Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x(t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.