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Stability of Abstract Evolution Equations
STABILITY O F ABSTRACT EVOLUTION EQUATIONS W. T. F. Blakeley* and A. J. Pritchard** *Department of Computing B Mathematical Sciences, The Polytechnic, Wolverhampton **School of Engineering Science, University of Warwick, Coventry
1 INTRODUCTION Identification isconcerned with obtaining a mathematical model of a dynanlical system. However in constructing a model of a complex system some effects or processes may be approximated or ignored, and so the model is not usually capable of describing all of the dynamical characteristics of the actual system. In this paper we consider two aspects of this nroblem. Firstly we examine the properties of existence, uniqueness and stability for nonlinear mathematical models, and secondly we determine a class of perturbations of the models for which these properties are conserved. We assume that the mathematical model is given by an abstract evolution equation of the form
where zEZ, tE[O,T]; Z is a Banach space and M is a non-linear operator on [O,T] x Z . 'Ye analyse (1.1) by assuming that he onerator N can he exnressed as the sum of a linear operator and a non-linear onerator i.e.
such that -A is the generator of a strongly continuous semi-group Tt on X . Hence the mild solution of
and if zoED(A), (1.4) is the strict solution of (1.3).
If also
then the origin is asymptotically stable in the large in the sense of Liauunov. In 32 we consider a linearized model of (1.1) which will be of the form
Although there are existence, uniqueness and stability theorems for (1.6) the conditions for these theorems may be difficult to verify, so me assume
where A is the same as in (1.2), (1.3) and BL(t) is a linearized version of the B(z,t) in (1.2). Moreover we choose A so that the properties (1.4), (1.5) are easy to obtain. We then use perturbation methods to find a set of sufficient conditions on B (t) so that the mild solution of (1.6) is well-defined and the origin is asympkotically stable in the large. Section 03 is concerned with perturbation theory of non-linear autonomous m-accretive operators. These results are important for two different reasons. By starting from the linearized model we can determine sufficient conditions
W. T . F . Blakeley
s o t h a t N is m - a c c r e t i v e a n d h e n c e t h e s o l u t i o n o f ( 1 . 1 ) is w e l l - d e f i n e d . Then b y u s i n g t h e s a m e t h e o r e m we a r e a b l e t o i m p o s e c o n d i t i o n s o n t h e u n i d e n t i f i e d p a r t o f t h e s y s t e m r e g a r d e d a s p e r t u r b a t i o n s of N so t h a t t h e a c t u a l s y s t e m h a s s i m i l a r p r o n e r t i e s t o t h a t o f t h e model. The i d e a s i n e a c h s e c t i o n are i l l u s t r a t e d w i t h e x a m p l e s . 2
PERTURBATIONS OF LINE4R SE'fI-GROUPS
I n t h i s s e c t i o n we e x a m i n e t h e s y s t e m
where A and B ( . ) a r e l i n e a r o p e r a t o r s . Our aim is t o a s s o c i a t e w i t h t h e o p e r a t o r A+B(.) aL n evolution operator !J(.,.) on A ( T ) = C ( t , s ) , 0 s < t g T? d e f i n e d by t (2.2) u ( ~ , s ) z= ~ ~ - -~ ~ ~ ~ ~z B ( o ) u ( o , s ) z d o 0
where -A is t h e i n f i n i t e s i m a l g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s se!ni-group T t on Z w i t h
W e s u p p o s e t h a t B L ( t ) b e l o n y s t o t h e c l a s s o f unbounded n e r t u r h a t i o n s w i t h t h e properties (2.4)
D(BL(t)) = Z
f o r each t ,
(2.5)
I I T ~ - ~ B ~ ( S4) Z 2 U I I Z I I, (t-s)"
t > s
ZETI(R~(S)):
o < a
< 1.
C o n d i t i o n s o n ?4,N,w,a w i l l b e o b t a i n e d s o t h a t t h e e v o l u t i o n o p e r a t o r I T ( t , s ) has the property (2.6)
llu(t,s)l
< ~ ~ e - ~ ' ( ~ , I- ~> )o
W e n o t e f i r s t t h a t ( 2 . 5 ) implies t h a t f o r any t > s t h e o n e r a t o r T
extension T
B (s) has
t-s L
R ( s ) t o a l l of Z with t-s L
and
E q u a t i o n ( 2 . 2 ) is now t a k e n a s t (2.9) I J ( ~ , S ) Z= T ~ - ~- z / T ~ - ~ B ~ ( P ) u ( o , s ) z ~ oo,
s
s < t
4
E4 t h e r e e x i s t s a
g
T.
