# American Institute of Mathematical Sciences

March  2020, 25(3): 841-857. doi: 10.3934/dcdsb.2019192

## Input-to-state stability of continuous-time systems via finite-time Lyapunov functions

 1 School of Mathematics and Physics, China University of Geosciences (Wuhan), 430074, Wuhan, China

* Corresponding author: Huijuan Li

Received  November 2018 Revised  March 2019 Published  September 2019

Fund Project: This work was partially supported by National Natural Science Foundation of China [NSFC11701533].

In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.

Citation: Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192
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