Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres
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.
KeywordsVariational problems of Herglotz type Higher-order variational problems Higher-order optimal control problems Riemannian cubic polynomials Euclidean sphere
Mathematics Subject Classification49K15 49S05 53B21 34H05
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.
- 2.Herglotz, G.: Berührungstransformationen. Lectures at the University of Göttingen, Göttingen (1930)Google Scholar
- 10.Santos, S.P.S., Martins, N., Torres, D.F.M.: An optimal control approach to Herglotz variational problems. Optimization in the natural sciences. Commun. Comput. Inf. Sci. (CCIS) 449, 107–117 (2015)Google Scholar
- 11.Santos, S.P.S., Martins, N., Torres, D.F.M.: Noether’s theorem for higher-order variational problems of Herglotz type. In: 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, vol. 2015, pp. 990–999 (2015)Google Scholar
- 28.Bloch, A.M, Crouch, P.E.: Reduction of Euler–Lagrange problems for constrained variational problems and relation with optimal control problems. In: Proceedings of the 33rd Conference on Decision and Control, Orlando, FL, pp. 2584–2590 (1994)Google Scholar
- 30.Bloch, A.M., Crouch, P.E.: On the equivalence of higher order variational problems and optimal control problems. In: Proceedings of the 35th Conference on Decision and Control, Kobe, Japan, pp. 1648–1653 (1996)Google Scholar
- 33.de León, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives. North-Holland Mathematics Studies, Nachbin, L. (ed.), vol. 112. Elsiever, Amsterdam (1985)Google Scholar
- 38.Abrunheiro, L. Camarinha, M., Clemente-Gallardo, J.: Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration. In: The Cape Verde International Days on Mathematics 2013, Praia, Cape Verde, Conference Papers in Mathematics, pp. 1–9. Hindawi Publishing Corporation (2013)Google Scholar