Linear Programming Based Routing Design for a Class of Positive Systems with Integral and Capacity Constraints*

We present techniques to design routing parameters for positive compartmental conservative systems. Such systems capture the dynamics of some material flowing through a network of interconnected reservoirs and have become popular, in particular, as models of air traffic flows. These techniques use Linear Programs (LP) to design static routing parameters for single destination networks with the following objectives: (a) minimize delay, (b) minimize delay and satisfy additional delay constraints which are formulated as integral constraints on the states of the network, and (c) satisfy capacity constraints. For each of these problems, we prove that the resulting closed loop systems are stable, positive, conservative and exhibit a user defined interconnection of sections. Additionally, problems (a) and (b) are shown to minimize delay over all choices of routing parameters such that the closed loop system exhibits these characteristics.