We study routing control in dynamic continuous-time networks in which each node is described by a linear internally positive system. We derive a linear programming (LP) characterization of optimal constant routing policies for the infinite-horizon routing problem under weighted l(1) constraints and objective. For finite horizon problems with piecewise constant state constraints, we provide sufficient feasibility conditions, which also take the form of an LP problem. We illustrate the solution for the finite horizon problem in an application example. Published by Elsevier Ltd