On Class of Linear Quadratic Non-cooperative Differential Games with Continuous Updating

  • Ildus Kuchkarov
  • Ovanes PetrosianEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)


The subject of this paper is a linear quadratic case of a differential game model with continuous updating. This class of differential games is essentially new, there it is assumed that at each time instant, players have or use information about the game structure defined on a closed time interval with a fixed duration. As time goes on, information about the game structure updates. Under the information about the game structure we understand information about motion equations and payoff functions of players. A linear quadratic case for this class of games is particularly important for practical problems arising in the engineering of human-machine interaction. The notion of Nash equilibrium as an optimality principle is defined and the explicit form of Nash equilibrium for the linear quadratic case is presented. Also, the case of dynamic updating for the linear quadratic differential game is studied and uniform convergence of Nash equilibrium strategies and corresponding trajectory for a case of continuous updating and dynamic updating is demonstrated.


Differential games with continuous updating Nash equilibrium Linear quadratic differential games 


  1. 1.
    Basar, T., Olsder, G.: Dynamic Noncooperative Game Theory. Academic Press, London (1995)zbMATHGoogle Scholar
  2. 2.
    Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)zbMATHGoogle Scholar
  3. 3.
    Dockner, E., Jorgensen, S., Long, N., Sorger, G.: Differential Games in Economics and Management Science. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  4. 4.
    Engwerda, J.: LQ Dynamic Optimization and Differential Games. Willey, New York (2005)Google Scholar
  5. 5.
    Gromova, E., Petrosian, O.: Control of information horizon for cooperative differential game of pollution control. In: 2016 International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference) (2016)Google Scholar
  6. 6.
    Kleimenov, A.: Non-antagonistic Positional Differential Games. Science, Ekaterinburg (1993)Google Scholar
  7. 7.
    Petrosian, O.: Looking forward approach in cooperative differential games. Int. Game Theory Rev. 18, 1–14 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Petrosian, O.: Looking forward approach in cooperative differential games with infinite-horizon. Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr. (4), 18–30 (2016)Google Scholar
  9. 9.
    Petrosian, O., Barabanov, A.: Looking forward approach in cooperative differential games with uncertain-stochastic dynamics. J. Optim. Theory Appl. 172, 328–347 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Petrosian, O., Nastych, M., Volf, D.: Non-cooperative differential game model of oil market with looking forward approach. In: Petrosyan, L.A., Mazalov, V.V., Zenkevich, N. (eds.) Frontiers of Dynamic Games, Game Theory and Management, St. Petersburg 2017. Birkhäuser, Basel (2018)CrossRefGoogle Scholar
  11. 11.
    Petrosian, O., Nastych, M., Volf, D.: Differential game of oil market with moving informational horizon and non-transferable utility. In: 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (2017)Google Scholar
  12. 12.
    Petrosyan, L., Murzov, N.: Game-theoretic problems in mechanics. Lith. Math. Collect. 3, 423–433 (1966)Google Scholar
  13. 13.
    Pontryagin, L.: On theory of differential games. Successes Math. Sci. 26, 4(130), 219–274 (1966)Google Scholar
  14. 14.
    Shevkoplyas, E.: Optimal solutions in differential games with random duration. J. Math. Sci. 199(6), 715–722 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yeung, D., Petrosian, O.: Cooperative stochastic differential games with information adaptation. In: International Conference on Communication and Electronic Information Engineering (2017)Google Scholar
  16. 16.
    Yeung, D., Petrosian, O.: Infinite horizon dynamic games: a new approach via information updating. Int. Game Theory Rev. 19, 1–23 (2017)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySaint-PetersburgRussia
  2. 2.National Research University Higher School of EconomicsSaint-PetersburgRussia

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