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.

翻译：本文研究了由对流-扩散-反应（ADR）方程描述的化学系统的最优边界控制问题。我们采用间断伽辽金有限元方法（DG-FEM）对控制偏微分方程进行空间离散，并通过多重打靶法直接离散化最优控制问题。时间离散及相应灵敏度计算采用显式单对角隐式龙格-库塔（ESDIRK）方法实现。ADR系统出现在过程系统工程中，其操作可通过非线性模型预测控制（NMPC）得到潜在改进。我们通过色谱案例研究展示了求解其最优控制问题（OCPs）的数值方法。制备型液相色谱是生物制药制造中重要的下游工艺。研究表明，针对间歇过程的多步洗脱轨迹可基于经济目标进行优化，相较于传统梯度洗脱轨迹展现出更优越的性能。