S
Theorem 2 . 1
W i t h I I T ~ - ~ B ~ ( s ) IgIi v e n b y ( 2 . 7 ) a n d llTtll
unique s o l u t i o n of ( 2 . 9 ) w i t h t h e following p r o p e r t i e s : U ( t , . ) i s s t r o n g l y c o n t i n u o u s on [ O , T ] and U ( . , s ) is s t r o n g l y c o n t i n u o u s on [ s , T l , U ( t , t ) = I , 0 < s < r S t < T. ( i i ) U(t,r)U(r,s) = U(t,s), (i)
P r c o f : - We c o n s t r u c t I J ( t , s ) b y s u c c e s s i v e a p p r o x i m a t i o n s s o t h a t Q1
U(t,s)z =
1 Un(t,s)z n=O
with
Stability of Abstract Evolution Equations
Taking norms we obtain
*
Writing IlU (t,s)zA = gn(t,s) n
(n 2 0)
we have
then following Micklin [I], it can be shown that g (t,s) is hounded hv hn(t,s) n where M[ ~ ~ ( l - a ) ] ~ do hn(t,~) = s (t-o)a-(n-l)(l-a) r(n-na)
i
1-a lh (t,s) is a power series in (t-s) , uniformly converTent for all t < w O n m to hence lgn(t,s) converges uniformly on [O,T]. Hence 1 U n (t,s) is convergent n= 0 0 on A(T) in the uniform topology. It is also a solution of (2.9) and U(t,t)=I. To show that the solution is unique we suppose that there is another solution Ul(t,s) and let
Then which is of the same form as (2.12). ent number of times we obtain t II~(t,s)zll 9 C IIR(p,s)zlldp
1
If we iterate this inequality a suffici-
(c
> 0)
S
so R(t,s)z
=
0 for all zEZ by Gronwall's inequality.
The semi-group property can be proved in a similar way. t U(t,r)U(r,s)z = Tt-rTr-s~ + T t-r jTt-pRL(o)U(o,s)z dp S t + I T ~ - ~ , B ~ ( P ' ) U ( P ' , ~ ) T J (do' ~ , S.) ~ r Using the semi-group property for Tt along with (2.8) and setting lJ(t,r)U(r,s) - U(t,s) = R(t,r,s) we have t R(t,r,s)z = I T ~ - ~ , B ~ ( o ~ ) R ( P ~ , ~do' ,s)z r and so by Gronwall's inequality we have R(t,r,s)z U(t,r)U(r,s) - U(t,s) = 0.
=
0,zEX and s
r
t. Hence
For the proof of continuity we write
and note that Tt-s is strongly continuous in s and t. sider only $(t,s)z.
Therefore we need conTake h >-0,tlE[s,T), tzE
W. T. F . Blakeley
and
using (2.8). Then l l $ ( t l + h , s ) z - @ ( t l , s ) z l + 0 a s h 0 by t h e s t r o n c c o n t i n t u i t y o f Tt a n d t h e f a c t t h a t 0 S a < 1 and ~ U ( o , s ) z l l is b o u n d e d . +
llb(t2,s)z
-
@(t2-h,s)zl1
+
0 as h
+
0 f o r similar reasons.
Now t a k e h > 0 , s l E [ O , T ) a n d s z ~ ( O , T l t h e n
and
Since U(o,sl)
=
U ( p , s ~ + h ) U ( s l + h , s l )a s h > 0 we h a v e
IIU(p,sl+h)z
-
U(p,sl)zll
s ~ ~ U ( o , s l + h ) lII l I - U ( s l + h , s i ) z l l
h e n c e I l @ ( t , s l + h ) z - @ ( t , s l ) z l l -+ 0 a s h + 0 b y t h e s t r o n g c o n t i n u i t y o f r J ( . , s ) o n [ s , T ] , t h e b o u n d e d n e s s o f IIU(t , s ) l l a n d t h e p r o p e r t y o f IITt-oBL(0)II . Il@(t,s2-h)z - @(t,s2)zll+ 0 a s h + 0 f o r s i m i l a r reasons. W e now h a v e t o c o n s i d e r t h e p r o b l e m o f o b t a i n i n g a n e s t i m a t e o f t h e f o r m ( 3 . 6 ) f o r I I U ( t , s ) l when RTtR < Mexp(-wt), o > 0 .
F o r s i m p l i c i t y i n t h e f o l l o w i n g we a s s u m e t h a t B i s i n d e n e n d e n t o f t s o t h a t t h e o p e r a t o r U ( t , s ) i s i n f a c t a s e m i - g r o u p U ( t , s ) = U t-s w h e r e tU z = T z - I T t q o B U o zdp t t o An e s t i m a t e o f lLJtll c o u l d b e f o u n d by summing t h e s e r i e s
b u t i n a p p l i c a t i o n s t h i s could-be d i f f i c u l t . An a l t e r n a t i v e a ~ p r o a c hi s t o i s d e t e r m i n e d by i t e r a t e t h e i n e q u a l i t y ( 2 . 1 2 ) N t i m e s o n l y where (2.14)
n(1-a) n(1-a)
-
a 2 0 a < o
for for
n 2 n < N.
Stability of Abstract Evolution Equations
T h e n we h a v e
w i t h g o ( t ) = 9e-wt, a n d we m y u s e G r o n w a l l ' s lemma t o o b t a i n
-N
-
0
-
where b ( t ) = I g n ( t ) and C = T 0
N(1-a)-a
[ Nr(1-a)]
T( ( g + 1 ) ( 1 - a )
S i n c e C > O t h i s r e s u l t h a s no u s e f o r o u r p u r p o s e s .
