Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
The Classical Calculus of Variations Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

2. The Hahn Quantum Variational Calculus

verfasst von : Agnieszka B. Malinowska, Delfim F. M. Torres

Erschienen in: Quantum Variational Calculus

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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 (Almeida and Torres 2010b; Carlson and Leitmann 2005a, b, 2008; Leitmann 2002, 2003) for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. Next, we prove a necessary optimality condition of Euler–Lagrange type for quantum variational problems involving Hahn’s derivatives of higher-order. Finally, we extend the previous results and obtain optimality conditions for generalized quantum variational problems with a Lagrangian depending on the free end-points. To illustrate the results, we provide several examples and discuss quantum versions of the Ramsey model and an adjustment model in economics.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel The Classical Calculus of Variations
Nächstes Kapitel The Power Quantum Calculus
Literatur
Aldwoah KA (2009) Generalized time scales and associated difference equations. Ph.D. Thesis, Cairo University, Cairo
Aldwoah KA, Hamza AE (2011) Difference time scales. Int J Math Stat 9(A11):106–125
Almeida R, Torres DFM (2009a) Hölderian variational problems subject to integral constraints. J Math Anal Appl 359(2):674–681
Almeida R, Torres DFM (2009b) Isoperimetric problems on time scales with nabla derivatives. J Vib Control 15(6):951–958
Almeida R, Torres DFM (2010a) Generalized Euler-Lagrange equations for variational problems with scale derivatives. Lett Math Phys 92(3):221–229
Almeida R, Torres DFM (2010b) Leitmann’s direct method for fractional optimization problems. Appl Math Comput 217(3):956–962
Annaby MH, Hamza AE, Aldwoah KA (2012) Hahn difference operator and associated Jackson-Nörlund integrals. J Optim Theory Appl 154(1):133–153MathSciNetCrossRefMATH
Arutyunov AV (2000) Optimality conditions—abnormal and degenerate problems. Kluwer Academic Publishers, DordrechtCrossRefMATH
Atici FM, McMahan CS (2009) 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):1–10MathSciNetMATH
Barro RJ, Sala-i-Martin X (1999) Economic growth. MIT Press, Cambridge
Caputo MR (2005) Foundations of dynamic economic analysis: optimal control theory and applications. Cambridge University Press, CambridgeCrossRef
Carlson DA (2002) An observation on two methods of obtaining solutions to variational problems. J Optim Theory Appl 114(2):345–361MathSciNetCrossRefMATH
Carlson DA, Leitmann G (2005a) Coordinate transformation method for the extremization of multiple integrals. J Optim Theory Appl 127(3):523–533
Carlson DA, Leitmann G (2005b) A direct method for open-loop dynamic games for affine control systems. Dynamic games: theory and applications. Springer, New York, pp 37–55
Carlson DA, Leitmann G (2008) Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations. J Global Optim 40(1–3):41–50MathSciNetCrossRef
Cresson J, Frederico GSF, Torres DFM (2009) Constants of motion for non-differentiable quantum variational problems. Topol Methods Nonlinear Anal 33(2):217–231MathSciNetMATH
Cruz PAF, Torres DFM, Zinober ASI (2010) A non-classical class of variational problems. Int J Math Model Numer Optim 1(3):227–236MATH
Ferreira RAC, Torres DFM (2008) Higher-order calculus of variations on time scales. Mathematical control theory and finance. Springer, Berlin, pp 149–159
Gouveia PDF, Torres DFM (2005) Automatic computation of conservation laws in the calculus of variations and optimal control. Comput Methods Appl Math 5(4):387–409MathSciNetMATH
Gouveia PDF, Torres DFM, Rocha EAM (2006) Symbolic computation of variational symmetries in optimal control. Control Cybernet 35(4):831–849MathSciNetMATH
Koornwinder TH (1994) Compact quantum groups and \(q\)-special functions. Representations of Lie groups and quantum groups (Trento, 1993). Longman Sci Tech, Harlow, pp 46–128
Leitmann G (2001a) On a class of direct optimization problems. J Optim Theory Appl 108(3):467–481
Leitmann G (2001b) Some extensions to a direct optimization method. J Optim Theory Appl 111(1):1–6
Leitmann G (2003) A direct method of optimization and its application to a class of differential games. Cubo Mat Educ 5(3):219–228MathSciNetMATH
Leitmann G (2004) 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):191–204MathSciNetMATH
Malinowska AB, Torres DFM (2009) Strong minimizers of the calculus of variations on time scales and the Weierstrass condition. Proc Est Acad Sci 58(4):205–212MathSciNetCrossRefMATH
Malinowska AB, Torres DFM (2010a) Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales. Appl Math Comput 217(3):1158–1162
Malinowska AB, Torres DFM (2010b) Natural boundary conditions in the calculus of variations. Math Methods Appl Sci 33(14):1712–1722
Malinowska AB, Torres DFM (2010c) The Hahn quantum variational calculus. J Optim Theory Appl 147(3):419–442
Sengupta JK (1997) Recent models in dynamic economics: problems of estimating terminal conditions. Int J Syst Sci 28:857–864CrossRefMATH
Silva CJ, Torres DFM (2006) Absolute extrema of invariant optimal control problems. Commun Appl Anal 10(4):503–515MathSciNetMATH
Torres DFM (2002) On the Noether theorem for optimal control. Eur J Control 8(1):56–63CrossRef
Torres DFM (2004a) Proper extensions of Noether’s symmetry theorem for nonsmooth extremals of the calculus of variations. Commun Pure Appl Anal 3(3):491–500
Torres DFM (2004b) 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
Torres DFM, Leitmann G (2008) Contrasting two transformation-based methods for obtaining absolute extrema. J Optim Theory Appl 137(1):53–59MathSciNetCrossRefMATH
van Brunt B (2004) The calculus of variations. Springer, New YorkMATH
Metadaten
Titel
The Hahn Quantum Variational Calculus
verfasst von
Agnieszka B. Malinowska
Delfim F. M. Torres
Copyright-Jahr
2014
Buch
Quantum Variational Calculus

Print ISBN: 978-3-319-02746-3

Electronic ISBN: 978-3-319-02747-0

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-02747-0_2

Premium Partner

    Bildnachweise