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

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.

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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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Correspondence to Luís Machado.

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Machado, L., Abrunheiro, L. & Martins, N. Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres. J Optim Theory Appl 182, 965–983 (2019). https://doi.org/10.1007/s10957-018-1424-0

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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