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The Lambert $$W$$ function and the spectrum of some multidimensional time-delay systems. (English) Zbl 1138.93026
Summary: We find an explicit expression for the eigenvalues of a retarded time-delay system with one delay, for the special case that the system matrices are simultaneously triangularizable, which includes the case where they commute. Using matrix function definitions we define a matrix version of the Lambert $$W$$ function, from which we form the expression. We prove by counter-example that some expressions in other publications on Lambert $$W$$ for time-delay systems do not always hold.

##### MSC:
 93B60 Eigenvalue problems 93C35 Multivariable systems, multidimensional control systems 34K35 Control problems for functional-differential equations
DDE-BIFTOOL
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##### References:
 [1] Asl, F.M.; Ulsoy, A.G., Analysis of a system of linear delay differential equations, Journal of dynamics system measure and control, 125, 215-223, (2003) [2] Bambi, M. (2006). Endogenous growth and time-to-build: The AK case. In 12th International conference on computers in economics and finance. EUI Working Paper, No. 2006/17. · Zbl 1181.91166 [3] Breda, D.; Maset, S.; Vermiglio, R., Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM journal on scientific computing, 27, 2, 482-495, (2005) · Zbl 1092.65054 [4] Chen, M., Abate, A., & Sastry, S. (2005). New congestion control schemes over wireless networks: Stability analysis. In Proceedings of the 16th IFAC world congress, Prague. [5] Chen, Y.; Moore, K., Analytical stability bound for a class of delayed fractional-order dynamical systems, Nonlinear dynamics, 29, 191-200, (2002) · Zbl 1020.34064 [6] Chen, Y.; Moore, K., Analytical stability bound for delayed second-order systems with repeating poles using Lambert function W, Automatica, 38, 891-895, (2002) · Zbl 1020.93019 [7] Corless, R.; Gonnet, G.H.; Hare, D.E.G.; Jeffrey, D.J.; Knuth, D.E., On the Lambert W function, Advances in computational mathematics, 5, 329-359, (1996) · Zbl 0863.65008 [8] Engelborghs, K. (2000). DDE-BIFTOOL: A Matlab package for bifurcation analysis of delay differential equations. Technical report, Department of Computer Science, K.U. Leuven. [9] Hale, J. K., & Lunel, S. M. V. (1993). Introduction to functional differential equations, Applied mathematical sciences (Vol. 99). Berlin: Springer. · Zbl 0787.34002 [10] Heffernan, J.M.; Corless, R.M., Solving some delay differential equations with computer algebra, Mathematical scientist, 31, 1, 21-34, (2006) · Zbl 1115.34074 [11] Higham, N. (2006). Functions of matrices. In L. Hogben (Ed.), Handbook of linear algebra. Boca Raton, FL: CRC Press. [12] Horn, R.; Johnson, C.R., Topics in matrix analysis, (1991), Cambridge University Press Cambridge, UK · Zbl 0729.15001 [13] Hövel, P.; Schöll, E., Control of unstable steady states by time-delayed feedback methods, Physical review E, 72, (2005) [14] Hwang, C.; Cheng, Y.-C., A note on the use of the Lambert W function in the stability analysis of time-delay systems, Automatica, 41, 1979-1985, (2005) · Zbl 1125.93440 [15] Jeffrey, D.J.; Hare, D.E.G.; Corless, R.M., Unwinding the branches of the Lambert W function, Mathematical scientist, 21, 1-7, (1996) · Zbl 0852.33001 [16] Kalmár-Nagy, T. (2005). A new look at the stability analysis of delay differential equations. In Proceedings of IDETC’05. [17] Pitcher, A. B., & Corless, R. M. (2005). Quasipolynomial root-finding: A numerical homotopy method. Technical report, Department of Applied Mathematics, University of Western Ontario. [18] Radjavi, H.; Rosenthal, P., Simultaneous triangularization, (2000), Springer New York · Zbl 0981.15007 [19] Shinozaki, H.; Mori, T., Robust stability analysis of linear time-delay systems by Lambert W function: some extreme point results, Automatica, 42, 10, 1791-1799, (2006) · Zbl 1114.93074 [20] Sontag, E.D., Mathematical control theory: deterministic finite dimensional systems, (1998), Springer New York · Zbl 0945.93001 [21] Wahi, P.; Chatterjee, A., Galerkin projections for delay differential equations, Transactions of the ASME, 127, 80-86, (2005) [22] Yi, S., & Ulsoy, A. G. (2006). Solution of a system of linear delay differential equations using the matrix Lambert function. In Proceedings of the 26th American control conference (pp. 2433-2438). Minneapolis. [23] Yi, S., Ulsoy, A. G., & Nelson, P. W. (2006). Analysis of systems of linear delay differential equations using the Lambert function and the Laplace transformation. Automatica, submitted for publication.
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