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 and Continuous Dynamical Systems, 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.

[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.

[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. 

[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. 

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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. 

[13]

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

[14]

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

[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. 

[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.

show all references

References:
[1]

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

[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.

[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. 

[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. 

[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.

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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. 

[13]

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

[14]

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

[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. 

[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.

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