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

1. 

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

Received  September 2015 Revised  February 2016 Published  July 2016

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.
Citation: Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1016/0022-247X(82)90243-8.  Google Scholar

[2]

R. M. Corless and D. J. Jeffrey, The Wright $\omega$ function,, Lecture Notes in Comput. Sci., 2385 (2002), 76.  doi: 10.1007/3-540-45470-5_10.  Google Scholar

[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.  doi: 10.1007/BF02124750.  Google Scholar

[4]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay equations. Functional, Complex, and Nonlinear Analysis,, Springer-Verlag, (1995).  doi: 10.1007/978-1-4612-4206-2.  Google Scholar

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M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion,, Phy. Rev. E, 67 (2003).  doi: 10.1103/PhysRevE.67.036204.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.98.114101.  Google Scholar

[7]

N. D. Hayes, Roots of the transcendental equation associated with a certain differential-difference equation,, J. London Math. Soc., 25 (1950), 226.   Google Scholar

[8]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985).  doi: 10.1017/CBO9780511810817.  Google Scholar

[9]

P. Hövel and E. Schöll, Control of unstable steady states by time-delayed feedback methods,, Phy. Rev. E, 72 (2005).   Google Scholar

[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.  doi: 10.1080/00207170010017275.  Google Scholar

[11]

H. Matsunaga, Delay-dependent and delay-independent stability criteria for a delay differential system,, Proc. Amer. Math. Soc., 136 (2008), 4305.  doi: 10.1090/S0002-9939-08-09396-9.  Google Scholar

[12]

K. Pyragas, Continuous control of chaos by self-controlling feedback,, Controlling chaos: Theoretical and practical methods in non-linear dynamics, (1996), 118.  doi: 10.1016/B978-012396840-1/50038-2.  Google Scholar

[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.  doi: 10.1016/j.automatica.2006.05.008.  Google Scholar

[14]

G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions,, Longman Scientific & Technical, (1989).   Google Scholar

[15]

J. Wei and C. Zhang, Stability analysis in a first-order complex differential equations with delay,, Nonlinear Anal., 59 (2004), 657.  doi: 10.1016/j.na.2004.07.027.  Google Scholar

[16]

E. M. Wright, Solution of the equation $z e^z = a$,, Bull. Amer. Math. Soc., 65 (1959), 89.  doi: 10.1090/S0002-9904-1959-10290-1.  Google Scholar

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.  doi: 10.1016/0022-247X(82)90243-8.  Google Scholar

[2]

R. M. Corless and D. J. Jeffrey, The Wright $\omega$ function,, Lecture Notes in Comput. Sci., 2385 (2002), 76.  doi: 10.1007/3-540-45470-5_10.  Google Scholar

[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.  doi: 10.1007/BF02124750.  Google Scholar

[4]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay equations. Functional, Complex, and Nonlinear Analysis,, Springer-Verlag, (1995).  doi: 10.1007/978-1-4612-4206-2.  Google Scholar

[5]

M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion,, Phy. Rev. E, 67 (2003).  doi: 10.1103/PhysRevE.67.036204.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.98.114101.  Google Scholar

[7]

N. D. Hayes, Roots of the transcendental equation associated with a certain differential-difference equation,, J. London Math. Soc., 25 (1950), 226.   Google Scholar

[8]

R. A. Horn and C. R. Johnson, Matrix Analysis,, Cambridge University Press, (1985).  doi: 10.1017/CBO9780511810817.  Google Scholar

[9]

P. Hövel and E. Schöll, Control of unstable steady states by time-delayed feedback methods,, Phy. Rev. E, 72 (2005).   Google Scholar

[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.  doi: 10.1080/00207170010017275.  Google Scholar

[11]

H. Matsunaga, Delay-dependent and delay-independent stability criteria for a delay differential system,, Proc. Amer. Math. Soc., 136 (2008), 4305.  doi: 10.1090/S0002-9939-08-09396-9.  Google Scholar

[12]

K. Pyragas, Continuous control of chaos by self-controlling feedback,, Controlling chaos: Theoretical and practical methods in non-linear dynamics, (1996), 118.  doi: 10.1016/B978-012396840-1/50038-2.  Google Scholar

[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.  doi: 10.1016/j.automatica.2006.05.008.  Google Scholar

[14]

G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions,, Longman Scientific & Technical, (1989).   Google Scholar

[15]

J. Wei and C. Zhang, Stability analysis in a first-order complex differential equations with delay,, Nonlinear Anal., 59 (2004), 657.  doi: 10.1016/j.na.2004.07.027.  Google Scholar

[16]

E. M. Wright, Solution of the equation $z e^z = a$,, Bull. Amer. Math. Soc., 65 (1959), 89.  doi: 10.1090/S0002-9904-1959-10290-1.  Google Scholar

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