Integral Form of Legendre-Gauss-Lobatto Collocation for Optimal Control

TL;DR

This paper introduces a novel integral form of Legendre-Gauss-Lobatto collocation for solving optimal control problems, deriving new optimality conditions and costate estimates, and demonstrating effectiveness on benchmark problems.

Contribution

It develops a new integral form and derivative-like form of LGL collocation, providing full-rank adjoint systems and costate estimates, with a simplified second integral form including an extra support point.

Findings

Derivation of first-order optimality conditions for the integral form.

Introduction of a derivative-like form with a full-rank adjoint system.

Successful application to benchmark optimal control problems.

Abstract

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.