- Email: [email protected]
Stability of perturbed neutral functional differential equations
Nonlineur Annl~sis, Theor). Methods & Applications, Vol. 2 No. 5, pp 563-571 0 Pergamon Press Ltd. 1978. Printed in Great Britain
STABILITY
OF PERTURBED
NEUTRAL
FUNCTIONAL
DIFFERENTIAL
EQUATIONS*
A. F. IZE lnstituto
de CiCncias MatemBticas
de Slo Carlos, Departamento de Matemitica, 13.560, SLo Carlos, Sbo Paula, Brazil
Universidade
de Sbo Paula,
and J. G. DOS REIS Faculdade
de Medicina
de Ribeirlo Preto, Departamento 14.100, Ribeirio Preto, Sk
de MatemLtica, Paulo, Brazil
Universidade
de Sbo Paulo.
(Received in revisedform 22 September 1977) Key words: Neutral functional differential equations, Lyapunov asymptotically stable, perturbed linear systems, uniformly
functions, uniformly stable, stable operator, exponentially
uniform stable.
INTRODUCTION
R” BE a real or complex n-dimensional C = C([ - r, 01, R”) be the space of continuous
LET
linear vector space with norm 1. I. For Y >, 0 let functions taking [-r, 0] into R” with I/4 I/ defend
by l/411 = _;y~<~ IM)l. Suppose t,isa real number define the operators
and g, f are continuous
functions
taking
[fo, 3n) x C + R” and
D(.): [to, ‘xX7) x C -+ R” by w+
for t E [T, #m), 4 E C. A functional
= A(L) 440) -
differential
equation
dtv 4)
is a system of the form
(1) where X, E C is defined by .x,(0) = .~_(t+ 8), -r d 0 d 0. For any Q,E C, r~E [to. m) a function x = .x(0,4) defined on [r~ - r, CT+ A) is said to be a solution of (1) on (a, 0 + A) with initial value 4 at CTif .x is continuous on [r~ - r, CJ + A) and relation (1) is satisfied on (6, 0 + A). We say that (1) is a neutral functional differential equation if det A(t) # 0, t E [t,, 00) and for every 4 E C, t E [t,, co), g(t, 4) = I!,. dp(t, 0) 4(e), where ,Uis a n x n matrix of bounded variation for 8 E [-r, 0] and there is a scalar function p(s) continuous non decreasing for s E [0, r], p(O) = 0 such that 0 [4JpL(4
@I
4@)
<
P(S)
IS --s * This research
was supported
by. CNPq,
sup -F
FAPESP
and FINEP. 563
I Of@
I.
(2)
564
A. F. IzB
AND
J. G.
DOS
RIMS
We assume that for each (a, 4) E (t,, ‘3) x C, (1) has a unique solution which depends continuously upon (T,4 and that every bounded solution of (1) is defined for t E [o, YJ). We assume also that f(t, 0) = 0. D@rition. We say that the solution .Y = 0 of (1) is uniformly stable if for every E > 0 there exists a 6 = 6(r:) > 0 such that for all (TE [to, I%), t, > - in, any solution x(0,4) of (1) with initial value &Jat 0, ((4(( < 6 satisfies (I-x,(g, 4)(( < r for t 3 (T. It is uniformly asymptotically stable if it is uniformly stable and for some fixed 6 > 0, for any q > 0 there exists a T = T(q) > 0 such that j/41/ d 6 implies I(x,(cJ,4)/I < q for t 3 o + T. 1. LYAPUNOV
x c + R is continuous
IfK+,,‘ZQ)
FUNCTIONALS
we define
1. Let f(t, 0) = 0, ,f(t, 4) locally Lipchitzian in 4, uniformly with respect to t, and for all 6 in C, [A(t) &O)( d kl/djl for t b 0. Assume that the null solution of (1) is uniform asymptotically stable and there exists a constant L such that THEOREM
IIxt(to, 4) - .xt(to, Y))I d eL(‘-*O)I(+ - Yl).
(3)
Then there exists positive definite functions u(r), b(r), c(r) for 0 < r < r-r, non decreasing, u(O) = b(0) = c(0) = 0 and a scalar functional V(t, +)defined and continuous for (t, Cp)E [to, tx) x C’, )I~11d 1, such that (a) (b) (4 (d)
4JW,
&I) d W, 4)
#?q)
G V(t, 4) G ##I/)
w,
4) G - o,(IIw?
