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June
2017, 7(2): 185-189.
doi: 10.3934/naco.2017012
The optimal stabilization of orbital motion in a neighborhood of collinear libration point
Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034 Russia |
In this paper we consider the special problem of stabilization of controllable orbital motion in a neighborhood of collinear libration point $L_2$ of Sun-Earth system. The modification of circular three-body problem -nonlinear Hill's equations, which describe orbital motion in a neighborhood of libration point is used as a mathematical model. Also, we used the linearized equations of motion. We investigate the problem of spacecraft arrival on the unstable invariant manifold. When a spacecraft reaches this manifold, it does not leave the neighborhood of $L_2$ by long time. The distance to the unstable invariant manifold is described by a special function of phase variables, so-called ''hazard function". The control action directed along Sun-Earth line.
Keywords:
Circular three-body problem,
libration point,
Hill's equations,
Bellman function,
control,
stabilization.
Mathematics Subject Classification:
Primary: 49J15, 37J25; Secondary: 70F07.
Citation:
Alexander Shmyrov, Vasily Shmyrov. The optimal stabilization of orbital motion in a neighborhood of collinear libration point. Numerical Algebra, Control and Optimization,
2017, 7
(2)
: 185-189.
doi: 10.3934/naco.2017012
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References:
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V. N. Afanas'yev, V. V. Kolmanovsky and V. R. Nosov,
Mathematical Theory of Control-Systems Design, Vysshaya Shkola, Moscow, 2003. |
| [2] |
G. Gomez, J. Llibre, R. Martinez and C. Simo,
Dynamics and mission design near libration points. Vol. 1. Fundamentals: The case of collinear libration points, World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 2001. |
| [3] |
J. Guckenheimer and P. Holmes,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
| [4] |
C. Simo and T. J. Stuchi,
Central stable/unstable manifolds and the destruction of KAM Tori in the Planar Hill Problem, Physica D, 140 (2000), 1-32.
doi: 10.1016/S0167-2789(99)00211-0. |
| [5] |
A. Shmyrov and V. Shmyrov, Controllable orbital motion in a neighborhood of collinear
libration point, Applied Mathematical Sciences, 8 (2014), 487–492. Available from: http://dx.doi.org/10.12988/ams.2014.312711.
doi: 10.12988/ams.2014.312711. |
| [6] |
A. Shmyrov and V. Shmyrov, On controllability region of orbital motion near L1, Applied
Mathematical Sciences, 9 (2015), 7229–7236. Available from: http://dx.doi.org/10.12988/ams.2015.510638.
doi: 10.12988/ams.2015.510638. |
| [7] |
A. Shmyrov and V. Shmyrov, The criteria of quality in the problem of motion stabilization in
a neighborhood of collinear libration point, Iternational Conference on "Stability and Control
Processes" in Memory of V. I. Zubov (SCP 2015), 4-9 October 2015, (2015), art. 7342135,
345–347. Available from: http://dx.doi.org/10.1109/SCP.2015.7342135.
doi: 10.1109/SCP.2015.7342135. |
| [8] |
A. Shmyrov and V. Shmyrov,
The estimation of controllability area in the problem of controllable movement in a neighborhood of collinear libration point
2015 International Conference on Mechanics -Seventh Polyakhov's Reading, 2-6 February 2015 (2015), art. 7106776. Available from: http://dx.doi.org/10.1109/POLYAKHOV.2015.7106776. |
| [9] |
V. I. Zubov,
Lectures on Control Theory, Nauka, Moscow, 1975. |
show all references
References:
| [1] |
V. N. Afanas'yev, V. V. Kolmanovsky and V. R. Nosov,
Mathematical Theory of Control-Systems Design, Vysshaya Shkola, Moscow, 2003. |
| [2] |
G. Gomez, J. Llibre, R. Martinez and C. Simo,
Dynamics and mission design near libration points. Vol. 1. Fundamentals: The case of collinear libration points, World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 2001. |
| [3] |
J. Guckenheimer and P. Holmes,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
| [4] |
C. Simo and T. J. Stuchi,
Central stable/unstable manifolds and the destruction of KAM Tori in the Planar Hill Problem, Physica D, 140 (2000), 1-32.
doi: 10.1016/S0167-2789(99)00211-0. |
| [5] |
A. Shmyrov and V. Shmyrov, Controllable orbital motion in a neighborhood of collinear
libration point, Applied Mathematical Sciences, 8 (2014), 487–492. Available from: http://dx.doi.org/10.12988/ams.2014.312711.
doi: 10.12988/ams.2014.312711. |
| [6] |
A. Shmyrov and V. Shmyrov, On controllability region of orbital motion near L1, Applied
Mathematical Sciences, 9 (2015), 7229–7236. Available from: http://dx.doi.org/10.12988/ams.2015.510638.
doi: 10.12988/ams.2015.510638. |
| [7] |
A. Shmyrov and V. Shmyrov, The criteria of quality in the problem of motion stabilization in
a neighborhood of collinear libration point, Iternational Conference on "Stability and Control
Processes" in Memory of V. I. Zubov (SCP 2015), 4-9 October 2015, (2015), art. 7342135,
345–347. Available from: http://dx.doi.org/10.1109/SCP.2015.7342135.
doi: 10.1109/SCP.2015.7342135. |
| [8] |
A. Shmyrov and V. Shmyrov,
The estimation of controllability area in the problem of controllable movement in a neighborhood of collinear libration point
2015 International Conference on Mechanics -Seventh Polyakhov's Reading, 2-6 February 2015 (2015), art. 7106776. Available from: http://dx.doi.org/10.1109/POLYAKHOV.2015.7106776. |
| [9] |
V. I. Zubov,
Lectures on Control Theory, Nauka, Moscow, 1975. |
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