We present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first‐order necessary conditions for optimality and leads to two‐point boundary value problems that may be solved by numerical techniques to arrive at optimal trajectories. We demonstrate our theoretical results with numerical simulations on the optimal trajectory generation of a wheeled inverted pendulum and an attitude control problem of a spacecraft on the Lie group SO(3).

中文翻译：

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矩阵李群上的频率约束几何庞特里亚金极大值原理

我们在矩阵李群上提出了几何离散时间庞特里亚金最大原理（PMP），除了在问题制定阶段直接对状态和控制动作进行点约束外，还对控制轨迹进行了频率约束。该PMP为优化提供了一阶必要条件，并导致两点边界值问题，可以通过数值技术解决这些问题，以获得最优轨迹。我们用数值模拟证明了我们的理论结果，这些数值模拟是关于倒立摆的最佳轨迹生成和李群SO（3）上航天器的姿态控制问题。