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Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres

  • Luís Machado
  • Lígia Abrunheiro
  • Natália Martins
Article

Abstract

The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere \(S^n\) following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler–Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold \(S^n\) such as the problem of finding cubic polynomials on \(S^n\). It also finds applicability on the dynamics of the simple pendulum in a resistive medium.

Keywords

Variational problems of Herglotz type Higher-order variational problems Higher-order optimal control problems Riemannian cubic polynomials Euclidean sphere 

Mathematics Subject Classification

49K15 49S05 53B21 34H05 

Notes

Acknowledgements

The work of Lígia Abrunheiro and Natália Martins was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project UID/MAT/04106/2013. Luís Machado acknowledges “Fundação para a Ciência e a Tecnologia” (FCT–Portugal) and COMPETE 2020 Program for financial support through project UID-EEA-00048-2013.

The authors would like to thank the reviewers for their valuable suggestions to improve the quality of the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Luís Machado
    • 1
    • 2
  • Lígia Abrunheiro
    • 3
  • Natália Martins
    • 4
  1. 1.Institute of Systems and RoboticsUniversity of CoimbraCoimbraPortugal
  2. 2.Department of MathematicsUniversity of Trás-os-Montes e Alto Douro (UTAD)Vila RealPortugal
  3. 3.Center for Research and Development in Mathematics and Applications (CIDMA), Higher Institute of Accounting and AdministrationUniversity of AveiroAveiroPortugal
  4. 4.Center for Research and Development in Mathematics and Applications (CIDMA), Department of MathematicsUniversity of AveiroAveiroPortugal

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