This paper proposes a novel impulsive thrust strategy guided by optimal continuous thrust strategy to address two-player orbital pursuit-evasion game under impulsive thrust control. The strategy seeks to enhance the interpretability of impulsive thrust strategy by integrating it within the framework of differential game in traditional continuous systems. First, this paper introduces an impulse-like constraint, with periodical changes in thrust amplitude, to characterize the impulsive thrust control. Then, the game with the impulse-like constraint is converted into the two-point boundary value problem, which is solved by the combined shooting and deep learning method proposed in this paper. Deep learning and numerical optimization are employed to obtain the guesses for unknown terminal adjoint variables and the game terminal time. Subsequently, the accurate values are solved by the shooting method to yield the optimal continuous thrust strategy with the impulse-like constraint. Finally, the shooting method is iteratively employed at each impulse decision moment to derive the impulsive thrust strategy guided by the optimal continuous thrust strategy. Numerical examples demonstrate the convergence of the combined shooting and deep learning method, even if the strongly nonlinear impulse-like constraint is introduced. The effect of the impulsive thrust strategy guided by the optimal continuous thrust strategy is also discussed.

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.