Stabilization of Nonlinear Systems by Gain-Limited Feedback Laws

TL;DR

This paper investigates the fundamental limits of stabilizing nonlinear systems with gain constraints on feedback, providing necessary conditions and sharp bounds that influence the design of feedback laws.

Contribution

It introduces a quantitative openness condition and a maximal openness rate to establish fundamental lower bounds on feedback growth for stabilization.

Findings

Derived a necessary condition for gain-limited stabilizability.

Established sharp no-go results for broad classes of nonlinear systems.

Illustrated how openness rates limit feedback growth near equilibrium.

Abstract

We study local stabilization of nonlinear control systems under explicit gain constraints on the feedback law. Using a quantitative refinement of Brockett's openness condition, we introduce the notion of a maximal continuous openness rate for the system vector field near equilibrium. Combining this with a local-section characterization of stabilizability, we derive a general necessary condition for the existence of gain-limited stabilizing feedback. This condition yields sharp no-go results for broad classes of nonlinear systems, including systems that are stabilizable only by nonsmooth feedback. Several examples illustrate how openness rates impose fundamental lower bounds on stabilizing feedback growth near an equilibrium point.