Sampled-data local exponential stabilization of nonlinear retarded systems by first order approximation methods and spline interpolation of measures

It is proved, for systems described by nonlinear retarded functional differential equations, that (local) exponential stability is guaranteed under fast sampling and spline approximation of stabilizing feedbacks obtained in continuous-time by first-order approximation methods. At the (high) price to consider just the local case, here we make rid of any kind of assumptions such as: the availability and exhibition of Lyapunov-Krasovskii functionals with suitable properties; the availability and exhibition of suitable steepest descent feedbacks of the state with suitable properties; global Lipschitz properties of the function describing the dynamics and of the state feedback. No assumptions are introduced besides standard Fréchet differentiability at the origin of the function describing the dynamics, and the availability of a linear state feedback for the stabilization of the first order approximating linear system, which a huge literature is devoted to with so many results.