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 $\Omega_f(r)=\sup\{\rho:\mathbb{B}_\rho(0)\subset f(\mathbb{B}_r(0,0))\}$, so that Brockett's condition becomes $\Omega_f(r)>0$ for all sufficiently small $r>0$. If a feedback $u$ satisfies $\|u(x)\|\leq d(\|x\|)$, then the openness profile of the closed-loop field $F_u(x)=f(x,u(x))$ satisfies $\Omega_{F_u}(r)\leq \Omega_f\!\left(\sqrt{r^2+d(r)^2}\right)$. Consequently, any prescribed lower openness rate for the closed-loop dynamics yields a necessary lower bound on the feedback growth. For systems with $\Omega_f(r)\lesssim r^q$, 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.