The Chebyshev–Legendre collocation method for a class of optimal control problems

In this paper, we derive the so-called Chebyshev–Legendre method for a class of optimal control problems governed by ordinary differential equations. We use Legendre expansions to approximate the control and state functions and we employ the Chebyshev–Gauss–Lobatto (CGL) points as the interpolating points. Thus the unknown variables of the equivalent nonlinear programming problems are the coefficients of the Legendre expansions of both the state and the control functions. We evaluate the function values at the CGL nodes via the fast Legendre transform. In this way, the fast Legendre transform can be utilized to save CPU calculation time. Some numerical examples are given to illustrate the applicability and high accuracy of the Chebyshev–Legendre method in solving a wide class of optimal control problem.

Keywords:

• Chebyshev–Legendre method
• Optimal control problems
• Legendre polynomials
• Chebyshev–Gauss–Lobatto points
• Nonlinear programming problems

Acknowledgements

The authors would like to thank Professor B. K. Alpert for providing the FLT program. We also want to express our gratitude to the anonymous referees and the Editor for their constructive comments, which helped the revision of this paper. The research was supported by the National Science Foundation of China (10471089) and the Science Foundation of Shanghai (04JC14062).