A Structure-Preserving Semi-Lagrangian Scheme for the Optimal Control of Production-Destruction Systems
Please login to view abstract download link
We present a novel framework for the optimal control of production-destruction systems. These systems of ordinary differential equations possess a linear invariant and describe a broad class of processes in which the components are simultaneously produced and depleted. The control problem is posed over a finite time horizon and solved via the dynamic programming principle, leading to a Hamilton–Jacobi–Bellman partial differential equation. To approximate its viscosity solution, we develop a GPU-accelerated Modified Patankar Semi-Lagrangian (MPSL) scheme that integrates positivity-preserving and conservative Patankar-type methods within the semi-Lagrangian setting. We validate the methodology on two representative models: an enzyme-catalysed biochemical reaction and a compartmental epidemic model with behavioural feedback. The results confirm that the MPSL scheme outperforms classical methods in preserving the system’s structural properties, while providing more accurate optimal controls.