Over the last two decades, pseudospectral methods based on Lagrange interpolants have flourished in solving trajectory optimization problems and their flight implementations. In a seemingly unjustified departure from these highly successful methods, a new starting point for trajectory optimization is proposed. This starting point is based on the recently-developed concept of universal Birkhoff interpolants. The new approach offers a substantial computational upgrade to the Lagrange theory in completely flattening the rapid growth of the condition numbers from O(N2) to O(1), where N is the number of grid points. In addition, the Birkhoff-specific primal-dual computations are isolated to a well-conditioned linear system even for nonlinear, nonconvex problems. This is part I of a two-part paper. In part I, a new theory is developed on the basis of two hypotheses. Other than these hypotheses, the theoretical development makes no assumptions on the choices of basis functions or the selection of grid points. Several covector mapping theorems are proved to establish the mathematical equivalence between direct and indirect Birkhoff methods. In part II of this paper (with Proulx), it is shown that a select family of Gegenbauer grids satisfy the two hypotheses required for the theory to hold. Numerical examples in part II illustrate the power and utility of the new theory.

翻译：近二十年来，基于拉格朗日插值的伪谱方法在解决轨迹优化问题及其飞行实现中蓬勃发展。在与这些极为成功的方法看似缺乏合理性的背离中，本文提出了一种新的轨迹优化起点。该起点基于最近发展的通用Birkhoff插值概念。新方法在完全将条件数的快速增长从O(N²)平坦化至O(1)方面，为拉格朗日理论提供了实质性的计算改进，其中N为网格点数。此外，即使对于非线性、非凸问题，Birkhoff特有的原始-对偶计算也被隔离为一个良态的线性系统。本文是两篇系列论文的第一部分。在第一部分中，基于两个假设发展了一种新理论。除这些假设外，理论发展不对基函数的选择或网格点的选取做任何假设。本文证明了若干余切映射定理，以建立直接与间接Birkhoff方法之间的数学等价性。在本文的第二部分（与Proulx合著）中，将展示选定的盖根鲍尔网格族满足该理论所需的两个假设。第二部分的数值算例将阐明新理论的强大功能与实用性。