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2015 | 35 | 1 | 89-100
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Exponential stability of nonlinear non-autonomous multivariable systems

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EN
Abstrakty
EN
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.
Twórcy
  • Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva 84105, Israel
Bibliografia
  • [1] D. Aeyels and J. Peuteman, A new asymptotic stability criterion for nonlinear time-variant differential equations, IEEE Trans. Autom. Control 43 (1998), 968-971. doi: 10.1109/9.701102
  • [2] A.Yu. Aleksandrov E.B. Aleksandrova and A.P. Zhabko, Stability analysis for a class of nonlinear nonstationary systems via averaging, Nonlinear Dyn. Syst. Theory 13 (2013), 332-343.
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  • [4] Yu.L. Daleckii and M.G. Krein, Stability of Solutions of Differential Equations in Banach Space, AMS, Providence, R.I., 1974.
  • [5] C.A. Desoer, Slowly varying systems ẋ=A(t)x, IEEE Transactions on Automatic Control 14 (1969), 780-781. doi: 10.1109/TAC.1969.1099336
  • [6] M.I. Gil', Explicit Stability Conditions for Continuous Systems, Lectures Notes In Control and Information Sciences, Vol. 314 (Springer Verlag, Berlin, 2005).
  • [7] M.I. Gil', Stability of linear nonautonomous multivariable systems with differentiable matrices, Systems & Control Letters 81 (2015), 31-33. doi: 10.1016/j.sysconle.2015.03.005
  • [8] Karafyllis, Iasson and Jiang, Zhong-Ping, Stability and Stabilization of Nonlinear Systems, Communications and Control Engineering Series (Springer-Verlag London, Ltd., London, 2011).
  • [9] M.R. Liberzon, Essays on the absolute stability theory, Automation and Remote Control 67 (2006), 1610-1644. doi: 10.1134/S0005117906100043
  • [10] Nikravesh, Seyed Kamaleddin Yadavar, Nonlinear Systems Stability Analysis. Lyapunov-based Approach (CRC Press, Boca Raton, FL, 2013). doi: 10.1201/b13731
  • [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
  • [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
  • [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
  • [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.
  • [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
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1168
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