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.
