The Hamiltonian property of the flow of singular trajectories

L. V. Lokutsievskii

Sbornik. Mathematics, Tome 205 (2014) no. 3, pp. 432-458

Résumé

Pontryagin's maximum principle reduces optimal control problems to the investigation of Hamiltonian systems of ordinary differential equations with discontinuous right-hand side. An optimal synthesis is the totality of solutions to this system with a fixed terminal (or initial) condition, which fill a region in the phase space one-to-one. In the construction of optimal synthesis, singular trajectories that go along the discontinuity surface $N$ of the right-hand side of the Hamiltonian system of ordinary differential equations, are crucial. The aim of the paper is to prove that the system of singular trajectories makes up a Hamiltonian flow on a submanifold of $N$. In particular, it is proved that the flow of singular trajectories in the problem of control of the magnetized Lagrange top in a variable magnetic field is completely Liouville integrable and can be embedded in the flow of a smooth superintegrable Hamiltonian system in the ambient space. Bibliography: 17 titles.

Détail

Keywords: singular trajectories, singular extremals, Hamiltonian systems, integrable and superintegrable systems
Mots-clés : Lagrange top.