W e c a n f i n d a b e t t e r e s t i m a t e i f we u s e t h e s e m i - g r o u p p r o p e r t y o f T i f y t h e e s t i m a t e o f 1g i v e n bv ( 2 . 7 ) and u s e d above. s = y t a n d u s i n g ( ~ - 3 ) we ~ find that
t o modFrom ( 2 . 8 1 w i t h
where
T h e o r e m 2 . 1 i s s t i l l v a l i d as t h e a d d i t i o n a l term i n t h e k e r n e l K ( t , o ) i s
e- B ( t - p )
w h i c h is b o u n d e d b y u n i t y .
T h e c o r r e s p o n d i n g gn ( t ) a r e q i v e n by
a n d ( 2 . 1 5 ) is r e p l a c e d by
W e now u s e G r o n w a l l ' s lemma t o e s t i m a t e e e t g ( t )
-
and have
g ( t ) e R t G ?leCt + b ( t ) e C t
N where b ( t ) = Ign(t)eBt 1
and
C
=
-
F(l-a)-a T
[ Nlr(l-a)] N+l
I'[ ( N + l ) ( 1 - a ) ]
T h u s i n p l a c e o f ( 2 . 1 6 ) we h a v e
-
C l e a r l y t h i s e s t i m a t e w i l l depend upon T u n l e s s a = 0 ,
w i l l b e o f n o u s e a s s u c h when we c o n s i d e r T NUtN
<
g ( t ) S File - w ~ t
t h e n u s i n g t h e semi-group 1 unT1
If y = log M1 then
s
~
m.
Houlever s i n c e we h a v e
O < t S T
p r o p e r t y o f lJt we c a n show :
e
-
~
l
~
~
. . . 4 , $, m+N i . . .
and so
W. T. F. Blakeley
S i n c e w l = 6-C, w l d e p e n d s o n T s o t h a t i n d e t e r m i n i n g c o n d i t i o n s o n t h e o p e r a t o r B , t h r o u g h t h e p a r a m e t e r N 1 , f o r t h e o r i g i n o f t h e ~ e r t u r b e ds y s t e m t o b e a s y k p t o t i c a l l y s t a b l e we m u s t o p t i r n i s e o n T . I f N i s s m a l l e n o u g h t h e n we c a n f i n d T such t h a t
s i n c e C a n d b ( T ) i n c r e a s e c o n t i n u o u s l y w i t h N 1 a n d T a n d a r e z e r o when N 1 = 0 . I n c o n t r a s t t o t h e a b o v e a n a l y s i s we w i l l now f i n d a stability c r i t e r i o n b y estimating d i r e c t l y t h e s o l u t i o n of t h e i n t e g r a l equation t
< ~ e - + ~N l ' I- ~h ( ~ >d o
(2.21)
h(t)
(2.22) where
h ( t ) = eBtl1u 11 t
0
and
( t - ~ ) ~ U ' = w-6
Now s u p p o s e we c a n f i n d a f u n c t i o n H ( t ) w h i c h s a t i s f i e s
then
w h e r e h o ( t ) is t h e s o l u t i o n o f
>
Moreover h o ( t ) is g r e a t e r t h a n any h ( t ) which s a t i s f i e s ( 2 . 2 1 ) h e n c e H ( t ) h ( t ) f o r a l l t > 0.
W e l o o k f o r E ( t ) i n t h e f o r m o f a n e x o n e n t i a l b o u n d iM1eQt a n d n o t e t h a t R m u s t h e p o s i t i v e s i n c e i f P ( t ) = b l l e - G t ( 0 > 0 ) , when r e s u b s t i t u t e i n t o ( 2 . 2 3 ) we r e q u i r e ~ 4 l e - ' ~> Ile - w ' t
e-or, + N F4 1 t -----do
1
0
>
f o r which t h e r e . s no s o l u t i o n i f a W i t h H ( t ) = Mlent (R > 0 ) we s o l v e
(t-0)
a
0 since the integral diverges a t t
1
t
Since
10
e R ~
0-d
do =
L
Rt
l
% e-'y
1-CY o
dy
ut > 0
< -e
Rt
1 m
je-'y 1-a o
1-a
dy
Rt
=
we c a n see t h a t ( 2 . 2 4 ) i s s a t i s f i e d i f R i s c h o s e n s o t h a t
If t h e r e e x i s t s 6 > 0 such t h a t
t h e n ( 2 . 2 5 ) is s a t i s f i e d f o r a l l t 2 T s u c h t h a t
e ol-a'r ( l - a ) ,
-
m.
Stability of Abstract Evolution Equations
a n d i f M16 Z such t h a t
hfl
then ( 2 . 2 5 ) is v a l i d f o r a l l t
> 0.
Hence we c a n f i n d Vl,P(>c))
< n.l,eRt
h(t)
a n e s t i m a t e f o r t h i s v a l u e o f 9 b e i n g ~ i v e nby (2.28)
1 > 52
1-a T(1-a).
