Brockett Openness Profiles and Gain-Limited Feedback Stabilization

Abstract

Brockett's necessary condition asserts that a continuously stabilizable nonlinear control system must have a vector field that is open at the equilibrium. We show that the quantitative data behind this openness condition constrains the possible growth of stabilizing feedbacks. To a system vector field $f$, we associate its openness profile $\hat{I}{\text{c◯}}_f(r)=\sup \{\ddot{I}:{\mathbb{B}}_{\ddot{I}}(0)\subset f({\mathbb{B}}_r(0,0))\}$, so that Brockett's condition becomes $\hat{I}{\text{c◯}}_f(r)>0$ for all sufficiently small $r>0$. If a feedback $u$ satisfies $\Vert u(x)\Vert \le d(\Vert x\Vert )$, then the openness profile of the closed-loop field $F_u(x)=f(x,u(x))$ satisfies $\hat{I}{\text{c◯}}_{F_u}(r)\le \hat{I}{\text{c◯}}_f\left(\sqrt{r^2+d(r{)}^2}\right)$. Consequently, any prescribed lower openness rate for the closed-loop dynamics yields a necessary lower bound on the feedback growth. For systems with $\hat{I}{\text{c◯}}_f(r)\lesssim r^q$, linear-rate closed-loop openness forces $d(r)\gtrsim r^{1/q}$, and this exponent is sharp in elementary polynomial examples. Thus Brockett's condition is not merely a binary topological obstruction; its quantitative profile governs gain requirements for stabilizing feedback.

15 pages, 1 figure. Substantially revised version with corrected exposition and streamlined proofs