations.