## 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)+A=1m-τi(t)tbi(t)e-αi(t-s)x(s)ds=0,x(t)+A=1m0t-τi(t)bi(t)e-αi(t-s)x(s)ds=0, $$\begin{&array;} begin{split} displaystyle x"'(t)+\sum_{i=1}{m} &limitst;-τ_{i}(t)}{t}b_{i}(t)\text{e}{-α{i}(t-s) x(s)text{d} s &=0, x"'(t)+\sum_{i=1}{m}\∫&limits;0 τi(t)b_i(t)text{e}{-α i(t-s)}x(s)\text{d} s &= 0, end{split} end{&array;}$$ with measurable essentially bounded b_{i}(t) and τ_{i}(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 τ_{i}(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.

Original language | English |
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Pages (from-to) | 1165-1176 |

Number of pages | 12 |

Journal | Mathematica Slovaca |

Volume | 69 |

Issue number | 5 |

DOIs | |

State | Published - 1 Oct 2019 |

## Keywords

- Cauchy function
- Exponential stability
- distributed delays
- distributed input control
- integro-differential equations
- stabilization

## All Science Journal Classification (ASJC) codes

- General Mathematics