  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 optimal cost function and resulting sparse state feedback for these
systems can be computed by linear programming. 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.
