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The Hahn Quantum Variational Calculus

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Abstract

We introduce the Hahn quantum variational calculus. Necessary and sufficient optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange problems, are studied. We also show the validity of Leitmann’s direct method for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. To illustrate the results, we provide several examples and discuss a quantum version of the well known Ramsey model of economics.

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References

  1. Almeida, R., Torres, D.F.M.: Hölderian variational problems subject to integral constraints. J. Math. Anal. Appl. 359(2), 674–681 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bangerezako, G.: Variational q-calculus. J. Math. Anal. Appl. 289(2), 650–665 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bangerezako, G.: Variational calculus on q-nonuniform lattices. J. Math. Anal. Appl. 306(1), 161–179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cresson, J., Frederico, G.S.F., Torres, D.F.M.: Constants of motion for non-differentiable quantum variational problems. Topol. Methods Nonlinear Anal. 33(2), 217–231 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002)

    MATH  Google Scholar 

  6. Hahn, W.: Über orthogonalpolynome, die q-differenzenlgleichungen genügen. Math. Nachr. 2, 4–34 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge Univ. Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  8. Jackson, F.H.: Basic integration. Q. J. Math., Oxf. Ser. (2) 2, 1–16 (1951)

    Article  MATH  Google Scholar 

  9. Bird, M.T.: On generalizations of sum formulas of the Euler-Maclaurin type. Am. J. Math. 58(3), 487–503 (1936)

    Article  MathSciNet  Google Scholar 

  10. Jagerman, D.L.: Difference Equations with Applications to Queues. Dekker, New York (2000)

    Book  MATH  Google Scholar 

  11. Jordan, C.: Calculus of Finite Differences, 3rd edn. Chelsea, New York (1965). Introduction by Carver, H.C.

    MATH  Google Scholar 

  12. Álvarez-Nodarse, R.: On characterizations of classical polynomials. J. Comput. Appl. Math. 196(1), 320–337 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Costas-Santos, R.S., Marcellán, F.: Second structure relation for q-semiclassical polynomials of the Hahn tableau. J. Math. Anal. Appl. 329(1), 206–228 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dobrogowska, A., Odzijewicz, A.: Second order q-difference equations solvable by factorization method. J. Comput. Appl. Math. 193(1), 319–346 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kwon, K.H., Lee, D.W., Park, S.B., Yoo, B.H.: Hahn class orthogonal polynomials. Kyungpook Math. J. 38(2), 259–281 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Petronilho, J.: Generic formulas for the values at the singular points of some special monic classical H q,ω-orthogonal polynomials. J. Comput. Appl. Math. 205(1), 314–324 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Aldwoah, K.A.: Generalized time scales and associated difference equations. Ph.D. Thesis, Cairo University (2009)

  18. Annaby, M.H., Hamza, A.E., Aldwoah, K.A.: Hahn difference operator and associated Jackson-Nörlund integrals. Preprint (2009)

  19. Kelley, W.G., Peterson, A.C.: Difference Equations, 2nd edn. Harcourt/Academic Press, San Diego (2001)

    MATH  Google Scholar 

  20. Fort, T.: The calculus of variations applied to Nörlund’s sum. Bull. Am. Math. Soc. 43(12), 885–887 (1937)

    Article  MathSciNet  Google Scholar 

  21. Leitmann, G.: A note on absolute extrema of certain integrals. Int. J. Non-Linear Mech. 2, 55–59 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  22. Carlson, D.A.: An observation on two methods of obtaining solutions to variational problems. J. Optim. Theory Appl. 114(2), 345–361 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Carlson, D.A., Leitmann, G.: Coordinate transformation method for the extremization of multiple integrals. J. Optim. Theory Appl. 127(3), 523–533 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Carlson, D.A., Leitmann, G.: A direct method for open-loop dynamic games for affine control systems. In: Dynamic Games: Theory and Applications, pp. 37–55. Springer, New York (2005)

    Chapter  Google Scholar 

  25. Carlson, D.A., Leitmann, G.: Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations. J. Glob. Optim. 40(1–3), 41–50 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Leitmann, G.: On a class of direct optimization problems. J. Optim. Theory Appl. 108(3), 467–481 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Leitmann, G.: Some extensions to a direct optimization method. J. Optim. Theory Appl. 111(1), 1–6 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Leitmann, G.: On a method of direct optimization. Vychisl. Tekhnol. 7, 63–67 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Leitmann, G.: A direct method of optimization and its application to a class of differential games. Cubo Mat. Educ. 5(3), 219–228 (2003)

    MathSciNet  MATH  Google Scholar 

  30. Leitmann, G.: A direct method of optimization and its application to a class of differential games. Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal. 11(2–3), 191–204 (2004)

