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Center for Nonlinear Studies |
Dissipative flow networks model flow of fluids or commodities across a network. The flow dynamics on edges are governed by non-linear dissipative partial differential equations. The dynamics on adjacent edges are coupled through Kirchhoff-Neumann boundary conditions that also account for the injection parameters at the nodes. We establish a monotonicity property which states that the ordering of the initial states (e.g. density) is preserved throughout the time evolution of the system whenever the nodal injection parameters also obey the same ordering. We show that the dynamic system resulting from an appropriate choice of spatial discretization of the system of PDEs inherits this monotonicity property and can be used within simulation and optimization. We also prove a monotonicity property for dissipative networks in steady state and establish a connection between the dynamic and steady state results. These results enable significant simplification in the representation and algorithms for robust optimization and control problems under uncertain nodal injections.