# American Institute of Mathematical Sciences

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 [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

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
 [1] Stefan Ruschel, Serhiy Yanchuk. The Spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321 [2] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [3] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [4] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [5] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [6] Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 [7] Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108 [8] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [9] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [10] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [11] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [12] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [13] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [14] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [15] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [16] Sergio Conti, Georg Dolzmann. Optimal laminates in single-slip elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 1-16. doi: 10.3934/dcdss.2020302 [17] Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 [18] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [19] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [20] Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

2019 Impact Factor: 1.338