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Volume 15, Issue 1
Finite-Time Stability and Instability of Nonlinear Impulsive Systems

Guihua Zhao & Hui Liang

Adv. Appl. Math. Mech., 15 (2023), pp. 49-68.

Published online: 2022-10

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  • Abstract

In this paper, the finite-time stability and instability are studied for nonlinear impulsive systems. There are mainly four concerns. 1) For the system with stabilizing impulses, a Lyapunov theorem on global finite-time stability is presented. 2) When the system without impulsive effects is globally finite-time stable (GFTS) and the settling time is continuous at the origin, it is proved that it is still GFTS over any class of impulse sequences, if the mixed impulsive jumps satisfy some mild conditions. 3) For systems with destabilizing impulses, it is shown that to be finite-time stable, the destabilizing impulses should not occur too frequently, otherwise, the origin of the impulsive system is finite-time instable, which are formulated by average dwell time (ADT) conditions respectively. 4) A theorem on finite-time instability is provided for system with stabilizing impulses. For each GFTS theorem of impulsive systems considered in this paper, the upper boundedness of settling time is given, which depends on the initial value and impulsive effects. Some numerical examples are given to illustrate the theoretical analysis.

  • AMS Subject Headings

34A37, 93D40

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COPYRIGHT: © Global Science Press

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@Article{AAMM-15-49, author = {Zhao , Guihua and Liang , Hui}, title = {Finite-Time Stability and Instability of Nonlinear Impulsive Systems}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {15}, number = {1}, pages = {49--68}, abstract = {

In this paper, the finite-time stability and instability are studied for nonlinear impulsive systems. There are mainly four concerns. 1) For the system with stabilizing impulses, a Lyapunov theorem on global finite-time stability is presented. 2) When the system without impulsive effects is globally finite-time stable (GFTS) and the settling time is continuous at the origin, it is proved that it is still GFTS over any class of impulse sequences, if the mixed impulsive jumps satisfy some mild conditions. 3) For systems with destabilizing impulses, it is shown that to be finite-time stable, the destabilizing impulses should not occur too frequently, otherwise, the origin of the impulsive system is finite-time instable, which are formulated by average dwell time (ADT) conditions respectively. 4) A theorem on finite-time instability is provided for system with stabilizing impulses. For each GFTS theorem of impulsive systems considered in this paper, the upper boundedness of settling time is given, which depends on the initial value and impulsive effects. Some numerical examples are given to illustrate the theoretical analysis.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0381}, url = {http://global-sci.org/intro/article_detail/aamm/21125.html} }
TY - JOUR T1 - Finite-Time Stability and Instability of Nonlinear Impulsive Systems AU - Zhao , Guihua AU - Liang , Hui JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 49 EP - 68 PY - 2022 DA - 2022/10 SN - 15 DO - http://doi.org/10.4208/aamm.OA-2021-0381 UR - https://global-sci.org/intro/article_detail/aamm/21125.html KW - Impulsive systems, finite-time stability, finite-time instability, nonlinear systems. AB -

In this paper, the finite-time stability and instability are studied for nonlinear impulsive systems. There are mainly four concerns. 1) For the system with stabilizing impulses, a Lyapunov theorem on global finite-time stability is presented. 2) When the system without impulsive effects is globally finite-time stable (GFTS) and the settling time is continuous at the origin, it is proved that it is still GFTS over any class of impulse sequences, if the mixed impulsive jumps satisfy some mild conditions. 3) For systems with destabilizing impulses, it is shown that to be finite-time stable, the destabilizing impulses should not occur too frequently, otherwise, the origin of the impulsive system is finite-time instable, which are formulated by average dwell time (ADT) conditions respectively. 4) A theorem on finite-time instability is provided for system with stabilizing impulses. For each GFTS theorem of impulsive systems considered in this paper, the upper boundedness of settling time is given, which depends on the initial value and impulsive effects. Some numerical examples are given to illustrate the theoretical analysis.

Zhao , Guihua and Liang , Hui. (2022). Finite-Time Stability and Instability of Nonlinear Impulsive Systems. Advances in Applied Mathematics and Mechanics. 15 (1). 49-68. doi:10.4208/aamm.OA-2021-0381
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