Similitudes et différences entre la théorie du contrôle des systèmes chaotiques à temps discret et à temps continu

Abstract

International audienceFor linear systems, the basic notions of control theory, such as commandability, 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 loss 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 loss of linearity in the input for maps; more precisely, even if the behavior of the state at iteration k+1 depends linearly on the input, this does not guarantee that at iteration k+2 the dependence remains linear. Basically, the composition and differentiation of nonlinear functions generate totally different topological and structural behaviours. This contribution is devoted to the comparison between the structural properties for commandability, observability and flatness for these two types of dynamical system