# Stabilization under round robin scheduling of control inputs in nonlinear systems

## TL;DR

This paper proves that in nonlinear control systems, stabilizing feedback controllers can be sparsified through round-robin scheduling of control inputs, ensuring stability when the scheduling is sufficiently fast, with the basin of attraction preserved.

## Contribution

It introduces a novel analysis of stability under round-robin scheduled control input sparsification in nonlinear systems, extending switched systems stability theory.

## Key findings

- Fast round-robin scheduling preserves stability and basin of attraction.
- Local asymptotic stabilization is achievable under mild conditions.
- Numerical examples illustrate subtle stability features.

## Abstract

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.