Geodesic
Singular regimes in controlled systems with multidimensional control in a~polyhedron
Izvestiya. Mathematics , Tome 78 (2014) no. 5, pp. 1006-1027.

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We study Hamiltonian systems that are affine in a multidimensional control varying in a polyhedron $\Omega$. Quite often, a crucial role in the study of the global behaviour of solutions of such systems is played by special trajectories and the geometry of their neighbourhoods. We prove a theorem on the structure of the output of optimal trajectories to a first-order singular trajectory in a neighbourhood of this trajectory (and of the exit from it) for systems with holonomic control. We also prove that in a neighbourhood of a first-order singular trajectory, a Lagrangian surface is woven in a special way from the trajectories of the system that are singular with respect to the faces of $\Omega$. We suggest a simple way to find explicitly first-order special trajectories with respect to the faces of $\Omega$. As a result, we describe a complete picture of the optimal synthesis obtained by the successive conjugation of first-order singular extremals.
Keywords: optimal control, singular trajectories, multidimensional control, optimal synthesis. 
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L. V. Lokutsievskii. Singular regimes in controlled systems with multidimensional control in a~polyhedron. Izvestiya. Mathematics , Tome 78 (2014) no. 5, pp. 1006-1027. http://geodesic.mathdoc.fr/item/IM2_2014_78_5_a6/

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