This paper addresses the input-to-state stability (ISS) of infinite-dimensional systems by introducing a novel notion named generalized ISS-Lyapunov functional (GISS-LF) and the corresponding ISS Lyapunov theorem. Unlike the classical ISS-Lyapunov functional (ISS-LF) that must be positive definite, a GISS-LF can be positive semidefinite. Moreover, such a functional considers not only the relationship with elements in the state space but also takes into account the elements in the input space via a family of certain functionals. Consequently, this notion provides more options in constructing Lyapunov functionals for the ISS assessment of infinite-dimensional systems. In particular, we provide a positive answer to the open question raised by A. Mironchenko and C. Prieur, “Input-to-state stability of infinite-dimensional systems: recent results and open questions”, (Mironchenko and Prieur, 2020), regarding the existence of a coercive ISS-LF for the heat equation with Dirichlet boundary disturbances. To demonstrate the application of the proposed method, which we refer to as the generalized Lyapunov method, we present two examples, showing how to construct GISS-LFs by using positive semidefinite and non-coercive functionals for nonlinear parabolic equations defined over higher dimensional domains with Dirichlet boundary disturbances, and to derive small-gain conditions for guaranteeing the ISS with respect to distributed in-domain disturbances for coupled nonlinear degenerate parabolic equations, which contain ordinary differential equations as special cases.

中文翻译：

---

用于无限维系统输入到状态稳定性的广义 Lyapunov 泛函


本文通过引入一个名为广义 ISS-Lyapunov 泛函 （GISS-LF） 的新概念和相应的 ISS Lyapunov 定理来解决无限维系统的输入到状态稳定性 （ISS）。与必须是正定的经典 ISS-Lyapunov 泛函 （ISS-LF） 不同，GISS-LF 可以是正半定的。此外，这样的函数不仅考虑了与状态空间中元素的关系，还通过一系列特定的函数考虑了输入空间中的元素。因此，这个概念为构建 Lyapunov 泛函提供了更多选择，用于无限维系统的 ISS 评估。特别是，我们对 A. Mironchenko 和 C. Prieur 提出的开放性问题“无限维系统的输入到状态稳定性：最近的结果和开放性问题”（Mironchenko 和 Prieur，2020 年）提供了肯定的回答，该问题涉及具有狄利克雷边界干扰的热方程是否存在强制性 ISS-LF。为了演示所提出的方法（我们称之为广义 Lyapunov 方法）的应用，我们提供了两个示例，展示了如何通过使用正半定和非矫顽泛函来构建 GISS-LFs，用于在具有狄利克雷边界扰动的更高维域上定义的非线性抛物线方程，并推导出小增益条件来保证 ISS 相对于耦合非线性简并抛物线的分布式域内扰动方程，其中包含常微分方程作为特殊情况。