This is part II of a two-part paper. Part I presented a universal Birkhoff theory for fast and accurate trajectory optimization. The theory rested on two main hypotheses. In this paper, it is shown that if the computational grid is selected from any one of the Legendre and Chebyshev family of node points, be it Lobatto, Radau or Gauss, then, the resulting collection of trajectory optimization methods satisfy the hypotheses required for the universal Birkhoff theory to hold. All of these grid points can be generated at an $\mathcal{O}(1)$ computational speed. Furthermore, all Birkhoff-generated solutions can be tested for optimality by a joint application of Pontryagin's- and Covector-Mapping Principles, where the latter was developed in Part~I. More importantly, the optimality checks can be performed without resorting to an indirect method or even explicitly producing the full differential-algebraic boundary value problem that results from an application of Pontryagin's Principle. Numerical problems are solved to illustrate all these ideas. The examples are chosen to particularly highlight three practically useful features of Birkhoff methods: (1) bang-bang optimal controls can be produced without suffering any Gibbs phenomenon, (2) discontinuous and even Dirac delta covector trajectories can be well approximated, and (3) extremal solutions over dense grids can be computed in a stable and efficient manner.

翻译：本文是两篇系列论文的第二部分。第一部分提出了用于快速精确轨迹优化的通用伯克霍夫理论，该理论基于两个主要假设。本文证明，如果计算网格选自勒让德-切比雪夫节点点族中的任意一类（无论是洛巴托点、拉道点还是高斯点），则由此产生的轨迹优化方法集合均满足通用伯克霍夫理论成立所需的假设条件。所有这些网格点均可通过$\mathcal{O}(1)$计算速度生成。此外，所有伯克霍夫生成的解均可通过庞特里亚金原理与协向量映射原理（后者在第一部分中提出）的联合应用进行最优性检验，更重要的是，这种最优性检验无需借助间接方法，甚至无需显式构建应用庞特里亚金原理后产生的完整微分代数边值问题。本文通过数值问题求解来阐明上述所有思想。所选示例特别突出了伯克霍夫方法的三个实用特性：（1）能生成无吉布斯现象的bang-bang最优控制；（2）可精确近似不连续甚至狄拉克δ协向量轨迹；（3）能在稳定高效的方式下计算密集网格上的极值解。