## Abstract

In this paper, a new approach to stability of integro-differential equations x^{″}(t)+β _{1} ^{t} ∫ _{t-τ1(t)}e-^{α1(t-s)}x(s)ds+β _{2} ^{t} ∫ _{t-τ2(t)}e-^{α2(t-s)}x(s)ds=0 and x^{″}(t)+β_{1} _{0}∫ ^{t-τ1(t)}e-^{α1(t-s)}x(s)ds +β_{2} _{0} ∫ ^{t-τ2(t)}e-^{α2(t-s)}x(s)ds=0 is proposed. Under corresponding conditions on the coefficients α_{1}, α_{2}, β_{1} and β_{2} the first equation is exponentially stable if the delays τ_{1} (t) and τ_{2} (t) are large enough and the second equation is exponentially stable if these delays are small enough. On the basis of these results, assertions on stabilization by distributed input control are proven. It should be stressed that stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper demonstrate that this is not the case.

Original language | English |
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Pages (from-to) | 91-105 |

Number of pages | 15 |

Journal | Mathematical Modelling of Natural Phenomena |

Volume | 12 |

Issue number | 6 |

DOIs | |

State | Published - 2017 |

## Keywords

- Distributed delays
- Distributed inputs
- Exponential stability
- Integro-differential equations
- Stabilization