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This paper presents a new method of optimal trajectory design for formation flying. Under linearized assumptions and a quadratic performance index, we consider an attractive set for optimal control based on the linear quadratic regulator theory. An attractive set is defined as a set of all initial states to reach a desired state for a given cost. In particular, we define attractive sets for two problems: a fixed final-state, fixed final-time problem and an infinite-time problem. The properties of the two problems are investigated by plotting the attractive sets for each problem. Our results reveal that the L1-norm of the finite-time problem is close to that of the infinite-time problem even though the flight time is much smaller.

Optimal Trajectory Design of Formation Flying based on Attractive Sets