(w,&
-
u~,w[
&)I) d k/i4 - Y’(I
for all t 2 0, 4, Y in C, J/4/l, l(Yll < rl. Proof. From
Cruz and Hale [l, Lemma
7.11, there exists functions
p(u), r(u) such that
(i) p(u) is continuous, monotone increasing for u 2 0, p(0) = 0 (ii) u(u) continuous, monotone decreasing for u 3 0, with lim v(u) = 0 such that if u-1 lI4lld f-1’ and t2a t 2 (T.
IJX,b> &I) G P(JJ4II) 4t - 4,
(4)
We can assume also that v(u) has a continuous negative derivative and u(O) = 1. From the properties of P(U) there exists a function a(u) defined, continuously differentiable and positive on u 2 0, a(O) = 0, CL(U) strictly increasing lim x(u) = CT,such that tl- X_ u(u) = exp[ - H(U)]. Suppose
q(u) is a bounded,
continuously
q(u) > 0, q’(u) < d(u) for u > 0, q’ monotone
differentiable decreasing.
function on [0, m) such that q(0) = 0, Let /? > I(k) = k where IA(t) 4(O)l < k
Stability
of perturbed
neutral
functional
differential
565
equations
and define
(5) Since D satisfies (2),
and (a) is satisfied.(b),(c) and(d) follows step by step the argument being that V(t, 4) is defined by (5). Theorem (a’) (b’) (c’)
1 is an alternative
for Theorem
given in [2] with the only change
7.1 of Cruz and Hale [ 1] which uses the hypotheses:
I]o(r, &(I d v(r, 4) d h(ll4ll) v(r, 4) d - C(I]o(r, 4)I)) ]Ur, 4) - Ur, ‘y)] G k II4 - YJII
where V(t, 4) = sup ()D(t + s, x,+ ,(t, 4)II eas. The principal
advantage
of this definition
of V(t, 4)
is condition (c’) $%h is very useful in the study of perturbed systems. By the other hand 114112 0 does not imply necessarily that V(t, 4) B n(ll$ I/) for some positive increasing function a with n(O) = 0, what is quite often needed in the applications. With the definition of V(t, 4) given in Theorem 1 above this condition is satisfied. However, the Condition (3), that holds in general for linear systems, is not easy to verify for general non linear systems and it is an open question to know if it is true for system (1) withf Lipchitzian 2. STABILITY
Consider
OF
PERTURBED
LINEAR
SYSTEM
the system
$ a4 where D(t, 4) and f(t, 4) are bounded operator. Consider now the perturbed
l’J = fk l’t)
linear continuous system.
(6)
functionals
and D is a uniformly
zqt,
XJ = f(t, x,).
System (7) can be transformed
(7)
into the system
$ D(t,XJ = f(t, x1) + h(t, x,) where only the f function
is perturbed
h(t, xt) = To study the stability
properties
stable
(8)
and
fct,x1) -
j-k xJ -
$ [Dk .xJ -
D(t, x,)1.
of system (8) we will need the following.
A. F !ZCAND J. G. DOS REIS
566 LEMMA 1. Let x,(t,, 4) be a solution L such that
of (8) and y,(tO, 4) a solution
Jl.~~(t,,4) - _r,(t,, 4)I/ d [eL(t-to) -
of (6), then there exists a constant
I] lir(t. x~)/,
t 3 t,.
(9)
It is obvious that (9) is satisfied for f = t,. Suppose it is not true for every t 3 t,. Let t, be the supremum of the t’s for which inequality (9) is satisfied. From the continuity of both sides of (9) there exists a sequence {t,j, t, + 0 for n + _*, tn small enough such that I/.Y,,+ J& From
4) - yt, + ,,(t,, +)I/ > [@I + ‘,-‘o) -
Hale [3, p. 3541, there is a 11, small enough
11 (Mr, .xt)l.
such that if 0 < t, < 11,
Thus flS f,,
1
Ih(s. XJ ds > l(eLCtl+l,,P’o) -
1 - p&J c f,
I)
h(t,
+
t,,.stI i ,,,I.
Hence
1 - dho) tl
Ih(.s,s,J 3 (eLctl + r*~-‘O’-- 1) h(t, + t,. s,, + ,,,)
sup + I,,
and ii;;; 1,,-0,
1
sup lh(s, SJ ’ - P(ho) rl
>
iiiii f,, - 0 +
(e Uf, + f,,- tn1 t”
-
I’(h(t,
+rn>.~tl
,,,, 1).
