Sub-Lagrangians and sub-Hamiltonians on affine bundles. (English. Russian original) Zbl 1192.49049
J. Math. Sci., New York 161, No. 2, 261-282 (2009); translation from Sovrem. Mat. Prilozh. 61 (2008).
Summary: Sub-Lagrangians and sub-Hamiltonians are defined on anchored affine bundles as a natural extension of sub-Riemannian metrics. A duality considered between regular sub-Lagrangians and sub-Hamiltonians, gives the same solution of the Euler-Lagrange and Hamilton equations. Using the Pontryagin maximum principle, we prove that a similar situation of sub-Riemannian minimizers is encountered in this case, i.e., for a positive-definite sub-Lagrangian (sub-Hamiltonian), the locally arc-minimizing curves are either regular ones (as solutions of the Euler-Lagrange and Hamilton equations) or abnormal minimizers.
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 49N15 | Duality theory (optimization) |
| 58E25 | Applications of variational problems to control theory |
Keywords:
sub-Lagrangians; sub-Hamiltonian; sub-Riemannian metrics; duality; Pontryagin maximum principleReferences:
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