    MathSciNet  MATH  Google Scholar 

  31. Malinowska, A.B., Torres, D.F.M.: Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales. Appl. Math. Comput. (2010, in press). DOI:10.1016/j.amc.2010.01.015

  32. Silva, C.J., Torres, D.F.M.: Absolute extrema of invariant optimal control problems. Commun. Appl. Anal. 10(4), 503–515 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Torres, D.F.M., Leitmann, G.: Contrasting two transformation-based methods for obtaining absolute extrema. J. Optim. Theory Appl. 137(1), 53–59 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wagener, F.O.O.: On the Leitmann equivalent problem approach. J. Optim. Theory Appl. 142(1), 229–242 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Aldwoah, K.A., Hamza, A.E.: Difference time scales. Int. J. Math. Stat. 9(A11), 106–125 (2011)

    Google Scholar 

  36. Jackson, F.H.: On q-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)

    MATH  Google Scholar 

  37. Fort, T.: Finite Differences and Difference Equations in the Real Domain. Clarendon, Oxford (1948)

    MATH  Google Scholar 

  38. Nörlund, N.: Vorlesungen über Differencenrechnung. Springer, Berlin (1924)

    Google Scholar 

  39. Bohner, M.: Calculus of variations on time scales. Dyn. Syst. Appl. 13(3–4), 339–349 (2004)

    MathSciNet  MATH  Google Scholar 

  40. van Brunt, B.: The Calculus of Variations. Springer, New York (2004)

    MATH  Google Scholar 

  41. Arutyunov, A.V.: Optimality Conditions—Abnormal and Degenerate Problems. Kluwer Acad. Publ., Dordrecht (2000)

    MATH  Google Scholar 

  42. Torres, D.F.M.: On the Noether theorem for optimal control. Eur. J. Control 8(1), 56–63 (2002)

    Article  Google Scholar 

  43. Torres, D.F.M.: Proper extensions of Noether’s symmetry theorem for nonsmooth extremals of the calculus of variations. Commun. Pure Appl. Anal. 3(3), 491–500 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  44. Torres, D.F.M.: Carathéodory equivalence Noether theorems, and Tonelli full-regularity in the calculus of variations and optimal control. J. Math. Sci. (N. Y.) 120(1), 1032–1050 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  45. Gouveia, P.D.F., Torres, D.F.M.: Automatic computation of conservation laws in the calculus of variations and optimal control. Comput. Methods Appl. Math. 5(4), 387–409 (2005)

    MathSciNet  MATH  Google Scholar 

  46. Gouveia, P.D.F., Torres, D.F.M., Rocha, E.A.M.: Symbolic computation of variational symmetries in optimal control. Control Cybern. 35(4), 831–849 (2006)

    MathSciNet  MATH  Google Scholar 

  47. Atici, F.M., McMahan, C.S.: A comparison in the theory of calculus of variations on time scales with an application to the Ramsey model. Nonlinear Dyn. Syst. Theory 9(1), 1–10 (2009)

    MathSciNet  MATH  Google Scholar 

  48. Barro, R.J., Sala-i-Martin, X.: Economic Growth. MIT Press, Cambridge (1999)

    Google Scholar 

  49. Aulbach, B., Hilger, S.: A unified approach to continuous and discrete dynamics. In: Qualitative Theory of Differential Equations, Szeged, 1988. Colloq. Math. Soc. János Bolyai, vol. 53, pp. 37–56. North-Holland, Amsterdam (1990)

    Google Scholar 

  50. Nottale, L.: The theory of scale relativity. Int. J. Mod. Phys. A 7(20), 4899–4936 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Computer Science, Białystok University of Technology, 15-351, Białystok, Poland

    A. B. Malinowska

  2. Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    D. F. M. Torres

Authors
  1. A. B. Malinowska
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  2. D. F. M. Torres
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Corresponding author

Correspondence to D. F. M. Torres.

Additional information

Communicated by D. A. Carlson.

This work was partially supported by the Portuguese Foundation for Science and Technology (FCT) through the Center for Research and Development in Mathematics and Applications (CIDMA). The first author is currently a researcher at the University of Aveiro with the support of Białystok University of Technology, via a project of the Polish Ministry of Science and Higher Education “Wsparcie miedzynarodowej mobilnosci naukowcow”. The authors are grateful to Natália Martins for reading a preliminary version of the paper, and for many useful remarks.

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Malinowska, A.B., Torres, D.F.M. The Hahn Quantum Variational Calculus. J Optim Theory Appl 147, 419–442 (2010). https://doi.org/10.1007/s10957-010-9730-1

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