Measure openness
Replace a yes-or-no condition
The profile Ωf(r) retains the scale-dependent information hidden inside topological openness at the equilibrium.
Paper project · Nonlinear control
Brockett Openness Profiles and Gain-Limited Feedback Stabilization
Overview
Brockett’s condition has a quantitative shape.
Brockett’s necessary condition says that a nonlinear control system admitting continuous stabilizing feedback must have a vector field that is open at the equilibrium. The paper records how much openness is available at each scale through Ωf(r), the radius of the largest centered output ball guaranteed by an input ball of radius r.
A feedback law with growth bounded by d(r) can only sample a correspondingly enlarged portion of state-control space. Comparing the original and closed-loop openness profiles therefore turns a desired closed-loop rate into a necessary lower bound on feedback growth.
Track the gain budget
Feedback enlarges the sampled radius
If ‖u(x)‖≤d(‖x‖), the closed-loop profile is bounded by the original profile at radius √(r²+d(r)²).
Read off rate constraints
Openness limits low-gain stabilization
When Ωf(r) grows like a q-th power of r, linear-rate closed-loop openness requires feedback growth on the order of the 1/q power of r; polynomial examples make the exponent sharp.
Interactive necessary-condition laboratory
Openness constrains the gain envelope.
Move the open-loop rate, desired closed-loop rate, gain envelope, and inspection scale. The exact radial example shows when the theorem rules a target out—and why a quadratic openness profile demands square-root rather than linear feedback growth near the origin.
Citation
Use or reference this work.
The citation below uses the current arXiv record.
BibTeX citation
@article{christopherson2026brockett,
title = {Brockett Openness Profiles and Gain-Limited Feedback Stabilization},
author = {Christopherson, Bryce A. and Jafari, Farhad},
year = {2026},
eprint = {2602.17847},
archivePrefix = {arXiv},
primaryClass = {math.OC},
url = {https://arxiv.org/abs/2602.17847}
}