Then , _
Lth) Ih(t, , xr,)l 3
L eL”l -‘o’
Ik(f,. .x1,)1
0 and if we take L > l/(1 - p(h,)) we have a contradiction THEOREM 2. lf the solution y E 0 of (6) is uniformly asymptotically stable there exists a scalar functional ~(t, 4) defined and continuous for (t, 4) E [to, 1~) x C such that
Proqf
Let p = max((k
+ p(r))“, 1) and define
Ur, 4) = B [S From
the uniform
asymptotic
,R tI~t+i(r~ 4)I)’ ds + ;;!
stability
of the solution
/IY~+,(r, m)ll’_i.
y E 0, [3], there are constants
cyand k,
Stability
of perturbed
neutral
functional
differential
equations
567
such that (lyt(t,,, $)(I < k, e-““-‘“’ ((d(I. It is obvious that W, 4) > (I$(/’ and V(t, 4) 3 JD(t, 4))‘, since (o(t. $)I2 G k2j14112 + p2(r) 1(+(1’ G 8~~~~~’d W, 4). Let (T 2 0, then (I.Y~+&,&II d K, can e-asl(411. Hence V(t, 4) = fik, e2=0
ee2”“du
+ &,e2anl/~112
k, e2”“~~~(~2= k,(14(12
= fl(1 + l/(24)
and (b) is proved.
+ / sup (lY,+Jt, &)(I2 - sup IlYt+s(4 S>O SBO
d
4Jll’
-
lbttsk
42)Il”l
+ SUPI[IIIJi+s(tt wl12
-
llJi+,k
d4II’ll
G
s
s
om I[ll~t+sk
$,)ll2~
oa I(Il1’*+,k
x ((lYltsk
Mll
+ lIYt+&
ds
42m
dJ,)l/ - IlYt+s@, 42)IOds
+ sup(((ly,+,k $1)+ ll!‘t+s(t~ 42)lOI. s 20
d
koeane-zn(l\~I\l
+ koean(l14111
d k;e2=“#hll
+ (l~21))koeaae-as(ll~l
+ ll~211~~o~““~~““~Il~~II
+ lld4(1141
-
-
d4
- 4~11)~~
114211)
-a)
Hence
Iv?, 4,) - W, 4,)I Q P kzeZan WII and (c) is proved.
+ ll42ll)wll
-
lIm,i,(l
- &)
568
A. F. Izk AND J. G. DOS REIS
To prove (d) since
=
Oc b$o~ s
and
2is a decreasing llYf+s(~O’~)II
qt, l’,) <
Eii h- l h-O+
and the theorem THEOREM 3.
4)1i2
ds
+
w++s(to>
4)))’
f
function
it follows that
,:,bs(to~6)/I’ [S
ds - fi[~~@~, &I(’ f
ds]
is proved.
Assume
that (IW> 4) - D(4 $)I) + IIf(4 4) - f(4 &(I d rll4ll
(IO)
where y is chosen small enough in such a way that y+ E C,. Then if the solution y E 0 of (6) is uniformly asymptotically stable, the solution x = 0 of (7) is uniformly asymptotically stable. Proof: The solutions
of (7) are the solutions ;
of
Wr, xt) = f(4 .xt) + 44 .xJ
and from (10) IN $)I d ~ll4ll. Let .x,(t,, 4) be a solution of (7) for (141) small enough and t’*(r) = J’(r, q(t,, 4)). F rom (a) and (d) of Theorem 2 we have Ti;;i v*(t h-rO+
+ h) h
V(t, $) as in Theorem
v*(r)
dlim
f’+ + h, &+,,@Y@o, $‘I,) h
+ ‘;
v(t
h-+0+
where xt(to, 4) is a solution
vk
+ h>Xt+j,(t, x,(t,, 4,)) - v(t h
of (7) then.
X,(&Y 4,)
+ h>Pr+h(t>“t@,, 4,))
2. Define
Stability
v*o + 4
Ti;;;
w
-
+
neutral
functional
differential
equations
d - I(xt(t,, #)I 2
h
h-O+
of perturbed
v(t + h, x,+&YX,tt,, 4)))- v(t + h, &+,,k “t&y dd))
TiiT;
h
h-O+ From part (c) of Theorem v(t + h, X,+,,@, x,(t,, 4))) -
2 we have v(t + h. )?t+,@‘X&> 4))) d k(\(Xt+h(t, “&,
4)) + )?t+,,@,x,(t,, $))I\)
’ (IIXt+,,@Y X,(t,? 4)) - yt+,,@, Xt(tO,
dd)i,)’
Ij4’~+,,(t,x,(t,? &))I G k,IIx,(t,,
&)I.
