This paper addresses Lyapunov characterizations on input-to-state stability (ISS) of time-varying nonlinear systems with infinite delays. With novel ISS definitions in the case of nonlinear systems with infinite delays, we present several results on their ISS Lyapunov characterizations in the form of both ISS Lyapunov theorems and converse ISS Lyapunov theorems. It is shown that an infinite-delayed system is (locally) ISS if it has a (local) ISS Lyapunov functional, and conversely, there exists a (local) ISS Lyapunov functional if it is (locally) ISS. To prove the converse ISS Lyapunov theorems, we establish a key technical lemma bridging ISS/LISS and robust asymptotic stability of systems with infinite delays and two converse Lyapunov theorems concerning robust asymptotic stability of systems with infinite delays. Two distinctive advantages of this work are that a large class of infinite dimensional spaces are allowed and the results are established based on a more general Lipschitz condition, i.e., the right hand side Lipschitz (RS-L) condition. An example is provided for illustration of the obtained results.

本文讨论了具有无限延迟的时变非线性系统的输入到状态稳定性 (ISS) 的 Lyapunov 表征。在具有无限延迟的非线性系统的情况下，通过新颖的 ISS 定义，我们以 ISS Lyapunov 定理和逆 ISS Lyapunov 定理的形式展示了它们的 ISS Lyapunov 表征的几个结果。结果表明，如果无限延迟系统具有（局部）ISS Lyapunov 泛函，则它是（局部）ISS，反之，如果它是（局部）ISS，则存在（局部）ISS Lyapunov 泛函。为了证明逆 ISS Lyapunov 定理，我们建立了一个关键技术引理，桥接 ISS/LISS 和具有无限延迟的系统的稳健渐近稳定性和两个关于稳健的逆 Lyapunov 定理无限时滞系统的渐近稳定性。这项工作的两个显着优点是允许一大类无限维空间，并且结果是基于更一般的 Lipschitz 条件建立的，即右手边 Lipschitz (RS-L) 条件。提供了一个例子来说明获得的结果。