Efficient solutions to nonlinear optimal control problems using adaptive mesh orthogonal collocation methods

In a direct collocation method, a continuous-time optimal control problem is transcribed to a finite-dimensional nonlinear programming problem. Solving this nonlinear programming problem as efficiently as possible requires that sparsity at both the first- and second-derivative levels be exploited. In this dissertation the first and second derivative nonlinear programming problem sparsity is exploited using Gaussian quadrature orthogonal collocation at Legendre-Gauss-Radau points. Furthermore, a variable-order mesh refinement method is developed that allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. This mesh refinement employs a relative error estimate 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. This relative error estimate is 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 sub-intervals. Finally, a reusable software package is described that efficiently computes solutions to multiple phase optimal control problems. This software package exploits the sparse structure of the Radau collocation method, while also implementing the aforementioned variable-order mesh refinement method.