On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay

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October 2016, 36(10): 5657-5679. doi: 10.3934/dcds.2016048

On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay

Junya Nishiguchi ^1,

1.	Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

Received September 2015 Revised February 2016 Published July 2016

Abstract
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A transcendental equation $\lambda + \alpha - \beta\mathrm{e}^{-\lambda\tau} = 0$ with complex coefficients is investigated. This equation can be obtained from the characteristic equation of a linear differential equation with a single constant delay. It is known that the set of roots of this equation can be expressed by the Lambert W function. We analyze the condition on parameters for which all the roots have negative real parts by using the ``graph-like'' expression of the W function. We apply the obtained results to the stabilization of an unstable equilibrium solution by the delayed feedback control and the stability condition of the synchronous state in oscillator networks.

Keywords: Delay differential equations, exponential stability, equilibrium solutions., single constant delay.

Mathematics Subject Classification: Primary: 34K20, 34K25; Secondary: 93C2.

Citation: Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048

References:

[1]	K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627. doi: 10.1016/0022-247X(82)90243-8.
[2]	R. M. Corless and D. J. Jeffrey, The Wright $\omega$ function, Lecture Notes in Comput. Sci., 2385 (2002), Springer, Berlin, 76-89. doi: 10.1007/3-540-45470-5_10.
[3]	R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750.
[4]	O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay equations. Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.
[5]	M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phy. Rev. E, 67 (2003), 036204. doi: 10.1103/PhysRevE.67.036204.
[6]	B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett., 98 (2007), 114101. doi: 10.1103/PhysRevLett.98.114101.
[7]	N. D. Hayes, Roots of the transcendental equation associated with a certain differential-difference equation, J. London Math. Soc., 25 (1950), 226-232.
[8]	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. doi: 10.1017/CBO9780511810817.
[9]	P. Hövel and E. Schöll, Control of unstable steady states by time-delayed feedback methods, Phy. Rev. E, 72 (2005), 046203.
[10]	H. Kokame, K. Hirata, K. Konishi and T. Mori, State difference feedback for stabilizing uncertain steady states of non-linear systems, Internat. J. Control, 74 (2001), 537-546. doi: 10.1080/00207170010017275.
[11]	H. Matsunaga, Delay-dependent and delay-independent stability criteria for a delay differential system, Proc. Amer. Math. Soc., 136 (2008), 4305-4312. doi: 10.1090/S0002-9939-08-09396-9.
[12]	K. Pyragas, Continuous control of chaos by self-controlling feedback, Controlling chaos: Theoretical and practical methods in non-linear dynamics, (1996), 118-123. doi: 10.1016/B978-012396840-1/50038-2.
[13]	H. Shinozaki and T. Mori, Robust stability analysis of linear time-delay systems by Lambert $W$ function: Some extreme point results, Automatica, 42 (2006), 1791-1799. doi: 10.1016/j.automatica.2006.05.008.
[14]	G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical, Harlow, 1989.
[15]	J. Wei and C. Zhang, Stability analysis in a first-order complex differential equations with delay, Nonlinear Anal., 59 (2004), 657-671. doi: 10.1016/j.na.2004.07.027.
[16]	E. M. Wright, Solution of the equation $z e^z = a$, Bull. Amer. Math. Soc., 65 (1959), 89-93. doi: 10.1090/S0002-9904-1959-10290-1.

show all references

References:

[1]	K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86 (1982), 592-627. doi: 10.1016/0022-247X(82)90243-8.
[2]	R. M. Corless and D. J. Jeffrey, The Wright $\omega$ function, Lecture Notes in Comput. Sci., 2385 (2002), Springer, Berlin, 76-89. doi: 10.1007/3-540-45470-5_10.
[3]	R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750.
[4]	O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay equations. Functional, Complex, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.
[5]	M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phy. Rev. E, 67 (2003), 036204. doi: 10.1103/PhysRevE.67.036204.
[6]	B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Refuting the odd-number limitation of time-delayed feedback control, Phys. Rev. Lett., 98 (2007), 114101. doi: 10.1103/PhysRevLett.98.114101.
[7]	N. D. Hayes, Roots of the transcendental equation associated with a certain differential-difference equation, J. London Math. Soc., 25 (1950), 226-232.
[8]	R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985. doi: 10.1017/CBO9780511810817.
[9]	P. Hövel and E. Schöll, Control of unstable steady states by time-delayed feedback methods, Phy. Rev. E, 72 (2005), 046203.
[10]	H. Kokame, K. Hirata, K. Konishi and T. Mori, State difference feedback for stabilizing uncertain steady states of non-linear systems, Internat. J. Control, 74 (2001), 537-546. doi: 10.1080/00207170010017275.
[11]	H. Matsunaga, Delay-dependent and delay-independent stability criteria for a delay differential system, Proc. Amer. Math. Soc., 136 (2008), 4305-4312. doi: 10.1090/S0002-9939-08-09396-9.
[12]	K. Pyragas, Continuous control of chaos by self-controlling feedback, Controlling chaos: Theoretical and practical methods in non-linear dynamics, (1996), 118-123. doi: 10.1016/B978-012396840-1/50038-2.
[13]	H. Shinozaki and T. Mori, Robust stability analysis of linear time-delay systems by Lambert $W$ function: Some extreme point results, Automatica, 42 (2006), 1791-1799. doi: 10.1016/j.automatica.2006.05.008.
[14]	G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, Longman Scientific & Technical, Harlow, 1989.
[15]	J. Wei and C. Zhang, Stability analysis in a first-order complex differential equations with delay, Nonlinear Anal., 59 (2004), 657-671. doi: 10.1016/j.na.2004.07.027.
[16]	E. M. Wright, Solution of the equation $z e^z = a$, Bull. Amer. Math. Soc., 65 (1959), 89-93. doi: 10.1090/S0002-9904-1959-10290-1.

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