# Exponential stability of nonlinear non-autonomous multivariable systems

• Volume: 35, Issue: 1, page 89-100
• ISSN: 1509-9407

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## Abstract

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We consider nonlinear non-autonomous multivariable systems governed by differential equations with differentiable linear parts. Explicit conditions for the exponential stability are established. These conditions are formulated in terms of the norms of the derivatives and eigenvalues of the variable matrices, and certain scalar functions characterizing the nonlinearity. Moreover, an estimate for the solutions is derived. It gives us a bound for the region of attraction of the steady state. As a particular case we obtain absolute stability conditions. Our approach is based on a combined usage of the properties of the "frozen" Lyapunov equation, and recent norm estimates for matrix functions. An illustrative example is given.

## How to cite

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Michael I. Gil'. "Exponential stability of nonlinear non-autonomous multivariable systems." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 35.1 (2015): 89-100. <http://eudml.org/doc/276528>.

@article{MichaelI2015,
abstract = { We consider nonlinear non-autonomous multivariable systems governed by differential equations with differentiable linear parts. Explicit conditions for the exponential stability are established. These conditions are formulated in terms of the norms of the derivatives and eigenvalues of the variable matrices, and certain scalar functions characterizing the nonlinearity. Moreover, an estimate for the solutions is derived. It gives us a bound for the region of attraction of the steady state. As a particular case we obtain absolute stability conditions. Our approach is based on a combined usage of the properties of the "frozen" Lyapunov equation, and recent norm estimates for matrix functions. An illustrative example is given. },
author = {Michael I. Gil'},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {nonlinear nonautonomous systems; exponential stability; absolute stability},
language = {eng},
number = {1},
pages = {89-100},
title = {Exponential stability of nonlinear non-autonomous multivariable systems},
url = {http://eudml.org/doc/276528},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Michael I. Gil'
TI - Exponential stability of nonlinear non-autonomous multivariable systems
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2015
VL - 35
IS - 1
SP - 89
EP - 100
AB - We consider nonlinear non-autonomous multivariable systems governed by differential equations with differentiable linear parts. Explicit conditions for the exponential stability are established. These conditions are formulated in terms of the norms of the derivatives and eigenvalues of the variable matrices, and certain scalar functions characterizing the nonlinearity. Moreover, an estimate for the solutions is derived. It gives us a bound for the region of attraction of the steady state. As a particular case we obtain absolute stability conditions. Our approach is based on a combined usage of the properties of the "frozen" Lyapunov equation, and recent norm estimates for matrix functions. An illustrative example is given.
LA - eng
KW - nonlinear nonautonomous systems; exponential stability; absolute stability
UR - http://eudml.org/doc/276528
ER -

## References

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11. [11] J. Peuteman and D. Aeyels, Exponential stability of nonlinear time-varying differential equations and partial averaging, Math. Control Signals Syst. 15 (2013), 42-70. doi: 10.1007/s004980200002 Zbl1010.34035
12. [12] J. Peuteman and D. Aeyels, Exponential stability of slowly time-varying nonlinear systems, Math. Control Signals Syst. 15 (2013), 202-228. doi: 10.1007/s004980200008 Zbl1019.93050
13. [13] Rionero, Salvatore, On the nonlinear stability of nonautonomous binary systems, Nonlin. Anal. 75 (2012), 2338-2348. doi: 10.1016/j.na.2011.10.032 Zbl1251.37026
14. [14] V.A. Yakubovich, The application of the theory of linear periodic Hamiltonian systems to problems of absolute stability of nonlinear systems with a periodic nonstationary linear part, Vestn. Leningr. Univ. Math. 20 (1987), 59-65. Zbl0652.34019
15. [15] R.E. Vinograd, An improved estimate in the method of freezing, Proc. Amer. Soc. 89 (1983), 125-129. doi: 10.1090/S0002-9939-1983-0706524-1 Zbl0525.34040

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