In this paper, a new approach to stability of integro-differential equations x (t) + β1 t t−τ 1 (t) e −α 1 (t−s) x (s) ds + β2 t t−τ 2 (t) e −α 2 (t−s) x (s) ds = 0 and x (t) + β1 t−τ 1 (t) 0 e −α 1 (t−s) x (s) ds + β2 t−τ 2 (t) 0 e −α 2 (t−s) x (s) ds = 0 is proposed. Under corresponding conditions on the coefficients α1, α2, β1 and β2 the first equation is exponentially stable if the delays τ1 (t) and τ2 (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.

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 $$\begin{array}{} \begin{split} \displaystyle x'''(t)+\sum_{i=1}^{m}\int\limits_{t-\tau_{i}(t)}^{t}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &=0, \\ x'''(t)+\sum_{i=1}^{m}\int\limits_{0}^{t-\tau _{i}(t)}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &= 0, \end{split} \end{array}$$ 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.