Abstract
This paper is devoted to the stability of neutral type functional differential equations whose principal terms are small in a certain sense. We derive the explicit conditions for the exponential and absolute stabilities, as well as for the L p-stability. Besides, solution estimates for the considered equations are established. They provide bounds for the regions of attraction of steady states. We also consider some classes of equations with neutral type linear parts and nonlinear causal map**s. These equations include differential, differential-delay, integro-differential, and other traditional equations. The main methodology presented in the paper is based on a combined usage of the recent norm estimates for matrix-valued functions with the generalized Bohl–Perron principle for neutral type functional differential equations. Our approach enables us to apply the well-known results of the theory of matrices to the stability analysis.
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References
Ardjouni, A., Djoudi, A.: Fixed points and stability in neutral nonlinear differential equations with variable delays. Opusc. Math. 32(1), 519–529 (2012)
Azbelev, N.V., Simonov, P.M.: Stability of differential equations with aftereffects. stability. Control Theory Methods Applications, vol. 20. Taylor & Francis, London (2003)
Bellen, A., Guglielmi, B., Ruehli, A.E.: Methods for linear systems of circuits delay differential equations of neutral type. IEEE Trans. Circuits Syst. 46, 212–216 (1999)
Berezansky, L., Braverman, E.: On exponential stability of a linear delay differential equation with an oscillating coefficient. Appl. Math. Lett. 22(12), 1833–1837 (2009)
Berezansky, L., Braverman, E., Domoshnitsky, A.: Stability of the second order delay differential equations with a dam** term. Differ. Equ. Dyn. Syst. 16(3), 185–205 (2008)
Bylov, B.F., Grobman, B.M., Nemyckii, V.V., Vinograd, R.E.: The Theory of Lyapunov Exponents. Nauka, Moscow (1966) (in Russian)
Cahlon, B., Schmidt, D.: Necessary conditions and algorithmic stability tests for certain higher odd order neutral delay differential equations. Dyn. Syst. Appl. 20(2–3), 223–245 (2011)
Chen, Y., Xue, A., Lu, R., Zhou, S.: On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations. Nonlinear Anal. Theory Methods Appl. 68, 2464–2470 (2011)
Daleckii, Y.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. American Mathematical Society, Providence (1974)
Demidenko, G.V.: Stability of solutions to linear differential equations of neutral type. J. Anal. Appl. 7(3), 119–130 (2009)
Dunford, N., Schwartz, J.T.: Linear Operators: Part I. Interscience Publishers, New York (1966)
Fridman, E.: New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems. Syst. Control Lett. 43, 309–319 (2001)
Gil’, M.I.: Operator functions and localization of spectra. Lecture Notes in Mathematics, vol. 1830. Springer, Berlin (2003)
Gil’, M.I.: Difference equations in normed spaces: stability and oscillations. North-Holland, Mathematics Studies, vol. 206. Elsevier, Amsterdam (2007)
Gil’, M.I.: Exponential stability of nonlinear neutral type systems. Arch. Control Sci. 22(2), 125–143 (2012)
Gil’, M.I.: Stability of vector functional differential equations: a survey. Quaestiones Mathematicae 35, 1–49 (2012)
Gil’, M.I.: Stability of Vector Differential Delay Equations. Birkhäuser, Basel (2013)
Gil’, M.I.: On Aizerman’s type problem for neutral type systems. Eur. J. Control 19, 113–117 (2013)
Gil’, M.I.: The generalized Bohl–Perron principle for the neutral type Vector functional differential equations. Math. Control Signals Syst. (MCSS) 25(1), 133–145 (2013)
Gil’, M.I.: Estimates for fundamental solutions of neutral type functional differential equations. Int. J. Dyn. Syst. Differ. Equ. 4(4), 255–273 (2013)
Halanay, A.: Differential Equations: Stability, Oscillation, Time Lags. Academic, New York, (1966)
Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New-York (1993)
Han, Q.-L.: A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems. Automatica 44, 272–277 (2008)
Kolmanovskii, V.B., Nosov, V.R.: Stability of Functional Differential Equations. Academic, London (1986)
Nam, P.T., Phat, V.N.: An improved stability criterion for a class of neutral differential equations. Appl. Math. Lett. 22, 31–35 (2009)
Park, J.H., Won, S.: Stability analysis for neutral delay-differential systems. J. Franklin Inst. 337, 67–75 (2000)
Sun, Y., Wang, L.: Note on asymptotic stability of a class of neutral differential equations. Appl. Math. Lett. 19, 949–953 (2006)
Walther, H-O: More on linearized stability for neutral equations with state-dependent delays. Differ. Equ. Dyn. Syst. 19(4), 315–333 (2011)
Wang, X., Li, S., Xu, D.: Globally exponential stability of periodic solutions for impulsive neutral-type neural networks with delays. Nonlinear Dyn. 64, 65–75 (2011)
Wu, M., He, Y., She, J.-H.: New delay-dependent stability criteria and stabilizing method for neutral systems. IEEE Trans. Autom. Control 49, 2266–2271 (2004)
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Gil’, M. (2014). Stability of Neutral Type Vector Functional Differential Equations with Small Principal Terms. In: Pardalos, P., Rassias, T. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1124-0_10
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DOI: https://doi.org/10.1007/978-1-4939-1124-0_10
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