U s i n g ( 2 . 2 2 ) a n d ( 2 . 2 8 ) we now h a v e a n e s t i m a t e o f
w h e r e R1-a
kJtl
i n t h e form
F o r s t a h i l i t y we r e q u i r e R < P s o o n u s i n g ( 2 . 1 8 ) we
> NIT(l-a).
n e e d R < -!X l+Y
(2.30) i.e. Rut
N1
NlT(1-a)
<
= ~ r ~ ( l + y ) s" ol ( 2 . 3 0 ) becomes
The p a r a m e t e r y i s s t i l l a t o u r d i s p o s a l . when y = - s o t h a t t h e b e s t e s t i m a t e i s
The maximum v a l u e o f
occurs 1-Y
CL
(2.31)
MNr(1-a)
< w
1-a 1-a a a (1-a)
.
We w i l l d e m o n s t r a t e t h e a p p l i c a b i l i t y o f t h e s e r e s u l t s b y c o n s i d e r i n g two e x a m p l e s . The f i r s t i s t h e o n e - d i m e n s i o n a l d i f f u s i o n p r o c e s s , w h e r e t h e u n n e r t u r b e d s y s t e m is z(x,O) = z ( x ) . 0
z ( 0 , t ) = z ( 1 , t ) = 0 a n d Z = L2 [ 0 , 1 1 . A b s t r a c t i n g t h i s e q u a t i o n i t is e a s y t o show t h a t t h e s o l u t i o n is g i v e n i n terms o f a s e m i - g r o u p Tt w h e r e ca
TtzO
=
1 2 e - n 2 m 2 ts i n nmx n=l
1 I s i n n.rryzo(y)dy 0
< e - ~ 2. t Hence t h e c l a s s o f p e r t u r b a t i o n o p e r a t o r s we c a n a l l o w and II T II t L(Z) m u s t s a t i s f y ( 2 . 5 ) a n d i f t h e p e r t u r b e d s e m i - g r o u n is t o h e n e g a t i v e l y e x p o n e n t i a l l y bounded w e r e q u i r e ( 2 . 3 1 ) . For examnle i f Bz = a z X t h e n s i n c e i t i s e a s y t o show t h a t
we r e q u i r e la1 <
me -- 2 . 0 6 f o r a s y m p t o t i c s t a b i l i t y o f t h e n e r t u r b e d s y s t e m . 2
F o r o u r s e c o n d e x a m p l e we c o n s i d e r t h e h e a t c o n d u c t i o n p r o b l e m i n R"
w i t h Z = L 2 ( R ) w h e r e R is a n o p e n b o u n d e d s e t i n Rn w i t h b o u n d a r y
az
r.
On
r
we and t h e
r l i s i n s u l a t e d s o t h a t - -- = 0 on r 1 i s k e p t a t a m b i e n t t e m p e r a t u r e w h i c h we t a k e t o b e z e r o . U n d e r c e r t a i n s m o o t h n e s s c o n d i t i o n s o n R a n d T i t is p o s s i b l e t o show t h a t t h e s o l u t i o n is g i v e n i n terms o f a s e m i - g r o u p Tt s u c h t h a t assume t h a t p a r t o f t h e boundary
rest o f t h e b o u n d a r y
IITtII
r/rl
< ~le-~'
f o r some
w > O.
W. T . F. B l a k e l e y
Now l e t u s a s s u m e t h a t t h e b o u n d a r y n a r t T I i s n o t p e r f e c t l y i n s u l a t e d a n d i n fact
t h e n we r e g a r d t h i s new p r o b l e m a s a p e r t u r b a t i o n o f t h e o r i g i n a l p r o b l e m . U s i n v t h e m e t h o d s o f C u r t a i n a n d P r i t c h a r d [ 21 we c a n r e f o r m u l a t e t h e ~ r o b l e m s o t h a t t h e a n a l y s i s o f t h i s s e c t i o n c a n b e a p p l i e d . Vow B w i l l map ~ ' ( r l ) f u n c t i o n s i n t o a l a r g e r s p a c e t h a n L ' ( P ) , however t h e semi-groun Tt is s m o o t h i n g i n t h a t we a r e a b l e t o s h o w t h a t
T h e r e f o r e we w i l l h a v e e x i s t e n c e a n d u n i q u e n e s s o f t h e m i l d s o l u t i o n o f t h e p e r t u r b e d problem and s t a b i l i t y w i l l f o l l o w i f
3
PERTURBATION THEOREMS FOR NON-LINEAR OPERATORS
I n t h i s s e c t i o n w e assume t h a t A and B a r e n o n - l i n e a r accretive. Definition
A is m-accretive
o p e r a t o r s w i t h A rn-
i f A is a c c r e t i v e i . e . f o r e a c h y,zED(A),X > 0 .
