Monotone Order Properties for Control of Nonlinear Parabolic PDE on Graphs

TL;DR

This paper establishes conditions under which monotone order properties are preserved in nonlinear parabolic PDE systems on graphs, with implications for control and monitoring of networked flow systems.

Contribution

It provides novel conditions ensuring the preservation of monotone orderings in PDE systems on graphs, including boundary and initial condition considerations.

Findings

Order preservation depends on initial and boundary conditions

First crossing of solutions occurs at graph vertices when order is not preserved

Results applicable to fluid flow control and network monitoring

Abstract

We derive conditions for the propagation of monotone ordering properties for a class of nonlinear parabolic partial differential equation (PDE) systems on metric graphs. For such systems, PDE equations with a general nonlinear dissipation term define evolution on each edge, and balance laws create Kirchhoff-Neumann boundary conditions at the vertices. Initial conditions, as well as time-varying parameters in the coupling conditions at vertices, provide an initial value problem (IVP). We first prove that ordering properties of the solution to the IVP are preserved when the initial conditions and time-varying coupling law parameters at vertices are appropriately ordered. In addition, we prove that when monotone ordering is not preserved, the first crossing of solutions occurs at a graph vertex. We consider the implications for robust optimal control formulations and real-time monitoring involving uncertain dynamic flows on networks, and discuss application to subsonic compressible fluid flow with energy dissipation on physical networks.