Abstract: Both geometric and dynamic methods are widely applied to GPS positioning. Thedynamic method may be further subdiveded into semdynamic (short-arc) and dynamic(long-arc) modes. In this paper, the mathematical models for GPS dynamic positioningin long-arc are developed, including the state equations for both satellite orbits and stationtrajectories, and GPS observation equations. The approaches to optimal control andkalman filtering are introduced to this analysis in order to reflect the statistical andphysical performance of the problem. Finally, the optimal smoothing formulation ofpost-fitted orbits are given.