The calculus of variations is used to develop the necessary theory and derive the optimality conditions for a spacecraft to transfer between a set of initial and final conditions, while minimizing a combination of fuel consumption and a function of the estimation error covariance matrix associated with the spacecraft state. The theory is developed in a general manner that allows for multiple observers, moving observers, covariance associated with an arbitrary frame, a wide variety of observation types, multiple gravity bodies, and uncertainties in the spacecraft equations of motion based on the thrusting status of the engine. A series of example trajectories from low Earth orbit (LEO) to a near geosynchronous Earth orbit (GEO) shows that either the trace of the covariance at the final time or the integral of the trace of the covariance matrix associated with the error in the Cartesian position and velocity can be reduced significantly with a small increase in the fuel consumption. An additional example illustrates the covariance associated with the semimajor axis can be significantly reduced for a transfer from Earth orbit to lunar orbit. This example illustrates multiple, moving observers as well as a transfer in a multi-body gravitational field.
Zimmer, Scott; Ocampo, Oscar; and Bishop, Robert H., "Reducing Orbit Covariance for Continuous Thrust Spacecraft Transfers" (2010). Mechanical Engineering Faculty Research and Publications. 21.