Infinite Horizon Problems in the Calculus of Variations

The Role of Transformations with an Application to the Brachistochrone Problem
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Abstract

In this paper we consider a class of infinite horizon variational problems resulting from a transformation of singular variational problems. Herein we assume that the objective is convex. The problem setting implies a weighted Sobolev space as state space. For this class of problems we establish necessary optimality conditions in form of a Pontryagin type maximum principle. A duality concept of convex analysis is provided and used to establish sufficient optimality conditions. We apply the theoretical results proven to the problem of the Brachistochrone.

Keywords

Weighted functional spaces Calculus of variations Infinite horizon 

Mathematics Subject Classification (2010)

49J40 49K15 46N10 

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References

  1. 1.
    Aseev, S.M., Kryazhimskii, A.V.: The Pontryagin Maximum Principle and optimal economic growth problems. Proc. Steklov. Inst. Math. 257, 1–255 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Aseev, S.M., Veliov, V.M.: Maximum principle for problems with dominating discount. Dyn. Contin. Discret. Impuls. Syst. Ser. B 19(1–2b), 43–63 (2012)MATHGoogle Scholar
  3. 3.
    Aubin, J.P., Clarke, F.H.: Shadow prices and duality for a class of optimal control problems. SIAM J. Conrol Optim. 17(5), 567–586 (1979)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Balder, E.J.: An existence result for optimal economic growth problems. J. Math. Anal. Appl. 95, 195–213 (1983)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Balder, E.J.: The Brachistochrone problem made elementary. http://www.math.uu.nl/people/balder/ (2002)
  6. 6.
    Carlson, D.A., Haurie, A.B., Leizarowitz, A.: Infinite Horizon Optimal Control. Springer, New York (1991)CrossRefMATHGoogle Scholar
  7. 7.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, New York (2013)CrossRefMATHGoogle Scholar
  8. 8.
    Coleman, R.: A Detailed Analysis of the Brachistochrone Problem. arXiv:1001.2181v2[math.OC] (2012)
  9. 9.
    Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Wiley-Interscience, New York (1988)MATHGoogle Scholar
  10. 10.
    Elstrodt, J.: Maß und Integrationstheorie. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  11. 11.
    Grass, D., Caulkins, J.P., Feichtinger, G., Tragler, G., Behrens, D.A.: Optimal Control of Nonlinear Processes. Springer, Berlin (2008)CrossRefMATHGoogle Scholar
  12. 12.
    Feichtinger, G., Hartl, R.F.: Optimale Kontrolle ökonomischer Prozesse. de Gruyter, Berlin (1986)CrossRefMATHGoogle Scholar
  13. 13.
    Garg, D., Hager, W.W., Rao, A.V.: Pseudospectral methods for solving infinite-horizon optimal control problems. Automatica 47, 829–837 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Halkin, H.: Necessary conditions for optimal control problems with infinite horizons. Econometrica 42, 267–272 (1979)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Goldstine, H.H.: A History of the Calculus of Variations. Springer, New York (1980)MATHGoogle Scholar
  16. 16.
    Hall, AC: The Analysis and Synthesis of Linear Servomechanism. The Technology Press, M.I.T., Cambridge (1943)Google Scholar
  17. 17.
    Ioffe, A.D., Tichomirow, V.M.: Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979)Google Scholar
  18. 18.
    Ito, K., Kunisch, K.: Receding horizon optimal control for infinite dimensional systems. ESAIM: Control Optim. Calc. Var. 8, 741–760 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kalman, R.E.: Contribution to the theory of optimal control. Bol. Soc. Matem. Mex. 5, 102–119 (1960)MathSciNetGoogle Scholar
  20. 20.
    Kosmol, P.: Bemerkungen zur Brachistochrone. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 54, 91–94 (1984)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kufner, A.: Weighted Sobolev Spaces. Wiley, Chichester (1985)MATHGoogle Scholar
  22. 22.
    Lykina, V.: An existence theorem for a class of infinite horizon optimal control problems. J. Optim. Theory Appl. 69(1), 50–73 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lykina, V., Pickenhain, S., Wagner, M.: Different interpretations of the improper integral objective in an infinite horizon control problem. Math. Anal. Appl. 340, 498–510 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Magill, M.J.P.: Pricing infinite horizon programs. J. Math. Anal. Appl. 88, 398–421 (1982)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Michel, P.: On the transversality condition in infinite horizon optimal problems. Econometrica 50(4), 975–985 (1982)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pickenhain, S.: On adequate transversality conditions for infinite horizon optimal control problems – a famous example of Halkin. In: Crespo Cuaresma, J., Palokangas, T., Tarasyev, A. (eds.) Dynamic Systems, Economic Growth, and the Environment, pp. 3–22. Springer, Berlin (Dynamic Modelling and Econometrics in Economics and Finance 12) (2010)Google Scholar
  27. 27.
    Pickenhain, S.: Hilbert space treatment of optimal control problems with Infinite Horizon. In: Bock, H.G., Phu, H.X., Rannacher, R., Schloeder, J.P. (eds.) Modelling, Simulation and Optimization of Complex Processes - HPSC 2012, pp. 169–182. Springer, Berlin (2014)Google Scholar
  28. 28.
    Pickenhain, S.: Infinite horizon optimal control problems in the light of convex analysis in Hilbert spaces. J. Set-Valued Var. Anal. 23(1), 169–189 (2015)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Pickenhain, S., Burtchen, A., Kolo, K., Lykina, V.: An indirect pseudospectral method for linear-quadratic infinite horizon optimal control problems. Optimization 65(3), 609–633 (2016)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Pickenhain, S., Lykina, V., Wagner, M.: On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems. Control. Cybern. 37(2), 451–468 (2008)MathSciNetMATHGoogle Scholar
  31. 31.
    Lykina, V., Pickenhain, S.: Weighted functional spaces in infinite horizon optimal control problems: a systematic analysis of hidden opportunities and advantages. J. Math. Anal. Appl. 454(1), 195–218 (2017)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ramsey, F.P.: A mathematical theory of savings. Econ. J. 152(38), 543–559 (1928)CrossRefGoogle Scholar
  33. 33.
    Sontag, E.D.: Mathematical Control Theory. Springer, Berlin (1990)CrossRefMATHGoogle Scholar
  34. 34.
    Sussmann, H.J., Willems, J.C.: 300 years of optimal control: from the brachistochrone to the maximum principle. IEEE Control. Syst. Mag. 32–44 (1997)Google Scholar
  35. 35.
    Troutman, J.L.: Variational Calculus and Optimal Control, Optimization with Elementary Convexity. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  36. 36.
    Wiener, N.: Extrapolation, Interpolation and Smoothing of Stationary Time Series. MIT Press, Cambridge (1949)MATHGoogle Scholar
  37. 37.
    Yosida, K.: Functional Analysis. Springer, New York (1974)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.BTU Cottbus-SenftenbergCottbusGermany

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