Lyapunov stability and Lyapunov functions of infinite dimensional systems

Abstract

Sufficient conditions are found for the existence of positive definite functions of state which are nonincreasing in time along any trajectory of an autonomous system. The class of systems considered is quite general, and no restriction is made concerning the dimension of the state space or separability of effects of state and input of the subsystems. If certain other relations between the norm of interest on the state space and the positive definite functions are established, Lyapunov or in some cases asymptotic stability in the large can be established. The sufficiency part of the Kalman-Yacubovich lemma as applied to the same problem, is extended to include infinite dimensional systems. That is, it is shown that if the Popov criterion is satisfied, then a Lyapunov function of the Lur'e type exists, even in the infinite dimensional case.