Instability of Nonlinear Infinite-Dimensional Feedback Systems Using Lyapunov Functionals.

摘要

Sufficient conditions are given for the L2 instability of a broad class of nonlinear, time-varying feedback systems. The system under consideration is assumed to be decomposed into two subsystems; one passive and nonlinear, not necessarily memoryless, and the other unstable and linear, not necessarily finite-dimensional. The main results essentially state that the feedback system is unstable if the linear subsystem is strictly passive and bounded on a proper subset of L2. The results apply both to instability in the input-output sense and to instability of unforced systems. The principal conceptual tools of the analysis are a Lyapunov function and a state, both of which are defined on the linear subsystem in a manner not depending on the dimensionality of the system. As an application of these results, an instability counterpart to the circle criterion is presented which applies to a class of systems more general than those of previous results. The conditions of this counterpart imply, in addition to L2 instability, global state space instability. Conditions are also given under which the Lyapunov function defined in the analysis may be used to establish asymptotic stability in the large. (Author)