Forward and Converse Lyapunov Theorems for Discrete Dynamical Systems

This paper derives the necessary and sufficient conditions for the Lyapunov function such that the equilibrium point of a dynamical system is stable or Globally Asymptotic Stable (GAS). The paper shows that continuity of Lyapunov function at the equilibrium point is the only necessary and sufficient condition for stability. We shows that a certain type of converse theorem can not be proved with continuous Lyapunov function. The Tower of Babel ‘TOB’ is given as an example of stable dynamical system in which no continuous Lyapunov function exists for this system.