Control of the Continuity Equation with a Non Local Flow

TL;DR

This paper develops a mathematical framework for controlling weak solutions to the continuity equation with non local flow, including models like supply chains and pedestrian dynamics, establishing well-posedness and optimality conditions.

Contribution

It proves well-posedness, differentiability of solutions, and derives optimality conditions for the control of non local flow models in the continuity equation.

Findings

Proved well-posedness of the class of equations

Established differentiability of solutions with respect to initial data

Derived necessary optimality conditions for control

Abstract

This paper focuses on the optimal control of weak (i.e. in general non smooth) solutions to the continuity equation with non local flow. Our driving examples are a supply chain model and an equation for the description of pedestrian flows. To this aim, we prove the well posedness of a class of equations comprising these models. In particular, we prove the differentiability of solutions with respect to the initial datum and characterize its derivative. A necessary condition for the optimality of suitable integral functionals then follows.