Multi-Dimensional Laplace Transforms and Non-Linear Problems

Abstract

1) IntroductionAs long ago as 1910 an explicit input-output relationship for systems described by a large class of ordinary non-linear differential equations was obtained by Volterra(l) in the form now known as a Volterra series, viz.(1.1) m More recently it has been applied to electrical network and control problems by, amongst others, Wienerv", Barrett'<', Brilliantv", Georgev" and Flake(6). The convergence of this series solution has been discussed by Volterra, Barrett and Brilliant. The last showed that convergence occurred in physical time-invariant systems composed of (a) non-linear subsystems with no memory (i.e. the output of the subsystem depends only on the instantaneous value of the input); (b) linear subsystems with, in general, memory. This class of system is very general and the only excluded class appears to be a hysteretic system. This however leaves open the rate of convergence and for violent non-linearities it could be that a prohibitively large number of terms of the Volterra series would be needed to give an adequate approximation.George'P! explored the use of the Volterra series in the building block approach to control systems and demonstrated the usefulness ofthe multi-dimensional Laplace transform (M.D.L.T.) which can be defined as follows. On introducing a set of artificial time variables t l, t 2, ... , t m into a function x(t) (x(t) == 0 for t < 0) such that x(t) == x(t l, t 2, ... , t m) whenever t = t l = t 2 = ... = t m,The m-dimensional inverse is then given by !l'-I X(SI' S2' ... , sm) = (2 1 .)mf ... f X(SI, S2, ... , sm)exp(sit l + ... + Sm tm> dS I .•. ds.; m 1tJ I, i; (1.3) with tI = t2 = ... = t.. = t and l, the infinite line Res, = c i (a positive constant) > Re(poles ofx treated as a function of S;). at WESTERN MICHIGAN UNIVERSITY on June 5, 2016 ije.sagepub.com Downloaded from