Sparse Optimal Control for Nonlinear Trajectory Design in Three-Body Problem

Abstract

This paper proposes a direct method for the minimization of the l¹ norm of the control inputs of the trajectory design in the three-body problem. This method is based on the sparse optimal control, which considers minimizing the thrust arcs. The minimum thrust arc appears as a result of the minimization of the l¹ norm in the formulation of the optimal trajectory problem. The sparse optimal control can be reduced to a convex optimization because the objective function and constraints represent convex sets by assuming linear dynamics. However, the equations of motion of the three-body problem are nonlinear and highly unstable so that it is difficult to find the optimal trajectory by a standard convex optimization problem. To satisfy the nonlinearity condition, this paper constructs the successive sparse optimal control based on the sparse optimal control. This method derives the l¹ minimum solution that satisfies the nonlinearity condition by solving the sparse optimal control iteratively. As an application, the transfer between Lyapunov orbits in the Sun-Earth system is solved and it is verified that the proposed method can perform even for the unstable dynamics. This example in the three-body problem also demonstrates the instability of the dynamics can be suppressed by efficiently utilizing the stable and unstable manifold of underlying dynamics. Then the solution is compared with the optimal solution obtained by nonlinear programming.