Stabilization of Nonlinear Systems by Gain-Limited Feedback Laws

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.

💡 Research Summary

The paper addresses a practically motivated question: when a nonlinear control system must be stabilized by a feedback law whose magnitude is limited by a prescribed function of the state, under what conditions does such a gain‑limited feedback exist? The authors focus on local asymptotic stabilization of systems of the form (\dot x = f(x,u)) around an equilibrium ((x_e,0)) and introduce a non‑constant “gain‑limit” function (d(x)) that bounds the admissible control norm in a neighbourhood of the equilibrium. While classical results guarantee that any stabilizable system can be stabilized with arbitrarily small control on a sufficiently small ball (provided the feedback is continuous), this ignores the growth rate of the feedback near the equilibrium, which is often the critical factor in real applications (e.g., actuator saturation, power limits).

To capture this growth‑rate issue, the authors refine Brockett’s well‑known openness condition. Brockett’s theorem states that a necessary condition for the existence of a continuous stabilizing feedback is that the mapping ((x,u)\mapsto f(x,u)) be open at the equilibrium. Openness, however, is a binary property and does not quantify how “strongly” the image of a small ball expands. The paper therefore defines continuous openness: a mapping (f) is continuously open at a point (x^\ast) with rate function (g) if for all sufficiently small radii (r) one has
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