Dynamic Games and Applications

, Volume 8, Issue 2, pp 254–279 | Cite as

Potential Differential Games

  • Alejandra Fonseca-Morales
  • Onésimo Hernández-Lerma


This paper introduces the notion of a potential differential game (PDG), which roughly put is a noncooperative differential game to which we can associate an optimal control problem (OCP) whose solutions are Nash equilibria for the original game. If this is the case, there are two immediate consequences. Firstly, finding Nash equilibria for the game is greatly simplified, because it is a lot easier to deal with an OCP than with the original game itself. Secondly, the Nash equilibria obtained from the associated OCP are automatically “pure” (or deterministic) rather than “mixed” (or randomized). We restrict ourselves to open-loop differential games. We propose two different approaches to identify a PDG and to construct a corresponding OCP. As an application, we consider a PDG with a certain turnpike property that is obtained from results for the associated OCP. We illustrate our results with a variety of examples.


Differential games Nash equilibria Potential games Optimal control Maximum principle 

Mathematics Subject Classification

91A23 91A10 49N70 49N90 34H05 



Funding was provided by CONACyT (Grant No. 221291).


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Alejandra Fonseca-Morales
    • 1
  • Onésimo Hernández-Lerma
    • 1
  1. 1.Mathematics DepartmentCINVESTAV-IPNMexico CityMexico

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