We know that
From
Lemma
above and’condition
(10) we have
IIXt+&? X&? 4)) - 4’l+&, Xt(tOY4)) < (e” -
1) II& “t)\l
< (e” -
1) YllX&Y &(I
and then
11% +hk XtttOy&)[I G teLh- 1)IIdt,xc)ll+ k&k,, $)I(. Hence Iv(t + h, X,+&9 x,(t,, 4))) -
v(t + h, X, +,,@Y X,&,, &))I
d (k Y(e” -
1) II@,,
4411
+ 2k,kllx,(t,,
4411)(eLh -
= IIxt(to, 4)II’ (k (e” -
1) rlj.+,,
4111
1) Y + 2k,k) (eLh - 1) Y
= jIxt(to, $)]I’ (k (e” - 1) (eLh -
1) y* + 2kok(eLh - 1) y).
Hence fi
h-l
1 v(t
+
h,
Xt+,,(t,
x,@,,
4,))
-
v(t
+
h,
Xt+h(t5
X,@,,
&))I
h+O+ <
/(xt(to, +)I(’ Tiiii h-‘(k(eLh - 1) (e& - 1) y2 + 2kko(eLh - 1) y(
= 2k k,,Ly~~~,;t,~qi)~~ 2. Thus if we take y < 1/(2kk,L) l’*(t
G
h+O+
+ h) - V*(t)
h
Q -
)Ixt(to,
#II2
+ 2&&y
Ijx,(to,
4)112
Ilxttt,,4)112 G’+,b - 1) = -Y1 IIx,kJ,9)112Y Yl > 0.
=
Then system (7) is uniformly
asymptotically
stable.
569
A.
570
F. IzB AND
J. G.
DOS REIS
Another simple proof that implies in particular that the solutions stable is the following. From conditions (b) and (d) of Theorem 2, fi;;; v*(t
+ h) -
v*(t)
d -y,v*(t),
h
Yl =
v*tt + “h’- v*@)<
Lfi-
of (7) are exponentially
l/K,
_yl
v*(t) or log V*(t
E
+ h) - log V*(t)
d
h
h-O+
log v*(t)
- log v*(t,)
< -yl(t
-Y1
- to)
or (j.X,(t,, 4)II’ < V*(t) < V*(t,) K e-rl(t-to) and the theorem THEOREM 4.
is proved.
Assume
that
If@, 4) - fk Then if the solution uniform asymptotically
4)I + ID@, 4) - o(t, @)I G s(t) (1411,j%)
4’ = 0 of (6) is uniform stable.
Proof: As in Theorem I/X,+,,k @,,
< K e-yt(t-to) 11411”
asymptotically
4 we have from inequality
4)) - &+hk “&
dt < ;I’C
stable the solution
(9)
$))[I < (eLh -
1) /I& -xt)II
< (eLh -
I)&)
(I”&> +)I)
and
and therefore IIXt+h(t, .x&3 $))I[ d I(eLh -
1) &) Ilx#03 4)II + ~oll.~t(~o>@)II.
Thus we have 1v(t + h, x, + h(t, .x,@,, 4,)) -
v(t + h>?‘t+ h(h “$,,
d (k(eLh + 2k,I(x,(t,,
&,)I
1) g(t) )I”&, $)I1 4)/I) (eLh -
= II.xJt,, 4))1 ’ (k (eLh -
1) g(t) I/.#,, 1) (eLh -
4)II
1) g2(t) + 2ko(eLh -
1) g(t)).
(11) .X = 0 of (7) is
Stability
of perturbed
neutral
functional
differential
571
equations
Hence Izl
K’[IV(t
+ h, x,, Jf, X&, 44)) -
Jqt + h, Y,+Jk “&I,
h-O+
4N)II
Therefore fi
v*(t
+ h) -
v*(t)
h
h+O+
d -
IIXrUo. 4)1/2 + (I”&, =
owdt)
-
<
(k,g(t) -
4)Ij’ 1)
II.+,,
k’s(t) 4111’
1) V*(t).
Thus we have )I.+)? 4)(I G v*(t) < V*(to)exp[kl~~ < k exp - (t implies that the solution tically stable.
which
Acknol~,/cdgements-We
thank
gWds]ew(--(t
to))
to) ~~~~~.
x E 0 of (7) is exponentially
A. A. Freiria
-
for interesting
comments
stable and therefore
uniform
asympto-
on the paper
REFERENCES 1. CKUZ M. A. & HALE J. K., Stability of functional differential equations of neutral type. J. d(f/. Eqns 7 (2) 334-355 (1970). 2. HALE J. K. and CRUZ M. A., Asymptotic behavior of neutral functional differ-ential equations. Archs rational merh. Analysis34,331-353 (1969), 3. HALE J. K., Ordinary Differential Equations. Wiley Interscience, New York (1969).
