Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

TL;DR

This paper models a re-entrant manufacturing system using a scalar conservation law, proving existence, uniqueness, stability of solutions, and establishing an optimal control framework to match out-flux with forecast demand.

Contribution

It extends previous models by incorporating both local and nonlocal velocity functions, providing rigorous mathematical analysis and an optimal control solution.

Findings

Existence and uniqueness of weak solutions established.

Solution and out-flux stability with respect to initial and boundary data.

Optimal control minimizing the difference between out-flux and demand proven to exist.

Abstract

In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of \cite{CKWang}, the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in $L^{\infty}$. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the $L^p$-sense with $p\in [1,\infty)$, the difference between the actual out-flux and a forecast demand over a fixed time period.