Variable-Order Mesh Refinement for Solving Optimal Control Problems Using an Integral Legendre-Gauss-Radau Collocation Method

A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre-Gauss-Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a Legendre-Gauss-Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into subintervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into subintervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed-order methods.