# Exponential stability of nonlinear non-autonomous multivariable systems

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2015)

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

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topMichael 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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- [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
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- [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
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