Journal of Applied Mathematics and Stochastic Analysis 6, Number 2, Summer 1993, 153-160
A NOTE ON CONTROLLABILITY OF NEUTRAL VOLTERRA
INTEGRODIFFERENTIAL SYSTEMS x
K. BALACttANDRAN and P. BALASUBRAMANIAM
Bharathiar University
Department of Mathematics
Coimbatore 61 046
Tamil Nadu, INDIA
ABSTRACT
Sufficient conditions for the controllability of nonlinear neutral
Volterra integrodifferential systems are established. The results are
obtained using the Schauder fixed point theorem.
Key words:
point method.
Controllability, neutral Volterra systems, fixed
AMS (MOS) subject classifications:
93B05.
1. INTRODUCTION
The theory of functional differential equations has been studied by several authors [6,
9, 11, 15]. The problem of controllability of linear neutral systems has been investigated by
Banks and Kent [3], and Jacobs and Langenhop [12]. Motivation for physical control systems
and its importance in other fields can be found in
[11, 13]. Angell [1] and Chukwu [4]
discussed the functional controllability of nonlinear neutral systems, and Underwood and
Chukwu [16] studied the null controllability for such systems. Further, Chukwu [5] considered
the Euclidian controllability of a neutral system, with nonlinear base. Onwuatu [14] discussed
the problem for nonlinear systems of neutral functional differential equations with limited
controls.
Gahl
[10]
derived a set of sufficient conditions for the controllability of nonlinear
neutral systems through the fixed point method developed by Dauer
Balachandran
[2]
[7].
Recently,
established sufficient conditions for the controllability of nonlinear neutral
Volterra integrodifferential systems and infinite delay neutral Volterra systems. In this paper
we shall derive a new set of sufficient conditions for the controllability of nonlinear neutral
Volterra integrodifferentia! systems by suitably adopting the technique of Do
11eceived: November,
[8].
1991. Revised: March, 1993.
Printed in the U.S.A. (C) 1993 The Society of Applied Mathematics, Modeling and Simulation
153
K. BALACHANDRAN and P. BAISUBRAMANIAM
154
2. PRELIMINAlES
Let J = [0,
1], tl > 0 and let Q be the Banach space of all continuous functions
(z, u): J x JR" x R "
with the norm defined by
II (, )II
we I I
D: J--R
n
=,p
tEJ
m
Iz(t) l.
=
I 11 + !1 ,-, I
Define the norm of a continuous n x m matrix valued function
R by
m
I D(t)I!
ma.
max
3= 1
where
di:
are the elements of
tEJ
D.
Consider the linear neutral Volterra integrodifferential system of the form
t
dt[z(t) -/C(t-s)z(s)ds-g(t)]- Az(t) + / G(t- s)x(s)ds + B(t)u(t)
0
(1)
0
and the nonlinear system
0
= Az(t) +
/ O(t- s)z(s)ds + B(t)u(t) + f(t,z(t), u(t)),
0
where z(t)E
B(t)
ln,u(t) Rm, C(t) and G(t)
are n xn continuous matrix valued functions and
A a constant n x n matrix and f and g are,
respectively, continuous and absolutely continuous vector functions.
We consider the
controllability on a bounded interval J of system (2).
is a continuous n x m matrix valued functions,
[18]: A function x:JR n
problem (1) or (2) through (0, z(0))on J, if
Definition 1
(i)
(ii)
z is continuous on
is said to be a solution of the initial value
J,
[z(t)- f C(t- s)z(s)ds- g(t)] is absolutely continuous on J,
0
(iii)
(1)
or
(2) holds almost everywhere on J.
Definition 2:
:1
Rn there
The system
exists a control function
(2) is said to be controllable on J if for every :(0),
u(t) defined on jr such that the solution of (2) satisfies
A Note on Controllability of Neutral Volterra Integrodifferential Systems
155
The solution of (1) can be written, as in [17], in the form
x(t) = Z(t)[x(0)- g(0)] + g(t) +
f 2(t-
s)g(s)ds +
2(t s) = -Si’(oz
s) and Z(t)
s)B(s)u(s)ds,
0
0
where
/ Z(t
is an n x n continuously differentiable matrix satisfying the
equation
0
J
Z(0) = I and the solution of nonlinear system (2)
is given by
t[Z(t)- J,C(t
with
s)Z(s)ds] = AZ(t)
G(t- s)Z(s)ds
0
z(t) = Z(t)[z(O) g(O)} + g(t) +
f 2(t-
s)g(s)ds
0
J
+ Z(t,s)[B(s)u(s) + f(s,z(s), u(s))]ds.
0
Define the matrix W by
t
W(t) =
/ Z(t- s)B(s)B*(s)Z*(t- s)ds,
0
where the
denotes the transpose matrix. We know that system
only if W is nonsingular
(1)
is controllable on J if and
[2].
