A Bound for the Joint Spectral Radius of Operators in a Hilbert Space

Michael Gil'

doi:10.32323/ujma.543952

EN

A Bound for the Joint Spectral Radius of Operators in a Hilbert Space

Abstract

We suggest a bound for the joint spectral radius of a finite set of operators in a Hilbert space. In appropriate situations that bound enables us to avoid complicated calculations and gives a new explicit stability test for the discrete time switched systems. The illustrative example is given. Our results are new even in the finite dimensional case.

Keywords

Joint spectral radius,Hilbert space,Discrete time switched systems

References

[1] G. C. Rota and G. Strang. A note on the joint spectral radius, Proc. Netherlands Acad., 22 (1960) 379-381.
[2] R. Jungers, The Joint Spectral Radius: Theory and Applications, Springer, Berlin, 2009.
[3] V. Kozyakin, On the computational aspects of the theory of joint spectral radius, Dokl. Akad. Nauk, 427 (2009), 160-164, in Russian, translation in Doklady Mathematics, 80 (2009), 487-491.
[4] X. Dai, Y. Huang and M. Xiao, Almost sure stability of discrete-time switched linear systems: A topological point of view, SIAM J. Control Optim., 47 (2008), 2137-156.
[5] J.-W. Lee and G. E. Dullerud, Uniform stabilization of discrete-time switched and Markovian jump linear systems, Automatica, 42 (2), (2006) 205-218.
[6] D. Liberzon, Switching in Systems and Control. Birkh´auser, Boston, 2003.
[7] H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Transac. on Automatic Control, 54 (2), (2009) 308-322.
[8] R. Shorten, F. Wirth, O. Mason, K.Wul and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.
[9] W. Xiang and J. Xiao, Convex sufficient conditions on asymptotic stability and l2 gain performance for uncertain discrete-time switched linear systems. IET Control Theory Appl. 8 (3), (2014), 211-218.
[10] G. Zhai and X. Xu, A unified approach to stability analysis of switched linear descriptor systems under arbitrary switching, Int. J. Appl. Math. Comput. Sci., 20 (2), (2010), 249-259.

[11] L. Zhang, Y. Zhu, P. Shi, Q. Lu, Time-Dependent Switched Discrete-Time Linear Systems: Control and Filtering Springer International Publishing, Switzerland, 2016.
[12] C. Heil and G. Strang, Continuity of the joint spectral radius: Applications to wavelets, in ”Linear Algebra for Signal Processing,” IMA Vol. Math. Appl. 69, Springer-Verlag, New York, (1995), 51-61.
[13] I. D. Morris, A rapidly-converging lower bound for the joint spectral radius via multiplicative ergodic theory, Adv. Math., 225 (2010), 3425-3445.
[14] V. Kozyakin, On accuracy of approximation of the spectral radius by the Gelfand formula, Linear Algebra Appl., 431 (2009), 2134-2141.
[15] V. Kozyakin, A relaxation scheme for computation of the joint spectral radius of matrix sets, J. Difference Equ. Appl., 17 (2), (2011) 185-201.
[16] V. D. Blondel and Y. Nesterov, Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices, SIAM J. Matrix Anal. Appl., 31 (2009), 865-876.
[17] P.A. Parrilo and A. Jadbabaie, Approximation of the joint spectral radius using sum of squares, Linear Algebra Appl., 428 (2008), 2385-2402
[18] J. N. Tsitsiklis and V. D. Blondel, The Lyapunov exponent and joint spectral radius of pairs of matrices are hard when not impossible—to compute and to approximate, Math. Control, Signals and Systems, 10 (1997), 31-40.
[19] T. Eisner, Stability of Operators and Operator Semigroups, Operator Theory: Advances and Applications Vol. 209, Birkh˝auser Verlag, Basel, 2010.
[20] M.I. Gil’. Difference Equations in Normed Spaces. Stability and Oscillations, North-Holland, Mathematics Studies 206, Elsevier, Amsterdam, 2007.
[21] M.I. Gil’, Operator Functions and Operator Equations, World Scientific, New Jersey, 2018.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Michael Gil'
0000-0002-6404-9618
Israel

Publication Date

June 28, 2019

Submission Date

March 25, 2019

Acceptance Date

May 13, 2019

Published in Issue

Year 2019 Volume: 2 Number: 2

DOI

https://doi.org/10.32323/ujma.543952

IZ

https://izlik.org/JA67GP29JP

APA

Gil’, M. (2019). A Bound for the Joint Spectral Radius of Operators in a Hilbert Space. Universal Journal of Mathematics and Applications, 2(2), 94-99. https://doi.org/10.32323/ujma.543952

AMA

1.Gil’ M. A Bound for the Joint Spectral Radius of Operators in a Hilbert Space. Univ. J. Math. Appl. 2019;2(2):94-99. doi:10.32323/ujma.543952

Chicago

Gil’, Michael. 2019. “A Bound for the Joint Spectral Radius of Operators in a Hilbert Space”. Universal Journal of Mathematics and Applications 2 (2): 94-99. https://doi.org/10.32323/ujma.543952.

EndNote

Gil’ M (June 1, 2019) A Bound for the Joint Spectral Radius of Operators in a Hilbert Space. Universal Journal of Mathematics and Applications 2 2 94–99.

IEEE

[1]M. Gil’, “A Bound for the Joint Spectral Radius of Operators in a Hilbert Space”, Univ. J. Math. Appl., vol. 2, no. 2, pp. 94–99, June 2019, doi: 10.32323/ujma.543952.

ISNAD

Gil’, Michael. “A Bound for the Joint Spectral Radius of Operators in a Hilbert Space”. Universal Journal of Mathematics and Applications 2/2 (June 1, 2019): 94-99. https://doi.org/10.32323/ujma.543952.

JAMA

1.Gil’ M. A Bound for the Joint Spectral Radius of Operators in a Hilbert Space. Univ. J. Math. Appl. 2019;2:94–99.

MLA

Gil’, Michael. “A Bound for the Joint Spectral Radius of Operators in a Hilbert Space”. Universal Journal of Mathematics and Applications, vol. 2, no. 2, June 2019, pp. 94-99, doi:10.32323/ujma.543952.

Vancouver

1.Michael Gil’. A Bound for the Joint Spectral Radius of Operators in a Hilbert Space. Univ. J. Math. Appl. 2019 Jun. 1;2(2):94-9. doi:10.32323/ujma.543952