Journal of Dynamical and Control Systems

, Volume 11, Issue 2, pp 157–176 | Cite as

Attainable Set of a Nonlinear Controlled Microeconomic Model

  • E. V. Grigorieva
  • E. N. Khailov
Original Article


In this paper, a nonlinear dynamical controlled system of differential equations describing the process of production and sales of a perishable consumer good is studied. The price of the good is the control parameter. Attainable sets of the system are investigated. It is proved that only bang-bang controls with at most one switching can lead trajectories to the boundary of an attainable set. Thus, this boundary is formed by the union of two one-parameter curves; the parameter is the moment of switching. Attainable sets for different parameters of the model are constructed with the use of a computer program written in MAPLE. Possible economic applications are discussed.

Key words and phrases:

Attainable set nonlinear controlled system microeconomic dynamical model 


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  1. 1.
    1. P. S. Alexandrov, Elementary concepts of topology. Dover Publications (1977).Google Scholar
  2. 2.
    2. S. N. Avvakumov and Yu. N. Kiselev, Qualitative study and algorithms in the mathematical model of innovation diffusion. J. Math. Sci. 116 (2003), No. 6, 3657–3672.Google Scholar
  3. 3.
    3. F. L. Chernousko, State estimation for dynamical systems. CRS Press, Boca Raton, Florida (1994).Google Scholar
  4. 4.
    4. F. Colonius and D. Szolnoki, Algorithms for computing reachable sets and control sets. In: Proc. IFAC Symposium on Nonlinear Control Systems (NOLCOS 2001), 4–6 July 2001, St. Petersburg, Russia, pp. 756–761.Google Scholar
  5. 5.
    5. A. V. Dmitruk, A generalized estimate on the number of zeros for solutions of a class of linear differential equations. SIAM J. Control Optim. 30 (1992), No. 5, 1087–1091.Google Scholar
  6. 6.
    6. A. A. Gorski, I. G. Kolpakova, and B. Ya. Lokshin, A dynamic model of a manufacturing, storage, and marketing processes. J. Comput. System Sci. 31 (1993), 153–157.Google Scholar
  7. 7.
    7. ———, A dynamic model of production, storage, and sale of daily demand goods. J. Comput. System Sci. 37 (1998), 137–143.Google Scholar
  8. 8.
    8. E V. Grigorieva and E. N. Khailov, On the attainability set for a nonlinear system in the plane. Moscow Univ., Comput. Math. Cyb. (2001), No. 4, 27–32.Google Scholar
  9. 9.
    9. O. Hajek, Control theory in the plane. Lect. Notes Control Inform. Sci. 153 (1991).Google Scholar
  10. 10.
    10. P. Hartman, Ordinary differential equations. John Wiley & Sons, New York (1964).Google Scholar
  11. 11.
    11. E. B. Lee and L. Markus, Foundations of optimal control theory. John Wiley & Sons, New York (1970). Google Scholar
  12. 12.
    12. A. Yu. Levin, Nonoscillating solution of the equation x(n) + p1(t) x(n−1) + ··· + pn (t)x = 0. Usp. Mat. Nauk 24 (1969), No. 2, 43–96.Google Scholar
  13. 13.
    13. A. I. Panasyuk and V. I. Panasyuk, Asymptotic turnpike optimization of control systems. Nauka, Minsk (1986).Google Scholar
  14. 14.
    14. T. Partasarathy, On global univariance theorems. Lect. Notes Math. 977 (1983).Google Scholar
  15. 15.
    15. I. Shigeo, A note on global implicit function theorems. IEEE Trans. Circuits Systems 32 (1985), No. 5, 503–505.Google Scholar
  16. 16.
    16. D. Szolnoki, Set oriented methods for computing reachable sets and control sets. Discrete Contin. Dynam. Syst. Ser. B 3 (2003), No. 3, 361–382.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceTexas Woman’s UniversityDentonUSA
  2. 2.Department of Computer Mathematics and CyberneticsMoscow State UniversityMoscowRussia

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