It is clear that x can be obtained if there exist continuous functions x(.) and u(.)
such that
u(t)
B*(t)Z*(t 1 t)W
l(tl)[X Z(tl)(X(O
t1
f (t
I
--8)g(8)ds-
0
g(O)) g(tl)
/ Z(t -8)f(a,x(s), t(s))ds]
0
and
z(t) = Z(t)[z(O) g(O)] + g(t) +
/ 2(t
s)g(s)ds
0
+
/ Z(t- s)[B(s)u(s) + f(s,x(s),u(s))lds.
0
Now, we will find conditions for the existence of such
(i = 1, 2, ., q) then [1 a [[ (i = 1, 2, ., q)
is the
L
norm
(4)
x( and u(). If a E LI(J)
of ai(s (i = 1,2,...,q). That is,
K. BAI.ACHANDRAN and P. BALASUBRAMANIAM
156
t1
I c, il
=
/0
()
ds (i
= 1, 2,..., q).
Next, for our convenience, let us introduce the following notations:
K = max{ I
k
= max( II z(t- s)B(s)II t, 1},
I W-’(ta)II I Z(t
a1
= 3k{ I B*(s)Z*(tl s)II
b
= 3K II
c
= max{ai, bi}
d1
= 3k I B*(s)Z*(tl s)II I W- l(tl)I1[I a
!1,
s)II
I I }, (i = 1,2,...,q)
(i = 1, 2,..., q)
1,2,...,q)
(i
/
II z(ta)II
I(0)- g(0)
+ g(t) +
d2
d
-
I
3
I Z(t)il
I(0)-- g(0)
/
g(tx)
/
/ (tx
/0 2(tl
s)g(s)ds [],
s)g(s)ds I,
0
= maz{dl, d2}.
3. MAIN RESULTS
Now,
we will prove the following main theorem, which is a generalization of Theorem
2 of Balachandran
[2].
Let measurable functions i:R n x Rm---*R +
functions oi: J--,R + (i 1,2,..., q) be such that
Theorem:
(i
1,2,...,q) and L 1
q
cq(t)i(z,u) for every (t,z, u) e J x R n x R m.
f(t,z, u) <
i=1
Then the controllability
of(t)
implies the controllability
q
of (2) if
cisup{i(z u): I (, u)II _< })--
lrnoosup(ri=1
Proofi
Define T: QQ by
T(x, u) = (y, v),
where
v(t) = B*(t)Z*(t I t)W- (t)[z
Z(tl)(Z(O
g(0))- g(tl)
A Note on Controllability of Neutral Volterra Integrodifferential Systems
157
(6)
u() = z()[(0)- (0)] + () +
f 2(-
)()d
0
+ / Z(t s)[S(s)v(s) + f(s,z(s), u(s))]ds.
(7)
0
Based on our assumptions, T is continuous. Clearly the solutions
are fixed points of
u(-) and z(.
to
(3) and (4)
T. We will prove the existence of such fixed points by using the Schauder
fixed point theorem.
Let i(r) = supIi(x u): I (z, u)I]
-< r}.
Since
(5) holds, there exists ro > 0 such that
q
c(o) + d <
i=l
Now, let
Q’o = {(’ ’)
If (x, u) E
Q: I (, ,)II _< "o}.
Qro, from (6) and (7), we have
_< II B*(t)Z*(tl t)II II w- 1(tl)II[I
I I
t
t
0
0
+ I Z(tl)II I(O)- a(O)
q
q
(dl/3k) + (1/3k)
aii(ro)
i-1
q
_< (1/3k)(d +
cii(ro)
i=l
< (ro/3k) _< (to/3)
and
II u I
/ 2(t- s)g(s)ds
_< I z(t)II I(O)- (o) + a(t) +
0
+
i
q
I Z(t-.)B(.)II
I
0
0
i=1
q
<_ (dlal +
i-I
K. BALACHANDRAN and P. BALASUBRAMANIAM
158
q
_< (d/3) + k I[ v [[ + (1/3)
Z ci’]"i(ro
i=1
q
c(0))+ k II v I
(1/3)(d +
i=1
<_ (to/3) + (to/3) = 2(ro/3 ).
Hence T maps Qr into itself. Further, it is easy to see that T(Qr) is equicontinuous
for all r > 0 IS]. By the Ascoli-Arzela theorem, T(Qr0 is compact in Q. Since Qr0 is
nonempty, closed, bounded and convex, by the Schauder fixed point theorem, solutions of (3)
and (4) exist. Hence the proof is complete.
To apply the above theorem we have to construct ci’s and
satisfied.
These constructions are different for different situations.
construction of ai’s and
i’s is easily achieved by taking q
1, a 1
a
i’s
such that
However,
(5)
is
an obvious
1 and
Ca(,u) = (x,u) sup{ f(t,z,u) l’t e J}.
In this case (5) holds if
Now,
we will state a
was proved by Balachandran
Corolly:
If the
corollary which is a particular case of the above theorem and it
[2].
continuous
function f satisfies the
lira
uniformly in E J and
if system (1)
condition
If(t,z,u)l =0
is controllable on
J, then system (2) is controllable
on
J.
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