The translation of the vehicle with a controlled thrust vector to a given landing location with minimal fuel consumption

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Abstract

The problem of vehicle’s translation to a certain landing location above the surface of the planet is considered. Using the Pontryagin maximum principle, the optimal control problem is reduced to a boundary value problem for a system of nonlinear differential equations. A qualitative analysis of the optimal phase trajectories of the system is carried out, their properties are established, illustrated by the results of numerical modeling. The domains in the plane of phase variables are analytically described, from which it is possible to achieve a terminal set. A synthesis of optimal control is constructed.

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About the authors

N. A. Oryol

Lomonosov Moscow State University

Author for correspondence.
Email: nikita.orel@math.msu.ru
Russian Federation, Moscow

O. Yu. Cherkasov

Shenzhen MSU-BIT University; Lomonosov Moscow State University

Email: oyuche@yandex.ru
China, Shenzhen; Moscow, Russia

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Supplementary files

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2. Fig. 1. Setting the task of selecting a landing site.

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3. Fig. 2. Phase portraits of the system (2.11): a - a = 0.5, b - a = 1.

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4. Fig. 3. Mutual arrangement of the terminal set and constraints u = ±u in the plane (u, v) at a = 1 in the case of free v (T).

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5. Fig. 4. In support of statements 1 and 2.

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6. Fig. 5. Results of numerical modelling at x (0) = 0, v (0) = 0, T = 0.5; 1; 1.5: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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7. Fig. 6. Results of numerical modelling at x (0) = 0, v (0) = 1.5, T = 0.5; 1; 1.5: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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8. Fig. 7. Results of numerical modelling at x (0) = 0, v (0) = 0.3, T = 2, u = 0.5; 1: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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9. Fig. 8. Results of numerical modelling at x (0) = 0, v (0) = 1.5, T = 1.44, u = 0.5: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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10. Fig. 9. Results of numerical modelling at x (0) = 0, v (0) = v (T) = 0, T = 1; 1.5; 2; 3: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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11. Fig. 10. Results of numerical modelling at x (0) = 0, v (0) = 0.3, v (T) = 0, T = 1; 1.5; 2: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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12. Fig. 11. Results of numerical modelling at x (0) = 0, v (0) = 1.5, v (T) = 0.7, T = 1; 1.5: a – trajectories in the (u, v) plane, b – trajectories in the (t, u) plane, c – trajectories in the (t, v) plane, d – trajectories in the (t, x) plane.

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13. Fig. 12. Mutual arrangement of the terminal set and constraints u = ±u in the plane (u, v) at a = 1 in the case of fixed v (T).

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14. Fig. 13. Results of numerical modelling at x (0) = 0, v (0) = v (T) = 0, T = 2: a - trajectories in the (u, v) plane, b - trajectories in the (t, u) plane, c - trajectories in the (t, v) plane, d - trajectories in the (t, x) plane.

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