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We derive a monotonicity property for general, transient flows of a commoditytransferred throughout a network, where the flow is characterized by densityand mass flux dynamics on the edges with density continuity and mass balanceconditions at the nodes. The dynamics on each edge are represented by a generalsystem of partial differential equations that approximates subsoniccompressible fluid flow with energy dissipation. The transferred commodity maybe injected or withdrawn at any of the nodes, and is propelled throughout thenetwork by nodally located compressors. These compressors are controllableactuators that provide a means to manipulate flows through the network, whichwe therefore consider as a control system. A canonical problem requirescompressor control protocols to be chosen such that time-varying nodalcommodity withdrawal profiles are delivered and the density remains withinstrict limits while an economic or operational cost objective is optimized. Inthis manuscript, we consider the situation where each nodal commoditywithdrawal profile is uncertain, but is bounded within known maximum andminimum time-dependent limits. We introduce the monotone parameterized controlsystem property, and prove that general dynamic dissipative network flowspossess this characteristic under certain conditions. This property facilitatesvery efficient formulation of optimal control problems for such systems inwhich the solutions must be robust with respect to commodity withdrawaluncertainty. We discuss several applications in which such control problemsarise and where monotonicity enables simplified characterization of systembehavior.

Monotonicity of Actuated Flows on Dissipative Transport Networks