On the Lp(p≥1) stability of a class of nonlinear systems

The purpose of this letter is to initiate a study of the question of Lp(p≥1) stability of a larger class of nonlinear feedback systems whose forward loop is represented by a truncated Volterra series of the form (Ax)(t) = Σn = 1^{m < ∞}∫0^t∫0^tKn(t; τ1, ... τn) Πi = 1ⁿx(τi)dτi. It is demonstrated that under suitable conditions the open-loop system A is continuous and boundaed, and maps Lp[0, ∞) into Lp[0, ∞). It is shown that if the corresponding (unity gain) feedback system has a solution in some Lebesgue class Lp(0 > p ≥ ∞), then the output of the feedback system belongs to the same class as the input.