An adaptive mesh refinement method for indirectly solving optimal control problems

The indirect solution of optimal control problems (OCPs) with inequality constraints and parameters is obtained by solving the two-point boundary value problem (BVP) involving index-1 differential-algebraic equations (DAEs) associated with its first-order optimality conditions. This paper introduces an adaptive mesh refinement method based on a collocation method for solving the index-1 BVP-DAEs. The paper first derives a method to estimate the relative error between the numerical solution and the exact solution. The relative error estimate is then used to guide the mesh refinement process. The mesh size is increased when the estimated error within a mesh interval is beyond the numerical tolerance by either increasing the order of the approximating polynomial or dividing the interval into multiple subintervals. In the mesh interval where the error tolerance has been met, the mesh size is reduced by either decreasing the degree of the approximating polynomial or merging adjacent mesh intervals. An efficient parallel implementation of the method is implemented using Python and CUDA. The paper presents three examples which show that the approach is more computationally efficient and robust when compared with fixed-order methods.