Asymptotic and Finite Time Semistability for Nonlinear Discrete-Time Systems

This paper focuses on semistability and finite time semistability analysis of discrete-time dynamical systems having a continuum of equilibria. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we build on the theories of semistability and finite-time semistability for continuous-time dynamical systems to develop a rigorous framework for discrete semistability and discrete finite-time semistability. Specifically, Lyapunov and converse Lyapunov theorems for semistability and finite time semistability are developed, and the regularity properties of the Lyapunov function establishing finite time semistability are shown to be related to the settling time function capturing the finite settling time behavior of the dynamical system.