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Several Integral Inequalities

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Stability and Control of Nonlinear Time-varying Systems
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Abstract

In this chapter, several new integral inequalities are presented, which are effective in dealing with the integrodifferential inequalities whose variable exponents are greater than 1. Compared with existed integral inequalities, those proposed here can be applied to more complicated differential equations.

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References

  1. 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.

    MathSciNet  MATH  Google Scholar 

  2. Hong J. A result on stability of time-varying delay differential equation. Acta Math Sin. 1983;26(3):257–61.

    Google Scholar 

  3. Si L. Boundness, stability of the solution of time-varying delay neutral differential equation. Acta Math Sin. 1974;17(3):197–204.

    Google Scholar 

  4. Si L. Stability of delay neutral differential equations. Huhhot: Inner Mongolia Educational Press; 1994. p. 106–41.

    Google Scholar 

  5. Alekseev VM. An estimate for the perturbation of the solution of ordinary differential equation. Vestnik Moskovskogo Universiteta. Seriya I. Matematika, Mekhanika. 1961;2:28–36.

    Google Scholar 

  6. Li Y. Boundness, stability and error estimate of the solution of nonlinear different equation. Acta Math Sin. 1962;12(1):28–36 In Chinese.

    Google Scholar 

  7. Brauer F. Perturbations of nonlinear systems of differential equations. J Math Anal Appl. 1966;14:198–206.

    Article  MathSciNet  Google Scholar 

  8. Elaydi S, Rao M, Rama M. Lipschitz stability for nonlinear Volterra integro differential systems. Appl Math Comput. 1988;27(3):191–9.

    MathSciNet  MATH  Google Scholar 

  9. Giovanni A, Sergio V. Lipschitz stability for the inverse conductivity problem. Adv Appl Math. 2005;35(2):207–41.

    Article  MathSciNet  Google Scholar 

  10. Hale JK. Ordinary differential equations. Interscience, New York: Wiley; 1969.

    MATH  Google Scholar 

  11. Jiang F, Meng F. Explicit bounds on some new nonlinear integral inequalities with delay. J Comput Appl Math. 2007;205(1):479–86.

    Article  MathSciNet  Google Scholar 

  12. Soliman AA. Lipschitz stability with perturbing Liapunov functionals. Appl Math Lett. 2004;17(8):939–44.

    Article  MathSciNet  Google Scholar 

  13. Soliman AA. On Lipschitz stability for comparison systems of differential equations via limiting equation. Appl Math Comput. 2005;163(3):1061–7.

    MathSciNet  MATH  Google Scholar 

  14. Huang L. Stability theory. Beijing: Peking University Press; 1992. p. 235–83.

    Google Scholar 

  15. Horn RA, Johnson CR. Topics in matrix analysis. Cambridge University press; 1991.

    Google Scholar 

  16. Kumpati SN, Jeyendran B. A common lyapunov function for stable LTI systems with commuting A-matrixes. IEEE Trans Autom Control. 1994;39(12):2469–71.

    Article  Google Scholar 

  17. Lee SH, Kim TH, Lim JT. A new stability analysis of switched systems. Automatica. 2000;36:917–22.

    Article  MathSciNet  Google Scholar 

  18. Liberzon D, Hespanta JP, Morse AS. Stability of switched systems: a Lie-algebraic condition. Syst Control Lett N-Holl. 1999;37:117–22.

    Article  MathSciNet  Google Scholar 

  19. Mareada KS, Balakrishan J. A common Lyapunov function for stable LTI systems with commuting \(A\)-martices. IEEE Trans Autom Control. 1994;39(12):2469–71.

    Article  Google Scholar 

  20. Molchanov AP, Pyatnitskiy YS. Criteria of asymptotic stability of differential and difference inclusions encountered in control theory. Syst Control Lett N-Holl. 1989;13:59–64.

    Article  MathSciNet  Google Scholar 

  21. La Salle J, Lefschetz S. Stability by Lyapunov’s direct method. New York, N.Y.: Academic Press; 1961.

    MATH  Google Scholar 

  22. Polanski K. On absolute stability analysis by polyhydric Lypunov functions. Automatica. 2000;36:573–8.

    Article  MathSciNet  Google Scholar 

  23. Schmitendorf WE, Barmish BR. Null controllability of linear systems with constrained controls. SIAM J Control Optim. 1980;18:327–45.

    Article  MathSciNet  Google Scholar 

  24. Shevitz D, Paden B. Lyapunov stability theory of non-smooth systems. In: Proceeding of the 32nd conference on decision and control, San Antonio, Texas; 1993. p. 416–421.

    Google Scholar 

  25. Tatsushi O, Yasuyuki F. Two conditions concerning common quadratic lyapunov functions for linear systems. IEEE Trans Autom Control. 1997;42(5):719–21.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

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

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Correspondence to Shuli Guo .

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Guo, S., Han, L. (2018). Several Integral Inequalities. In: Stability and Control of Nonlinear Time-varying Systems. Springer, Singapore. https://doi.org/10.1007/978-981-10-8908-4_8

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  • DOI: https://doi.org/10.1007/978-981-10-8908-4_8

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  • Print ISBN: 978-981-10-8907-7

  • Online ISBN: 978-981-10-8908-4

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