Kybernetika 54 no. 4, 718-735, 2018

Delay-dependent stability of Runge-Kutta methods for linear neutral systems with multiple delays

Guang-Da HuDOI: 10.14736/kyb-2018-4-0718

Abstract:

In this paper, we are concerned with stability of numerical methods for linear neutral systems with multiple delays. Delay-dependent stability of Runge-Kutta methods is investigated, i. e., for delay-dependently stable systems, we ask what conditions must be imposed on the Runge-Kutta methods in order that the numerical solutions display stability property analogous to that displayed by the exact solutions. By means of Lagrange interpolation, Runge-Kutta methods can be applied to neutral differential systems with multiple delays. Based on the argument principle, sufficient conditions for delay-dependent stability of Runge-Kutta methods combined with Lagrange interpolation are presented. Numerical examples are given to illustrate the main results.

Keywords:

Runge-Kutta method, neutral differential systems with multiple delays, delay-dependent stability, Lagrange interpolation, argument principle

Classification:

65L05, 65L07, 65L20

References:

  1. A. Bellen and M. Zennaro: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford 2003.   DOI:10.1093/acprof:oso/9780198506546.001.0001
  2. J. W. Brown and R. V. Churchill: Complex Variables and Applications. McGraw-Hill Companies, Inc. and China Machine Press, Beijing 2004.   CrossRef
  3. J. K. Hale and S. M. Verduyn Lunel: Strong stabilization of neutral functional differential equations. IMA J. Math. Control Info. 19 (2002), 5-23.   DOI:10.1093/imamci/19.1\_and\_2.5
  4. G. D. Hu: Stability criteria of linear neutral systems with distributed delays. Kybernetika 47 (2011), 273-284.   CrossRef
  5. G. D. Hu and B. Cahlon: Estimations on numerically stable step-size for neutral delay differential systems with multiple delays. J. Comput. Appl. Math. 102 (1999), 221-234.   DOI:10.1016/s0377-0427(98)00215-5
  6. G. D. Hu, G. D. Hu and X. Zou: Stability of linear neutral systems with multiple delays: boundary criteria. Appl. Math. Comput. 148 (2004), 707-715.   DOI:10.1016/s0096-3003(02)00929-3
  7. C. Huang and S. Vandewalle: An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM J. Scientific Computing 25 (2004), 1608-1632.   DOI:10.1137/s1064827502409717
  8. L. W. Johnson, R. Dean Riess and J. T. Arnold: Introduction to Linear Algebra. Prentice-Hall, Englewood Cliffs 2000.   CrossRef
  9. E. I. Jury: Theory and Application of $z$-Transform Method. John Wiley and Sons, New York 1964.   CrossRef
  10. A. V. Kim and A. V. Ivanov: Systems with Delays. Scrivener Publishing LLC, Salem, Massachusetts 2015.   DOI:10.1002/9781119117841
  11. V. B. Kolmanovskii and A. Myshkis: Introduction to Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht 1999.   DOI:10.1007/978-94-017-1965-0
  12. J. D. Lambert: Numerical Methods for Ordinary Differential Systems. John Wiley and Sons, New York 1999.   CrossRef
  13. P. Lancaster and M. Tismenetsky: The Theory of Matrices with Applications. Academic Press, Orlando 1985.   CrossRef
  14. W. Michiels and S. Niculescu: Stability, Control and Computation for Time Delay Systems: An Eigenvalue Based Approach. SIAM, Philadelphia 2014.   DOI:10.1137/1.9781611973631
  15. H. Tian and J. Kuang: The stability of the $\theta$-methods in numerical solution of delay differential equations with several delay terms. J. Comput. Appl. Math. 58 (1995), 171-181.   DOI:10.1016/0377-0427(93)e0269-r
  16. T. Vyhlidal and P. Zitek: Modification of Mikhaylov criterion for neutral time-delay systems. IEEE Trans. Automat. Control 54 (2009), 2430-2435.   DOI:10.1109/tac.2009.2029301
  17. W. Wang: Nonlinear stability of one-leg methods for neutral Volterra delay-integro-differential equations. Math. Comput. Simul. 97 (2014), 147-161.   DOI:10.1016/j.matcom.2013.08.004