STABILITY
OF PERTURBED
NEUTRAL
FUNCTIONAL
DIFFERENTIAL
EQUATIONS*
A. F. IZE lnstituto
de CiCncias MatemBticas
de Slo Carlos, Departamento de Matemitica, 13.560, SLo Carlos, Sbo Paula, Brazil
Universidade
de Sbo Paula,
and J. G. DOS REIS Faculdade
de Medicina
de Ribeirlo Preto, Departamento 14.100, Ribeirio Preto, Sk
de MatemLtica, Paulo, Brazil
Universidade
de Sbo Paulo.
(Received in revisedform 22 September 1977) Key words: Neutral functional differential equations, Lyapunov asymptotically stable, perturbed linear systems, uniformly
functions, uniformly stable, stable operator, exponentially
uniform stable.
INTRODUCTION
R” BE a real or complex n-dimensional C = C([ - r, 01, R”) be the space of continuous
LET
linear vector space with norm 1. I. For Y >, 0 let functions taking [-r, 0] into R” with I/4 I/ defend
by l/411 = _;y~<~ IM)l. Suppose t,isa real number define the operators
and g, f are continuous
functions
taking
[fo, 3n) x C + R” and
D(.): [to, ‘xX7) x C -+ R” by w+
for t E [T, #m), 4 E C. A functional
= A(L) 440) -
differential
equation
dtv 4)
is a system of the form
(1) where X, E C is defined by .x,(0) = .~_(t+ 8), -r d 0 d 0. For any Q,E C, r~E [to. m) a function x = .x(0,4) defined on [r~ - r, CT+ A) is said to be a solution of (1) on (a, 0 + A) with initial value 4 at CTif .x is continuous on [r~ - r, CJ + A) and relation (1) is satisfied on (6, 0 + A). We say that (1) is a neutral functional differential equation if det A(t) # 0, t E [t,, 00) and for every 4 E C, t E [t,, co), g(t, 4) = I!,. dp(t, 0) 4(e), where ,Uis a n x n matrix of bounded variation for 8 E [-r, 0] and there is a scalar function p(s) continuous non decreasing for s E [0, r], p(O) = 0 such that 0 [4JpL(4
@I
4@)
<
P(S)
IS --s * This research
was supported
by. CNPq,
sup -F
FAPESP
and FINEP. 563
I Of@
I.
(2)
564
A. F. IzB
AND
J. G.
DOS
RIMS
We assume that for each (a, 4) E (t,, ‘3) x C, (1) has a unique solution which depends continuously upon (T,4 and that every bounded solution of (1) is defined for t E [o, YJ). We assume also that f(t, 0) = 0. D@rition. We say that the solution .Y = 0 of (1) is uniformly stable if for every E > 0 there exists a 6 = 6(r:) > 0 such that for all (TE [to, I%), t, > - in, any solution x(0,4) of (1) with initial value &Jat 0, ((4(( < 6 satisfies (I-x,(g, 4)(( < r for t 3 (T. It is uniformly asymptotically stable if it is uniformly stable and for some fixed 6 > 0, for any q > 0 there exists a T = T(q) > 0 such that j/41/ d 6 implies I(x,(cJ,4)/I < q for t 3 o + T. 1. LYAPUNOV
x c + R is continuous
IfK+,,‘ZQ)
FUNCTIONALS
we define
1. Let f(t, 0) = 0, ,f(t, 4) locally Lipchitzian in 4, uniformly with respect to t, and for all 6 in C, [A(t) &O)( d kl/djl for t b 0. Assume that the null solution of (1) is uniform asymptotically stable and there exists a constant L such that THEOREM
IIxt(to, 4) - .xt(to, Y))I d eL(‘-*O)I(+ - Yl).
(3)
Then there exists positive definite functions u(r), b(r), c(r) for 0 < r < r-r, non decreasing, u(O) = b(0) = c(0) = 0 and a scalar functional V(t, +)defined and continuous for (t, Cp)E [to, tx) x C’, )I~11d 1, such that (a) (b) (4 (d)
4JW,
&I) d W, 4)
#?q)
G V(t, 4) G ##I/)
w,
4) G - o,(IIw?
(w,&
-
u~,w[
&)I) d k/i4 - Y’(I
for all t 2 0, 4, Y in C, J/4/l, l(Yll < rl. Proof. From
Cruz and Hale [l, Lemma
7.11, there exists functions
p(u), r(u) such that
(i) p(u) is continuous, monotone increasing for u 2 0, p(0) = 0 (ii) u(u) continuous, monotone decreasing for u 3 0, with lim v(u) = 0 such that if u-1 lI4lld f-1’ and t2a t 2 (T.
