A Gauss--Lobatto quadrature method for solving optimal control problems

Abstract

This paper proposes a direct approach for solving optimal control problems. The time domain is divided into multiple subdomains, and a Lagrange interpolating polynomial using the Legendre--Gauss--Lobatto points is used to approximate the states and controls. The state equations are enforced at the Legendre--Gauss--Lobatto nodes in a nonlinear programming implementation by partial Gauss--Lobatto quadrature in each subdomain. The final state in each subdomain is enforced by a full Gauss--Lobatto quadrature. The Bolza cost functional is naturally approximated using Gauss--Lobatto quadrature across all subdomains.