a n d i f t h e r a n g e o f I + X A is t h e w h o l e o f 7 f o r some X > 0 . K a t o [ 3 ] h a s shown t h a t i n t h i s c a s e
h a s a u n i q u e s o l u t i o n f o r e a c h z ED(A) i f Z i s u n i f o r m l y c o n v e x a n d h a s a l s o d e r i v e d s e v e r a l c o n d i t i o n s o n t h 8 o p e r a t o r R s u c h t h a t A+B i s m - a c c r e t i v e . Unf o r t u n a t e l y a l l o f t h e s e c o n d i t i o n s r e q u i r e t h a t B is a c c r e t i v e and t h i s is r e s t r i c t i v e i n a n v l i c a t i o n s s i n c e i t i m p l i e s t h a t t h e p e r t u r b a t i o n is s t a b i l i z i n g . Our a i m is t o r e l a x t h i s c o n d i t i o n . T h e r e s u l t o f K a t o t h a t we u s e i s t h e following:Let A be a single-valued m-accretive operator, B be a singleTheorem 3.1. v a l u e d a c c r e t i v e o p e r a t o r on a r e f l e x i v e Banach s p a c e Z w i t h n ( B ) 3 D(A). If A 0 = BO = O a n d i f f o r e a c h zo€D(A) t h e r e a r e a n e i g h b o u r h o o d U o f zo a n d c o n s t a n t s a ' , b' such t h a t (3.3 )
II Bx-Byll
< a'll x-yll
t h e n A+B i s m - a c c r e t i v e
+ blllAx-Ayll
f o r x , ycD(A)nU
if b' < 1.
W e d e r i v e t h e f o l l o w i n g s e q u e n c e o f lemmas.
Lemma 3 . 1 .
I? b e as i n T h e o r e m 3 . 1 . b u t w i t h ( 3 . 3 ) r e p l a c e d b y
Let A,
IIBx-Byll
G a'llx-yll
t h e n A+B is m - a c c r e t i v e I n t h e l i m i t as n (3.5 )
+
IIBx-Byll
if b'
+ b'Il(2-
2
( 2 n - 1 ) )(Ax-Ay)
+ (1-
1
2(2n-1)
) ( Ex-By ) ll
< 1
t h i s c o n d i t i o n becomes G a'llx-yll
+ b'll2(Ax-Ay)
+ (Bx-Fy)ll.
Proof S p l i t t h e o p e r a t o r A+B b y w r i t i n g i t a s P + P B + ( l - S ) B , w h e r e 0 < ? < 1. A p p l y i n g K a t o ' s r e s u l t , T h e o r e m 3 . 1 . , r e g a r d i n g PR as a n e r t u r b a t i o n o f A t h e n (1-B)B a s a p e r t u r b a t i o n o f A+RR we f i n d t h a t A+B i s m - a c c r e t i v e i f t h e r e e x i s t c o n s t a n t s a , a l l b , b l w i t h e a c h b < 1 s u c h t h a t f o r x,y€D(A)nU
II R(Bx-Rv)ll
G all x-vll
+
bll Ax-Ayll
Stability of Abstract Evolution Equations If (3.7) holds true then we have
1 IVe require h < 1 which gives (3.6) if we set a = l-?-b,c a n d h = l - Fb c-hl i.e. Pbl < 1-3-bl? which gives - . - 1 / ( 1 + _ ? I , ) Since b l < 1 11.e can choose 6 = 13 ' Thus (3.6) and (3.7) can be replaced by the single condition II Rx-Byll 1I-$3a X-yll + bill $(Ax-Av) + ~(Bx-Ry)ll ~,yFQ(8)'7~1.
We note that (3.4) with n
=
1 is identical to (3.8).
To prove (3.4) with n = 2 we let A+yR+(l-y)B where 0 < Y < 1 and derive two conditions similar to (3.6) and (3.7) using (3.8) instead of Kato's condition (3.3) These conditions are
(1-v)ll Rx-Byll < +l3a
x-yll + hill {(AX-by)
1+2v
+ T(R~-Ey)ll
u
If (3.10) is valid then
'fa 1 vh 1 , b = . e which is identical to (3.9) if a = b b 1--f- -'(l+y) 2 Y 2-b1 b require b < 1 i.e. 1-y --'(l+~) > y which glves y < ;i~-i;;. Since hl < 1 we can 2 With this value of v (3.10) becomes (3.4) with n = 2. choose y = :. The general result can be obtained by an induction argument. The accretiveness condition on R can be relaxed to obtain the following result. Lemma 3.2. Let A be single-valued and m-accretive and A+B single-valued and If An = BO = 0 and there exists non-negative conaccretive with D(B)>D(A). stants a,b with b < 1 and an a , 0 < n < 1 such that
then A+aB is m-accretive Proof:- Set A+aB = (1-n)A + a(A+B) for 0 S a < 1 then by Lemma 1 , accretive if there exists an a > 0 , b < 1 such that
A+%R
is m-
which is (3.11). We now consider these results when Z is a Hilhert snace H. condition for accretiveness (3.1) can be replaced by Re > 0
(3.12)
In this case the
for all x,y€D(A)
where<.,. > denotes the inner product on H. Lemma 3.3 Let A be single-valued and m-accretive, A+B be single valued and accretive on a Hilbert space H with D(B)>D(A)>O. If A 0 = BO = 0 and there exist a, b > 1 such that
II Bx-Byll
(3.13) then A +
2 l+b2 B
S
a211x-y11
is m-accretive.
