Christophe LETELLIER
16/10/2023

Jean-Pierre Barbot

**Abstract**

or linear systems, the basic notions of control theory, such as controllability, observability, detectability and stabilizability, are the same for discrete-time and continuous-time systems. The situation is quite different for nonlinear systems such as chaotic systems, particularly due to the lack of continuity for discrete-time systems. The first consequence of this discontinuity is the existence of one-dimensional discrete autonomous chaotic systems. The differences between the dynamics of nonlinear systems made of maps and ordinary differential equations lead to very different properties. The most obvious
is the lack of linearity in the input for maps ; more precisely, even if the behavior of the state at the (*k* + 1)th iteration depends linearly on the input, this does not guarantee that at the iteration (*k* + 2), the dependence remains linear. Basically, the composition and differentiation of nonlinear functions produce totally different topological and structural behaviours. This contribution is devoted to the comparison between the structural properties for observability and flatness for these two types of dynamical system. For the sake of space constraints, controllability, stabilizability and detectability are not discussed below, but it is important to note that a flat system is structurally stabilizable, controllable and detectable.