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We consider a basic model of a dynamical distribution network, modeled as adirected graph with storage variables corresponding to every vertex and flowinputs corresponding to every edge, subject to unknown but constant inflows andoutflows. In [1] we showed how a distributed proportionalintegral controllerstructure, associating with every edge of the graph a controller state,regulates the state variables of the vertices, irrespective of the unknownconstant inflows and outflows, in the sense that the storage variables convergeto the same value (load balancing or consensus). In many practical cases, theflows on the edges are constrained. The main result of [1] is a sufficient andnecessary condition, which only depend on the structure of the network, forload balancing for arbitrary constraint intervals of which the intersection hasnonempty interior. In this paper, we will consider the question about how todecide the steady states of the same model as in [1] with given networkstructure and constraint intervals. We will derive a graphic condition, whichis sufficient and necessary, for load balancing. This will be proved by aLyapunov function and the analysis the kernel of incidence matrix of thenetwork. Furthermore, we will show that by modified PI controller, the storagevariable on the nodes can be driven to an arbitrary point of admissible set.

A graphic condition for the stability of dynamical distribution networks with flow constraints