This work considers the development of a numerical-analytical procedure for computing optimal time-fixed low-thrust limited-power transfers between arbitrary orbits. It is assumed that Earth’s gravitational field is described by the main three zonal harmonics J₂, J₃ and J₄. The optimization problem is formulated as a Mayer problem of optimal control with the state variables defined by the Cartesian elements—components of the position vector and the velocity vector—and a consumption variable that describes the fuel spent during the maneuver. Pontryagin Maximum Principle is applied to determine the optimal thrust acceleration. A set of classical orbital elements is introduced as a new set of state variables by means of an intrinsic canonical transformation defined by the general solution of the canonical system described by the undisturbed part of the maximum Hamiltonian. The proposed procedure involves the development of a two-stage algorithm to solve the two-point boundary value problem that defines the transfer problem. In the first stage of the algorithm, a neighboring extremals method is applied to solve the “mean” two-point boundary value problem of going from an initial orbit to a final orbit at a prescribed final time. This boundary value problem is described by the mean canonical system that governs the secular behavior of the optimal trajectories. The maximum Hamiltonian function that governs the mean canonical system is computed by applying the classic concept of “mean Hamiltonian”. In the second stage, the well-known Newton–Raphson method is applied to adjust the initial values of adjoint variables when periodic terms of the first order are included. These periodic terms are recovered by computing the Poisson brackets in the transformation equations, which are defined between the original set of canonical variables and the new set of average canonical variables, as described in Hori method. Numerical results show the main effects on the optimal trajectories due to the zonal harmonics considered in this study.

这项工作考虑了用于计算任意轨道之间最佳时间固定低推力有限功率传输的数值分析程序的开发。假设地球引力场由三个主要的纬向谐波J ₂、J ₃和J _{4 描述}. 优化问题被表述为最优控制的 Mayer 问题，其状态变量由笛卡尔元素定义——位置向量和速度向量的分量——以及描述机动过程中消耗的燃料的消耗变量。庞特里亚金最大原理用于确定最佳推力加速度。通过由最大哈密顿量的未扰动部分描述的正则系统的一般解定义的内在正则变换，将一组经典轨道元素作为一组新的状态变量引入。所提出的程序涉及开发一个两阶段算法来解决定义传输问题的两点边值问题。在算法的第一阶段，应用相邻极值法解决在规定的最终时间从初始轨道到最终轨道的“平均”两点边值问题。这个边界值问题由控制最佳轨迹的长期行为的平均规范系统描述。控制平均正则系统的最大哈密顿函数是通过应用“平均哈密顿量”的经典概念来计算的。在第二阶段，当包含一阶周期项时，应用众所周知的 Newton-Raphson 方法来调整伴随变量的初始值。这些周期项是通过计算变换方程中的泊松括号来恢复的，如 Hori 方法中所述，这些方程是在原始典型变量集和新的平均典型变量集之间定义的。