Brockett Openness Profiles and Gain-Limited Feedback Stabilization
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arXiv
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| Natura: | Preprint |
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2026
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| author | Christopherson, Bryce Jafari, Farhad |
| author_facet | Christopherson, Bryce Jafari, Farhad |
| contents | 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 $Ω_f(r)=\sup\{ρ:\mathbb{B}_ρ(0)\subset f(\mathbb{B}_r(0,0))\}$, so that Brockett's condition becomes $Ω_f(r)>0$ 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 $Ω_{F_u}(r)\leq Ω_f\!\left(\sqrt{r^2+d(r)^2}\right)$. Consequently, any prescribed lower openness rate for the closed-loop dynamics yields a necessary lower bound on the feedback growth. For systems with $Ω_f(r)\lesssim r^q$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_17847 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Brockett Openness Profiles and Gain-Limited Feedback Stabilization Christopherson, Bryce Jafari, Farhad Optimization and Control 93D15, 93C10 (Primary), 34D20, 54C60 (Secondary) 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 $Ω_f(r)=\sup\{ρ:\mathbb{B}_ρ(0)\subset f(\mathbb{B}_r(0,0))\}$, so that Brockett's condition becomes $Ω_f(r)>0$ 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 $Ω_{F_u}(r)\leq Ω_f\!\left(\sqrt{r^2+d(r)^2}\right)$. Consequently, any prescribed lower openness rate for the closed-loop dynamics yields a necessary lower bound on the feedback growth. For systems with $Ω_f(r)\lesssim r^q$, 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. |
| title | Brockett Openness Profiles and Gain-Limited Feedback Stabilization |
| topic | Optimization and Control 93D15, 93C10 (Primary), 34D20, 54C60 (Secondary) |
| url | https://arxiv.org/abs/2602.17847 |