Combination of non-classical pseudospectral and direct methods for the solution of brachistochrone problem

Abstract

In this paper, we propose a combination of non-classical pseudospectral and direct methods to find the solution of brachistochrone problem. The method converts the optimal control problem of brachistochrone, into a sequence of quadratic programming problems. To this end, the quasilinearization method is used to replace the nonlinear optimal control problem into a sequence of constrained linear-quadratic optimal control problems; then each of the state variables is approximated by a weighted interpolation function based on the non-classical orthogonal polynomials. The method gives the information of the quadratic programming problems explicitly (the Hessian and the gradient of the cost function). Using this method, the solution of the brachistochrone problem is compared with those in the literature.

Keywords:

• non-classical pseudospectral method
• direct method
• brachistochrone problem
• quadratic programming

2000 AMS Subject Classifications :

• 49J15
• 49J40
• 47J30

Acknowledgements

The authors are very grateful to the anonymous referees and also Prof. E.H. Twizell (editor) for carefully reading the paper and for their comments and suggestions.