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We derive conditions for monotonicity properties that characterize generalflows of a commodity over a network, where the flow is described by potentialand flow dynamics on the edges, as well as potential continuity andKirchhoff-Neumann mass balance requirements at nodes. The transported commoditymay be injected or withdrawn at any of the network nodes, and its movementthroughout the network is controlled by nodal actuators. For a class ofdissipative nonlinear parabolic partial differential equation (PDE) systems onnetworks, we derive conditions for monotonicity properties in steady-stateflow, as well as for propagation of monotone ordering of states with respect totime-varying boundary condition parameters. In the latter case, initialconditions, as well as time-varying parameters in the coupling conditions atvertices, provide an initial boundary value problem (IBVP). We prove thatordering properties of the solution to the IBVP are preserved when the initialconditions and the parameters of the time-varying coupling law areappropriately ordered. Then, we prove that when monotone ordering is notpreserved, the first crossing of solutions occurs at a network node. Weconsider the implications for robust optimization and optimal controlformulations and real-time monitoring of uncertain dynamic flows on networks,and discuss application to subsonic compressible fluid flow with energydissipation on physical networks. The main result and monitoring policy aredemonstrated for gas pipeline test networks and a case study using datacorresponding to a real working system. We propose applications of this generalresult to the control and monitoring of natural gas transmission networks.

Monotonicity Properties of Physical Network Flows and Application to Robust Optimal Allocation