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，基于拉格朗日插值函数的伪谱方法在轨迹优化问题求解及其飞行实现中蓬勃发展。在看似缺乏依据地偏离这些高度成功的方法之际，本文提出了一种新的轨迹优化起点。该起点基于近期发展的通用Birkhoff插值概念。新方法通过将条件数的快速增长从O(N²)彻底平缓至O(1)（其中N为网格点数），为拉格朗日理论带来了实质性的计算提升。此外，即便对于非线性非凸问题，Birkhoff特有的原始-对偶计算也被隔离到良态线性系统中。本文为两篇论文中的第一部分。在第一部分中，基于两个假设发展了一种新理论。除这两个假设外，理论发展过程未对基函数选择或网格点选取作任何假设。通过证明若干协向量映射定理，建立了直接Birkhoff方法与间接Birkhoff方法之间的数学等价性。在本文第二部分（与Proulx合作）中，将证明特定Gegenbauer网格族满足本理论所需的两个假设。第二部分中的数值算例将展示新理论的强大功能与实用性。