Fair and sparse solutions in network-decentralised flow control

We proposed network-decentralised control strategies, in which each actuator can exclusively rely on local information, without knowing the network topology and the external input, ensuring that the flow asymptotically converges to the optimal one with respect to the p-norm. For 1<∞, the flow converges to a unique constant optimal up*. We show that the state converges to the optimal Lagrange multiplier of the optimisation problem. Then, we consider networks where the flows are affected by unknown spontaneous dynamics and the buffers need to be driven exactly to a desired set-point. We propose a network-decentralised proportional-integral controller that achieves this goal along with asymptotic flow optimality; now it is the integral variable that converges to the optimal Lagrange multiplier. The extreme cases p=1 and p=∞ are of some interest since the former encourages sparsity of the solution while the latter promotes fairness. Unfortunately, for p=1 or p=∞ these strategies become discontinuous and lead to chattering of the flow, hence no optimality is achieved. We then show how to approximately achieve the goal as the limit for p1 or p∞.