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The dynamic-routing problem in a packet-switching telecommunication network is addressed by a receding-horizon approach. The nodes of the network must accomplish the following tasks: (i) generating routing decisions to minimize the expected total delay, spent by messages in the network, on the basis of local information and possibly of some data received from the neighboring nodes; (ii) computing their routing strategies by measuring local variables and exchanging a small amount of data with other nodes. The first task leads to regard the nodes as the cooperating decision makers of a team organization. The second task calls for a computationally distributed algorithm. To solve this team optimal control problem two main approximating choices have been done: (1) the use of a receding-horizon framework, and (2) the use of a given structure for each decision maker, in which a finite number of parameters have to be determined to optimize the packet routing. Among the various possible fixed-structure functions, feed-forward neural networks have been chosen for their powerful approximation capabilities. The neural approximation of such team-optimal routing strategies is computed in a numerical example, and used in network routing simulations performed by means of NS-2, in order to show the feasibility and effectiveness of the methodology.