Nontangency-Based Lyapunov Tests for Convergence in Discrete-Time Dynamical Systems

This paper focuses on stability analysis of discrete-time dynamical systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. In this paper, we develop Lyapunov-based tests for convergence using the notion of nontangency between the vector field and invariant subsets of the level and sublevel sets of the Lyapunov function and its difference. Specifically, we study nontangency of the vector field to the set of equilibria and provide sufficient conditions for convergence. Moreover, a relationship between Lyapunov stability of an equilibrium point and convergence is provided.