# American Institute of Mathematical Sciences

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

Received  October 2016 Revised  April 2017 Published  June 2017

Fund Project: The authors are supported by SPbSU grant 9.37.345.2015.

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.

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
##### 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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##### 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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