Coordination Concepts in Multilevel Hierarchical Differential Games

This paper presents possible extensions of the usual differential game theory to the case where the players, say the supremal players do not completely govern the plant, but control it only via infimal players who have their own pay-off functions. One assumes that the internal structure of the game is given in the sense that the supremal players cannot define arbitrary decompositions for the system. One further assumes that these supremal players have given programming terminal objectives which they desire to attain. In this framework, the supremal players can only select suitable functions, which are the coordination controls, to coordinate the system for the best. It follows that the problem so defined is simultaneously an optimum control problem and a programming one. The direct coordination mode, the open-loop indirect coordination mode and the closed-loop indirect coordination mode are defined and are mainly based upon a direct reference to the overall system, together with a two-level optimization process. The explicit equations are given in the important case where the comparison criterion is defined in the form of an Euclidean norm, and an illustrative example is solved. This approach would be interesting to apply to the problem of learning and self-learning in optimum control systems for instance, via gradient techniques.