Brockett 开性剖面与增益受限反馈镇定
Brockett Openness Profiles and Gain-Limited Feedback Stabilization
AI总结：
本文通过引入开性剖面量化Brockett必要条件，证明系统向量场的开性剖面约束了镇定反馈的增益增长，并给出了反馈增益的下界。
摘要：
Brockett 必要条件断言，一个连续可镇定的非线性控制系统必须有一个在平衡点处开的向量场。我们表明，这种开性条件背后的定量数据限制了镇定反馈的可能增长。对于系统向量场 $f$，我们关联其开性剖面 $Ω_f(r)=\sup\{ρ:\mathbb{B}_ρ(0)\subset f(\mathbb{B}_r(0,0))\}$，使得 Brockett 条件变为对所有足够小的 $r>0$ 有 $Ω_f(r)>0$。如果反馈 $u$ 满足 $\|u(x)\|\leq d(\|x\|)$，则闭环场 $F_u(x)=f(x,u(x))$ 的开性剖面满足 $Ω_{F_u}(r)\leq Ω_f\!\left(\sqrt{r^2+d(r)^2} ight)$。因此，任何对闭环动态的指定下开性率都会产生反馈增长的必要下界。对于具有 $Ω_f(r)\lesssim r^q$ 的系统，线性率的闭环开性迫使 $d(r)\gtrsim r^{1/q}$，并且该指数在初等多式例子中是尖锐的。因此，Brockett 条件不仅仅是一个二元的拓扑障碍；其定量剖面决定了镇定反馈的增益要求。
英文摘要：
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.
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