Null Controllability of Nonlinear Neutral Volterra Integrodifferential Systems with Infinite Delay

In this paper, sufficient conditions for the null controllability of nonlinear neutral Volterra integrodifferential systems with infinite delay are developed. It is shown that if the uncontrolled system is uniformly asymptotically stable, and if the linear system is controllable, then the nonlinear infinite neutral system is null controllable.


Introduction
It is well known that the future state of realistic models in the natural sciences, economics and engineering depend not only on the present but on the past state and the derivative of the past state [7].
Such models which contain past information are called hereditary system.
Neutral functional differential equations are characterized by a delay in the derivative.Equations of this type have applications in many areas of applied mathematics [10].For instance, it is well known Electronic Journal.http://www.math.spbu.ru/user/diffjournal49 that the mixed initial-boundary hyperbolic partial differential equations which arise in the study of lossless transmission lines can be replaced by neutral differential systems [3].This equivalence has been the basis of a number of investigations on the stability of distributed networks which is of immense importance to systems models in the control of fluctuation of current.Furthermore, most economic models for the control of capital stock from an initial capital endowment to the desired stock are governed by neutral systems [8].
Controllability is the property of being able to steer between two arbitrary points in the state space.On the other hand, null controllability is the property of being able to steer all points exactly to the origin.
This has important connections with the concept of stabilizability.
Investigation into the controllability of functional differential systems to the origin has attracted great attention in recent years with the growing interest in disease control models in which the number of infected individuals is desired to be controlled to zero [11].Balachandran and Dauer [4] studied the null controllability of nonlinear infinite delay systems with time varying multiple delays in control whereas Balachandran and Leelamani [6] investigated the null controllability of neutral evolution integrodifferential systems with infinite delay.Iyai [12] discussed the Euclidean null controllability of infinite neutral differential systems.Sinha [15] derived a set of sufficient conditions for the nullcontrollability of nonlinear infinite delay systems with restrained controls.Onwuatu [14] studied the null controllability of nonlinear infinite neutral system.Eke [9] established a set of conditions for the null controllability of linear control systems.Umana and Nse [17] studied the null controllability of nonlinear integrodifferential systems with delays whereas Umana [16] discussed the relative null controllability of linear systems with multiple delays in state and control.
The aim of this paper is to study the null controllability of nonlinear neutral Volterra integrodifferential systems with infinite delay on a finite interval 01 [ , ] J t t  , 10 tt  , when its linear system is assumed controllable and its uncontrolled system is uniformly asymptotically stable with the assumption that the perturbation function satisfies some smoothness and growth conditions.
Electronic Journal.http://www.math.spbu.ru/user/diffjournal50 Volterra integrodifferential equations occur often in applied mathematics [3].For instance, a simplified model for compartmental systems with pipes is represented by nonlinear neutral Volterra integrodifferential equation.Compartmental models are frequently used in theoretical epidemiology, physiology, population dynamics, analysis of ecosystems, and chemical reaction kinetics [5].

Preliminaries
Let n and m be positive integers, R the real line   ,   .We denote by n R , the space of real n  tuples with the Euclidean norm defined by  .If J is any interval of R , the usual Lebesgue space of square integrable functions from   be a given real number and let   be the Banach space of functions which are continuous and bounded on Consider the nonlinear neutral Volterra integrodifferential systems with infinite delay of the following form

t C t s x s ds g t Ax t G t s x s ds dt
where the initial function  is continuous and bounded on n R and where nn  continuously differentiable matrix valued function whose elements are square integrable on ( , 0]  , () Gt is an nn  continuous matrix whose elements are square integrable on ( , 0]  , and () Bt is a continuous nm  matrix function, A is a constant nn  matrix and f and g are respectively continuous and absolutely continuous n  vector functions.
Balachandran [1], Balachandran and Balasubramaniam [2] and Balachandran and Dauer [3] studied the controllability of the system (2.1) using the Schauder fixed point theorem.Our aim in this paper is to study the null controllability problem for the system (2.1).We shall show that if the free system The solution of (2.4) can be written as in Wu [18]: for some suitably chosen 10 t t t  , and Then x is a solution of (2.1) corresponding to u with initial state 0 ()

Conclusion
The paper contains sufficient conditions for the null controllability of nonlinear neutral Volterra integrodifferential systems with infinite delay.These conditions are given with respect to the uniform asymptotic stability of the free linear base system and the controllability of the linear controllable base system, with the assumption that the perturbation function f satisfies some smoothness and growth conditions.
The system (2.1) is said to be null controllable on J if for each Differential Equations and Control Processes, № 2,2011 Electronic Journal.http://www.math.spbu.ru/user/diffjournal52 where ᴛ denotes matrix transpose.Definition 2.1: [13]s closed, bounded and convex, by Riesz's theorem[13], it is relatively compact under the transformation T .The Schauder theorem implies that is a solution pair of the set of integral equations (3.4) and (3.5), hence, the system (2.1) is Euclidean null controllable. T