Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8829ArticlesСтатьиResearch ArticleAnalytical Approach to Analysis of Extremal Trajectories and Stability of Programmed MotionАналитический синтез экстремальных траекторий и устойчивость программного движенияAzimovD MАзимовДильмурат МухамаджановичDepartment of Mechanical EngineeringИнженерно-механический факультетazimov@hawaii.eduMukharlyamovR GМухарлямовРоберт ГарабшевичDepartment of Theoretical MechanicsКафедра теоретической механикиrobgar@mail.ruUniversity of Hawaii at ManoaГавайский университет МаноаPeoples’ Friendship University of RussiaРоссийский университет дружбы народов150420124NO4 (2012)№4 (2012)879508092016Copyright ©; 2012, Азимов Д.М., Мухарлямов Р.Г.2012Азимов Д.М., Мухарлямов Р.Г.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/8829Analytical approach to analysis of optimal trajectories of a rocket and method of determining control actions for stable programmed motion are considered. It is shown that the analytical trajectory solutions can be used to test the existence of conjugate points on extremals. Control laws that can provide stable motion are discussed.Рассматриваются задачи аналитического построения оптимальных траекторий ракеты и соответствующих управляющих воздействий, обеспечивающих устойчивое программное движение. Доказывается возможность использования аналитического решения для определения сопряжённых точек экстремальных траекторий. Приводится анализ законов управления устойчивым движением.variation problemautonomous space guidancerocketLa-grange multipliersRiccati equationJacobi conditionconjugate pointsstable motionprogrammed motioncontrolвариационная задачаавтономное управление в пространстверакетамножители Лагранжауравнение Риккатиусловия Якобисопряженные точкиустойчивое движениепрограммное движениеуправление1.Lawden D. F. Optimal Trajectories for Space Navigation. — London: Butterworths, 1963. — Pp. 55–99.2.Azimov D. M. Analytical Solutions for Intermediate Thrust Arcs of Rocket Trajectories in a Newtonian Field // Journal of Applied Mathematics and Mechanics. — 1996. — Vol. 60, No 3. — Pp. 421–427.3.Azimov D. M. Active Legs of Extremal Trajectories in Linear Central Field // Journal of Automation and Remote Control. Springer Science. — 2005. — Vol. 65, No 10. — Pp. 1533–1551.4.Marchal M. J., Winn C. B. Survey Paper-Synthesis of the Analytical Results on Optimal Transfers Between Keplerian Orbits ONERA Tire a Part. — 1967.5.Milyutin A. A., Osmolovski N. P. Calculus of Variations and Optimal Control // Translations in Mathematical Monographs. — Rhode Island: Providence, 1998. — Vol. 180. — Pp. 7–148.6.Mesherskii I. V. Works on Mechanics of Bodies with Variable Mass. — M.-L.: GITTL, 1949. — 275 p.7.Galiullin A. S. Methods of Solving Inverse Problems of Dynamics. — M.: Nauka, 1986. — 224 p.8.Bozis G., Ichtiaroglou S. Existence and Construction of Dynamical Systems having a Prescribed Integral of Motion-an Inverse Problem // Inverse Problems. — 1987. — Vol. 3. — Pp. 213–227.9.Baumgarte J. Stabilization of Constraints and Integrals of Motion in Dynamical Systems // Comp. Math. Appl. Mech. Eng. — 1972. — Vol. 1. — Pp. 1–16.10.Stabilization of Constrained Mechanical Systems with DAEs and Invariant Manifolds / U. M. Ascher, H. Chin, L. R. Petzold, S. Reich // J. Mechanics of Structures and Machines. — 1995. — Vol. 23. — Pp. 135–158.11.Mukralyamov R. G. On Construction of Systems of Differential Equations of Mechanical Systems’ Motion // Differential Equations. — 2003. — Vol. 3, No 23. — Pp. 343–353.12.Bishop R. H., Azimov D. M. Analytical Trajectories for Extremal Motion with Low-Thrust Exhaust-Modulated Propulsion // AIAA Journal of Spacecraft and Rockets. — 2001. — Vol. 381, No 61. — Pp. 897–903.13.Hull D. G. Optimal Control Theory for Applications. — New-York: Springer-Verlag New-York, Inc., 2003. — Pp. 258–301.14.Beletski V. V., Egorov V. A., Ershov M. G. Analysis of Trajectories of Interplanetary Missions with Constant Power Propulsions. — M: Nauka, 1964. — Vol. 3, Pp. 507–522.