IJX,b> &I) G P(JJ4II) 4t - 4,
(4)
We can assume also that v(u) has a continuous negative derivative and u(O) = 1. From the properties of P(U) there exists a function a(u) defined, continuously differentiable and positive on u 2 0, a(O) = 0, CL(U) strictly increasing lim x(u) = CT,such that tl- X_ u(u) = exp[ - H(U)]. Suppose
q(u) is a bounded,
continuously
q(u) > 0, q’(u) < d(u) for u > 0, q’ monotone
differentiable decreasing.
function on [0, m) such that q(0) = 0, Let /? > I(k) = k where IA(t) 4(O)l < k
Stability
of perturbed
neutral
functional
differential
565
equations
and define
(5) Since D satisfies (2),
and (a) is satisfied.(b),(c) and(d) follows step by step the argument being that V(t, 4) is defined by (5). Theorem (a’) (b’) (c’)
1 is an alternative
for Theorem
given in [2] with the only change
7.1 of Cruz and Hale [ 1] which uses the hypotheses:
I]o(r, &(I d v(r, 4) d h(ll4ll) v(r, 4) d - C(I]o(r, 4)I)) ]Ur, 4) - Ur, ‘y)] G k II4 - YJII
where V(t, 4) = sup ()D(t + s, x,+ ,(t, 4)II eas. The principal
advantage
of this definition
of V(t, 4)
is condition (c’) $%h is very useful in the study of perturbed systems. By the other hand 114112 0 does not imply necessarily that V(t, 4) B n(ll$ I/) for some positive increasing function a with n(O) = 0, what is quite often needed in the applications. With the definition of V(t, 4) given in Theorem 1 above this condition is satisfied. However, the Condition (3), that holds in general for linear systems, is not easy to verify for general non linear systems and it is an open question to know if it is true for system (1) withf Lipchitzian 2. STABILITY
Consider
OF
PERTURBED
LINEAR
SYSTEM
the system
$ a4 where D(t, 4) and f(t, 4) are bounded operator. Consider now the perturbed
l’J = fk l’t)
linear continuous system.
(6)
functionals
and D is a uniformly
zqt,
XJ = f(t, x,).
System (7) can be transformed
(7)
into the system
$ D(t,XJ = f(t, x1) + h(t, x,) where only the f function
is perturbed
h(t, xt) = To study the stability
properties
stable
(8)
and
fct,x1) -
j-k xJ -
$ [Dk .xJ -
D(t, x,)1.
of system (8) we will need the following.
A. F !ZCAND J. G. DOS REIS
566 LEMMA 1. Let x,(t,, 4) be a solution L such that
of (8) and y,(tO, 4) a solution
Jl.~~(t,,4) - _r,(t,, 4)I/ d [eL(t-to) -
of (6), then there exists a constant
I] lir(t. x~)/,
t 3 t,.
(9)
It is obvious that (9) is satisfied for f = t,. Suppose it is not true for every t 3 t,. Let t, be the supremum of the t’s for which inequality (9) is satisfied. From the continuity of both sides of (9) there exists a sequence {t,j, t, + 0 for n + _*, tn small enough such that I/.Y,,+ J& From
4) - yt, + ,,(t,, +)I/ > [@I + ‘,-‘o) -
Hale [3, p. 3541, there is a 11, small enough
11 (Mr, .xt)l.
such that if 0 < t, < 11,
Thus flS f,,
1
Ih(s. XJ ds > l(eLCtl+l,,P’o) -
1 - p&J c f,
I)
h(t,
+
t,,.stI i ,,,I.
Hence
1 - dho) tl
Ih(.s,s,J 3 (eLctl + r*~-‘O’-- 1) h(t, + t,. s,, + ,,,)
sup + I,,
and ii;;; 1,,-0,
1
sup lh(s, SJ ’ - P(ho) rl
>
iiiii f,, - 0 +
(e Uf, + f,,- tn1 t”
-
I’(h(t,
+rn>.~tl
,,,, 1).
