On the -Boundedness of Solutions of Nonlinear Functional Equations

Abstract

Let ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> 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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) denote the subset of ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\eqalignno{g &amp;= f + KQf \cr g &amp;= Kf + Qf}$</tex> in which K and Q are causal operators, with K linear and Q nonlinear, g ε ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> , and f is a solution belonging to ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> . 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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) and approaches zero as t → ∞ for all initial conditions, and, in addition, it is possible to show that if f ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞), then f(t) → 0 as t → ∞. In this paper we attack the stability problem by deriving conditions under which g ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) and f ε ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> imply that f ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞). From an engineering viewpoint, the assumption that f ε L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> 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 <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</inf> in fact belongs to L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2N</inf> (0, ∞) and approaches zero as t → ∞.