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Tr. Mat. Inst. Steklova, 2002, Volume 236, Pages 174–196 (Mi tm288)  

The Structure of Optimal Synthesis in the Vicinity of Singular Manifolds for Problems Affine with Respect to Control

M. I. Zelikin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A classification of phase portraits of optimal synthesis in the vicinity of universal singular manifolds is considered for systems of constant rank that are affine with respect to the control. Both the phase state and the control are assumed to be multidimensional. The classification is based on the order of singular extremals and on the involutiveness (or noninvolutiveness) of the velocity indicatrix. It is shown that the synthesis of optimal trajectories is a fibered space over the base $W$ formed by singular optimal trajectories; the fibers consist of nonsingular optimal trajectories. For a multidimensional control, the singular manifold $W$ is stratified. In the involutive case, the fibers are one-dimensional. In the noninvolutive case, the fibers are multidimensional and contain trajectories with switching at increasing frequency (chattering trajectories); the dimension of the fibers and the structure of the field of trajectories inside the fibers depend on the order of the singular extremals. Application of the theory developed to classical problems of the mechanics of controlled systems and to the evaluation of exact constants in Kolmogorov-type inequalities for derivatives is described.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 236, 164–185

Bibliographic databases:
UDC: 517.977
Received in December 2000

Citation: M. I. Zelikin, “The Structure of Optimal Synthesis in the Vicinity of Singular Manifolds for Problems Affine with Respect to Control”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 236, Nauka, MAIK Nauka/Inteperiodika, M., 2002, 174–196; Proc. Steklov Inst. Math., 236 (2002), 164–185

Citation in format AMSBIB
\Bibitem{Zel02}
\by M.~I.~Zelikin
\paper The Structure of Optimal Synthesis in the Vicinity of Singular Manifolds for Problems Affine with Respect to Control
\inbook Differential equations and dynamical systems
\bookinfo Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko
\serial Tr. Mat. Inst. Steklova
\yr 2002
\vol 236
\pages 174--196
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm288}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1931018}
\zmath{https://zbmath.org/?q=an:1024.49022}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 236
\pages 164--185


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