An hp mesh refinement method for optimal control using discontinuity detection and mesh size reduction

A variable-order adaptive mesh refinement method for solving optimal control problems is described. The method employs orthogonal collocation at Legendre-Gauss-Radau points. The mesh refinement method uses a previously derived convergence rate in order to modify the mesh. First, in regions where the solution is not sufficiently smooth, the method employs mesh interval refinement to place mesh points near discontinuities in the solution. Next, in regions where the solution is smooth the method increases the degree of the approximating polynomial. Furthermore, the method can decrease the size of the mesh in one of two ways. First, by representing the state using a power series approximation it is possible to decrease the required polynomial degree when it is determined that the coefficients of the highest powers of the power series are insignificant in comparison to the mesh refinement accuracy tolerance. Second, mesh intervals can be combined if it is determined that the power series representations are the same in two or more adjacent mesh intervals. Finally, the method is described in detail and is applied successfully to an example from the open literature.