On using disconnected level sets Lyapunov functions in the context of sampled-data systems

The main objective of this paper is to give an interpretation of using non-convex and disconnected level sets Lyapunov functions in the stability analysis of discrete time systems obtained by the discretization of a continuous time Lur´e system. For simplicity reasons, Euler discretization scheme is used to illustrate the features of the proposed method. The main result of this paper shows that it is possible to build, for the original continuous time system, a sequence of bounded connected sets that converges to the origin using this type of Lyapunov functions. To this end, sufficient LMI conditions ensuring the stability of the discrete-time model and an upper bound on the error between the sampled state and the continuous trajectory are used to prove the proposed results. An example will be considered to illustrate this questioning.