Numerical radius for the asymptotic stability of delay differential equations

References

• Bellen A, Zennaro M. Numerical methods for delay differential equations. Oxford: Oxford University Press; 2003.
   Google Scholar
• Hale J, Verduyn Lunel S. Introduction to functional-differential equations. New York (NY): Springer-Verlag; 1993.
   Google Scholar
• Kuang J-X, Cong Y. Stability of numerical methods for delay differential equations. Beijing: Science Press; 2006.
   Google Scholar
• Kuang Y. Delay differential equations with applications in population dynamics. Boston (MA): Academic Press; 1993.
   Google Scholar
• Al-Mutib A. Stability properties of numerical methods for solving delay differential equations. J Comput Appl Math. 1984;10:71–79.
   Web of Science ®Google Scholar
• Huang C, Vandewalle S. An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM J Sci Comput. 2004;25:1608–1632.
   Web of Science ®Google Scholar
• in’t Hout K, Spijker M. Stability analysis of numerical methods for delay differential equations. Numer Math. 1991;59:807–814.
   Web of Science ®Google Scholar
• Jackiewicz Z. Asymptotic stability analysis of θ-methods for functional differential equations. Numer Math. 1984;43:389–396.
   Web of Science ®Google Scholar
• Koto T. A stability property of A-stable neutral Runge–Kutta methods for systems of delay differential equations. BIT. 1994;34:262–267.
   Web of Science ®Google Scholar
• Liu M, Spijker M. The stability of the θ-methods in the numerical solution of delay differential equations. IMA J Numer Anal. 1990;10:31–48.
   Web of Science ®Google Scholar
• Mori T, Fukuma N, Kuwahara M. Simple stability criteria for single and composite linear systems with time delays. Int J Control. 1981;34:1175–1184.
   Web of Science ®Google Scholar
• Qiu L, Mitsui T, Kuang J-X. The numerical stability of the θ-methods for delay differential equations with many variable delays. J Comput Math. 1999;17:523–532.
   Web of Science ®Google Scholar
• Sun L. Stability analysis for delay differential equations with multidelays and numerical examples. Math Comp. 2006;75:151–165.
   Web of Science ®Google Scholar
• Torelli L. Stability of numerical methods for delay differential equations. J Comput Appl Math. 1989;25:15–26.
   Web of Science ®Google Scholar
• Wang S. Further results on stability of X(t) = AX(t) + BX(t – τ). Syst Control Lett. 1982;19:165–168.
   Web of Science ®Google Scholar
• Watanabe D, Roth M. The stability of difference formulas for delay differential equations. SIAM J Numer Anal. 1985;22:132–145.
   Web of Science ®Google Scholar
• Goldberg M, Tadmor E. On the numerical radius and its applications. Linear Algebra Appl. 1982;42:263–284.
   Web of Science ®Google Scholar
• Horn R, Johnson C. Topics in matrix analysis. New York (NY): Cambridge University Press; 1991.
   Google Scholar
• Dahlquist G. Stability and error bounds in the numerical integration of ordinary differential equations [PhD thesis]. Uppsala: Almqvist & Wiksell; 1958.
   Google Scholar
• Lozinskiĭ S. Error estimates for the numerical integration of ordinary differential equations, part I. Izv Vyss Uceb Zaved Matematika. 1958;6:52–90.
   Google Scholar
• Johnson C. Numerical determination of the field of values of a general complex matrix. SIAM J Numer Anal. 1978;15:595–602.
   Web of Science ®Google Scholar
• Mathias R. Matrix completions, norms and Hadamard products. Proc Amer Math Soc. 1993;117:905–918.
   Web of Science ®Google Scholar
• Golub G, Van Loan CF. Matrix computations. 4th ed. Baltimore (MD): Johns Hopkins University Press; 2013.
   Google Scholar
• Braconnier T, Higham N. Computing the field of values and pseudospectra using the Lanczos method with continuation. BIT. 1996;36:422–440.
   Web of Science ®Google Scholar
• He C, Watson G. An algorithm for computing the numerical radius. IMA J Numer Anal. 1997;17:329–342.
   Web of Science ®Google Scholar
• Mengi E, Overton M. Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix. IMA J Numer Anal. 2005;25:648–669.
   Web of Science ®Google Scholar
• Uhlig F. Geometric computation of the numerical radius of a matrix. Numer Alg. 2009;52:335–353.
   Web of Science ®Google Scholar
• Watson G. Computing the numerical radius. Linear Algebra Appl. 1996;234:163–172.
   Web of Science ®Google Scholar
• Mengi E, Overton M. MATLAB code numr.m. Software for robust stability measures. 2009. Available from: http://www.cs.nyu.edu/mengi/robuststability.html.
   Google Scholar
• Uhlig F. MATLAB m-file, NumRadius.m. 2008. Available from: http://www.auburn.edu/uhligfd/mfiles/NumRadius.m.
   Google Scholar
• Trefethen L, Battles Z. Chebfun, a collection of MATLAB algorithms; 2002. Available from: http://web.comlab.ox.ac.uk/projects/chebfun/.
   Google Scholar
• Jin X, Wei Y. Numerical linear algebra and its applications. Beijing: Science Press; 2004; Oxford: Alpha Science International Ltd.; 2012; 2nd ed., Beijing: Science Press; 2015.
   Google Scholar
• Lu L. Numerical stability of the θ-methods for systems of differential equations with several delay terms. J Comput Appl Math. 1991;34:291–304.
   Web of Science ®Google Scholar
• Yang B, Qiu L, Mitsui T. GPG-stability of Runge–Kutta methods for generalized delay differential systems. Comput Math Appl. 1997;37:89–97.
   Web of Science ®Google Scholar
• Li C-K, Tam B, Wu P. The numerical range of a nonnegative matrix. Linear Algebra Appl. 2002;350:1–23.
   Web of Science ®Google Scholar
• Ascher U, Petzold L. The numerical solution of delay-differential-algebraic equations of retarded and neutral type. SIAM J Numer Anal. 1995;32:1635–1657.
   Web of Science ®Google Scholar
• Mitsui T. Stability analysis of numerical methods for systems of neutral delay-differential equations. BIT. 1995;35:504–515.
   Web of Science ®Google Scholar
• Kuang J-X, Tian H, Mitsui T. Asymptotic and numerical stability of systems of neutral differential equations with many delays. J Comput Appl Math. 2009;223:614–625.
   Web of Science ®Google Scholar
• Tian H, Yu Q, Kuang J. Asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. SIAM J Numer Anal. 2011;49:608–618.
   Web of Science ®Google Scholar
• Vyhlídal T, Zítek P. Modification of Mikhaylov criterion for neutral time-delay systems. IEEE Trans Autom Control. 2009;54:2430–2435.
   Web of Science ®Google Scholar
• Wang W, Wen L, Li S. Stability of linear multistep methods for nonlinear neutral delay differential equations in Banach space. J Comput Appl Math. 2010;233:2423–2437.
   Web of Science ®Google Scholar
• Zhang C, Vandewalle S. Stability analysis of Runge–Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J Numer Anal. 2004;24:193–214.
   Web of Science ®Google Scholar