# Article

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Keywords:
neutral systems with multiple delays; delay-dependent stability; linear multi-step method; Lagrange interpolation; argument principle
Summary:
In this paper, we are concerned with numerical methods for linear neutral systems with multiple delays. For delay-dependently stable neutral systems, we ask what conditions must be imposed on linear multi-step methods in order that the numerical solutions display stability property analogous to that displayed by the exact solutions. Combining with Lagrange interpolation, linear multi-step methods can be applied to the neutral systems. Utilizing the argument principle, a sufficient condition is derived for linear multi-step methods with preserving delay-dependent stability. Numerical examples are given to illustrate the main results.
References:
[1] Aleksandrov, A. Y., Hu, G. D., Zhabko, A. P.: Delay-independent stability conditions for some classes of nonlinear systems. IEEE Trans. Automat. Control 59 (2014), 2209-2214. DOI 10.1109/tac.2014.2299012 | MR 3245263
[2] Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford 2003. DOI 10.1093/acprof:oso/9780198506546.001.0001 | MR 1997488
[3] Brown, J. W., Churchill, R. V.: Complex Variables and Applications. McGraw-Hill Companies, Inc. and China Machine Press, Beijing 2004. MR 0112948
[4] Fridman, E.: Introduction to Time-Delay Systems: Analysis and Control. Birkhäuser, New York 2014. DOI 10.1007/978-3-319-09393-2 | MR 3237720
[5] Hale, J. K., Lunel, S. M. Verduyn: Strong stabilization of neutral functional differential equations. IMA J. Math. Control Inform. 19 (2002), 5-23. DOI 10.1093/imamci/19.1_and_2.5 | MR 1899001
[6] Han, Q. L.: Stability criteria for a class of linear neutral systems with time-varying discrete and distributed delays. IMA J. Math. Control Inform. 20 (2003), 371-386. DOI 10.1093/imamci/20.4.371 | MR 2028146
[7] Hu, G. D.: Delay-dependent stability of Runge-Kutta methods for linear neutral systems with multiple delays. Kybernetika 54 (2018), 718-735. DOI 10.14736/kyb-2018-4-0718 | MR 3863252
[8] Hu, G. D., Cahlon, B.: Estimations on numerically stable step-size for neutral delay differential systems with multiple delays. J. Compt. Appl. Math. 102 (1999), 221-234. DOI 10.1016/s0377-0427(98)00215-5 | MR 1674027
[9] Hu, G. D., Liu, M.: Stability criteria of linear neutral systems with multiple delays. IEEE Trans. Automat. Control 52 (2007), 720-724. DOI 10.1109/tac.2007.894539 | MR 2310053
[10] Huang, C., Vandewalle, S.: 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 | MR 2087328
[11] Johnson, L. W., Riess, R. Dean, Arnold, J. T.: Introduction to Linear Algebra. Prentice-Hall, Englewood Cliffs 2000.
[12] Jury, E. I.: Theory and Application of $z$-Transform Method. John Wiley and Sons, New York 1964.
[13] Kim, A. V., Ivanov, A. V.: Multistep numerical methods for fuctional difefrential equations. Math. Comput. Simul. 45 (1998), 377-384. DOI 10.1016/s0378-4754(97)00117-1 | MR 1622949
[14] Kolmanovskii, V. B., Myshkis, A.: Introduction to Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht 1999. DOI 10.1007/978-94-017-1965-0 | MR 1680144
[15] Lambert, J. D.: Numerical Methods for Ordinary Differential Systems. John Wiley and Sons, New York 1999. MR 1127425
[16] Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications. Academic Press, Orlando 1985. MR 0792300
[17] Maset, S.: Instability of Runge-Kutta methods when applied to linear systems of delay differential equations. Numer. Math. 90 (2002), 555-562. DOI 10.1007/s002110100266 | MR 1884230
[18] Medvedeva, I. V., Zhabko, A. P.: Synthesis of Razumikhin and Lyapunov-Krasovskii approaches to stability analysis of time-delay systems. Automatica 51 (2015), 372-377. DOI 10.1016/j.automatica.2014.10.074 | MR 3284791
[19] Tian, H., Kuang, J.: 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 | MR 1343634
[20] Vyhlidal, T., Zitek, P.: Modification of Mikhaylov criterion for neutral time-delay systems. IEEE Trans. Automat. Control 54 (2009), 2430-2435. DOI 10.1109/tac.2009.2029301 | MR 2562848

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