gpops-ii uses a class of methods referred to as formula_10-adaptive gaussian quadrature collocation where the collocation points are the nodes of a gauss quadrature (in this case, the legendre-gauss-radau [lgr] points). the mesh consists of intervals into which the total time interval formula_11 in each phase is divided, and lgr collocation is performed in each interval. because the mesh can be adapted such that both the degree of the polynomial used to approximate the state formula_12 and the width of each mesh interval can be different from interval to interval, the method is referred to as an formula_10-adaptive method (where "formula_14" refers to the width of each mesh interval, while "formula_15" refers to the polynomial degree in each mesh interval). the lgr collocation method has been developed rigorously in refs., while formula_10-adaptive mesh refinement methods based on the lgr collocation method can be found in refs., .