We proposed network-decentralized 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 < p < ∞ , the flow converges to a unique constant optimal up∗. We show that the state converges to the optimal Lagrange multiplier of the optimization 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-decentralized 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 p 1 or p ∞.

A well-known property of linear resistive electrical networks is that the current distribution minimizes the total dissipated power. When the circuit includes resistors with nonlinear monotonic characteristic, the current distribution minimizes in general a different functional. We show that, if the nonlinear characteristic is a threshold-like function and the current generator is concentrated in a single point, as in the case of lightning or dielectric discharge, then the current flow is concentrated along a single path, which is a minimum path to the ground with respect to the threshold. We also propose a dynamic model that explains and qualitatively reproduces the lightning transient behavior: initial generation of several plasma branches and subsequent dismissal of all branches but the one reaching the ground first, which is the optimal one.

To control the flow in a dynamical network where the nodes are associated with buffer variables and the arcs with controlled flows, we consider a network-decentralised strategy such that each arc controller makes its decision exclusively based on local information about the levels of the buffers that it connects. We seek a flow control law that asymptotically minimises a cost specified in terms of a weighted L₁-norm. This approach has the advantage of providing a solution that is generally sparse, because it uses a limited number of controlled flows. In particular, in the presence of a resource demand applied on a single node, the asymptotic flow is concentrated along the shortest path.

This paper proposes a decentralised explicit (closed-form) iterative formula that solves convex programming problems with linear equality constraints and interval bounds on the decision variables. In particular, we consider a team of decision agents, each setting the value of a subset of the variables, and a team of information agents, in charge of ensuring that the equality constraints are fulfilled. The structure of the constraint matrix imposes a communication pattern between decision and information agents, which can be represented as a bipartite graph. We associate each information agent with an integral variable and each decision agent with a saturated function, which takes the interval bounds into account, and we design a decentralised dynamic mechanism that globally converges to the optimal solution. Under mild conditions, the convergence is shown to be exponential. We also provide a discrete-time algorithm, based on the Euler system, and we give an upper bound for the step parameter to ensure convergence. Although the considered optimisation problem is static, we show that the proposed scheme can be successfully applied to find the optimal solution of network-decentralised dynamic control problems.