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

Abstract

In this paper we consider controllable orbital motion in a neighborhood of the first collinear libration point L1 of the Sun-Earth system. This libration point is unstable. For a long stay of the spacecraft in this area of space required the control action. We model the motion by equations circular restricted three-body problem. At the same time, we use non-linear approximation of these equations, so-called Hills equations and linearized equations. For solution of the problem of stabilization of motion, we use the model of linear-quadratic optimization. This model offers a standard approach for the construction of stabilizing control laws. In this work, we present an original family of quadratic functionals, which were built with the help of the special linear function of the phase variables, so-called “hazard function”. The increase of module of this function module mean departure of a spacecraft from a neighborhood of the libration point and the decrease of this module corresponds to the stabilization of motion. For the represented family of functionals we have built the Bellman function and showed that the control damps square of hazard function. Numerical simulations of the orbital motion with obtained controls is realized in the nonlinear model of Hills equations and in model of circular three-body problem.