  We study stability of multivariable control-affine nonlinear systems under
sparsification of feedback controllers. Sparsification in our context refers to
the scheduling of the individual control inputs one at a time in rapid periodic
sweeps over the set of control inputs, which corresponds to round-robin
scheduling. We prove that if a locally asymptotically stabilizing feedback
controller is sparsified via the round-robin scheme and each control action is
scaled appropriately, then the corresponding equilibrium of the resulting
system is stabilized when the scheduling is sufficiently fast; under mild
additional conditions, local asymptotic stabilization of the corresponding
equilibrium can also be guaranteed. Moreover, the basin of attraction for the
equilibrium of scheduled system also remains same as the original system under
sufficiently fast switching. Our technical tools are derived from optimal
control theory, and our results also contribute to the literature on the
stability of switched systems in the fast switching regime. Illustrative
numerical examples depicting several subtle features of our results are
included.

Optimization and Control;Systems and Control;Dynamical Systems