Integral Form of Legendre-Gauss-Lobatto Collocation for Optimal Control
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908414520590336 |
|---|---|
| author | Abadia-Doyle, Gabriela Hager, William W. Rao, Anil V. |
| author_facet | Abadia-Doyle, Gabriela Hager, William W. Rao, Anil V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13938 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integral Form of Legendre-Gauss-Lobatto Collocation for Optimal Control Abadia-Doyle, Gabriela Hager, William W. Rao, Anil V. Optimization and Control 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. |
| title | Integral Form of Legendre-Gauss-Lobatto Collocation for Optimal Control |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2506.13938 |