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Year 2019, Volume: 2 Issue: 2, 94 - 99, 28.06.2019
https://doi.org/10.32323/ujma.543952

Abstract

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.

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

Year 2019, Volume: 2 Issue: 2, 94 - 99, 28.06.2019
https://doi.org/10.32323/ujma.543952

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.

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.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Michael Gil' 0000-0002-6404-9618

Publication Date June 28, 2019
Submission Date March 25, 2019
Acceptance Date May 13, 2019
Published in Issue Year 2019 Volume: 2 Issue: 2

Cite

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 Gil’ M. A Bound for the Joint Spectral Radius of Operators in a Hilbert Space. Univ. J. Math. Appl. June 2019;2(2):94-99. doi:10.32323/ujma.543952
Chicago Gil’, Michael. “A Bound for the Joint Spectral Radius of Operators in a Hilbert Space”. Universal Journal of Mathematics and Applications 2, no. 2 (June 2019): 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 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, 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 2019), 94-99. https://doi.org/10.32323/ujma.543952.
JAMA 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, 2019, pp. 94-99, doi:10.32323/ujma.543952.
Vancouver Gil’ M. A Bound for the Joint Spectral Radius of Operators in a Hilbert Space. Univ. J. Math. Appl. 2019;2(2):94-9.

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