Ma, Jian; Zheng, Baodong; Zhang, Chunrui A matrix method for determining eigenvalues and stability of singular neutral delay-differential systems. (English) Zbl 1244.93123 J. Appl. Math. 2012, Article ID 749847, 11 p. (2012). Summary: The eigenvalues and the stability of a singular neutral differential system with single delay are considered. Firstly, by applying the matrix pencil and the linear operator methods, new algebraic criteria for the imaginary axis eigenvalue are derived. Second, practical checkable criteria for the asymptotic stability are introduced. Cited in 2 Documents MSC: 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory PDFBibTeX XMLCite \textit{J. Ma} et al., J. Appl. Math. 2012, Article ID 749847, 11 p. (2012; Zbl 1244.93123) Full Text: DOI References: [1] J. K. Hale, E. F. Infante, and F. S. P. Tsen, “Stability in linear delay equations,” Journal of Mathematical Analysis and Applications, vol. 105, no. 2, pp. 533-555, 1985. · Zbl 0569.34061 [2] E. Jarlebring and M. E. Hochstenbach, “Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations,” Linear Algebra and its Applications, vol. 431, no. 3-4, pp. 369-380, 2009. · Zbl 0569.34061 [3] S. L. Campbell and V. H. Linh, “Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions,” Applied Mathematics and Computation, vol. 208, no. 2, pp. 397-415, 2009. · Zbl 1170.65063 [4] Z. Gao, “PD observer parametrization design for descriptor systems,” Journal of the Franklin Institute, vol. 342, no. 5, pp. 551-564, 2005. · Zbl 1169.65079 [5] Z. Gao and S. X. Ding, “Actuator fault robust estimation and fault-tolerant control for a class of nonlinear descriptor systems,” Automatica, vol. 43, no. 5, pp. 912-920, 2007. · Zbl 1141.93317 [6] Z. Gao, T. Breikin, and H. Wang, “Reliable observer-based control against sensor failures for systems with time delays in both state and input,” IEEE Transactions on Systems, Man, and Cybernetics A, vol. 38, no. 5, pp. 1018-1029, 2008. [7] T. E. Simos, “Closed Newton-Cotes trigonometrically-fitted formulae of high order for long-time integration of orbital problems,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1616-1621, 2009. · Zbl 1117.93019 [8] S. Stavroyiannis and T. E. Simos, “Optimization as a function of the phase-lag order of nonlinear explicit two-step P-stable method for linear periodic IVPs,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2467-2474, 2009. · Zbl 1171.65449 [9] T. E. Simos, “Exponentially and trigonometrically fitted methods for the solution of the Schrödinger equation,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1331-1352, 2010. · Zbl 1192.65111 [10] W. Zhu and L. R. Petzold, “Asymptotic stability of linear delay differential-algebraic equations and numerical methods,” Applied Numerical Mathematics, vol. 24, no. 2-3, pp. 247-264, 1997. · Zbl 1192.65111 [11] J. Louisell, “A matrix method for determining the imaginary axis eigenvalues of a delay system,” Institute of Electrical and Electronics Engineers, vol. 46, no. 12, pp. 2008-2012, 2001. · Zbl 0879.65060 [12] W. Jiang, The Degenerate Differential Systems with Delay, Anhui University, Hefei, China, 1998. · Zbl 0938.34068 [13] D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, “Vector spaces of linearizations for matrix polynomials,” SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 4, pp. 971-1004, 2006. · Zbl 1007.34078 [14] 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 1132.65027 [15] E. Jarlebring, “On critical delays for linear neutral delay systems,” in Proceedings of the European Control Conference (ECC ’07), Kos, Greece, 2007. · Zbl 1138.93026 [16] L. Li and Y. Yuan, “Dynamics in three cells with multiple time delays,” Nonlinear Analysis. Real World Applications, vol. 9, no. 3, pp. 725-746, 2008. · Zbl 1091.93026 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.