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Oscillatority conditions for nonlinear systems with delay. (English) Zbl 1151.34054

The authors propose sufficient conditions of oscillatority in the sense of Yakubovich for nonlinear systems with time delay. By using a circadian oscillations model and a blood cells production model as illustrative examples, they showe that the amplitude bounds obtained from analytical calculations agree with those from computer simulations.

MSC:

34K13 Periodic solutions to functional-differential equations
34K11 Oscillation theory of functional-differential equations
93C23 Control/observation systems governed by functional-differential equations
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References:

[1] Z.-P. Jiang, A. R. Teel, and L. Praly, “Small-gain theorem for ISS systems and applications,” Mathematics of Control, Signals, and Systems, vol. 7, no. 2, pp. 95-120, 1994. · Zbl 0836.93054 · doi:10.1007/BF01211469
[2] R. Sepulchre, M. Janković, and P. V. Kokotović, Constructive Nonlinear Control, Communications and Control Engineering Series, Springer, Berlin, Germany, 1997. · Zbl 1067.93500
[3] D. V. Efimov and A. L. Fradkov, “Excitation of oscillations in nonlinear systems under static feedback,” in Proceedings of the 43rd IEEE Conference on Decision and Control (CDC ’04), vol. 3, pp. 2521-2526, Nassau, Bahamas, December 2004.
[4] A. L. Fradkov, “Feedback resonance in nonlinear oscillators,” in Proceedings of the 5th European Control Conference (ECC ’99), Karlsruhe, Germany, August-September 1999.
[5] A. L. Fradkov, “Physics and control: exploring physical systems by feedback,” in Proceedings of the 5th IFAC Symposium on Nonlinear Control Systems (NOLCOS ’01), A. B. Kurzhanski and A. L. Fradkov, Eds., pp. 1421-1427, Elsevier, St. Petersburg, Russia, July 2002.
[6] G. A. Leonov, I. M. Burkin, and A. I. Shepeljavyi, Frequency Methods in Oscillation Theory, vol. 357 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996. · Zbl 0844.34005
[7] S. Martínez, J. Cortés, and F. Bullo, “Analysis and design of oscillatory control systems,” IEEE Transactions on Automatic Control, vol. 48, no. 7, pp. 1164-1177, 2003. · Zbl 1364.93655 · doi:10.1109/TAC.2003.814104
[8] V. A. Yakubovich, “A frequency theorem in control theory,” Siberian Mathematical Journal, vol. 14, no. 2, pp. 265-289, 1973. · Zbl 0271.93017 · doi:10.1007/BF00967952
[9] V. A. Yakubovich, “Frequency conditions of oscillations in nonlinear control systems with one single-valued or hysteresis-type nonlinearity,” Automation and Remote Control, vol. 36, no. 12 part 1, pp. 1973-1985, 1975. · Zbl 0342.93025
[10] V. B. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, vol. 463 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. · Zbl 0917.34001
[11] V. B. Kolmanovskii and V. R. Nosov, Stability and Periodic Regimes of Control Systems with Aftereffect, Nauka, Moscow, Russia, 1981. · Zbl 0457.93002
[12] T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, vol. 178 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1985. · Zbl 0635.34001
[13] F. Mazenc and S.-I. Niculescu, “Lyapunov stability analysis for nonlinear delay systems,” Systems & Control Letters, vol. 42, no. 4, pp. 245-251, 2001. · Zbl 0974.93059 · doi:10.1016/S0167-6911(00)00093-1
[14] J. Mallet-Paret and G. R. Sell, “The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay,” Journal of Differential Equations, vol. 125, no. 2, pp. 441-489, 1996. · Zbl 0849.34056 · doi:10.1006/jdeq.1996.0037
[15] J. D. Murray, Mathematical Biology. Vol. 1: An Introduction, vol. 17 of Interdisciplinary Applied Mathematics, Springer, New York, NY, USA, 3rd edition, 2002. · Zbl 1006.92001 · doi:10.1007/b98868
[16] G. Enciso and E. D. Sontag, “On the stability of a model of testosterone dynamics,” Journal of Mathematical Biology, vol. 49, no. 6, pp. 627-634, 2004. · Zbl 1067.92029 · doi:10.1007/s00285-004-0291-5
[17] D. Angeli and E. D. Sontag, “An analysis of a circadian model using the small-gain approach to monotone systems,” in Proceedings of the 43rd IEEE Conference on Decision and Control (CDC ’04), vol. 1, pp. 575-578, Nassau, Bahamas, December 2004.
[18] A. Goldbeter, “A model for circadian oscillations in the Drosophila period protein (PER),” Proceedings of the Royal Society of London-B. Biological Sciences, vol. 261, no. 1362, pp. 319-324, 1995.
[19] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. · Zbl 0777.34002
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