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Stabilization of third order differential equation by delay distributed feedback control with unbounded memory

  • Alexander Domoshnitsky EMAIL logo , Irina Volinsky and Anatoly Polonsky
Published/Copyright: October 5, 2019
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 69 Issue 5

Abstract

There are almost no results on the exponential stability of differential equations with unbounded memory in mathematical literature. This article aimes to partially fill this gap. We propose a new approach to the study of stability of integro-differential equations with unbounded memory of the following forms

x‴(t)+∑i=1m∫t−τi(t)tbi(t)e−αi(t−s)x(s)ds=0,x‴(t)+∑i=1m∫0t−τi(t)bi(t)e−αi(t−s)x(s)ds=0,

with measurable essentially bounded bi(t) and τi(t), i = 1, …, m.

We demonstrate that, under certain conditions on the coefficients, integro-differential equations of these forms are exponentially stable if the delays τi(t), i = 1, …, m, are small enough. This opens new possibilities for stabilization by distributed input control. According to common belief this sort of stabilization requires first and second derivatives of x. Results obtained in this paper prove that this is not the case.

MSC 2010: Primary 45M10; 45J05; 45D05; 34K20; 45F05
Keywords: exponential stability; stabilization; integro-differential equations; distributed delays; distributed input control; Cauchy function

  1. (Communicated by Michal Fečkan)

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Received: 2018-11-23
Accepted: 2019-01-19
Published Online: 2019-10-05
Published in Print: 2019-10-25

© 2019 Mathematical Institute Slovak Academy of Sciences

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  4. Natural extension of EF-spaces and EZ-spaces to the pointfree context
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https://doi.org/10.1515/ms-2017-0298
Keywords for this article
exponential stability; stabilization; integro-differential equations; distributed delays; distributed input control; Cauchy function

Articles in the same Issue

  1. Regular papers
  2. Professor Anatolij Dvurečenskij
  3. A generating theorem of punctured surface triangulations with inner degree at least 4
  4. Natural extension of EF-spaces and EZ-spaces to the pointfree context
  5. On weak isometries in directed groups
  6. On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
  7. On the extensions of discrete valuations in number fields
  8. On certain classes of generalized derivations
  9. Nilpotent elements in medial semigroups
  10. Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
  11. Sharp coefficient bounds for starlike functions associated with the Bell numbers
  12. Sufficient conditions for Carathéodory functions and applications to univalent functions
  13. Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
  14. Linear directional differential equations in the unit ball: solutions of bounded L-index
  15. On oscilatory fourth order nonlinear neutral differential equations – IV
  16. Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
  17. Lerch’s theorem on nabla time scales
  18. IK-convergence of sequences of functions
  19. Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
  20. Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
  21. On the amenability of a class of Banach algebras with application to measure algebra
  22. A bivariate Kumaraswamy-exponential distribution with application
  23. Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
  24. Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
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