REMARKS ON THE CONTROLLABILITY OF NONLINEAR PERTURBATIONS OF VOLTERRA INTEGRODIFFERENTIAL SYSTEMS

Sufficient conditions for the complete controllability of nonlinear perturbations of Volterra integrodifferential systems with implicit derivative are established. The results generalize the results of Dauer and Balachandran [9] and are obtained through the notions of condensing map and measure of noncompactness of 
a set.


Introduction
The controllability of perturbed nonlinear systems has been studied by several authors [2-4, 7- 9] with the help of fixed point theorems.Dacka [6] introduced a new method of analysis to study the controllability of nonlinear systems with implicit derivative based on the measure of noncom- pactness of a set and Darbo's fixed point theorem.This method has been extended to a larger class of perturbed systems by Balachandran [2,3].Anichini et al. [1] studied the problem through the notions of condensing map and measure of noncompactness of a set.They used the fixed point theorem due to Sadovskii [11].In this note, we shall study the controllability of non- linear perturbations of Volterra integrodifferential systems with implicit derivative by suitably adopting the technique of Anichini et al. [1].The results generalize the results of Dauer and Bala- chandran [9].

Prehminaries
We first summarize some facts concerning condensing maps; for definitions and results about the measure of noncompactness and related topics, the reader can refer to the paper of Dacka [6].
Let X be a subset of a Banach space.An operator T:X--X is called condensing if, for any bounded subset E in X with #(E) 0, we have #(T(E))< #(E), where #(E) denotes the mea- sure of noncompactness of the set E as defined in [11].
We observe that, as a consequence of the properties of #, if an operator T is the sum of a com- pact and a condensing operator, then T itself is a condensing operator.Further, if the operator P.XX satisfies the condition Px-Py[ <klx-yl for x, yEX, with 0<k<l, then the operator P has a fixed point property.However, the condition Px-Pyl < x-yl for z, y E X is insufficient to ensure that P is a condensing map or that P will admit a fixed point (Browder [5]).The fixed point property holds in the condensing case (Sadovskii [11]).
Let Ca(J denote the space of continuous R n valued functions on the interval J. x Cn(J and h > 0, let For O(x,h) sup{Ix(t)-x(s)l; t,s G J with and write O(E,h)sup O(x,h), so that O(E,. is the modulus of continuity of a bounded set E. xEE Set Oo(E -lhmoO(E,h ).Assume that gt is the set of functions w" R + R + that are right contin- uous and nondecreasing such that w(r) < r, for r > 0. Let J -[to, t 1].
Lamina 1: [11] Let X C Cn(J and let an d 7 be functions defined on [0, t 1-to] such that limt(s)-limT(s)-O.If a transformation T:X---*Cn(J maps bounded sets into bounded sets and x X with w , then T is a condensing mapping. Lemma2: [1, 11] Let X C Cn([t0,tl]), let I [O, 1], and let S c X be a bounded closed con- vex set.Let H:I xS---X be a continuous operator such that, for any I, the map H(a,.): - Finally, it is possible to show that for any bounded and equicontinuous set.E in cln(J), the following relations holds: where DE-{2;x E E}. #cln(E)-#I(E)-#(DE)-#cn(DE

Main Results
Consider the nonlinear perturbations of the Volterra integrodifferential system of the form &(t) g(t,x) + / h(t,s,x(s))ds + B(t,x(t))u(t) o + f(t,x(t),&(t),(Sx)(t),u(t)),..., J [to, t1] x(to) Xo, where the operator S is defined by ()(t) / (t, , ())d.Here, x(t) R', u(t) R TM and the functions g,h,f,B and k satisfy the following hypotheses: i) g: J RnR n is continuous and continuously differentiable with respect to x. ii) h: J J Rn---+R n is continuous and continuously differentiable with respect to x.
iii) B(t,x(t)) is a continuous family of matrices on J Rn. (1) Let x(t, to, Xo) be the unique solution of the equation existing on some interval J. Define (t) (t, ) + / (t, , ()) o a(t, to, o) (t, (t, to, o)) and H(t, s, to, Xo) hx(t s, x(s, to, Xo)).Then X(t, to, Xo) ox (t, to, Xo) exist and is the solution of 9(t) G(t, to, xo)y(t + / H(t,s;to, xo)Y(s)ds o such that X(to, to, Xo)-I.
Then the solution of the equation ( 1)is given by [10] (t) (t, to, o) + f x(t, , ())[B(, )u() + f(, (), (), (S)(), u())]d o + / f IX(t, n(t, to s where R(t, s; to, x0) is the solution of the equation OR(t J s; to, Xo) + R(t s; to, xo)G(s to, Xo) + R(t, r; to, xo)H(r s; to, xo)dr 0 8 such that R(t, t; to, Xo) I on the interval t o _ s _ t and to; t o, o) x(t, o, o).We say the system (1) is completely controllable on J if, for any Xo, X 1 E Rn, there exists a continuous control function u(t) defined on J such that the solution of (1) satisfies x(tl)x 1.

to s
The main results concerning the controllability of the system (1) is given in the following theorem.
Theorem: Let the system (1) satisfy all the above conditions (i) to (v) and assume the addi- tional conditions I](t,x,,Sx, u) -0, (a) lim sup x (b) there exists a continuous nondecreasing function o: R + R + with (r) < r, such ha f(t,x,y, Sx, u)-f(t,x,z, Sx, u) < w( ly-zl for all (t,x,y, Sx, u) e Jxn3nxn TM (c) there exists a positive constant 6 such that detW(to, tl,X _ ( for all x. Then the system (1) is completely controllable on J.
Then, for suitable positive constants a, b, c we can write I1-[ea-exp(-b)]lxl / lel <_ lul / I1 / lel /, so we divide by I + I + I1 and from the arbitrariness of c, we get the existence of a ball S in Cm(J cln(J)sufficiently large such that I-cT(r) > 0 for r] (u,x) E cOS.