Numerical stability test of neutral delay differential equations.

*(English)*Zbl 1166.65356Summary: The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function.

On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

##### MSC:

65L07 | Numerical investigation of stability of solutions to ordinary differential equations |

34K20 | Stability theory of functional-differential equations |

34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |

34K40 | Neutral functional-differential equations |

##### Keywords:

numerical examples; stability; Lambert W function; iterative algorithm; neutral delay differential equations##### Software:

DDE-BIFTOOL
PDF
BibTeX
XML
Cite

\textit{Z. H. Wang}, Math. Probl. Eng. 2008, Article ID 698043, 10 p. (2008; Zbl 1166.65356)

**OpenURL**

##### References:

[1] | G. Stépán and Z. Szabó, “Impact induced internal fatigue cracks,” in Proceedings of the ASME Design Engineering Technical Conferences (DETC ’99), Las Vegas, Nev, USA, September 1999. |

[2] | A. Bellen, N. Guglielmi, and A. E. Ruehli, “Methods for linear systems of circuit delay differential equations of neutral type,” IEEE Transactions on Circuits and Systems, vol. 46, no. 1, pp. 212-216, 1999. · Zbl 0952.94015 |

[3] | A. G. Balanov, N. B. Janson, P. V. E. McClintock, R. W. Tucker, and C. H. T. Wang, “Bifurcation analysis of a neutral delay differential equation modelling the torsional motion of a driven drill-string,” Chaos, Solitons and Fractals, vol. 15, no. 2, pp. 381-394, 2003. · Zbl 1037.34075 |

[4] | Z. N. Masoud, M. F. Daqaq, and N. A. Nayfeh, “Pendulation reduction on small ship-mounted telescopic cranes,” Journal of Vibration and Control, vol. 10, no. 8, pp. 1167-1179, 2004. |

[5] | D. A. W. Barton, Dynamics and bifurcations of non-smooth delay equations, Ph.D. dissertation, University of Bristol, Bristol, UK, 2006. |

[6] | Z. N. Masoud and A. H. Nayfeh, “Sway reduction on container cranes using delayed feedback controller,” Nonlinear Dynamics, vol. 34, no. 3-4, pp. 347-358, 2003. · Zbl 1041.70502 |

[7] | Y. N. Kyrychko, K. B. Blyuss, A. Gonzalez-Buelga, S. J. Hogan, and D. J. Wagg, “Real-time dynamic substructuring in a coupled oscillator-pendulum system,” Proceedings of the Royal Society of London A, vol. 462, no. 2068, pp. 1271-1294, 2006. · Zbl 1149.70322 |

[8] | Y. N. Kyrychko, S. J. Hogan, A. Gonzalez-Buelga, and D. J. Wagg, “Modelling real-time dynamic substructuring using partial delay differential equations,” Proceedings of the Royal Society of London A, vol. 463, no. 2082, pp. 1509-1523, 2007. · Zbl 1131.74021 |

[9] | Z. N. Masoud, A. H. Nayfeh, and D. T. Mook, “Cargo pendulation reduction of ship-mounted cranes,” Nonlinear Dynamics, vol. 35, no. 3, pp. 299-311, 2004. · Zbl 1068.70542 |

[10] | Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. · Zbl 0777.34002 |

[11] | Y. X. Qin, Y. Q. Liu, L. Wang, and Z. X. Zhen, Stability of Motion of Dynamical Systems with Time Lag, Science Press, Beijing, China, 2nd edition, 1989. |

[12] | G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, vol. 210 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1989. · Zbl 0686.34044 |

[13] | S.-I. Niculescu, Delay Effects on Stability. A Robust Control Approach, vol. 269 of Lecture Notes in Control and Information Sciences, Springer, London, UK, 2001. · Zbl 0997.93001 |

[14] | H. Y. Hu and Z. H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer, Berlin, Germany, 2002. · Zbl 1035.93002 |

[15] | J. M. Krodkiewski and T. Jintanawan, “Stability improvement of periodic vibration of multi-degree-of-freedom systems by means of time-delay control,” in Proceedings of the International Conference on Vibration, Noise and Structural Dynamics, vol. 1, pp. 340-351, Venice, Italy, April, 1999. |

[16] | Z. H. Wang and H. Y. Hu, “Calculation of the rightmost characteristic root of retarded time-delay systems via Lambert W function,” to appear in Journal of Sound and Vibration. |

[17] | R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics, vol. 5, no. 4, pp. 329-359, 1996. · Zbl 0863.65008 |

[18] | C. Hwang and Y.-C. Cheng, “A note on the use of the Lambert W function in the stability analysis of time-delay systems,” Automatica, vol. 41, no. 11, pp. 1979-1985, 2005. · Zbl 1125.93440 |

[19] | H. Shinozaki and T. Mori, “Robust stability analysis of linear time-delay systems by Lambert W function: some extreme point results,” Automatica, vol. 42, no. 10, pp. 1791-1799, 2006. · Zbl 1114.93074 |

[20] | M. Y. Fu, A. W. Olbrot, and M. P. Polis, “Robust stability for time-delay systems: the edge theorem and graphical tests,” IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 813-820, 1989. · Zbl 0698.93070 |

[21] | M. Y. Fu, A. W. Olbrot, and M. P. Polis, “The edge theorem and graphical tests for robust stability of neutral time-delay systems,” Automatica, vol. 27, no. 4, pp. 739-741, 1991. |

[22] | W. Michiels and T. Vyhlídal, “An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type,” Automatica, vol. 41, no. 6, pp. 991-998, 2005. · Zbl 1091.93026 |

[23] | K. Engelborghs, T. Luzyanina, and G. Samaey, “DDE-BIFTOOL v2.0: a Matlab package for the computation analysis of delay differential equations,” Tech. Rep. TW220, Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium, 2001. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.