Convergence Results of Galerkin Optimal Control with Legendre Test Function∗∗The work and views expressed in this paper are those of the authors and do not reflect the oficial policy or position of the U.S. Department of the Army, the U.S. Department of the Navy, the U.S. Department of Defense or the U.S. Government.

Abstract

A Galerkin-based family of numerical formulations is presented for solving nonlinear optimal control problems. This paper introduces a family of direct methods that calculate optimal trajectories by discretizing the system dynamics using Galerkin numerical techniques and approximate the cost function with Gaussian quadrature. In this numerical approach, the analysis is based on L 2 -norms. An important result in the theoretical foundation is that the feasibility and consistency theorems are proved for problems with piecewise continuous controls. Galerkin methods may be formulated in a number of ways that allow for efficiency and/or improved accuracy while solving a wide range of optimal control problems with a variety of state and control constraints. Numerical formulations using Legendre test functions are derived. One formulation allows for a weak enforcement of boundary conditions, which imposes end conditions only up to the accuracy of the numerical approximation itself. Finally, a numerical example is shown to demonstrate the exponential convergence of Galerkin optimal control.