Lyapunov Theory for Discrete Time Systems

In this work, we present the equivalent of many theorems available for continuous time systems. In particular, the theory is applied to Averaging Theory and Separation of time scales. In particular the proofs developed for Averaging Theory and Separation of time scales departs from those typically used in continuous time systems that are based on twice differentiable change of variables and the multiple use of the Implicit Function Theorem and Mean Value Theorem. More specifically, by constructing a suitable Lyapunov function only Lipschitz conditions are necessary. Finally, it is shown that under mild condition on the so-called "interconnection conditions" the proposed tools can guarantee semi-global exponential stability rather than the more stringent local exponential stability typically found in the literature