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：本文是两部分论文的第二部分。第一部分提出了一种用于快速精确轨迹优化的通用Birkhoff理论，该理论基于两个主要假设。本文证明，若计算网格选自Legendre和Chebyshev节点点族（包括Lobatto、Radau或Gauss类型），则由此产生的轨迹优化方法集合满足通用Birkhoff理论成立所需的全部假设。所有这些网格点均可通过$\mathcal{O}(1)$的计算速度生成。此外，所有Birkhoff生成的解可通过联合应用Pontryagin原理与协态映射原理（后者已在第一部分中提出）进行最优性检验。更重要的是，最优性检验无需借助间接法，甚至无需显式构造由Pontryagin原理导出的完整微分代数边值问题。本文通过数值算例阐明上述思想。所选示例特别突出了Birkhoff方法的三个实用特性：（1）可产生无吉布斯现象的bang-bang最优控制；（2）能够良好近似不连续甚至狄拉克δ协态轨迹；（3）可在稠密网格上稳定高效地计算极值解。