Inverse Design and Boundary Controllability for the Chromatography System

Abstract

We consider the prototypical example of the \(2\times 2\) liquid chromatography system and characterize the set of initial data leading to a given attainable profile at \(t=T\). For profiles that are not attainable at time T, we study a non-smooth optimization problem: recovering the initial data that lead as close as possible to the target in the \(L^2\)-norm. We then study the system on a bounded domain and use a boundary control to steer its dynamics to a given trajectory. Finally, we implement a suitable finite volumes scheme to illustrate these results and show its numerical convergence. Minor modifications of our arguments apply to the Keyfitz–Kranzer system.