On two discrete-time system stability concepts and supermartingales

Abstract

A random discrete-time system {xn}, N = 0, 1, 2, ... is called stochastically stable if for every [epsilon] > 0 there exists a [lambda] > 0 such that the probability P[(supn || xn ||) > [epsilon]] P[|| x0 || > [lambda]] V([middle dot]) satisfies the supermartingale definition on {V(xn)} in a neighborhood of the origin; earlier proofs of stochastic stability require additional restrictions. A criterion for xn --> 0 almost surely is developed. It consists of a global inequality on {U(xn)} stronger than the supermartingale defining inequality, but applied to a U([middle dot]) that need not be a Lyapunov function. The existence of such a U([middle dot]) is exhibited for a stochastically unstable nontrivial stochastic system. This indicates that our criterion for xn --> 0 is "tight," and that the two stability concepts studied are substantially distinct.