October  2017, 37(10): 5319-5335. doi: 10.3934/dcds.2017231

Relations between Bohl and general exponents

1. 

Silesian University of Technology Faculty of Automatic Control, Electronics and Computer Science Akademicka 16 Street, 44-100 Gliwice, Poland

2. 

Institute of Mathematics of National Academy of Sciences of Belarus Surganova 11, Minsk, Belarus

3. 

Belarus State Economic University Partyzanski praspiekt 26, 220070 Minsk, Belarus

* Corresponding author: Michał Niezabitowski

Received  February 2017 Revised  April 2017 Published  June 2017

In the paper we study the problem of the influence of the parametric uncertainties on the Bohl exponents of discrete time-varying linear system. We obtain formulas for the computation of the exact boundaries of lower and upper mobility for the supremum and infimum of the Bohl exponents under arbitrary small perturbations of system coefficients matrices on the basis of the transition matrix.

Citation: Artur Babiarz, Adam Czornik, Michał Niezabitowski, Evgenij Barabanov, Aliaksei Vaidzelevich, Alexander Konyukh. Relations between Bohl and general exponents. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5319-5335. doi: 10.3934/dcds.2017231
References:
[1]

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications Marcel Dekker, New York, 2000.  Google Scholar

[2]

A. BabiarzA. Czornik and M. Niezabitowski, On the number of upper Bohl exponents for diagonal discrete time-varying linear system, Journal of Mathematical Analysis and Applications, 429 (2015), 337-353.  doi: 10.1016/j.jmaa.2015.04.022.  Google Scholar

[3]

E. A. Barabanov and A. V. Konyukh, Bohl exponents of linear differential systems, Memoirs on Differential Equations and Mathematical Physics, 24 (2001), 151-158.   Google Scholar

[4]

E. A. Barabanov and A. V. Konyukh, The exact extreme boundaries of mobility of Bohl exponents of solution to linear differential system under small perturbations of its coefficient matrix, Vesti NAN Belarusi. Ser. Phys i Matem. Nauk, 3 (2015), 1-15.   Google Scholar

[5]

P. Bohl, The structure of the fundamental matrices of R-systems with almost periodic coefficients, Journal fur die Reine und Angewandte Mathematik, 144 (1914), 284-313.  doi: 10.1515/crll.1914.144.284.  Google Scholar

[6]

B. F. Bylov, Almost reducible system of differential equations, Sibirskii Matematicheski-Zhurnal, 3 (1962), 333-359.   Google Scholar

[7]

A. CzornikJ. Klamka and M. Niezabitowski, About the number of the lower Bohl exponents of diagonal discrete linear time-varying systems, Proceedings of 11th IEEE International Conference on Control Automation (ICCA), (2014), 461-466.  doi: 10.1109/ICCA.2014.6870964.  Google Scholar

[8]

A. Czornik and M. Niezabitowski, Alternative formulae for lower general exponent of discrete linear time-varying systems, Journal of the Franklin Institute, 352 (2015), 399-419.  doi: 10.1016/j.jfranklin.2014.11.003.  Google Scholar

[9]

A. Czornik, The relations between the senior upper general exponent and the upper Bohl exponents, Proceedings of 19th International Conference on the Methods and Models in Automation and Robotics (MMAR), (2014), 897-902.  doi: 10.1109/MMAR.2014.6957476.  Google Scholar

[10]

Y. L. Daletskii and M. G. Krein, Stability of Solutions to Differential Equations in Banach Spaces American Mathematical Society, 1974.  Google Scholar

[11]

N. A. Izobov, Lyapunov Exponents and Stability (Stability Oscillations and Optimization of Systems) Cambridge Scientific Publishers, 2013. Google Scholar

[12]

V. M. Millionshchikov, The structure of the fundamental matrices of R-systems with almost periodic coefficients, Dokl. Akad. Nauk SSSR, 171 (1966), 288-291.   Google Scholar

[13]

V. M. Millionshchikov, Instability of the characteristic indices of statistically regular systems, Mat. Zametki, 2 (1967), 315-318.   Google Scholar

[14]

