Boundedness of spacecraft hovering under dead-band control in time-invariant systems

In this paper, we derive sufficient conditions for local and global boundedness of spacecraft motion inside a prescribed region subject to a dead-band hovering thrust control law in time-invariant Lagrangian dynamical systems. Using the conservative properties of the system, we define a zero-velocity restriction on the spacecraft motion, then show that a dead-band controller exists that bounds nearby trajectories arbitrarily close to the desired hovering position. The minimum number of independent directions that the dead band must restrict is well defined as a function of hovering position and divides the position space into distinct dynamical regions. We present numerical plots of these regions in the two-body, restricted three-body, and Hill problems and find hovering near a central body usually requires dead-band control in only one or two independent directions to be bounded. The effects of uncertainty in the initial hovering state on the zero-velocity surface are evaluated and the largest allowable perturbations in the Jacobi constant that maintain boundedness are formulated.