A formula for the lower Bohl exponent of discrete time-varying systems

• Autorzy: Czornik Adam , Simek Krzysztof

Identyfikatory

DOI

10.24425/acs.2023.146282

• Języki publikacji: EN

Abstrakty

EN

In this note, a formula for the lower Bohl exponent of a discrete system with variable coefficients and weak variation was proved. This formula expresses the Bohl exponent through the eigenvalues of the coefficient matrix. Based on these formulas a necessary and sufficient condition for an uniform exponential instability of such systems is also presented.

Słowa kluczowe

EN

time-varying systems; asymptotic stability; Bohl exponent

• Wydawca: Polish Academy of Sciences, Committee of Automatic Control and Robotics
• Czasopismo: Archives of Control Sciences
• Rocznik: 2023
• Tom: Vol. 33, No. 2
• Strony: 413--424
• Opis fizyczny: Bibliogr. 20 poz., wzory

Twórcy

autor

Czornik Adam

• Silesian University of Technology, Faculty of Automatic Control, Electronics and Computer Science, Akademicka Street 16, 44-101 Gliwice, Poland

autor

Simek Krzysztof

• Silesian University of Technology, Faculty of Automatic Control, Electronics and Computer Science, Akademicka Street 16, 44-101 Gliwice, Poland

Bibliografia

• [1] R.P. Agarwal: Difference Equations and Inequalities. Theory, Methods, and Applications. Marcel Dekker, New York, 2000.
• [2] P.T. Anh, A. Babiarz, A. Czornik and T.S. Doan: Proportional local assignability of the dichotomy spectrum of one-sided discrete time-varying linear systems. SIAM Journal of Control and Optimization, accepted for publication.
• [3] B. Aulbach and S. Siegmund: The dichotomy spectrum for noninvertible systems of linear difference equations. Journal of Nonlinear Differential Equations and Applications, 7(6), (2001), 895-913.
• [4] A. Babiarz, I. Banshchikova, A. Czornik, E. Makarov, M. Niezabitowski and S. Popova: Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems. IEEE Transactions on Automatic Control, 63(11), (2018), 3825-3837. DOI: 10.1109/TAC.2018.2823086.
• [5] A. Babiarz, E. Barabanov, A. Czornik, A. Konyukh, M. Niezabitowski and A. Vaidzelevich: Relations between Bohl and general exponents. Discrete and Continuous Dynamical Systems, 37(10), (2017), 5319-5335. DOI: 10.3934/dcds.2017231.
• [6] E.A. Barabanov and A.V. Konyukh: Bohl exponents of linear differentail systems, Memoirs on Differential equations and Mathematical Physics, 24, (2001), 151-158.
• [7] P. Bohl: Uber Differentialgleichungen. Journal fur Reine und Angewandte Mathematik, 1914(144), (1914), 284-313. DOI: 10.1515/crll.1914.144.284.
• [8] D. Cabada, K. Garcia, C. Guevara and H. Leiva: Controllability of time varying semilinear non-instantaneous impulsive systems with delay, and nonlocal conditions. Archives of Control Sciences, 32(2), (2022), 335-357. DOI: 10.24425/acs.2022.141715.
• [9] A. Czornik: Perturbation Theory for Lyapunov Exponents of Discrete Linear Systems. Wydawnictwa AGH, Kraków 2012.
• [10] A. Czornik and M. Niezabitowski: Alternative formulae for lower general exponent of discrete linear time-varying systems. Journal of the Franklin Institute, 352(1), (2015), 399-419. DOI: 10.1016/j.jfranklin.2014.11.003.
• [11] J. Daleckii and M.G. Krein: Stability of Solutions of Differential Equations in Banach Space. American Mathematical Society, Providence, 1974. DOI: 10.1090/mmono/043.
• [12] C. Desoer: Slowly varying discrete system xi+1=aixi. Electronics Letters, 6(11), (1970), 339-340. DOI: 10.1002/rnc.2835.
• [13] M.I. Gil: Difference Equations in Normed Spaces. Stability and Oscillations. North-Holland, Mathematics Studies, 206 Elsevier: Amsterdam, The Netherlands, 2007.
• [14] D. Hinrichsen and A.J. Prichard: Mathematical Systems Theory I. Springer-Verlag, Berlin, 2005.
• [15] R. Horn and C. Johnson: Matrix Analysis. Cambridge University Press, Cambridge. 1988. DOI: 10.1017/CBO9780511810817.
• [16] N.A. Izobov: Lyapunov Exponents and Stability. Cambridge Scientific Publishers, Cambridge, 2012.
• [17] El M. Magri, C. Amissi, L. Afifi and M. Lhous: On the minimum energy compensation for linear time-varying disturbed systems. Archives of Control Sciences, 32(4), (2022), 733-754. DOI: 10.24425/acs.2022.143669.
• [18] V.M. Millionschikov: On the spectral theory of nonautonomous linear systems of differential equations. Trudy Moskovskogo Matematicheskogo Obshchestva, 18 (1968), 147-186, (in Russian).
• [19] V.M. Millionshchikov: The instability of the singular exponents and the non-symmetry of the relation of reducibility of linear systems of differential equations. Differentsialnye Uravneniya, 5(4), (1969), 749-750 (in Russian). DOI: 10.1007/s10958-015-2551-x.
• [20] M.Rachik, M.Lhous and A.Tridane: On the asymptotic stability of non-linear discrete systems. Archives of Control Sciences, 10(3-4), (2000), 125-140.

Uwagi

1. The research of Adam Czornik was supported by the Polish National Agency for Academic Exchange according to the decision PPN/BEK/2020/1/00188/UO/00001.

2. Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).