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Discrete optimal control problems on networks and dynamic games with $$p$$ players. (English) Zbl 1080.91016
Summary: We consider a special class of discrete optimal control problems on networks. The dynamics of the system is described by a directed graph of passages. An additional integral-time cost criterion is given and the starting and final states of the system are fixed. The game-theoretical models for such a class of problems are formulated, and some theoretical results connected with the existence of the optimal solution in the sense of Nash are given. A polynomial-time algorithm for determining Nash equilibria is proposed. The results are applied to decision making systems and determining the optimal strategies in positional games on networks.
##### MSC:
 91A43 Games involving graphs 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 91A25 Dynamic games 93C55 Discrete-time control/observation systems