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Pseudospectra and stability radii for analytic matrix functions with application to time-delay systems. (English) Zbl 1108.15010

The authors start developing a unified theory of pseudospectra and stability radii for general analytic matrix functions, leading to computable formulae for arbitrary norms measuring the size of perturbations of the matrices \(A_i\) in
\[ \det\{\sum A_ip_i(\lambda)\}=0\; , \]
where \(p_i\) is an entire function. Application to stability radii, as well as computational issues are then discussed. Special properties of pseudospectra of a class of retarded delay differential equations are identified. These properties are related with the behavior of the eigenvalues. In the end some numerical examples are presented.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15A54 Matrices over function rings in one or more variables
34K20 Stability theory of functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34L05 General spectral theory of ordinary differential operators

Software:

DDE-BIFTOOL; Eigtool
PDFBibTeX XMLCite
Full Text: DOI HAL

References:

[1] Trefethen, L.; Embree, M., Spectra and Pseudospectra, the Behavior of Nonnormal Matrices and Operators (2005), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1085.15009
[2] Trefethen, L., Pseudospectra of linear operators, SIAM Rev., 39, 3, 383-406 (1997) · Zbl 0896.15006
[3] Hinrichsen, D.; Pritchard, A., Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness. Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness, Texts in Applied Mathematics, vol. 48 (2005), Springer Verlag: Springer Verlag Berlin · Zbl 1074.93003
[4] Tisseur, F.; Meerbergen, K., The quadratic eigenvalue problem, SIAM Rev., 43, 235-286 (2001) · Zbl 0985.65028
[5] Tisseur, F.; Higham, N., Structured pseudospectra for polynomial eigenvalue problems with applications, SIAM J. Matrix Anal. Appl., 23, 1, 187-208 (2001) · Zbl 0996.65042
[6] K. Green, T. Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, in press.; K. Green, T. Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, in press. · Zbl 1106.65068
[7] Michiels, W.; Roose, D., An eigenvalue based approach for the robust stabilization of linear time-delay systems, Internat. J. Control, 76, 7, 678-686 (2003) · Zbl 1039.93059
[8] Pappas, G.; Hinrichsen, D., Robust stability of linear systems described by higher order dynamic equations, IEEE Trans. Automat. Control, 38, 1430-1435 (1993) · Zbl 0788.93069
[9] Genin, Y.; Stefan, R.; Van Dooren, P., Real and complex stability radii of polynomial matrices, Linear Algebra Appl., 351-352, 381-410 (2002) · Zbl 1004.15019
[10] Zhou, K.; Doyle, J.; Glover, K., Robust and Optimal Control (1996), Prentice Hall: Prentice Hall Upper Saddle River, NJ
[11] Michiels, W.; Engelborghs, K.; Roose, D.; Dochain, D., Sensitivity to infinitesimal delays in neutral equations, SIAM J. Control Optim., 40, 4, 1134-1158 (2002) · Zbl 1016.34079
[12] Hale, J.; Verduyn Lunel, S., Introduction to Functional Differential Equations. Introduction to Functional Differential Equations, Applied Mathematical Sciences, vol. 99 (1993), Springer Verlag: Springer Verlag Berlin · Zbl 0787.34002
[13] Burke, J.; Lewis, A.; Overton, M., Optimization and pseudospectra, with applications to robust stability, SIAM J. Matrix Anal. Appl., 25, 80-104 (2003) · Zbl 1061.15007
[14] P. Van Dooren, V. Vermaut, On stability radii of generalized eigenvalue problems, in: Proceedings of the 1997 European Control Conference (ECC’97), Brussels, Belgium, 1997.; P. Van Dooren, V. Vermaut, On stability radii of generalized eigenvalue problems, in: Proceedings of the 1997 European Control Conference (ECC’97), Brussels, Belgium, 1997.
[15] Curtain, R.; Pritchard, A., Functional Analysis in Modern Applied Mathematics (1977), Academic Press: Academic Press London · Zbl 0448.46002
[16] Gallestey, E.; Hinrichsen, D.; Pritchard, A., Spectral value sets of closed linear operators, Proc. R. Soc. Lond. A, 456, 1397-1418 (2000) · Zbl 0982.47007
[17] Byers, R., A bisection method for measuring the distance of a stable matrix to the unstable matrices, SIAM J. Sci. Stat. Comp., 9, 9, 875-881 (1988) · Zbl 0658.65044
[18] Boyd, S.; Balakrishnan, V., A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its \(L_\infty \)-norm, Systems Control Lett., 15, 1-7 (1990) · Zbl 0704.93014
[19] Boyd, S.; Balakrishnan, V.; Kabamba, P., A bisection method for computing the \(H_\infty\) norm of a transfer matrix and related problems, Math. Control Signals Systems, 2, 207-219 (1989) · Zbl 0674.93020
[20] Y. Genin, P. Van Dooren, V. Vermaut, Convergence of the calculation of \(\mathcal{H}_{\operatorname{\infty;}} \); Y. Genin, P. Van Dooren, V. Vermaut, Convergence of the calculation of \(\mathcal{H}_{\operatorname{\infty;}} \)
[21] Golub, G.; Van Loan, C., Matrix Computations (1993), John Hopkins University Press: John Hopkins University Press Baltimore
[22] K. Engelborghs, T. Luzyanina, G. Samaey, DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations, TW Report 330, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, October 2001.; K. Engelborghs, T. Luzyanina, G. Samaey, DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations, TW Report 330, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, October 2001.
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