AI 中文总结
该研究针对全耦合向量值非参数系统，通过博弈论刻画反馈能力阈值，证明了阈值与博弈值\\(\Gamma_d\\)等价，给出了不同维度下的阈值关系及近邻反馈律的下界。
AI 中文摘要
我们研究了\\(\mathbb{R}^d\\)中离散时间系统\\(x_{t+1}=f(x_t)+u_t+w_{t+1}\\)的反馈镇定问题，其中\\(f\\)未知且存在任意有界扰动。对于标量被控对象，广义Lipschitz不确定性下的尖锐反馈能力阈值为\\(3/2+\sqrt{2}\\)。我们处理全耦合向量值系统，这类系统无法使用标量阶和区间递推，且耦合特性排除了逐坐标约化的可能。我们引入了响应历史逃逸博弈，其获胜条件要求在规定斜率下存在有限包络且状态半径无界。Borel确定性和斜率单调性给出了独立定义的博弈值\\(\Gamma_d\\)。我们证明被控对象问题存在有限严格反馈能力阈值，并将其与\\(\Gamma_d\\)对应。若\\(L<\Gamma_d\\)，则存在一个因果反馈律可镇定不确定性类中的所有被控对象，使其抵御所有有界扰动序列；若\\(L>\Gamma_d\\)，则对每个因果反馈律，都存在同一类中的一个被控对象和一个有界扰动序列，使得闭环状态序列无界。阈值等价性通过一致亚临界响应博弈控制器和基于Kirszbraun-Valentine延拓定理的希尔伯特空间实现引理证明。显式近邻反馈律在每个有限维都给出严格大于1的下界，包括\\(\Gamma_2\ge 2/\sqrt{3}\\)。维度单调性给出\\(\Gamma_d\le\Gamma_1\\)，与标量理论对比得\\(\Gamma_1=3/2+\sqrt{2}\\)。
英文摘要
We study feedback stabilization for the discrete-time system $x_{t+1}=f(x_t)+u_t+w_{t+1}$ in $\mathbb{R}^d$ with unknown $f$ and arbitrary bounded disturbances. For scalar plants, the sharp feedback capability threshold under generalized Lipschitz uncertainty is $3/2+\sqrt{2}$. We treat fully coupled vector-valued systems, where scalar order and interval recursion are unavailable and coupling precludes a coordinatewise reduction. We introduce a response-history escape game in which the adversary seeks a finite envelope and an unbounded state radius. Borel determinacy ensures that exactly one player has a winning strategy at each slope. We prove that the same player wins from every finite response history, and slope monotonicity gives an independently defined game value $Γ_d$. We prove that $Γ_d$ is finite and is the strict feedback capability threshold for the plant problem. If $L<Γ_d$, one causal feedback law stabilizes every plant in the uncertainty class against every bounded disturbance sequence. If $L>Γ_d$, for every causal feedback law there exist a plant in the same class and a bounded disturbance sequence such that the closed-loop state sequence is unbounded. The proof uses one controller for all subcritical slopes and a realization in a Hilbert space based on the Kirszbraun--Valentine extension theorem. An explicit nearest-neighbor law gives a lower bound above one in every finite dimension, including $Γ_2\ge 2/\sqrt{3}$. Dimension monotonicity gives $Γ_d\leΓ_1$, and comparison with the scalar theory yields $Γ_1=3/2+\sqrt{2}$.
Comments22 pages
