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

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.