Then , _
Lth) Ih(t, , xr,)l 3
L eL”l -‘o’
Ik(f,. .x1,)1
0 and if we take L > l/(1 - p(h,)) we have a contradiction THEOREM 2. lf the solution y E 0 of (6) is uniformly asymptotically stable there exists a scalar functional ~(t, 4) defined and continuous for (t, 4) E [to, 1~) x C such that
Proqf
Let p = max((k
+ p(r))“, 1) and define
Ur, 4) = B [S From
the uniform
asymptotic
,R tI~t+i(r~ 4)I)’ ds + ;;!
stability
of the solution
/IY~+,(r, m)ll’_i.
y E 0, [3], there are constants
cyand k,
Stability
of perturbed
neutral
functional
differential
equations
567
such that (lyt(t,,, $)(I < k, e-““-‘“’ ((d(I. It is obvious that W, 4) > (I$(/’ and V(t, 4) 3 JD(t, 4))‘, since (o(t. $)I2 G k2j14112 + p2(r) 1(+(1’ G 8~~~~~’d W, 4). Let (T 2 0, then (I.Y~+&,&II d K, can e-asl(411. Hence V(t, 4) = fik, e2=0
ee2”“du
+ &,e2anl/~112
k, e2”“~~~(~2= k,(14(12
= fl(1 + l/(24)
and (b) is proved.
+ / sup (lY,+Jt, &)(I2 - sup IlYt+s(4 S>O SBO
d
4Jll’
-
lbttsk
42)Il”l
+ SUPI[IIIJi+s(tt wl12
-
llJi+,k
d4II’ll
G
s
s
om I[ll~t+sk
$,)ll2~
oa I(Il1’*+,k
x ((lYltsk
Mll
+ lIYt+&
ds
42m
dJ,)l/ - IlYt+s@, 42)IOds
+ sup(((ly,+,k $1)+ ll!‘t+s(t~ 42)lOI. s 20
d
koeane-zn(l\~I\l
+ koean(l14111
d k;e2=“#hll
+ (l~21))koeaae-as(ll~l
+ ll~211~~o~““~~““~Il~~II
+ lld4(1141
-
-
d4
- 4~11)~~
114211)
-a)
Hence
Iv?, 4,) - W, 4,)I Q P kzeZan WII and (c) is proved.
+ ll42ll)wll
-
lIm,i,(l
- &)
568
A. F. Izk AND J. G. DOS REIS
To prove (d) since
=
Oc b$o~ s
and
2is a decreasing llYf+s(~O’~)II
qt, l’,) <
Eii h- l h-O+
and the theorem THEOREM 3.
4)1i2
ds
+
w++s(to>
4)))’
f
function
it follows that
,:,bs(to~6)/I’ [S
ds - fi[~~@~, &I(’ f
ds]
is proved.
Assume
that (IW> 4) - D(4 $)I) + IIf(4 4) - f(4 &(I d rll4ll
(IO)
where y is chosen small enough in such a way that y+ E C,. Then if the solution y E 0 of (6) is uniformly asymptotically stable, the solution x = 0 of (7) is uniformly asymptotically stable. Proof: The solutions
of (7) are the solutions ;
of
Wr, xt) = f(4 .xt) + 44 .xJ
and from (10) IN $)I d ~ll4ll. Let .x,(t,, 4) be a solution of (7) for (141) small enough and t’*(r) = J’(r, q(t,, 4)). F rom (a) and (d) of Theorem 2 we have Ti;;i v*(t h-rO+
+ h) h
V(t, $) as in Theorem
v*(r)
dlim
f’+ + h, &+,,@Y@o, $‘I,) h
+ ‘;
v(t
h-+0+
where xt(to, 4) is a solution
vk
+ h>Xt+j,(t, x,(t,, 4,)) - v(t h
of (7) then.
X,(&Y 4,)
+ h>Pr+h(t>“t@,, 4,))
2. Define
Stability
v*o + 4
Ti;;;
w
-
+
neutral
functional
differential
equations
d - I(xt(t,, #)I 2
h
h-O+
of perturbed
v(t + h, x,+&YX,tt,, 4)))- v(t + h, &+,,k “t&y dd))
TiiT;
h
h-O+ From part (c) of Theorem v(t + h, X,+,,@, x,(t,, 4))) -
2 we have v(t + h. )?t+,@‘X&> 4))) d k(\(Xt+h(t, “&,
4)) + )?t+,,@,x,(t,, $))I\)
’ (IIXt+,,@Y X,(t,? 4)) - yt+,,@, Xt(tO,
dd)i,)’
Ij4’~+,,(t,x,(t,? &))I G k,IIx,(t,,
&)I.
