Controllability of shadow reaction-diffusion systems

Hernández-Santamaría V., E. Zuazua. Controllability of shadow reaction-diffusion systems.  J Differ Equations, Vol. 268, No. 7, pp. 3781-3818 (2020). DOI: 10.1016/j.jde.2019.10.012

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 PDE-ODE system where one component solves a parabolic partial differential equation (PDE) and the other one an ordinary differential equation (ODE). This reduced system contains the essential dynamics of the original one.

We analyze these shadow systems from a controllability perspective and prove two types of results. First, by employing Carleman inequalities, ODE arguments and regularity results, we prove that the null controllability of the shadow model holds. This result, together with the effectiveness of the controls of the shadow system to control the full original dynamics for large values of the diffusivity parameter, is then 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. These results justify the eficiency of shadow systems not only for modeling the dynamics of reactiondiffusion system in the large di usivity limit of one of the equations, but at the control level too.

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