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

TL;DR

This paper introduces a novel semi-Lagrangian scheme for optimal control of production-destruction systems, combining theoretical development with numerical methods, and demonstrates its effectiveness through biochemical and disease modeling case studies.

Contribution

It develops a modified Patankar semi-Lagrangian scheme integrated with dynamic programming for controlling production-destruction systems, including a reconstruction algorithm for optimal controls.

Findings

Enhanced numerical stability over classical methods

Successful application to biochemical and epidemiological models

Improved accuracy in optimal control trajectories

Abstract

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 conservative scheme for the corresponding backward-in-time Hamilton-Jacobi-Bellman equation. Furthermore, we provide a suitable 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.