This paper presents an efficient and stable DNNs-based Radau pseudospectral method for the free-time elliptical orbit pursuit-evasion game based on the equivalent reconstruction of the game model. Firstly, the relative dynamics equations are established by adding the nonlinear terms caused by the eccentricity to the Hill–Clohessy–Wilshire equations. Then the original pursuit-evasion problem is deduced to a 4-dimensional one-sided optimal control problem (OCP) based on the equivalent reconstruction. Secondly, in order to apply the deep neural networks (DNNs) to map the relationship between the OCP and the solution, the normalization of costates is introduced to eliminate the non-uniqueness of solutions when generating samples for training DNNs. Thirdly, the DNNs-based Radau pseudospectral method is proposed where the DNNs output the guesses of solutions to the derived OCP and the Radau pseudospectral method optimizes the histories of controls obtained by the guesses to the convergence. The simulation results demonstrate that the proposed method converges more stably and decreases the calculation time greatly compared with the traditional indirect method.

本文基于博弈模型的等价重构，提出了一种高效稳定的基于 DNNs 的 Radau 伪谱方法，用于自由时间椭圆轨道追逃博弈。首先，通过将由偏心引起的非线性项添加到 Hill-Clohessy-Wilshire 方程中来建立相对动力学方程。然后将原始的追逃问题推导为基于等效重构的4维单边最优控制问题（OCP）。其次，为了应用深度神经网络 (DNNs) 来映射 OCP 和解之间的关系，在生成用于训练 DNN 的样本时，引入了协变量的归一化以消除解的非唯一性。第三，提出了基于 DNN 的 Radau 伪谱方法，其中 DNN 输出对派生的 OCP 的解的猜测，而 Radau 伪谱方法优化了通过对收敛的猜测获得的控制历史。仿真结果表明，与传统的间接方法相比，该方法收敛更稳定，计算时间大大减少。