研究姿态机动的时间最优控制时, 通常只考虑最优解为Bang-Bang控制的情况, 假设了开关函数仅在有限个孤立点为零, 本文对假设之外的奇异最优控制进行探讨。以球对称光学敏捷卫星为对象, 根据姿态机动最优控制问题的数学模型, 证明了开关函数的特殊性质。在不同边界条件下, 结合一阶必要条件、广义勒让德-克莱伯西条件等, 分析了最优控制包含奇异区间的所有情况, 得到了奇异控制最优性的必要条件。结果表明, 对于起止角速度为0, 乃至起止角速度相同的所有工况, 可以直接排除奇异控制最优的可能性, 最优控制中各个分量在+1和-1之间切换; 而起止角速度相异时, 存在奇异控制最优的情况, 且奇异分量的开关函数在全过程均为0, 奇异阶数为无穷阶, 文中给出了相应算例予以佐证。

The possibility of singular optimal control in the time-optimal reorientation of an agile satellite with independent three-axis control was investigated in this paper. Generally, it is assumed that the switching functions take the value zero only at isolated points, which dictates that the controls are "bang-bang". However, some new results were found in this paper. An inertially symmetric rigid body was considered. Based on the optimum model, two propositions about the switching functions were derived. All possible optimal control strategies under different boundary conditions were identified. These included bang-bang solution, one control singular, two controls singular and all three controls singular. Using Pontryagin's principle and the Generalized Legendre-Clebsch condition, necessary conditions for optimality of singular control were obtained. The research illustrated that all three control components could not be simultaneously singular at any point on an optimal path. For rest to rest maneuvers, singular control was proved to be not optimal. Actually, for the boundary conditions that the initial and final angular velocities were equal, the possibility of singular optimal control could be completely dismissed. The components of the optimal control switch between +1 and -1. On the other hand, the cases of singular optimal control exist while the final angular velocity was different from the initial one, the switching function of the singular component remained at zero all the time, and the order of the singular control was infinite. Numerical examples of time-optimal solutions with one control singular and two controls singular were presented.