Thomas Parisini, University of Trieste, Italy

Abstract: In engineering and economic systems, many situations may occur, in which a process is influenced by the presence of several decision makers (DM). Different degrees of cooperation and different degrees of distribution of available information among the decision makers are possible.

In this lecture, we consider the case where various DMs share different information patterns but they make decisions aimed at the accomplishment of a common goal, i.e., the minimization of the same cost functional. Such an organization can be accurately described within the well known framework of ``team theory''. Typical examples of team organizations can be encountered in: communication and computer networks extending in large geographical areas where several routing nodes cooperate on the overall performance; large-scale traffic systems when a metropolitan area is divided into relatively independent sectors; large-scale freeway systems; production plants in which mobile robots moving semi-finished products have to be coordinated; geographically distributed systems for the production of electrical energy; etc.

A general approach to the solution of a team optimal decision problem has not yet been presented in the literature. Therefore, in this lecture we give up looking for optimal solutions to a general team optimal control problem, and propose a technique to obtain suboptimal (but approximate to any degree of accuracy) solutions. This is accomplished by constraining the control functions to take on a fixed structure with a certain number of ``free'' parameters to be optimized according to the Extended Ritz Method (ERIM). Among the various possible fixed-structure functions, feedforward neural networks are chosen for their powerful approximation capabilities and because these functional structures allow for a simple distributed computation of the local control strategies by stochastic approximation techniques. The neural control methodology is worked out on two important benchmark problems. A simple team within the LQG framework is first considered, where two decision makers with scalar information are present. When the problem admits a known optimal solution, our approach has demonstrated to be able to approximate it. Quite satisfactory results were obtained also in a case (the well-known Witsenhausen counterexample) where the optimal solution has not yet been found (it is however known that it exists). Then, dynamic routing in communication networks is considered. A nonlinear discrete-time dynamic model is given for a store-and-forward packet switching network in which the routing nodes play the role of cooperating DMs of a team. The resulting problem does not verify either the LQG hypotheses or the partially nestedness assumption on the information structure. The ERIM has demonstrated to be effective also in this application.