This work considers infinite-horizon optimal control of positive linear systems applied to the case of network routing problems. We demonstrate the equivalence between Stochastic Shortest Path (SSP) problems and optimal control of a certain class of linear systems. This is used to construct a heuristic search framework for linear positive systems inspired by existing methods for SSP. We propose a heuristics-based algorithm for {efficiently finding local solutions to the analyzed class of optimal control problems with a given initial state and positive linear dynamics. } By leveraging the bound on optimality in each state provided by the heuristics, we also derive a novel distributed algorithm for calculating local controllers within a specified performance bound, with a distributed condition for termination. More fundamentally, the results allow for analysis of the conditions for explicit solutions to the Bellman equation utilized by heuristic search methods.
Ohlin et al. (Mon,) studied this question.