  This paper considers the optimal boundary control of chemical systems
described by advection-diffusion-reaction (ADR) equations. We use a
discontinuous Galerkin finite element method (DG-FEM) for the spatial
discretization of the governing partial differential equations, and the optimal
control problem is directly discretized using multiple shooting. The temporal
discretization and the corresponding sensitivity calculations are achieved by
an explicit singly diagonally-implicit Runge Kutta (ESDIRK) method. ADR systems
arise in process systems engineering and their operation can potentially be
improved by nonlinear model predictive control (NMPC). We demonstrate a
numerical approach for the solution to their optimal control problems (OCPs) in
a chromatography case study. Preparative liquid chromatography is an important
downstream process in biopharmaceutical manufacturing. We show that multi-step
elution trajectories for batch processes can be optimized for economic
objectives, providing superior performance compared to classical gradient
elution trajectories.
