Abstract
This paper aims to review some uniform stability results for impulsive systems. For the review, we classify the models of impulsive systems into time-based impulsive systems and state-based ones, including continuous-time impulsive systems, discrete-time impulsive systems, stochastic impulsive systems, and impulsive hybrid systems. According to these models, we review, respectively, the related stability concepts and some representative results focused on uniform stability, including the results on uniform asymptotic stability, input-to-state stability (ISS), KLL -stability (uniform stability expressed by KLL -functions), event-stability, and event-ISS. And we formulate some questions for those not fully developed aspects on uniform stability at each subsection.
Highlights
Impulsive systems have been widely studied in recent years due to the variety of applications in the fields such as mechanics, control technology, communication networks, robotics, biological population dynamics, power systems, etc
The first paper where impulses were introduced into differential equations is the article written by Milman and Myshkis [1] in 1960
The stability notions and the comparison principle approach for stochastic impulsive systems have been established in Theorems 9 and 11 by [32] and a more general SIS model was built in Equation (16) by [33]
Summary
Impulsive systems have been widely studied in recent years due to the variety of applications in the fields such as mechanics, control technology, communication networks, robotics, biological population dynamics, power systems, etc. Besides the above stated uniform stability, ISS, KLL-stability, and event-stability, concepts such as dissipativity [81,82,83,84,85], contraction [86], incremental stability [87], finite-time stability [88,89], stability for impulsive systems with different jump maps [90], stability for abstract impulsive differential equations, conditional stability (dichotomy, trichotomy), and impulsive control see, for example, [91,92,93,94,95,96,97], have been proposed and extensively studied.
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