A decentralized solution for the constrained minimum cost flow

In this paper we propose a decentralized solution to the problem of network stabilization, under flow constraints ensuring steady-state flow optimality. We propose a stabilizing strategy for network flow control with capacity constraints which drives the buffer levels arbitrarily close to a desired reference. This is a decentralized strategy optimizing the flow via the minimization of a quadratic cost of the control. A second problem characterized by non-fully connected networks is also considered, for which an exact network equilibrium is not possible. Here, the strategy, in the absence of constraints leads to a least square decentralized problem, but, unfortunately, in the presence of constraints, it does not solve the constrained least square, but the minimum-distance from the unconstrained least square solution. An extension is considered to networks with intrinsic linear dynamics and we prove that our strategy is flow-optimal, with respect to a certain cost, even in this case.