2025/06/16 by Abadia-Doyle, Gabriela, Hager, William W., Rao, Anil V.
Engineering · Mathematics · Physics and Astronomy · #Advanced Optimization Algorithms Research #FOS: Mathematics #Model Reduction and Neural Networks #Optimization and Control (math.OC) #Spacecraft Dynamics and Control
paper · pdf · doi:10.48550/arxiv.2506.13938
openalex publication_date 2025/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A new method is described for solving optimal control problems using direct collocation at Legendre-Gauss-Lobatto points. The approach of this paper employs a polynomial approximation of the right-hand side vector field of the differential equations and leads to the following important outcomes. First, the first-order optimality conditions of the LGL integral form are derived, which lead to a full-rank transformed adjoint system and novel costate estimate. Next, a derivative-like form of the LGL collocation method is obtained by multiplying the system by the inverse of an appropriate full-rank block of the integration matrix. The first-order optimality conditions of the LGL derivative-like form are then derived, leading to an equivalent full-rank transformed adjoint system and secondary novel costate estimate which is related to the costate estimate of the integral form via a linear transformation. Then, it is shown that a second integral form can be constructed by including an additional noncollocated support point, but such a point is superfluous and has no impact on the solution to the nonlinear programming problem. Finally, the method is demonstrated on two benchmark problems: a one-dimensional initial value optimal control problem with an analytic solution and a time-variant orbit raising optimal control problem.