A Parallel and Conservative Semi-Lagrangian Scheme for Optimal Control in Production-Destruction Processes

The mathematical modeling of various real-life phenomena often leads to the formulation of positive and conservative Production-Destruction differential Systems (PDS). Here we address a general finite horizon Optimal Control Problem (OCP) for PDS and delve into the properties of its continuous-time solution. Leveraging the dynamic programming approach, we recast the OCP as a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation, whose unique viscosity solution corresponds to the value function [1]. We then propose a parallel-in-space semi-Lagrangian approximation scheme for the HJB equation [3] and derive the optimal control in feedback form. Finally, to reconstruct the optimal trajectories of the controlled PDS, we employ unconditionally positive and conservative modified Patankar linear multistep methods [2]. [1] CRANDALL, M. G.; ISHII, H.; LIONS, P.-L. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 1992, 27.1: 1-67. [2] IZZO, G.; MESSINA, E.; PEZZELLA, M.; VECCHIO, A. TITOLO DA ACCERTARE LUNEDì. In preparation. [3] FALCONE, M.; FERRETTI, R. Semi-Lagrangian approximation schemes for linear and Hamilton—Jacobi equations. SIAM, 2013.