Integral form of Legendre-Gauss-Lobatto collocation for optimal control

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.