Abstract
In this paper, we discuss the semicontinuities of orbital and limit set maps in an impulsive semidynamical system and investigate their relationships with the stabilities of orbits. Actually, we only deal with infinite impulsive trajectories under the hypotheses that each prolongational set is compact in the phase space. We prove that if the limit set is stable (eventually stable or eventually weakly stable), then the corresponding limit set map is upper semicontinuous or lower semicontinuous, respectively. And if the limit set map is upper semicontinuous (lower semicontinuous), then the corresponding limit set is stable (eventually stable or eventually weakly stable, respectively). Furthermore, we give several sufficient conditions to guarantee that limit sets are minimal.
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The authors sincerely thank the referees for the many valuable suggestions and corrections. Also, the authors greatly appreciate the editor’s quick correspondence and excellent editorial work.
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Li, K., Ding, C., Wang, F. et al. Limit Set Maps in Impulsive Semidynamical Systems. J Dyn Control Syst 20, 47–58 (2014). https://doi.org/10.1007/s10883-013-9204-5
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DOI: https://doi.org/10.1007/s10883-013-9204-5