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Note on the persistent property of a feedback control system with delays. (English) Zbl 1187.34106

Summary: By developing some new analysis techniques, we show that the following feedback control system of differential equations with delays is permanent.

$\begin{array}{cc}& \frac{dN\left(t\right)}{dt}=r\left(t\right)N\left(t\right)\left[1-\frac{{N}^{2}\left(t-{\tau }_{1}\left(t\right)\right)}{{k}^{2}\left(t\right)}-c\left(t\right)u\left(t-{\tau }_{2}\left(t\right)\right)\right],\hfill \\ & \frac{du\left(t\right)}{dt}=-a\left(t\right)u\left(t\right)+b\left(t\right)N\left(t-{\tau }_{1}\left(t\right)\right),\hfill \end{array}$

where ${\tau }_{1},{\tau }_{2},a,b,c,r,k\in C\left(ℝ,\left(0,+\infty \right)\right)$ are $\omega$-periodic functions, which means that feedback control variable has no influence on the persistent property of the above system.

##### MSC:
 34K25 Asymptotic theory of functional-differential equations 93B52 Feedback control
##### Keywords:
feedback control; permanence
##### References:
 [1] Hale, J.: Theory of functional differential equations, (1977) [2] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1992) · Zbl 0752.34039 [3] Fan, G. H.; Li, Y. K.; Qin, M. C.: The existence of positive periodic solutions for periodic feedback control systems with delays, Zeitschrift für angewandte Mathematik und mechanik 84, No. 6, 425-430 (2004) [4] Gaines, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations, (1977) [5] Chen, F. D.; Liao, X. Y.; Huang, Z. K.: The dynamic behavior of N-species cooperation system with continuous time delays and feedback controls, Applied mathematics and computation 181, No. 2, 803-815 (2006) · Zbl 1102.93021 · doi:10.1016/j.amc.2006.02.007 [6] Chen, F. D.: Global stability of a single species model with feedback control and distributed time delay, Applied mathematics and computation 178, No. 2, 474-479 (2006) · Zbl 1101.92035 · doi:10.1016/j.amc.2005.11.062 [7] Fan, Y. H.; Wang, L. L.: Permanence for a discrete model with feedback control and delay, Discrete dynamics in nature and society 2008 (2008) · Zbl 1149.39003 · doi:10.1155/2008/945109 [8] Nie, L. F.; Peng, J. G.; Teng, Z. D.: Permanence and stability in multi-species non-autonomous Lotka–Volterra competitive systems with delays and feedback controls, Mathematical and computer modelling 49, No. 1–2, 295-306 (2009) · Zbl 1165.34373 · doi:10.1016/j.mcm.2008.05.004 [9] Chen, F. D.; Chen, X. X.; Shi, J. L.: Dynamic behavior of a nonlinear single species diffusive system, Advances in complex systems 8, No. 4, 399-417 (2005) · Zbl 1163.34360 · doi:10.1142/S021952590500049X [10] Chen, F. D.; Shi, J. L.: Periodicity in a logistic type system with several delays, Computer and mathematics with applications 48, No. 1–2, 35-44 (2004) · Zbl 1061.34050 · doi:10.1016/j.camwa.2004.02.001 [11] Chen, F. D.; Sun, D. X.; Shi, J. L.: Periodicity in a food-limited population model with toxicants and state dependent delays, Journal of mathematical analysis and applications 288, No. 1, 132-142 (2003) · Zbl 1087.34045 · doi:10.1016/S0022-247X(03)00586-9 [12] Chen, F. D.; Xie, X. D.: Permanence and extinction in nonlinear single and multiple species system with diffusion, Applied mathematics and computation 177, No. 1, 410-426 (2006) · Zbl 1090.92046 · doi:10.1016/j.amc.2005.11.019 [13] Li, Y.; Liu, P.; Zhu, L.: Positive periodic solutions of a class of functional differential systems with feedback controls, Nonlinear analysis 57, 655-666 (2004) · Zbl 1064.34049 · doi:10.1016/j.na.2004.03.006