INFORMATION AND CONTROL 19, 1--9 (1971) Distributed-Parameter Systems with Time-Delay CHRIS P. TSOKOS Department of Statistics and Statistical Laboratory, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 AND A. N. V. RAO Department of Mathematics University of Rhode Island, Kingston, Rhode Island 02881 The concern of this paper is with the asymptotic behavior of the solutions of some nonlinear systems of integro-differential equations that occur, quite often, in the study of nonlinear control systems with time-delay and/or dis- tributed-parameter elements. INTRODUCTION 1. The purpose of this paper is to investigate the absolute stability of control systems characterized by nonlinear integro-differential equations with time- delay. We consider the case that the relationship between the error signal and the state variable is given in the form of a convolution integral. Specifically, we investigate the systems characterized by the following difference- differential equations: (, x'(t) = Ax(t) -~- Bx(t -- -r) + b~[a(t)], wherein and ~(t) =z(o + fl ~(t - ~) x(,) d,, 0.1) t x'(t) : Ax(t) + Bx(t -- T) + f l(t -- u) q~[a(u)] du + b¢[a(t)], o 1 © 1971 by Academic Press, Inc. (l.2) 643/z9/z-I