We derive conditions for monotonicity properties that characterize general flows of a commodity over a network, where the flow is described by potential and flow dynamics on the edges, as well as potential continuity and the Kirchhoff–Neumann mass balance requirements at nodes. The transported commodity may be injected or withdrawn at any of the network nodes, and its movement throughout the network is controlled by nodal actuators. For a class of dissipative nonlinear parabolic partial differential equation (PDE) systems on networks, we derive conditions for monotonicity properties in steady-state flow, as well as for propagation of monotone ordering of states with respect to time-varying boundary condition parameters. In the latter case, initial conditions and time-varying parameters in the coupling conditions at vertices provide an initial boundary value problem (IBVP). We prove that ordering properties of the solution to the IBVP are preserved when the initial conditions and the parameters of the time-varying coupling law are appropriately ordered. Then, we prove that when monotone ordering is not preserved, the first crossing of solutions occurs at a network node. We consider the implications for robust optimization and optimal control formulations and real-time monitoring of uncertain dynamic flows on networks and discuss the application to subsonic compressible fluid flow with energy dissipation on physical networks. The main result and monitoring policy are demonstrated for gas pipeline test networks and a case study using data corresponding to a real working system. We propose applications of this general result to the control and monitoring of natural gas transmission networks.

中文翻译：

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物理网络流的单调性及其在鲁棒优化分配中的应用

我们推导出表征商品在网络上的一般流动的单调性条件，其中流动由边缘上的势能和流动动力学描述，以及势能连续性和节点处的基尔霍夫-诺依曼质量平衡要求。运输的商品可以在任何网络节点注入或取出，其在整个网络中的移动由节点执行器控制。对于网络上的一类耗散非线性抛物线偏微分方程 (PDE) 系统，我们推导了稳态流中单调性的条件，以及关于时变边界条件参数的状态单调排序的传播。在后一种情况下，顶点耦合条件中的初始条件和时变参数提供了初始边界值问题 (IBVP)。我们证明了当初始条件和时变耦合律的参数被适当排序时，IBVP 解的排序特性得以保留。然后，我们证明当不保留单调排序时，解的第一次交叉发生在网络节点。我们考虑了鲁棒优化和最优控制公式以及网络上不确定动态流动的实时监控的影响，并讨论了在物理网络上具有能量耗散的亚音速可压缩流体流动的应用。主要结果和监测策略针对天然气管道测试网络和使用与实际工作系统相对应的数据的案例研究进行了演示。