This paper establishes sufficient conditions for the controllability and null controllability of linear systems. The aim is to use the variation of constant formula to deduce our controllability grammiam, by exploiting the properties of the grammiam and the asymptotic stability of the free system, we achieved our results. @JASEM

In this study a Taylor's method is developed to find an approximate solution for initial value problem for nonlinear integro-differential equations of the Fredholm type. The method transforms the nonlinear integro-differential equation to a matrix equation which corresponds to a system of nonlinear equations with unknown coefficients.

Using Schaefer fixed point theorem, we establish a set of sufficient condition for the existence of mild solutions of nonlinear integrodifferential equations in Banach space.

Sufficient conditions for the Euclidean null controllability of non-linear delay systems with time varying multiple delays in the control and implicit derivative are derived. If the uncontrolled system is uniformly asymptotically stable and if the control system is controllable, then the non-linear infinite delay system is Euclidean null controllable. @JASEM. The control processes for many dynamic systems are often severely limited, for example, there may be delays in the control actuators. Models of systems with delays in the control occur in population studies. Most specifically models of systems with distributed delays in the control occur in the study of agricultural economics and population dynamics, Arstein (1982), Arstein and Tadmor (1982). In most biological populations the accumulation of metabolic products may inconvenience a population and this result in the fall of birth rate and increase in death rate. If it is assumed that total toxic action in the birth and death rates is expressed by an integral term in the logistic equation then an appropriate model is the integro-differential equation with infinite delays. Several authors have studied these systems and established sufficient conditions for the controllability and null controllability of these systems, Chukwu (1992), Gopalsany (1992). Chukwu (1980) showed that if the linear delay system () () t x t L t x , = & is uniformly asymptotically stable and () () () () t u t B x t L t x t + = , & is proper, then () () () () () () t u x t f t u t B x t L t x t t , , , + + = & is Euclidean null controllable, provided f satisfies certain growth and continuity condition. Sinha (1985) studied the non-linear infinite delay system () () () () () () () () ∫ ∞ − + + + + = 0 , , , t u x t f d t x A t u t B x t L t x t t θ θ θ & (1) and showed that (1) is Euclidean null controllable if the linear base system () () () () t u t B x t L t x t + = , & (2) is proper and the free system () () () () ∫ ∞ − ∞ + + = 0 , θ θ d t x A x t L t x t & (3) is uniformly asymptotically stable, provided that f satisfies some growth conditions. Balachandran and Dauer (1996) studied the null controllability of the non-linear system () () () () () () () () () () () () [ ] 0 0 0 , , , , t t t t x t u t x t f ds s t x s A t h u t B x t L t x n i i i t ∞ − ∈ = + + + + = ∑ ∫ = ∞ − φ & (4) Hale (1974) provided sufficient conditions for the stability of systems of the form

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