SIAM Journal on Control and Optimization, Volume 59, Issue 3, Page 1989-2010, January 2021.
Nonlinear robust stability theorems due to Georgiou and Smith use the concept of the nonlinear gap metric. While the link between gap metric uncertainty and the small-gain theorem is clearly drawn already in Georgiou and Smith's original seminal work, the contribution of this paper is to provide an extensive treatise on the relation between these robust stability type theorems and the nonlinear small-gain type theorems in multiple scenarios. Here a special case of the nonlinear small-gain theorem using gain functions is shown to be equivalent to a fundamental robust stability theorem of Georgiou and Smith for feedback systems. Since existence of solutions is a fundamental requirement of a feedback system and typically it is harder to establish than the uniqueness of solutions, we also show that both global and local forms of the nonlinear small-gain theorem which establish existence and boundedness properties simultaneously imply the corresponding types of Georgiou and Smith's robust stability theorem in gain function formulation. Moreover, if we only consider the boundedness for both theorems in the local setting, we show that the equivalence between them can be similarly established as in the global setting.

中文翻译：

---

Georgiou 和 Smith 的鲁棒稳定性类型定理与非线性小增益定理之间的联系

SIAM Journal on Control and Optimization，第 59 卷，第 3 期，第 1989-2010 页，2021 年 1 月。
由于 Georgiou 和 Smith 的非线性鲁棒稳定性定理使用非线性间隙度量的概念。虽然间隙度量不确定性和小增益定理之间的联系已经在 Georgiou 和 Smith 的原始开创性工作中清楚地描绘出来，但本文的贡献是对这些鲁棒稳定性类型定理与非线性小增益定理之间的关系提供了广泛的论述。多种场景下的增益类型定理。这里显示了使用增益函数的非线性小增益定理的一个特例，等效于 Georgiou 和 Smith 的反馈系统基本稳健稳定性定理。由于解决方案的存在是反馈系统的基本要求，并且通常比解决方案的唯一性更难建立，我们还表明，建立存在性和有界性的非线性小增益定理的全局和局部形式同时暗示了增益函数公式中相应类型的 Georgiou 和 Smith 鲁棒稳定性定理。此外，如果我们只考虑局部设置中两个定​​理的有界性，我们表明它们之间的等价性可以像在全局设置中一样建立。