# American Institute of Mathematical Sciences

October  2017, 14(5&6): 1399-1406. doi: 10.3934/mbe.2017072

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

 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China

* Corresponding author: Sanyi Tang

Received  April 25, 2016 Accepted  September 19, 2016 Published  May 2017

Fund Project: The first author is supported by the National Natural Science Foundation of China (NSFC 11631012,11471201), and by the Fundamental Research Funds for the Central Universities (GK201701001)

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.

Citation: Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1399-1406. doi: 10.3934/mbe.2017072
##### References:

show all references

##### References:
Illustrations of impulsive set, phase set and definition of impulsive semi-dynamical system for model (2). The parameter values are fixed as: $r=1, K=52, \beta=0.19, \eta=0.45, \omega=0.19, \delta=0.36, \tau=5, ET=35, \theta=0.7$
Three possible trajectories of model (2) with $x_{\Gamma_1}<(1-\theta)ET<ET<x_{\Gamma_2}$ for model (2). The parameter values are fixed as: $r=1, K=52, \beta=0.19, \eta=0.45, \omega=0.19, \delta=0.36, \tau=5, ET=35, \theta=0.7(A), 0.659(B)$ and $0.5(C)$
Continuity of Poincaré map and time function without impulse of model (2) for different $\theta$. The other parameter values are fixed as: $r=1, K=52, \beta=0.19, \eta=0.45, \omega=0.19, \delta=0.36, \tau=5, ET=35$
 [1] Bangyu Shen, Xiaojing Wang, Chongyang Liu. Nonlinear state-dependent impulsive system in fed-batch culture and its optimal control. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 369-380. doi: 10.3934/naco.2015.5.369 [2] Odo Diekmann, Karolína Korvasová. Linearization of solution operators for state-dependent delay equations: A simple example. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 137-149. doi: 10.3934/dcds.2016.36.137 [3] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [4] Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167 [5] Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 [6] Meng Zhang, Kaiyuan Liu, Lansun Chen, Zeyu Li. State feedback impulsive control of computer worm and virus with saturated incidence. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1465-1478. doi: 10.3934/mbe.2018067 [7] Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167 [8] Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded state-dependent delay equation. Conference Publications, 2001, 2001 (Special) : 56-65. doi: 10.3934/proc.2001.2001.56 [9] Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633 [10] Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003 [11] Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014 [12] Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065 [13] Madhu Jain, Sudeep Singh Sanga. Admission control for finite capacity queueing model with general retrial times and state-dependent rates. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2019073 [14] Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365 [15] Dalila Azzam-Laouir, Fatiha Selamnia. On state-dependent sweeping process in Banach spaces. Evolution Equations & Control Theory, 2018, 7 (2) : 183-196. doi: 10.3934/eect.2018009 [16] Paul-Eric Chaudru De Raynal. Weak regularization by stochastic drift : Result and counter example. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1269-1291. doi: 10.3934/dcds.2018052 [17] Carsten Collon, Joachim Rudolph, Frank Woittennek. Invariant feedback design for control systems with lie symmetries - A kinematic car example. Conference Publications, 2011, 2011 (Special) : 312-321. doi: 10.3934/proc.2011.2011.312 [18] M. Arisawa, P.-L. Lions. Continuity of admissible trajectories for state constraints control problems. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 297-305. doi: 10.3934/dcds.1996.2.297 [19] H. W. J. Lee, Y. C. E. Lee, Kar Hung Wong. Differential equation approximation and enhancing control method for finding the PID gain of a quarter-car suspension model with state-dependent ODE. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2019055 [20] Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267

2018 Impact Factor: 1.313