Formation flying along elliptical orbit using attractive sets of optimal control

This paper proposes a new method of optimal trajectory design for formation flying along an elliptical orbit. Under linearized assumptions and a quadratic performance index, the optimal cost is quadratic in the initial state. Attractive sets of optimal control are defined as contours of the optimal cost based on linear quadratic regulator theory. It describes a set of all initial states to reach a desired state by a given cost. By solving the optimal control problem for TH equations, the optimal cost is obtained as a time-periodic function. This paper develops the attractive set for a time-periodic system and procedure to draw the attractive set is shown. The advantage of using attractive sets for optimal trajectory design is that it can determine the optimal initial state immediately. The various shape of the periodic orbit and the attractive sets are demonstrated by weight parameters of optimal control theory.