Price Based Linear Quadratic Control Under Transportation Delay

Abstract

We study a simple transportation problem on a string graph. The objective is to regulate the node levels of some decaying quantity to optimize dynamical performance. This can be achieved by controlling the flows, which are subject to delay, between neighbouring nodes. The problem is considered from two perspectives. In the first (the social perspective), all nodes cooperate to find the flows that maximize the aggregated utility of the entire transportation network. In the second (the user perspective), the nodes instead try to maximimize their own utility. Our main contribution is to give an implementation of the feedback law that gives the social optimum, that only depends on the local states and a set of prices defined by a distributed update rule. These prices align the social and user optimum in a budget neutral way, and give all nodes no worse cost than if they were on their own.