Novel Bounds for Solutions of Nonlinear Differential Equations

A. A. Martynyuk

S.P. Timoshenko Institute of Mechanics of NAS of Ukraine, Kyiv, Ukraine.

DOI: 10.4236/am.2015.61018                

Abstract

In this paper the estimates for norms of solutions to nonlinear systems are obtained via an integral inequality. As an application we considered affine control systems and systems of equations for synchronization of motions.

Keywords

Nonlinear Systems, Novel Bounds for Solutions, Stability, Synchronization

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Bellman, R. (1953) Stability Theory of Differential Equations. Dover Publications, New York, 166 p.
[2]	Martynyuk, A.A., Lakshmikantham, V. and Leela, S. (1979) Stability of Motion: Method of Integral Inequalities. Naukova Dumka, Kiev. (In Russian)
[3]	Rao, M.R.M. (1980) Ordinary Differential Equations. Affiliated East-West Press Pvt Ltd., New Delhi-Madras, 266 p.
[4]	Gutowski, R. and Radziszewski, B. (1970) Asymptotic Behaviour and Properties of Solutions of a System of Non Linear Second Order Ordinary Differential Equations Describing Motion of Mechanical Systems. Archiwum Mechaniki Stosowanej, 6, 675-694.
[5]	Martynyuk, A.A. and Gutowski, R. (1979) Integral Inequalities and Stability of Motion. Naukova Dumka, Kiev. (In Russian)
[6]	Pachpatte, B.G. (1998) Inequalities for Differential and Integral Equations. Academic Press, San Diego.
[7]	Martynyuk, A., Chernetskaya, L. and Martynyuk, V. (2013) Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion. CRC Press, Boca Raton.
[8]	Louartassi, Y., Mazoudi, E.H.E. and Elalami, N. (2012) A New Generalization of Lemma Gronwall-Bellman. Applied Mathematical Sciences, 6, 621-628.
[9]	Rozo, M. (1971) Nonlinear Oscillations and Stability Theory. Nauka, Moscow. (In Russian)
[10]	Demidovich, B.P. (1967) Lectures on Mathematical Stability Theory. Nauka, Moscow. (In Russian)
[11]	Brauer, F. (1963) Bounds for Solutions of Ordinary Differential Equations. Proceedings of the American Mathematical Society, 14, 36-43. http://dx.doi.org/10.1090/S0002-9939-1963-0142829-0
[12]	Aleksandrov, A.Yu., Aleksandrova, E.B. and Zhabko, A.P. (2013) Stability Analysis of a Class of Nonlinear Nonstationary Systems via Averaging. Nonlinear Dynamics and Systems Theory, 13, 332-343.
[13]	N’Doye, I., Zasadzinski, M., Darouach, M., Radhy, N.-E. and Bouaziz, A. (2011) Exponential Stabilization of a Class of Nonlinear Systems: A Generalized Gronwall-Bellman Lemma Approach. Nonlinear Analysis, 74, 7333-7341. http://dx.doi.org/10.1016/j.na.2011.07.051
[14]	Babenko, S.V. and Martynyuk, A.A. (2013) Nonlinear Dynamic Inequalities and Stability of Quasi-Linear Systems on Time Scales. Nonlinear Dynamics and Systems Theory, 13, 13-24.
[15]	Bohner, M. and Martynyuk, A.A. (2007) Elements of Stability Theory of A. M. Lyapunov for Dynamic Equations on Time Scales. Nonlinear Dynamics and Systems Theory, 7, 225-251.
[16]	Martynyuk, A.A. (2012) Stability Theory of Solutions of Dynamic Equations on Time Scales. Phoenix, Kiev. (In Russian)