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Dynamical Systems and Microphysics by A. Blaquiére, F. Fer, A. Marzollo (eds.)

By A. Blaquiére, F. Fer, A. Marzollo (eds.)

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Thus, again there is a total of 12. 2n boundary conditions. Two-Person Zero-Sum Games In the preceding sections we considered dynamical systems controlled by a single agent. Now we present a brief discussion of a situation in which two agents, referred to as players, exert control over the evolution of the state of the system. 2) Control Ui(T)is selected by player i As before, we suppose that this choice is made on the basis of a player's knowledge of state xC,) by means of a feedback control (player's strategy) : Rn -+- q.

All the contributions at point x and at time t from point-clocks in this region at time t-llt are nearly in phase and do not cancel out. 2). " x-ullt (u+ou)llt " \. 2 t-llt t From the remark above and Assumption 5 it follows that only for point-clocks whose x4-coordinate is in the vicinity of x4-11t at time t-llt can we get important contributions at point x = (~,x4) and at time t. In other words, in the classical limit, the time x4 to which a small clock is pointing is approximately the same as the time t of observer 0, with the possible addition of a constant time-lag which can be disregarded.

1 u. + av* (x) at u. * (x) -at 1 1 1 1 ] 0 . 3) yields t u. 6) m i=l 14. 3), Concluding Remarks In the preceding sections we consider dynamical systems whose states evolve under the influence of a single controller or of two controllers (players), respectively. The choice of a cc~:1trol, or cf a control pair in the two player case, is presumed to be made with :"~=-l knowledge of the· system I s state; that is, a controller selects a ccn,:;:'ol value by means of a mapping from state into his control space.

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