+ b211Ax-Ay11
\d.
bf(2-a)/a = 1
and
T. F . Blakeley
then = ll A X - ~ ~ 1 1
II Ax-Ay+Rx-Byll
+ < Ax-Ay ,Rx-Rp > + < Bx-Ry ,Ax-4y > + b:lI Bx-Byll
+ ( 1 - h : )ll Bx-Byll 9
11 A X - ~ y l l + < Ax-Ay ,Bx-Ry > +
'Fx-Ry ll lu-Ay11
= a:ll x-yll =
1 + Ilb(Ax-Ay)
~ : I I X - ~ I I+
,Ax-4y > + b:ll3x-r3yll
+ h , (Bx-By)ll
~ : I I ~ - " ( A x - A+ ~ ()R X - F ~ ) I I
a
Now ( 3 . 1 4 ) i s v a l i d i f b 2 = l / b : , a 2 = a : / ( l - b : ) , u s i n g ( 3 . 1 3 ) , a n d s o we h a v e S i n c e b > 1 t h e n b l < 1 and s o a l l t h e c o n d i t i o n s o f ( 3 . 1 1 ) o f Lemma 3 . 2 . R is m - a c c r e t i v e . Lemma 3 . 2 . a r e s a t i s f i e d a n d we h a v e A+aR = A + l+b2
-
C o r o l l a r y W i t h t h e s a m e c o n d i t i o n s a s i n Lemma 3 . 3 . b u t w i t h ( I - € ) A + R a c c r e t i v e i n s t e a d o f A+B a c c r e t i v e a n d ( 3 . 1 3 ) r e p l a c e d h y
-
t h e n A+B i s m - a c c r e t i v e . Set B
Proof:-
=(%)B
2 a n d ----i= 1 - r . l+b
If
( l - ~ ) i l + gi s a c c r e t i v e t h e n s o i s
( 1 - E ) ( A + B ) a n d a l s o A+B. I l ~ x - % y l l< ~ a211x-y112 + ( 1 - ~ ~ ) 1 l A x - A y 1 1 ~
which is ( 3 . 1 3 ) .
A+(1-e)B i . e .
H e n c e b y Lemma 3 . 3 .
A+E
then
is r n - a c c r e t i v e .
Lemma 3 . 4 . L e t A b e s i n g l e - v a l u e d and m - a c c r e t i v e , A+R h e s i n g l e - v a l u e d and I f A 0 = BO = 0 a n ? a c c r e t i v e o n a r e a l H i l b e r t s p a c e H, w i t h n ( B ) > D ( A ) > O . t h e r e e x i s t s non-nogative c o n s t a n t s a ' , b ' > 1 such t h a t
1 ( 1 + 6 ) b , - R B is m - a c c r e t i v e . 1 b'-1 Proof:C h o o s e a s . t . 0 9 r < --, a n d l e t b = t h e n b < 1 and b 2-T -q- b' s i n c e b ' > 1. S e t a = a ' ( l + b ) t h e n s q u a r i n g ( 3 . 1 5 ) g i v e s
+
t h e n f o r a n y 6 > 0, A
o g
[ ( ) b
2
- 11 II AX-AyII
5
[a'll n-yll + b t ll A X - ~ y l l ]+ a
2abll x-711
+
< 1
- 2abll x-yll
21 h.-2 2 - 3 - 1 1 I\ An-Ayl 1 a'il x-rH + b 9 l l.4x-PgII 1
-
I
l l ~ ( ~ x - ~ y )+ l (lb 2 - 1 )ll Rx-!3yll
r
'3
which i m p l i e s
11 A X - ~ y l l + 2
< Ax-Ay,
+ b211*(~x-~y) a
a211x-y11
>
+ 11 R X - B ~ ~ I I + Rx-Byll
II Ax-Ay + Bx-Byll
i.e. s o b y Lemma 3 . 2 . Since b'
Bx-By
=
'
+ 2abll x-3-11. l l2--1 -(4~-rl~.)
S all x-yll
3
2-9 + bll -(Ax-4y)
+ Rx-PyII
+ Rx-By11
A+aB i s m - a c c r e t i v e .
1 12-0 l+b,Tb+l)
then
-i
=
( l + "1 ) b l - ' irith
=
a n d s i n c e h < 1,
"
W e now h a v e o u r f i n a l r e s u l t w h i c h is e s s e n t i a l l y a c o r o l l a r y t o Lemma 3 . 4 .
