Abstract:
In this paper, we consider the minimum time problem for a space rocket whose
dynamics is given by a control-affine system with drift. The admissible control set is a disc. We
study extremals in the neighbourhood of singular points of the second order. Our approach is
based on applying the method of a descending system of Poisson brackets and the Zelikin –
Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s
maximum principle. We show that in the neighbourhood of any singular point there is a family
of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time,
while the control performs an infinite number of rotations around the circle.
Keywords:
Hamiltonian system of Pontryagin’s maximum principle, singular extremal, control-affine system with drift, descending system of Poisson brackets, resolution of singularity, blow-up, coupled attitude orbit problem.
Citation:
Mariya I. Ronzhina, Larisa A. Manita, “Spiral-Like Extremals near a Singular Surface
in a Rocket Control Problem”, Regul. Chaotic Dyn., 28:2 (2023), 148–161
\Bibitem{RonMan23}
\by Mariya I. Ronzhina, Larisa A. Manita
\paper Spiral-Like Extremals near a Singular Surface
in a Rocket Control Problem
\jour Regul. Chaotic Dyn.
\yr 2023
\vol 28
\issue 2
\pages 148--161
\mathnet{http://mi.mathnet.ru/rcd1199}
\crossref{https://doi.org/10.1134/S1560354723020028}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4572230}
Linking options:
https://www.mathnet.ru/eng/rcd1199
https://www.mathnet.ru/eng/rcd/v28/i2/p148
This publication is cited in the following 5 articles:
M. I. Ronzhina, L. A. Manita, “Strukturnaya ustoichivost logarifmicheskikh spiralei v zadachakh upravleniya s osoboi ekstremalyu vtorogo poryadka”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 35:1 (2025), 117–128
M. I. Ronzhina, L. A. Manita, “Logarifmicheskie spirali v zadachakh optimalnogo upravleniya s upravleniem iz kruga”, Materialy Voronezhskoi mezhdunarodnoi vesennei matematicheskoi shkoly «Sovremennye metody kraevykh zadach.
Pontryaginskie chteniya—XXXIV», Voronezh, 3-9 maya 2023 g. Chast 4, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 233, VINITI RAN, M., 2024, 75–88
M. I Ronzhina, L. A Manita, “FAMILY OF LOGARITHMIC SPIRALS IN HAMILTONIAN SYSTEMS OF DIMENSION 8 WITH CONTROL IN A DISK”, Differencialʹnye uravneniâ, 60:11 (2024), 1531
M. I. Ronzhina, L. A. Manita, “Family of Logarithmic Spirals in Hamiltonian Systems
of Dimension 8 with Control in a Disk”, Diff Equat, 60:11 (2024), 1603
D. A. Tarkhov, D. A. Lavygin, O. A. Skripkin, M. D. Zakirova, T. V. Lazovskaya, “Optimal Control Selection for Stabilizing the Inverted Pendulum Problem Using Neural Network Method”, Opt. Mem. Neural Networks, 32:S2 (2023), S214
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