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

TL;DR

This paper analyzes a hyperbolic conservation law model for a re-entrant manufacturing system, establishing solution existence, regularity, and optimal control strategies for output regulation.

Contribution

It introduces a mathematical framework for modeling, analyzing, and controlling a complex manufacturing process with nonlocal velocity and boundary controls.

Findings

Proved existence and uniqueness of solutions for $L^1$-data.

Established regularity properties of solutions.

Identified and proved optimality of time-optimal control for state transitions.

Abstract

This article studies a hyperbolic conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. Characteristic features are the nonlocal character of the velocity and that the influx and outflux constitute the control and output signal, respectively. We prove the existence and uniqueness of solutions for $L^1$-data, and study their regularity properties. We also prove the existence of optimal controls that minimizes in the $L^2$-sense the mismatch between the actual and a desired output signal. Finally, the time-optimal control for a step between equilibrium states is identified and proven to be optimal.