hal-01228523
https://hal-ensta-paris.archives-ouvertes.fr/hal-01228523
arxiv:1312.4491
doi:10.1007/s13235-014-0133-z
[ENSTA] ENSTA Paris
[INSMI] CNRS-INSMI - INstitut des Sciences Mathématiques et de leurs Interactions
[UMA_ENSTA] Unité de Mathématiques Appliquées (UMA)
[TDS-MACS] Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes
[UNIV-PARIS-SACLAY] Université Paris-Saclay
[ENSTA-SACLAY] ENSTA-SACLAY
[ANR] ANR
Cournot Maps for Intercepting Evader Evolutionsby a Pursuer
Aubin, Jean-Pierre
Chen, Luxi
Desilles, Anna
[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]
[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]
ART
Pursuer–Evader interception games · Variable temporal windows · Agestructured dynamical systems
Viability constraint
Intertemporal optimality
Cournot–McKendrick valuation functions
Instead of studying evolutions governed by an evolutionary system starting at agiven initial state on a prescribed future time interval, finite or infinite, we tackle the problemof looking both for a past interval [T − D, T ] of duration D and for the viable evolutionsarriving at a prescribed terminal state at the end of the temporal window (and thus telescopingif more than one such evolutions exist). Hence, given time-duration dependent evolutionarysystem and viability constraints, as well as time dependent departure constraints, the Cournotmap 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 fromwhich at least one viable evolution will reach the given terminal state x at T . Cournot mapscan be used by a Pursuer to intercept an evader’s evolution in dynamic game theory. Afterproviding some properties of Cournot maps are next investigated, above all, the regulationmap piloting the viable evolutions at each time and for each duration from the beginningof the temporal window up to terminal time. The next question investigated is the selectionof controls or regulons in the regulation map whenever several of them exist. Selectionprocesses 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), orintertemporal. In this case, viable evolutions are required to optimize someprescribed 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 securityvehicle 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). Thesoftware of the Cournot algorithm computes the minimal duration and the moment when thePursuer leaves its round to reach the detected location as soon as possible and how to proceedby embedding in the Pursuer system the graph of the feedback map governing the evolutionof the Pursuer vehicle.
2015-09-01
en
Dynamic Games and Applications
Springer Verlag