The paper deals with the functional differential equation
$\begin{equation*} y'(t)+∈t_0^{∞}μ_0(ds)\, y(t-s)+\sum_{k=1}^∞ e^{iω_kt}∈t_0^{∞}μ_k(ds)\, y(t-s)=f(t), \end{equation*}$
where the functions $y$ and $f$ take their values in a Hilbert space, $ω_k∈\mathbb{R}$ , $μ_k$ are bounded operator-valued measures concentrated on $[0, +∞)$ , and $\sum_{k=1}^∞\Vertμ_k\Vert < ∞$ . It is shown that the equation is stable provided the unperturbed equation $y'(t)+\int{{_0^{∞}}}μ_0(ds)\, y(t-s)=f(t)$ is at least strictly passive (and consequently stable) and a special estimate holds; this estimate is certainly true if $|ω_k|$ are sufficiently large.
Citation: |
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