  We derive a monotonicity property for general, transient flows of a commodity
transferred throughout a network, where the flow is characterized by density
and mass flux dynamics on the edges with density continuity and mass balance
conditions at the nodes. The dynamics on each edge are represented by a general
system of partial differential equations that approximates subsonic
compressible fluid flow with energy dissipation. The transferred commodity may
be injected or withdrawn at any of the nodes, and is propelled throughout the
network by nodally located compressors. These compressors are controllable
actuators that provide a means to manipulate flows through the network, which
we therefore consider as a control system. A canonical problem requires
compressor control protocols to be chosen such that time-varying nodal
commodity withdrawal profiles are delivered and the density remains within
strict limits while an economic or operational cost objective is optimized. In
this manuscript, we consider the situation where each nodal commodity
withdrawal profile is uncertain, but is bounded within known maximum and
minimum time-dependent limits. We introduce the monotone parameterized control
system property, and prove that general dynamic dissipative network flows
possess this characteristic under certain conditions. This property facilitates
very efficient formulation of optimal control problems for such systems in
which the solutions must be robust with respect to commodity withdrawal
uncertainty. We discuss several applications in which such control problems
arise and where monotonicity enables simplified characterization of system
behavior.