V. M. Millionshchikov, A proof of the attainability of the central exponents of linear systems, Sibirsk. Mat. vz., 10 (1969), 99-104.   Google Scholar

[15]

K. P. Persidskii, To the stability theory of differential equations system integrals, Izv. Fiz. Mat. Ob-va pri Kazansk. Un-te, 8 (1936), 47-85.   Google Scholar

[16]

R. E. Vinograd, Simultaneous attainability of central Lyapunov and Bohl exponents for ODE linear systems, Proceedings of the American Mathematical Society, 88 (1983), 595-601.  doi: 10.1090/S0002-9939-1983-0702282-5.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications Marcel Dekker, New York, 2000.  Google Scholar

[2]

A. BabiarzA. Czornik and M. Niezabitowski, On the number of upper Bohl exponents for diagonal discrete time-varying linear system, Journal of Mathematical Analysis and Applications, 429 (2015), 337-353.  doi: 10.1016/j.jmaa.2015.04.022.  Google Scholar

[3]

E. A. Barabanov and A. V. Konyukh, Bohl exponents of linear differential systems, Memoirs on Differential Equations and Mathematical Physics, 24 (2001), 151-158.   Google Scholar

[4]

E. A. Barabanov and A. V. Konyukh, The exact extreme boundaries of mobility of Bohl exponents of solution to linear differential system under small perturbations of its coefficient matrix, Vesti NAN Belarusi. Ser. Phys i Matem. Nauk, 3 (2015), 1-15.   Google Scholar

[5]

P. Bohl, The structure of the fundamental matrices of R-systems with almost periodic coefficients, Journal fur die Reine und Angewandte Mathematik, 144 (1914), 284-313.  doi: 10.1515/crll.1914.144.284.  Google Scholar

[6]

B. F. Bylov, Almost reducible system of differential equations, Sibirskii Matematicheski-Zhurnal, 3 (1962), 333-359.   Google Scholar

[7]

A. CzornikJ. Klamka and M. Niezabitowski, About the number of the lower Bohl exponents of diagonal discrete linear time-varying systems, Proceedings of 11th IEEE International Conference on Control Automation (ICCA), (2014), 461-466.  doi: 10.1109/ICCA.2014.6870964.  Google Scholar

[8]

A. Czornik and M. Niezabitowski, Alternative formulae for lower general exponent of discrete linear time-varying systems, Journal of the Franklin Institute, 352 (2015), 399-419.  doi: 10.1016/j.jfranklin.2014.11.003.  Google Scholar

[9]

A. Czornik, The relations between the senior upper general exponent and the upper Bohl exponents, Proceedings of 19th International Conference on the Methods and Models in Automation and Robotics (MMAR), (2014), 897-902.  doi: 10.1109/MMAR.2014.6957476.  Google Scholar

[10]

Y. L. Daletskii and M. G. Krein, Stability of Solutions to Differential Equations in Banach Spaces American Mathematical Society, 1974.  Google Scholar

[11]

N. A. Izobov, Lyapunov Exponents and Stability (Stability Oscillations and Optimization of Systems) Cambridge Scientific Publishers, 2013. Google Scholar

[12]

V. M. Millionshchikov, The structure of the fundamental matrices of R-systems with almost periodic coefficients, Dokl. Akad. Nauk SSSR, 171 (1966), 288-291.   Google Scholar

[13]

V. M. Millionshchikov, Instability of the characteristic indices of statistically regular systems, Mat. Zametki, 2 (1967), 315-318.   Google Scholar

[14]

V. M. Millionshchikov, A proof of the attainability of the central exponents of linear systems, Sibirsk. Mat. vz., 10 (1969), 99-104.   Google Scholar

[15]

K. P. Persidskii, To the stability theory of differential equations system integrals, Izv. Fiz. Mat. Ob-va pri Kazansk. Un-te, 8 (1936), 47-85.   Google Scholar

[16]

R. E. Vinograd, Simultaneous attainability of central Lyapunov and Bohl exponents for ODE linear systems, Proceedings of the American Mathematical Society, 88 (1983), 595-601.  doi: 10.1090/S0002-9939-1983-0702282-5.  Google Scholar

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