Linear Discrete-Time Pareto-Nash-Stackelberg Control and Its Principles

  • Valeriu UngureanuEmail author
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 89)


A direct-straightforward method for solving linear discrete-time optimal control problem is applied to solve the control problem of a linear discrete-time system as a mixture of multi-criteria Stackelberg and Nash games. For simplicity, the exposition starts with the simplest case of linear discrete-time optimal control problem and, by sequential considering of more general cases, investigation finalizes with the highlighted Pareto-Nash-Stackelberg and set valued control problems. Different solution principles are compared and their equivalence is proved. We need to remark that there are other possible title variants of the considered models like, e.g., a multi-agent control problem of the Pareto-Nash-Stackelberg type. There is an appropriate approach in (Leitmann, Pickl and Wang Dynamic Games in Economics, Springer, Berlin, 205–217, 2014), [1]. A more simple and largely used title for such games is dynamic games, see e.g. (Başar, Olsder, Society for Industrial and Applied Mathematics, Philadelphia, 536, 1999), [2], (Long, A Survey of Dynamic Games in Economics, World Scientific, New Jersey, XIV–275, 2010), [3]. We insist on the above title in order to highlight both the game and control natures of the modelled real situations and processes. More the more we can refer in this context a Zaslavski’s recent monograph (Zaslavski, Discrete-Time Optimal Control and Games on Large Intervals, Springer, Switzerland, X+398, 2017, [4]) that uses an appropriate approach to the names of considered mathematical models.


  1. 1.
    Leitmann, G., S. Pickl, and Z. Wang. 2014. Multi-agent Optimal Control Problems and Variational Inequality Based Reformulations. In Dynamic Games in Economics, ed. J. Haunschmied, V.M. Veliov, and S. Wrzaczek, 205–217. Berlin: Springer.CrossRefGoogle Scholar
  2. 2.
    Başar, T., and G.J. Olsder. 1999. Dynamic Noncooperative Game Theory, 536. Philadelphia: Society for Industrial and Applied Mathematics.zbMATHGoogle Scholar
  3. 3.
    Long, N.V. 2010. A Survey of Dynamic Games in Economics. New Jersey: World Scientific, XIV–275 pp.Google Scholar
  4. 4.
    Zaslavski, A.J. 2017. Discrete-Time Optimal Control and Games on Large Intervals. Springer, Switzerland, X+398 pp.Google Scholar
  5. 5.
    Pontryagin, L.S., V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko. 1961. Mathematical theory of optimal processes. Moscow: Nauka, 393 pp. (in Russian).Google Scholar
  6. 6.
    Bellman, R. 1957. Dynamic Programming, 365. New Jersey: Princeton University Press.zbMATHGoogle Scholar
  7. 7.
    Vignaly, R., and M. Prandini. 1996. Minimum resource commitment for reachability specifications in a discrete time linear setting. IEEE Transactions on Automatic Control 62 (6): 3021–3028.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Von Neumann, J., and O. Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton, New Jersey: Annals Princeton University Press, (2nd edn., 1947), 674 pp.Google Scholar
  9. 9.
    Nash, J. 1951. Noncooperative games. Annals of Mathematics 54 (2): 280–295.CrossRefGoogle Scholar
  10. 10.
    Von Stackelberg, H. Marktform und Gleichgewicht (Market Structure and Equilibrium), Vienna: Springer Verlag, 1934, XIV+134 pp. (in German).Google Scholar
  11. 11.
    Leitmann, G. 1978. On Generalized Stackelberg Strategies. Journal of Optimization Theory and Applications 26: 637–648.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ungureanu, V. 2008. Solution principles for simultaneous and sequential games mixture. ROMAI Journal 4 (1): 225–242.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Isaacs, R. 1965. Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. Berlin: Wiley, XXIII+385 pp.Google Scholar
  14. 14.
    Chikriy, A.A. 1992. Conflict-controlled processes. Kiev: Naukova Dumka, 384 pp. (in Russian).Google Scholar
  15. 15.
    Lin, Ch.-S., Ch.-T. Chen, F.-Sh. Chen and W.-Zh. Hung. 2014. A Novel Multiperson Game Approach for Linguistic Multicriteria Decision Making Problems. Mathematical Problems in Engineering, Article ID 592326, 20 pp.Google Scholar
  16. 16.
    Ashmanov, S.A., and A.V. Timohov. 1991. The Optimization Theory in Problems and Exercises, 142–143. Moscow: Nauka.Google Scholar
  17. 17.
    Rockafellar, T. 1970. Convex Analysis, 468. Princeton: Princeton University Press.CrossRefzbMATHGoogle Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceMoldova State UniversityChișinăuMoldova

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