First-order necessary conditions for optimality reveal the Hamiltonian nature of optimal control problems. Regardless of the overwhelming awareness of this result, the implications that it entails have not been fully explored. We discuss how the symplectic structure of optimal control constrains the flow of sub-volumes in the phase space. Special emphasis is devoted to dynamics in the neighborhood of optimal trajectories and insight is gained into how errors in the initial states affect terminal conditions. Specifically, we prove that if the optimal trajectory does not satisfy a particular condition, then there exists a set of variations in the initial states yielding a greater error in norm when mapped to the terminal time through the state transition matrix. We relate this result to the sensitivity problem in solving indirect problems for optimal control.

中文翻译：

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哈密​​顿结构引起的最优控制问题的敏感性

最优性的一阶必要条件揭示了最优控制问题的哈密顿性质。不管对此结果的压倒性意识如何，它所带来的影响还没有得到充分的探讨。我们讨论了最优控制的辛结构如何约束子空间在相空间中的流动。特别强调了最佳轨迹附近的动力学，并获得了关于初始状态中的错误如何影响末端条件的见解。具体而言，我们证明如果最佳轨迹不满足特定条件，则在通过状态转换矩阵映射到终端时间时，初始状态中会存在一组变化，从而产生更大的范数误差。