Optimal control of linear cost networks

We present a method for optimal control with respect to a linear cost function for positive linear systems with coupled input constraints. We show that the Bellman equation giving the optimal cost function and resulting sparse state feedback for these systems can be stated explicitly, with the solution given by a linear program. Our framework admits a range of network routing problems with underlying linear dynamics. These dynamics can be used to model traditional graph-theoretical problems like shortest path as a special case, but can also capture more complex behaviors. We provide an asynchronous and distributed value iteration algorithm for obtaining the optimal cost function and control law.