> O
Stability of Abstract Evolution Equations
-
Theorem 3 . 2 . W i t h t h e same c o n d i t i o n s a s i n Lemma 3 . 4 . b u t w i t h ( 1 - E ) A + R a c c r e t i v e i n s t e a d o f A+B a c c r e t i v e a n d w i t h ( 3 . 1 5 ) r e p l a c e d h y
- -
(3.16)
<
IIBx-Byll
allx-yll
+ b'(1-E)IIAx-Ayll
t h e n A + B is m - a c c r e t i v e . Proof :-
-
1 (b,
Set B =
B = ( l - € ) R i n Lemma 3 . 4
W e i l l u s t r a t e t h i s r e s u l t by considering t h e equation (3.17)
z
t t + 2 S z t + z XXXX + u s i n h a z = 0 ,
x
0
<
1,
t
>
0,
cr > 0
w h i c h d e s c r i b e s t h e b e n d i n g o f a heam o n a n o n - l i n e a r f o u n d a t i o n ( S h a r m a a n d Dasgupta [ 4 ] ) . T h i s e q u a t i o n c a n b e w r i t t e n i n t h e a b s t r a c t form
by s e t t i n g w = [ z , v ] with inner product.
'1'
with v = z
t
and i n t r o d u c i n g t h e r e a l H i l h e r t s p a c e
1
(3.19)
>
I [z
=
~
~
+
V~I V Z z +
EP( V I~~ Z ~+ V ~ +Y 2I i) 2 z 1 z 2 1 d x
0
T h e o p e r a t o r A is f o r m a l l y d e f i n e d b y Aw = [ - v , 2 < v + r a n d D(A) = {*H t i v e because
v = z = v
: AtfiH,
=
<
XXXX
1"
=
XX
zxx = 0 a t x = 0 , 1 1 .
{ ( ~ ~ + z : ~ ) d >x 0
Then A is m-accre-
f o r a l l *o(A).
0
T h e p e r t u r b i n g t e r m is f o r m a l l y d e f i n e d by Bw = [ 0 , p s i n h a z ] T
>
hence
(yv s i n h a z +
=
E sinh az)dx 1
0,
o
I n o r d e r t o show t h a t t h e c o n d i t i o n s o f T h e o r e m 3 . 2 . a p p l y j n a n e i g h h o u r h o o d o f t h e e q u i l i b r i u m p o i n t z = 0 we o b s e r v e f i r s t t h a t i f y l > y z t h e n
cr a [ + v ( ~ T + ~ ~ ~ ~f g2( ~ + t y; +~y ?) v 2 + . ) + I s i n h a y ~- s i n h ay2 = Q ( Y I - Y Z ) 1 a a a" 4 < & ~ 1 - ~ 2 ) [ +2 5 ( ~ : + ~ ; ) + F ( Y I + Y Z ) I ,
+
...
L e t llw112 G k 2 t h e n u s i n g t h e m e t h o d s o f F r e u n d a n d P l a u t 151 m e c a n e s t a b l i s h the inequality k2
a
~ ' ( z ~ ~ + < z r2 ~ )c a2 xr 2
where 8
0
and
X + cos2L s i n h 22 c 2 = l E X 3 sinh - sin with
2r2 =
E
O Smax x G lIzl
5.
H e n c e f r o m ( 3 . 2 1 ) we h a v e sinh ayl
- s i n h nyp
C o n s i d e r now t h e o p e r a t o r ( 1 - E ) A + B
<
a(yl-y2)cosh IF
(yl 2 y 2 )
W. T . F . Blakeley
U s i n g ( 3 . 2 3 ) and t h e i n e q u a l i t y
we see t h a t i f ( 1 - c ) A + R is t o b e a c c r e t i v e t h e n we m u s t h e a b l e t o f i n d such t h a t
E
> 0
We a l s o h a v e
hence
Thus p r o v i d e d ( 3 . 2 5 ) c a n b e s a t i s f i e d a l l t h e c o n d i t i o n s o f Theorem 3 . 2 . a r e f u l f i l l e d s o t h a t A+B i s m - a c c r e t i v e i n a n e i g h b o u r h o o d o f z = 0 a n d h e n c e -(A+B) g e n e r a t e s a n o n - l i n e a r c o n t r a c t i o n s e m i - g r o u p . W e n o t e t h a t assuming 5 , ~l a n d a r e given then (3.25) e f f e c t i v e l y determines t h e size of t h e neinhb o u r h o o d f o r w h i c h t h e s y s t e m is s t a h l e t h r o u g h t h e v a l u e o f k (= c R ) . F u r t h e r m o r e i f k is s u c h t h a t t h e i n e q u a l i t y h o l d s i n ( 3 . 2 5 ) t h e n t h e r e is a c e r t a i n a l l o w a b l e c l a s s o f p e r t u r b a t i o n s of t h e n o n - l i n e a r s y s t e m which w i l l e n s u r e e x i s t e n c e , u n i q u e n e s s and s t a h i l i t y o f t h e p e r t u r b e d s v s t e m . T h i s class o f p e r t u r b a t i o n s w i l l h e d e t e r m i n e d hy a n w l i c a t l o n o f Theorew 3 . 2 . a g a i n w i t h A r e p l a c e d h y t h e o p e r a t o r A+F o f ( 3 . 1 8 ) . A f u r t h e r example i l l u s t r a t i n g Lenna 3 . 4 .
