About a family of criteria of quality in the problem of motion stabilization in a neighborhood of the collinear libration point

Аuthors

Shmyrov A. S.^*, Shmyrov V. A.^**

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

*e-mail: a.shmyrov@spbu.ru
**e-mail: v.shmyrov@spbu.ru

Abstract

Flying near collinear libration points has been implemented since 1978, when the spacecraft ISEE-3 was launched in the vicintyof collinear libration point L1 of Sun-Earth system. Many spacecrafts are widely known, in particular SOHO — solar tracking station. This work considers controlled orbital movement in near unstable collinear libration point L1 of the Sun-Earth system. We use Hill’s equations as a mathematical model. These equations are the non-linear approximation of circular problem of three bodies equations. We also use linearized equations of motion. Libration point is a model concept of circular restricted problem of three bodies, and getting to the libration point is not the purpose of such studies. The important characteristic of the orbital motion of a spacecraft near libration point is the special function of phase variables of «hazard function», which is presented in our article. A spacecraft does not get away quickly from the vicinity of the libration point, when module of the hazard function is small. At zero value of this function the spacecraft stays near libration point wthin the framework of linear approximation. In this paper, we present quadratic functionals, which were built with the help of «hazard function». The instability of the libration point L1 requires solving the problem of spacecraft motion stabilization. The standard methods of modeling of linear-quadratic optimization are often successful for such tasks exploration. For this widely studied model, we proposed the special family of functionals. We applied also sufficient conditions of optimality in view of these functionals, and obtained the stabilizing controls in the explicit analytic form. We simulate controllable orbital motion near collinear libration point with obtained controls. The graphs show that in the course of transition from the model of Hill’s equations to a more adequate model of restricted problem of three bodies, the qualitative nature of controllable motion of a spacecraft on the given period of time integration near L1 does not change.

Keywords:

control, stabilization, optimality sufficient condition, restricted problem of three bodies

References

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5. Shmyrov A., Shmyrov V., Controllable orbital motion in a neighborhood of collinear libration point, Applied Mathematical Sciences, vol. 8 (9-12), 2014. pp. 487–492.

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7. Shmyrov A., Shmyrov V. Method of Lyapunov functions for controllable Hamiltonian systems, 20th International W-orkshop on Beam Dynamics and Optimization, June 30 2014-July 4 2014, pp. 156.

8. Shmyrov A., Shmyrov V. The estimation of controllability area in the problem of controllable movement in a neighborhood of collinear libration point, International Conference on Mechanics — Seventh Polyakhov’s Reading, 2-6 Feb. 2015. pp.1-3.

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