Initial costates derived by near-optimal reference sequence and least-squares method

In this paper, we present a novel initial costates solver for initializing time-optimal trajectory problems in relative motion with continuous low thrust. The proposed solver consists of two primary components: training a Multilayer Perceptron (MLP) for generating reference sequence and Time of Flight (TOF) to the target, and deriving a system of linear algebraic equations for obtaining the initial costates. To overcome the challenge of generating training samples for the MLP, the backward generation method is proposed to obtain five different training databases. The training database and sample form are determined by analyzing the input and output correlation using the Pearson correlation coefficient. The best-performing MLP is obtained by analyzing the training results with various hyper-parameter combinations. A reference sequence starting from the initial states is obtained by integrating forward with the near-optimal control vector from the output of MLP. Finally, a system of linear algebraic equations for estimating the initial costates is derived using the reference sequence and the necessary conditions for optimality. Simulation results demonstrate that the proposed initial costates solver improves the convergence ratio and reduce the function calls of the shooting function. Furthermore, Monte-Carlo simulation illustrates that the initial costates solver is applicable to different initial velocities, demonstrating excellent generalization ability.