The minimum-time correction of the satellite's orbit


Ibragimov D. N.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The minimum-time problem of the correction of the movement of the satellite is considered in this paper. The orbit of the spacecraft is assumed to be circular. Correction Correction is carried out by means of low-thrust engines capable of generating acceleration in the radial and transverse directions. Controls are assumed to be ideal and impulsive. It is proved that the initial problem can be reduced to the minimum-time problem for the linear non-stationary system with discrete time and limited control.

The solution of the minimum-time problem for a non-stationary linear discrete-time system is based on the use of the class of sets of 0-controllability — such sets of states of the system from which the origin can be reached in N steps, starting with step k. For the case when the set of feasible controls is a strictly convex body with non-empty interior, an analytic description of the sets of 0-controllability is constructed: each set can be represented as the algebraic sum of strictly convex sets, i.e. it is also a strictly convex set.

The lemma on the uniqueness of the expansion of the boundary point of the sum of two strictly convex sets is proved. It is possible to construct a criterion for the optimality of the trajectory of the system and control on the basis of this lemma. The trajectory and controls is connected with each other by a system of conjugate vectors. The obtained results are formulated in the form of the maximum principle. It is also proved that the initial state of the conjugate system is a normal to the set of 0-controllabillity, whose boundary point is the initial state of the control system. For the case, when a set of feasible controls is an ellipsoid, an explicit form of optimal control is proposed.

It is proved that if the initial state is an internal point, then the maximum principle becomes incorrect, and optimal control is not unique. Nevertheless, an algorithm is proposed that makes it possible to reduce the given case to the considered one.

The obtained theoretical results are applied to the problem of correcting the satellite’s orbit. The results of the calculations are given in the table.


ideal impulse correction, linear discrete-time nonstationary system, minimum-time problem, maximum principle


  1. Bakhshiyan B.Ts., Nazirov R.R., El’yasberg P.E. Opredelenie i korrektsiya dvizheniya (Determing and correction of the movement), Moscow, Nauka, 1980, 360 p.

  2. Bakhshiyan B.Ts. Otsenivanie i korrektsiya parametrov dvizhushchikhsya sistem (Estimation and correction of parameters of the moving systems), Moscow, Institut Kosmicheskikh Issledovanii RAN, 2012, 72 p.

  3. Malyshev V.V., Krasil’shchikov M.N., Bobronnikov V.T. Sputnikovye sistemy monitoringa (Satellite monitoring systems), Mоscоw, Izd-vo MAI, 2000, 568 p.

  4. Lebedev A.A, Krasil’shchikov M.N., Malyshev V.V. Optimal’noe upravlenie dvizheniem kosmicheskikh letatel’nykh apparatov (Optimum control over the movement of spacecrafts), Moscow, Mashinostroenie, 1974, 200 p.

  5. Malyshev V.V., Kibzun A.I. Analiz i sintez vysokotochnogo upravleniya letatel’nymi apparatami (Analysis and synthesis of high-precision control of aircraft), Moscow, Mashinostroenie, 1987, 304 p.

  6. Reshetnev M.F., Lebedev A.A., Bartenev V.A. Upravlenie i navigatsiya iskusstvennykh sputnikov Zemli na okolokrugovykh orbitakh (Control and navigation of artificial Earth satellites in circumterrestrial orbits), Moscow, Mashinstroenie, 1988, 336 p.

  7. Azanov V.M., Kan Yu.S. Trudy instituta sistemnogo analiza, 2015, no 2, pp. 18-26.

  8. Ignatov A.N. Trudy MAI, 2016, no.87, available at:

  9. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko B.F. Matematicheskaya teoriya optimal’nykh protsessov (Mathematical theory of optimal processes), Moscow, Nauka, 1969, 393 p.

  10. Boltyanskii V.G. Matematicheskie metody optimal’nogo upravleniya (Mathematical methods of optimal control), Moscow, Nauka, 1969, 408 p.

  11. Blagodatskikh V.I. Vvedenie v optimal’noe upravlenie (Introduction to the optimal control), Moscow, Vysshaja shkola, 2001, 240 p.

  12. Tabak D., Kuo B. Optimal’noe upravlenie i matematicheskoe programmirovanie (Optimum control and mathematical programming), Moscow, Nauka, 1975, 280 p.

  13. Boltyanskii V.G. Optimal’noe upravlenie diskretnymi sistemami (Optimal control of the discrete-time systems), Moscow, Nauka, 1973, 448 p.

  14. Moiseev N.N. Elementy teorii optimal’nykh sistem (Elements of the theory of the optimal processes), Moscow, Nauka, 1975, 526 p.

  15. Propoi A.I. Elementy teorii optimal’nykh diskretnykh protsessov (Elements of the theory of the optimal discrete-time processes), Moscow, Nauka, 1973, 256 p.

  16. Bellman R. E. Dinamicheskoe programmirovanie (Dynamic programming), Moscow, Izdatel’stvo Inostrannoi Literatury, 1960, 400 p.

  17. Ibragimov D.N., Sirotin A.N. Avtomatika i telemekhanika, 2015, no.9, pp. 3-30.

  18. Ibragimov D.N. Trudy MAI, 2015, no. 83, available at:

  19. Ibragimov D.N. Trudy MAI, 2016, no.87, available at:

  20. Polovinkin E.S., Balashov M.V. Elementy vypuklogo i sil’no vypuklogo analiza (Elements of convex and strongly convex analysis), Moscow, FIZMATLIT, 2004, 416 p.

  21. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza (Elements of the theory of functions and functional analysis), Moscow, FIZMATLIT, 2012, 572 p.

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