Study of a One-Dimensional Optimal Control Problem with a Purely State-Dependent Cost

  • A. V. Dmitruk
  • A. K. VdovinaEmail author
Original Research


A one-dimensional optimal control problem with a state-dependent cost and a unimodular integrand is considered. It is shown that, under some standard assumptions, this problem can be solved without using the Pontryagin maximum principle, by simple methods of the classical analysis, basing on the Tchyaplygin comparison theorem. However, in some modifications of the problem, the usage of Pontryagin’s maximum principle is preferable. The optimal synthesis for the problem and for its modifications is obtained.


Optimal control One-dimensional state Unimodular integrand State-dependent cost Tchyaplygin comparison theorem Pontryagin maximum principle 

Mathematics Subject Classification

49K21 49K15 90C30 



This research was partially supported by the Russian Foundation for Basic Research under grants 14-01-00784 and 16-01-00585.


  1. 1.
    Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, R.V., Mischenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York-London (1962). (translated from Russian)Google Scholar
  2. 2.
    Tchaplygin, S.A.: A new method for approximate integration of differential equations. In: “S.A. Tchaplygin. Collected works”, pp. 307–322. Nauka, Moscow (1976) (in Russian)Google Scholar
  3. 3.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York-London (1964)zbMATHGoogle Scholar
  4. 4.
    Filippov, A.F.: On some questions of the optimal regulating theory. Vestnik Moscov. Univ. Ser. Math-Mech. 3(2), 25–32 (1962). English translation in SIAM J. on Control, 1(1), pp. 76–84 (1962)Google Scholar
  5. 5.
    Sethi, S.P.: Nearest feasible paths in optimal control problems: theory. Ex. Count. Ex. JOTA 23(4), 563–579 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Sanders, J.: A note on optimal fish harvest. JOTA 24(2), 361–369 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Clark, C.W., De Pree, J.D.: A simple linear model for the optimal exploration of renewable resourses. Appl. Math. Optim. 5, 181–196 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ashmanov, S.A.: Introduction to Mathematical Economics. Moscow State University, Moscow(1980) (in Russian)Google Scholar
  9. 9.
    Cesari, L.: Optimization Theory and Applications. Springer, Berlin (1983)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hartl, R.F., Feichtinger, G.: A new sufficient conditions for most rapid paths. JOTA 54(2), 403–411 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Seierstad, A,K.: Sydsaeter: Optimal Control Theory with Economic Applications. North-Holland, Amsterdam (1987)Google Scholar
  12. 12.
    Kamien, M.I., Schwartz, N.L.: Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. Elsevier, Amsterdam (1991)zbMATHGoogle Scholar
  13. 13.
    Weitzman, M.L.: Income, Wealth, and the Maximum Principle. Harvard University Press, Cambridge (2003)zbMATHGoogle Scholar
  14. 14.
    Sethi, S.P., Thompson, G.L.: Optimal Control Theory. Springer, Berlin (2005)Google Scholar
  15. 15.
    Geering, H.P.: Optimal Control with Engineering Applications. Springer, Berlin (2007)zbMATHGoogle Scholar
  16. 16.
    Demin, N.S., Kuleshova, E.V.: Control of single-sector economy over a finite time interval with allowance for employer consumption. Autom Remote Control 69(9), 1576–1590 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Weber, T.A.: Optimal Control Theory with Applications in Economics. MIT Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hritonenko, N., Yatsenko, Y.: Mathematical Modeling in Economics, Ecology and the Environment. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2016

Authors and Affiliations

  1. 1.Russian Academy of Sciences, Central Economics and Mathematics InstituteMoscowRussia
  2. 2.Lomonosov Moscow State UniversityMoscowRussia

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