The Tent Method in Finite-Dimensional Spaces

Part of the Systems & Control: Foundations & Applications book series (SCFA)


The Tent Method is shown to be a general tool for solving a wide spectrum of extremal problems. First, we show its workability in finite-dimensional spaces. Then topology is applied for the justification of some results in variational calculus. A short historical remark on the Tent Method is made and the idea of the proof of the Maximum Principle is explained in detail, paying special attention to the necessary topological tools. The finite-dimensional version of the Tent Method allows one to establish the Maximum Principle and to obtain a generalization of the Kuhn–Tucker Theorem in Euclidean spaces.


Maximum Principle Extremal Problem Variational Calculus Conditional Extremum Tucker Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Boltyanski, V. (1958), ‘The maximum principle in the theory of optimal processes’, Dokl. Akad. Nauk SSSR 119(6), 1070–1073 (in Russian). MathSciNetGoogle Scholar
  2. Boltyanski, V. (1975), ‘The tent method in the theory of extremal problems’, Usp. Mat. Nauk 30, 3–65 (in Russian). Google Scholar
  3. Boltyanski, V. (1985), ‘Separation of a system of convex cones in a topological vector space’, Dokl. Akad. Nauk SSSR 283(5), 1044–1047 (in Russian). MathSciNetGoogle Scholar
  4. Boltyanski, V. (1986), ‘The method of tents in topological vector spaces’, Dokl. Akad. Nauk SSSR 289(5), 1036–1039 (in Russian). MathSciNetGoogle Scholar
  5. Boltyanski, V., & Poznyak, A. (1999b), ‘Robust maximum principle in minimax control’, Int. J. Control 72(4), 305–314. MATHCrossRefGoogle Scholar
  6. Boltyanski, V., Martini, H., & Soltan, V. (1999), Geometric Methods and Optimization Problems, Kluwer Academic, Dordrecht. MATHGoogle Scholar
  7. Bressan, A., & Piccoli, B. (2007), Introduction to the Mathematical Theory of Control, Vol. 2 of Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield. MATHGoogle Scholar
  8. Dubovitski, A., & Milyutin, A. (1963), ‘Extremum problems with constrains’, Dokl. Akad. Nauk SSSR 149(4), 759–762 (in Russian). Google Scholar
  9. Dubovitski, A., & Milyutin, A. (1965), ‘Extremum problems with constrains’, Zh. Vychisl. Mat. Mat. Fiz. 5(3), 395–453. Google Scholar
  10. Fattorini, H.O. (1999), Infinite-Dimensional Optimization and Control Theory, Vol. 62 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge. MATHGoogle Scholar
  11. Feldbaum, A. (1953), ‘Optimal processes in systems of automatic control’, Avtom. Telemeh. 14(6), 712–728 (in Russian). Google Scholar
  12. Hilton, P. (1988), ‘A brief, subjective history of homology and homotopy theory in this century’, Math. Mag. 60(5), 282–291. CrossRefGoogle Scholar
  13. Kuhn, H., & Tucker, A. (1951), ‘Nonlinear programming’, in Proceedings of the Second Berkeley Symposium on Math. Statistics and Probability, University of California Press, Berkeley, pp. 481–492. Google Scholar
  14. McShane, E. (1978), ‘The calculus of variations from the beginning through optimal control theory’, in Optimal Control and Differential Equations (A.B. Schwarzkopf, W.G. Kelley & S.B. Eliason, eds.), University of Oklahoma Press, Norman. Google Scholar
  15. Neustadt, L. (1969), ‘A general theory of extremals’, J. Comput. Syst. Sci. 3(1), 57–92. MathSciNetMATHCrossRefGoogle Scholar
  16. Polyak, B.T. (1987), Introduction to Optimization, Optimization Software Publication Division, New York. Google Scholar
  17. Pontryagin, L.S., Boltyansky, V.G., Gamkrelidze, R.V., & Mishenko, E.F. (1969), Mathematical Theory of Optimal Processes, Nauka, Moscow (in Russian). Google Scholar
  18. Poznyak, A.S. (2008), Advanced Mathematical Tools for Automatic Control Engineers, Vol. 1: Deterministic Technique, Elsevier, Amsterdam. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Vladimir G. Boltyanski
    • 1
  • Alexander S. Poznyak
    • 2
  1. 1.CIMATGuanajuatoMexico
  2. 2.Automatic Control DepartmentCINVESTAV-IPNMéxicoMexico

Personalised recommendations