On the -Boundedness of Solutions of Nonlinear Functional Equations

Let ε N denote the set of N-vector-valued functions of t defined on 0, ∞) such that for any real positive number y, the square of the modulus of each component of any element is integrable on [0, y, and let L 2N (0, ∞) denote the subset of ε N with the property that the square of the modulus of each component of any element is integrable on [0, ∞). In the study of nonlinear physical systems, attention is frequently focused on the properties of one of the following two types of functional equations g &= f + KQf g &= Kf + Qf in which K and Q are causal operators, with K linear and Q nonlinear, g ε ε N, and f is a solution belonging to ε N. Typically, f represents the system response and g takes into account both the independent energy sources and the initial conditions at t = 0. It is often important to determine conditions under which a physical system governed by one of the above equations is stable in the sense that the response to an arbitrary set of initial conditions approaches zero (i. e. , the zero vector) as t → ∞. In a great many cases of this type, g belongs to L 2N (0, ∞) and approaches zero as t → ∞ for all initial conditions, and, in addition, it is possible to show that if f ε L 2N (0, ∞), then f (t) → 0 as t → ∞. In this paper we attack the stability problem by deriving conditions under which g ε L 2N (0, ∞) and f ε ε N imply that f ε L 2N (0, ∞). From an engineering viewpoint, the assumption that f ε L N is almost invariably a trivial restriction. As a specific application of the results, we consider a nonlinear integral equation that governs the behavior of a general control system containing linear time-invariant elements and an arbitrary finite number of time-varying nonlinear elements. Conditions are presented under which every solution of this equation belonging to ε N in fact belongs to L 2N (0, ∞) and approaches zero as t → ∞.