Cournot Maps for Intercepting Evader Evolutions by a Pursuer

Instead of studying evolutions governed by an evolutionary system starting at a given initial state on a prescribed future time interval, finite or infinite, we tackle the problem of looking both for a past interval \([T-D,T]\) of duration D and for the viable evolutions arriving at a prescribed terminal state at the end of the temporal window (and thus telescoping if more than one such evolutions exist). Hence, given time-duration dependent evolutionary system and viability constraints, as well as time dependent departure constraints, the Cournot map associates with any terminal time \(T\) and state \(x\) the durations \(D(T,x)\) of the intervals \([T-D(T,x),T]\), the starting (or initial) states at the beginning of the temporal window from which at least one viable evolution will reach the given terminal state \(x\) at \(T\). Cournot maps can be used by a Pursuer to intercept an evader’s evolution in dynamic game theory. After providing some properties of Cournot maps are next investigated, above all, the regulation map piloting the viable evolutions at each time and for each duration from the beginning of the temporal window up to terminal time. The next question investigated is the selection of controls or regulons in the regulation map whenever several of them exist. Selection processes are either time dependent, when the selection operates at each time, duration, and state for selecting a regulon satisfying required properties (for instance, minimal norm, minimal speed), or intertemporal. In this case, viable evolutions are required to optimize some prescribed intertemporal functional, as in optimal control. This generates value functions, the topics of the second part of this study. An example is provided: the Pursuer is a security vehicle making the rounds along a predetermined path, the departure tube, for reaching any network location where and when alarms sound to signal the location (of the evader). The software of the Cournot algorithm computes the minimal duration and the moment when the Pursuer leaves its round to reach the detected location as soon as possible and how to proceed by embedding in the Pursuer system the graph of the feedback map governing the evolution of the Pursuer vehicle.