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Mathematical Biosciences and Engineering (MBE)
 

On the continuity of the function describing the times of meeting impulsive set and its application

Pages: 1399 - 1406, Volume 14, Issue 5/6, October/December 2017      doi:10.3934/mbe.2017072

 
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Sanyi Tang - College of Mathematics and Information Science, Shaanxi Normal University, Xi'an, 710062, China (email)
Wenhong Pang - School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China (email)

Abstract: The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.

Keywords:  Impulsive semi-dynamical system, PoincarĂ© map, continuity, state-dependent feedback control, counter example.
Mathematics Subject Classification:  Primary: 34A37; Secondary: 47N60.

Received: April 2016;      Accepted: September 2016;      Available Online: May 2017.

 References