On the sampling of a linear analytic control system

In the present work we characterize the discrete time system which reproduces exactly the evolutions in the state of a given vector input linear analytic continuous time system driven by inputs which are constant on time intervals of fixed amplitude. This is achieved by comparing the Volterra series associated respectively to the sampled and continuous input state functionals. Moreover we give a compact Lie formula for the solution of a parametrized nonlinear differential equation which enables to characterize the nonlinear difference equation which solves the problem in terms of formal Lie series. On these bases, it becomes very natural to introduce a notion of approximated sampling of main efficiency in practical situations for computing purpose.