Integral form of Legendre-Gauss-Lobatto collocation for optimal control
Year: 2025
Type: article
Abstract: A new method is described for solving optimal control problems using direct collocation at Legendre-Gauss-Lobatto (LGL) points. The approach of this paper employs a polynomial approximation of the right-hand side vector field of the differential equations and provides a value for the control at both endpoints of an interval. Using this polynomial approximation of the right-hand side vector field, the LGL collocation method of this paper is then developed in integral form which leads to the following key 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. Next, a convergence analysis of the method in multiple-interval symplectic Runge-Kutta form is developed, and this convergence analysis includes a superconvergence property at the mesh points. Furthermore, 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.
Institution: University of Florida
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,100
Funders: National Science Foundation, Charles Stark Draper Laboratory, Applied Physics Laboratory, Johns Hopkins University, Office of Naval Research, Division of Civil, Mechanical and Manufacturing Innovation, Air Force Research Laboratory, USAF Phillips Laboratory, National Science Foundation Graduate Research Fellowship Program
Awards: G332097973, G8672529463, G8968403120
OpenAlex ID: W4415015501