SIAM Journal on Control and Optimization, Volume 59, Issue 4, Page 2570-2593, January 2021.
The switch point algorithm is a new approach for solving optimal control problems whose solutions are either singular or bang-bang or both singular and bang-bang, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. Problems in this class can often be reduced to an optimization over the switching points. Formulas are derived for the derivative of the objective with respect to the switch points, the initial costate, and the terminal time. All these derivatives can be computed simultaneously in just one integration of the state and costate dynamics. Hence, gradient-based unconstrained optimization techniques, including the conjugate gradient method or quasi-Newton methods, can be used to compute an optimal control. The performance of the algorithm is illustrated using test problems with known solutions and comparisons with other algorithms from the literature.

中文翻译：

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切换点算法

SIAM Journal on Control and Optimization，第 59 卷，第 4 期，第 2570-2593 页，2021 年 1 月。
切换点算法是一种求解最优控制问题的新方法，其解要么是奇异的，要么是 bang-bang 的，要么是奇异的又是 bang-bang 的，并且在最优控制的时间点上具有有限数量的跳跃不连续性解决方案结构变化。此类问题通常可以简化为对切换点的优化。公式是为目标关于切换点、初始坐标和终端时间的导数推导出来的。所有这些导数都可以在状态和辅助动力学的一个集成中同时计算。因此，基于梯度的无约束优化技术，包括共轭梯度法或拟牛顿法，可用于计算最优控制。