The optimal stabilization of orbital motion in a neighborhood of collinear libration point

Alexander Shmyrov; Vasily Shmyrov

doi:10.3934/naco.2017012

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2017, Volume 7, Issue 2: 185-189. Doi: 10.3934/naco.2017012

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The optimal stabilization of orbital motion in a neighborhood of collinear libration point

Alexander Shmyrov and
Vasily Shmyrov

Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034 Russia

Received: September 30, 2016

Revised: March 31, 2017

Published: October 2017

The authors are supported by SPbSU grant 9.37.345.2015.

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Abstract

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.

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Mathematics Subject Classification: Primary: 49J15, 37J25; Secondary: 70F07.

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