We know that
From
Lemma
above and’condition
(10) we have
IIXt+&? X&? 4)) - 4’l+&, Xt(tOY4)) < (e” -
1) II& “t)\l
< (e” -
1) YllX&Y &(I
and then
11% +hk XtttOy&)[I G teLh- 1)IIdt,xc)ll+ k&k,, $)I(. Hence Iv(t + h, X,+&9 x,(t,, 4))) -
v(t + h, X, +,,@Y X,&,, &))I
d (k Y(e” -
1) II@,,
4411
+ 2k,kllx,(t,,
4411)(eLh -
= IIxt(to, 4)II’ (k (e” -
1) rlj.+,,
4111
1) Y + 2k,k) (eLh - 1) Y
= jIxt(to, $)]I’ (k (e” - 1) (eLh -
1) y* + 2kok(eLh - 1) y).
Hence fi
h-l
1 v(t
+
h,
Xt+,,(t,
x,@,,
4,))
-
v(t
+
h,
Xt+h(t5
X,@,,
&))I
h+O+ <
/(xt(to, +)I(’ Tiiii h-‘(k(eLh - 1) (e& - 1) y2 + 2kko(eLh - 1) y(
= 2k k,,Ly~~~,;t,~qi)~~ 2. Thus if we take y < 1/(2kk,L) l’*(t
G
h+O+
+ h) - V*(t)
h
Q -
)Ixt(to,
#II2
+ 2&&y
Ijx,(to,
4)112
Ilxttt,,4)112 G’+,b - 1) = -Y1 IIx,kJ,9)112Y Yl > 0.
=
Then system (7) is uniformly
asymptotically
stable.
569
A.
570
F. IzB AND
J. G.
DOS REIS
Another simple proof that implies in particular that the solutions stable is the following. From conditions (b) and (d) of Theorem 2, fi;;; v*(t
+ h) -
v*(t)
d -y,v*(t),
h
Yl =
v*tt + “h’- v*@)<
Lfi-
of (7) are exponentially
l/K,
_yl
v*(t) or log V*(t
E
+ h) - log V*(t)
d
h
h-O+
log v*(t)
- log v*(t,)
< -yl(t
-Y1
- to)
or (j.X,(t,, 4)II’ < V*(t) < V*(t,) K e-rl(t-to) and the theorem THEOREM 4.
is proved.
Assume
that
If@, 4) - fk Then if the solution uniform asymptotically
4)I + ID@, 4) - o(t, @)I G s(t) (1411,j%)
4’ = 0 of (6) is uniform stable.
Proof: As in Theorem I/X,+,,k @,,
< K e-yt(t-to) 11411”
asymptotically
4 we have from inequality
4)) - &+hk “&
dt < ;I’C
stable the solution
(9)
$))[I < (eLh -
1) /I& -xt)II
< (eLh -
I)&)
(I”&> +)I)
and
and therefore IIXt+h(t, .x&3 $))I[ d I(eLh -
1) &) Ilx#03 4)II + ~oll.~t(~o>@)II.
Thus we have 1v(t + h, x, + h(t, .x,@,, 4,)) -
v(t + h>?‘t+ h(h “$,,
d (k(eLh + 2k,I(x,(t,,
&,)I
1) g(t) )I”&, $)I1 4)/I) (eLh -
= II.xJt,, 4))1 ’ (k (eLh -
1) g(t) I/.#,, 1) (eLh -
4)II
1) g2(t) + 2ko(eLh -
1) g(t)).
(11) .X = 0 of (7) is
Stability
of perturbed
neutral
functional
differential
571
equations
Hence Izl
K’[IV(t
+ h, x,, Jf, X&, 44)) -
Jqt + h, Y,+Jk “&I,
h-O+
4N)II
Therefore fi
v*(t
+ h) -
v*(t)
h
h+O+
d -
IIXrUo. 4)1/2 + (I”&, =
owdt)
-
<
(k,g(t) -
4)Ij’ 1)
II.+,,
k’s(t) 4111’
1) V*(t).
Thus we have )I.+)? 4)(I G v*(t) < V*(to)exp[kl~~ < k exp - (t implies that the solution tically stable.
which
Acknol~,/cdgements-We
thank
gWds]ew(--(t
to))
to) ~~~~~.
x E 0 of (7) is exponentially
A. A. Freiria
-
for interesting
comments
stable and therefore
uniform
asympto-
on the paper
REFERENCES 1. CKUZ M. A. & HALE J. K., Stability of functional differential equations of neutral type. J. d(f/. Eqns 7 (2) 334-355 (1970). 2. HALE J. K. and CRUZ M. A., Asymptotic behavior of neutral functional differ-ential equations. Archs rational merh. Analysis34,331-353 (1969), 3. HALE J. K., Ordinary Differential Equations. Wiley Interscience, New York (1969).