Abstract The present article gathers the analysis of non-asymptotic convergence rates (finite-time and fixed-time) with the property of input-to-state stability. Theoretical tools to determine this joint property are presented for the case where an explicit ISS Lyapunov function is known, and when it remains in implicit form (e.g. as a solution of an algebraic equation). For the case of finite-time input-to-state stability, necessary and sufficient conditions are given whereas for the fixed-time case only a sufficient condition is obtained. Academic examples and numerical simulations support the obtained results.

中文翻译：

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有限时间和固定时间输入到状态的稳定性：显式和隐式方法

摘要 本文收集了具有输入到状态稳定性的非渐近收敛率（有限时间和固定时间）的分析。在显式 ISS 李雅普诺夫函数已知的情况下，以及当它保持隐式形式（例如，作为代数方程的解）的情况下，提出了确定这种联合特性的理论工具。对于有限时间输入到状态稳定性的情况，给出了充要条件，而对于固定时间的情况，只获得了充分条件。学术实例和数值模拟支持了所得结果。