Journal of Mathematical Sciences

, Volume 217, Issue 6, pp 672–682 | Cite as

Mix of Controls and the Pontryagin Maximum Principle

  • E. R. AvakovEmail author
  • G. G. Magaril-Il’yaev


In this paper, necessary conditions for a minimum (the Pontryagin maximum principle) for an optimal control problem are proved on the basis of the concept of a mix, which enables one to reduce the study of the original problem to some approximation thereof, which is linear in the control. The study of the latter problem proves more simple.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control, Consultants Bureau, New York (1987).CrossRefzbMATHGoogle Scholar
  2. 2.
    E. R. Avakov, G. G. Magaril-Il’yaev, and V. M. Tikhomirov, “Lagrange’s principle in extremum problems with constraints,” Russ. Math. Surv., 68, No. 3, 401–433 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    G. G. Magaril-Il’yaev, “The Pontryagin maximum principle: statement and proof,” Dokl. Math., 85, No. 1, 14–17 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G. G. Magaril-Il’yaev and V. M. Tikhomirov, “Newton’s method, differential equations, and the Lagrangian principle for necessary extremum conditions,” Proc. Steklov Inst. Math., 262, 149–169 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    V. M. Tikhomirov, “Tangent space theorem and some of its modifications,” Optim. Upr., No. 7, 22–30 (1977).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Control Sciences, Russian Academy of SciencesMoscowRussia
  2. 2.Institute for Information Transmission Problems, Russian Academy of Sciences; Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations