Modified Patankar Semi-Lagrangian Scheme for the Optimal Control of Production-Destruction Systems

In this manuscript, we present a comprehensive theoretical and numerical framework for the control of production-destruction differential systems. The general finite horizon optimal control problem is formulated and addressed through the dynamic programming approach. We develop a parallel in space semi-Lagrangian scheme for the corresponding backward-in-time Hamilton-Jacobi-Bellman equation. Furthermore, we provide a suitable conservative reconstruction algorithm for optimal controls and trajectories. The application to two case studies, specifically enzyme catalyzed biochemical reactions and infectious diseases, highlights the advantages of the proposed methodology over classical semi-Lagrangian discretizations.