Input-to-state stability of neutral type systems

We consider the system ${\displaystyle ẋ\left(t\right)-\int {₀}^{\eta }dR̃(\tau )ẋ(t-\tau )={\int }_0^{\eta }dR(\tau )x(t-\tau )+\left[Fx\right]\left(t\right)+u\left(t\right)}$ (ẋ(t) ≡ dx(t)/dt), where x(t) is the state, u(t) is the input, R(τ),R̃(τ) are matrix-valued functions, and F is a causal (Volterra) mapping. Such equations enable us to consider various classes of systems from the unified point of view. Explicit input-to-state stability conditions in terms of the L²-norm are derived. Our main tool is the norm estimates for the matrix resolvents, as well as estimates for fundamental solutions of the linear parts of the considered systems, and the Ostrowski inequality for determinants.