ℓ1-Optimal Newton Method for Nonlinear ℓ1-Optimal Control Problems

This paper proposes a new Newton method for solving finite time ℓ
¹
-optimal control problems with a boundary condition on the terminal state. In the proposed algorithm, the error between the terminal state and its desired value is described as a nonlinear function of the input sequence. The algorithm then updates the input towards the ℓ
¹
-optimal Newton direction to find a root of the nonlinear error function. The resulting root is an ℓ
¹
-optimal input sequence satisfying the boundary condition. The advantage of this method is that the error converges at least linearly and at most quadratically in finding an ℓ
¹
-optimal feasible solution. Additionally, we propose a continuation based method that relaxes the convergence condition of the proposed algorithm. The applicability of the proposed method is confirmed through a numerical example of transfer orbit generation for a satellite.