A Preliminary Analysis of a Variable-Order Approach to Solving Optimal Control Problems Using Pseudospectral Methods

An iterative variable-order hp−adaptive pseudospectral method is presented for solving optimal control problems. In the method of this paper, the accuracy of the solution in each segment is assessed by computing the violations in the dis-cretized differential-algebraic equations at a large number of sample points in each segment. The accuracy on the subsequent mesh is then improved by redistributing the collocation points using a functional that depends upon the curvature of the trajectory. The algorithm iterates on the collocation point distribution until the accuracy criterion is satisfied to a user-specified tolerance. The method allows for different degree polynomial approximations in each segment, but the algorithm is designed in such a manner that the maximum polynomial degree in every segment remains small. It is found that a computationally efficient algorithm results from the ability to vary the degree of the polynomial in each segment while keeping the maximum degree bounded. Finally, an example is considered to demonstrate the effectiveness of the method. I