Lipschitz Stability Analysis on a Type of Nonlinear Perturbed System

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Abstract

In this chapter, the notions of uniform Lipschitz stability are generalized and the relations between these notions are analyzed. Several sufficient conditions about uniform Lipschitz asymptotic stability of nonlinear systems is established by the proposed integral inequalities. These sufficient conditions can be similarly generalized to linearly perturbed differential systems that appear in the literature. Finally, an example of uniform Lipschitz asymptotic stability of nonlinear differential systems is shown.

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Notes

Acknowledgements

This work is Supported by National Key Research and Development Program of China (2017YFF0207400).

References

  1. 1.
    Alekseev VM. An estimate for the perturbation of the solution of ordinary differential equation (in Russian). Vestinik Moskov Univ Serl Math Mekn. 1961;2:28–36.Google Scholar
  2. 2.
    Brauer F. Perturbations of nonlinear systems of differential equations. J Math Anal Appl. 1966;14:198–206.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dannan FM, Elaydi S. Lipschitz stability of nonlinear differential equation II. J Math Anal Appl. 1986;113:562–79.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dannan FM, Elaydi S. Lipschitz stability of nonlinear differential equation. J Math Anal Appl. 1989;143:517–29.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Elaydi S, Rao M, Rama M. Lipschitz stability for nonlinear volterra integrodifferential systems. Appl Math Comput. 1988;27(3):191–9.MathSciNetMATHGoogle Scholar
  6. 6.
    Gallo A, Piccirillo AM. About new analogies of GronwallCBellmanCBihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems. Nonlinear Anal Theory Methods Appl. 2007;67(5):1550–9.CrossRefGoogle Scholar
  7. 7.
    Giovanni A, Sergio V. Lipschitz stability for the inverse conductivity problem. Adv Appl Math. 2005;35(2):207–41.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guo S, Irene M, Si L, Han L. Several integral inequalities and their applications in nonlinear differential systems. Appl Math Comput. 2013;219:4266–77.MathSciNetMATHGoogle Scholar
  9. 9.
    Hale JK. Ordinary differential equations. New York: Wiley-Interscience; 1969.MATHGoogle Scholar
  10. 10.
    Jiang F, Meng F. Explicit bounds on some new nonlinear integral inequalities with delay. J Comput Appl Math. 2007;205(1):479–86.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Li Y. Boundness, stability and error estimate of the solution of nonlinear different equation. Acta Mathematia Sinica. 1962;12(1):28–36 (in Chinese).Google Scholar
  12. 12.
    Mitropolskiy YA, Iovane G, Borysenko SD. About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications. Nonlinear Anal Theory Methods Appl. 2007;66(10):2140–65.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Soliman AA. Lipschitz stability with perturbing liapunov functionals. Appl Math Lett. 2004;17(8):939–44.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Soliman AA. On Lipschitz stability for comparison systems of differential equations via limiting equation. Appl Math Comput. 2005;163(3):1061–7.MathSciNetMATHGoogle Scholar
  15. 15.
    Wang W. A generalized retarded Gronwall-like inequality in two variables and applications to BVP. Appl Math Comput. 2007;191(1):144–54.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ye H, Gao J, Ding Y. A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl. 2007;328(2):1075–81.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yoshizawa T. Stability theory and the existence of periodic solutions and almost periodic solutions. New York: Springer; 1975.CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Key Laboratory of Complex System Intelligent Control and Decision, School of AutomationBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Cardiovascular Internal Medicine of Nanlou Branch, National Clinical Research Center for Geriatric DiseasesChinese PLA General HospitalBeijingChina

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