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Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations. (English) Zbl 1170.65063
The authors introduce a new type of eigenvalue problem, the polynomial two-parameter eigenvalue problem, with the quadratic two-parameter eigenvalue problem as a special case to analyze the asymptotic stability of delay-differential equations. This framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods. They also recognize a few new matrix pencil variants to determine the asymptotic stability of delay-differential equations.

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)
34K25 Asymptotic theory of functional-differential equations
15A22 Matrix pencils
15A18 Eigenvalues, singular values, and eigenvectors
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[1] Atkinson, F.V., Multiparameter eigenvalue problems, (1972), Academic Press New York · Zbl 0555.47001
[2] Bellman, R.; Cooke, K.L., Differential-difference equations, (1963), Academic Press · Zbl 0118.08201
[3] Chen, J., On computing the maximal delay intervals for stability of linear delay systems, IEEE trans. automat. control, 40, 6, 1087-1093, (1995) · Zbl 0840.93074
[4] Chen, J.; Gu, G.; Nett, C.N., A new method for computing delay margins for stability of linear delay systems, Systems control lett., 26, 2, 107-117, (1995) · Zbl 0877.93117
[5] Chiasson, J., A method for computing the interval of delay values for which a differential-delay system is stable, IEEE trans. automat. control, 33, 12, 1176-1178, (1988) · Zbl 0668.34074
[6] Ergenc, A.F.; Olgac, N.; Fazelina, H., Extended Kronecker summation for cluster treatment of LTI systems with multiple delays, SIAM J. control optim., 46, 1, 143-155, (2007) · Zbl 1136.93029
[7] H. Fassbender, N. Mackey, S. Mackey, C. Schröder, Structured polynomial eigenproblems related to time delay systems, TU Braunschweig, 2008, Preprint. · Zbl 1188.15007
[8] Fu, P.; Niculescu, S.-I.; Chen, J., Stability of linear neutral time-delay systems: exact conditions via matrix pencil solutions, IEEE trans. automat. control, 51, 6, 1063-1069, (2006) · Zbl 1366.34091
[9] Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, Control engineering, (2003), Birkhäuser Boston, MA · Zbl 1039.34067
[10] Gu, K.; Niculescu, S.-I., Survey on recent results in the stability and control of time-delay systems, J. dynam. syst.-T. ASME, 125, 158-165, (2003)
[11] Hale, J.; Infante, E.F.; Tsen, F.-S., Stability in linear delay equations, J. math. anal. appl., 105, 533-555, (1985) · Zbl 0569.34061
[12] Hochstenbach, M.E.; Košir, T.; Plestenjak, B., A jacobi – davidson type method for the two-parameter eigenvalue problem, SIAM J. matrix anal. appl., 26, 2, 477-497, (2005) · Zbl 1077.65036
[13] E. Jarlebring, On critical delays for linear neutral delay systems, in: Proc. Europ. Contr. Conf, 2007.
[14] E. Jarlebring, The Spectrum of Delay-Differential Equations: Numerical Methods, Stability and Perturbation, Ph.D. thesis, TU Braunschweig, 2008. · Zbl 1183.34001
[15] Jarlebring, E., Critical delays and polynomial eigenvalue problems, J. comput. appl. math., 224, 1, 296-306, (2009) · Zbl 1166.65040
[16] Kamen, E.W., On the relationship between zero criteria for two-variable polynomials and asymptotic stability of delay differential equations, IEEE trans. automat. control, 25, 983-984, (1980) · Zbl 0458.93046
[17] Kamen, E.W., Linear systems with commensurate time delays: stability and stabilization independent of delay, IEEE trans. automat. control, 27, 367-375, (1982) · Zbl 0517.93047
[18] Kharitonov, V., Robust stability analysis of time delay systems: a survey, Annual rev. control, 23, 23, 185-196, (1999)
[19] Louisell, J., A matrix method for determining the imaginary axis eigenvalues of a delay system, IEEE trans. automat. control, 46, 12, 2008-2012, (2001) · Zbl 1007.34078
[20] Mackey, S.; Mackey, N.; Mehl, C.; Mehrmann, V., Structured polynomial eigenvalue problems: good vibrations from good linearizations, SIAM J. matrix anal. appl., 28, 1029-1051, (2006) · Zbl 1132.65028
[21] Mackey, S.; Mackey, N.; Mehl, C.; Mehrmann, V., Vector spaces of linearizations for matrix polynomials, SIAM J. matrix anal. appl., 28, 971-1004, (2006) · Zbl 1132.65027
[22] Michiels, W.; Niculescu, S.-I., ()
[23] A. Muhič, B. Plestenjak, On quadratic and singular two-parameter eigenvalue problems, University of Ljubljana, 2008, Preprint.
[24] Niculescu, S.-I., Stability and hyperbolicity of linear systems with delayed state: a matrix-pencil approach, IMA J. math. control inform., 15, 4, 331-347, (1998) · Zbl 0918.93046
[25] Niculescu, S.-I., ()
[26] S.-I. Niculescu, P. Fu, J. Chen, On the stability of linear delay-differential algebraic systems: exact conditions via matrix pencil solutions, in: Proceedings of the 45th IEEE Conference on Decision and Control, 2006.
[27] Richard, J.-P., Time-delay systems: an overview of some recent advances and open problems, Automatica, 39, 10, 1667-1694, (2003) · Zbl 1145.93302
[28] Tisseur, F.; Meerbergen, K., The quadratic eigenvalue problem, SIAM rev., 43, 2, 235-286, (2001) · Zbl 0985.65028
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