is g i v e n i n [ 61
References [ I ] I l i c k l i n S . G . " I n t e g r a l E q u a t i o n s " Pergamon P r e s s 1 9 6 4 . 121 C u r t a i n R . a n d P r i t c h a r d A.J. "An a b s t r a c t t h e o r y f o r u n b o u n d e d c o n t r o l a c t i o n f c r d i s t r i b u t e d p a r a m e t e r s y s t e m s " I i n i v e r s i t y o f lvarwick r o n t r o l C e n t r e R e p o r t No. 3 9 . [ 3 ] K a t o T. " A c c r e t i v e o p e r a t o r s a n d n o n - l i n e a r e v o l u t i o n e q u a t i o n s i n B a n a c h s p a c e s " P r o c . Symp. P u r e ? l a t h . A m e r . \ l a t h . S o c . 2 p t 1 1 3 8 - 1 6 1 ( 1 9 7 0 ) . [ 4 ] Sharma S . a n d D a s g u p t a S."The b e n d i n g p r o b l e m o f a x i a l l y - c o n s t r a i n e d beams o n n o n - l i n e a r e l a s t i c f o u n d a t i o n s " I n t . J . S o l i d s S t r u c t u r e s 11 8 5 3 - 8 5 9 (1975). I 5 1 F r e u n d L . B . a n d P l a u t R.H."An e n e r g y - d i s p l a c e m e n t i n e q u a l i t y a p r s l i c a h l e t o p r o b l e m s i n t h e d y n a m i c s t a b i l i t y o f s t r u c t u r e s " J . A p p l . Mech. 5 3 6 - 7 ( 1 9 7 1 ) [ 6 1 P r i t c h a r d A . J . a n d B l a k e l e y W.T. F . " P e r t u r b a t i o n r e s u l t s a n d t h e i r a p p l i c a t i o n s t o problems i n s t r u c t u r a l dynamics" L e c t u r e Notes i n nfathematics V o l . 503 4 3 8 - 4 4 9 S p r i n g e r - V e r l a g ( 1 9 7 6 ) .
.
1 INTRODUCTION Identification isconcerned with obtaining a mathematical model of a dynanlical system. However in constructing a model of a complex system some effects or processes may be approximated or ignored, and so the model is not usually capable of describing all of the dynamical characteristics of the actual system. In this paper we consider two aspects of this nroblem. Firstly we examine the properties of existence, uniqueness and stability for nonlinear mathematical models, and secondly we determine a class of perturbations of the models for which these properties are conserved. We assume that the mathematical model is given by an abstract evolution equation of the form
where zEZ, tE[O,T]; Z is a Banach space and M is a non-linear operator on [O,T] x Z . 'Ye analyse (1.1) by assuming that he onerator N can he exnressed as the sum of a linear operator and a non-linear onerator i.e.
such that -A is the generator of a strongly continuous semi-group Tt on X . Hence the mild solution of
and if zoED(A), (1.4) is the strict solution of (1.3).
If also
then the origin is asymptotically stable in the large in the sense of Liauunov. In 32 we consider a linearized model of (1.1) which will be of the form
Although there are existence, uniqueness and stability theorems for (1.6) the conditions for these theorems may be difficult to verify, so me assume
where A is the same as in (1.2), (1.3) and BL(t) is a linearized version of the B(z,t) in (1.2). Moreover we choose A so that the properties (1.4), (1.5) are easy to obtain. We then use perturbation methods to find a set of sufficient conditions on B (t) so that the mild solution of (1.6) is well-defined and the origin is asympkotically stable in the large. Section 03 is concerned with perturbation theory of non-linear autonomous m-accretive operators. These results are important for two different reasons. By starting from the linearized model we can determine sufficient conditions
W. T . F . Blakeley
s o t h a t N is m - a c c r e t i v e a n d h e n c e t h e s o l u t i o n o f ( 1 . 1 ) is w e l l - d e f i n e d . Then b y u s i n g t h e s a m e t h e o r e m we a r e a b l e t o i m p o s e c o n d i t i o n s o n t h e u n i d e n t i f i e d p a r t o f t h e s y s t e m r e g a r d e d a s p e r t u r b a t i o n s of N so t h a t t h e a c t u a l s y s t e m h a s s i m i l a r p r o n e r t i e s t o t h a t o f t h e model. The i d e a s i n e a c h s e c t i o n are i l l u s t r a t e d w i t h e x a m p l e s . 2
PERTURBATIONS OF LINE4R SE'fI-GROUPS
I n t h i s s e c t i o n we e x a m i n e t h e s y s t e m
where A and B ( . ) a r e l i n e a r o p e r a t o r s . Our aim is t o a s s o c i a t e w i t h t h e o p e r a t o r A+B(.) aL n evolution operator !J(.,.) on A ( T ) = C ( t , s ) , 0 s < t g T? d e f i n e d by t (2.2) u ( ~ , s ) z= ~ ~ - -~ ~ ~ ~ ~z B ( o ) u ( o , s ) z d o 0
where -A is t h e i n f i n i t e s i m a l g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s se!ni-group T t on Z w i t h
W e s u p p o s e t h a t B L ( t ) b e l o n y s t o t h e c l a s s o f unbounded n e r t u r h a t i o n s w i t h t h e properties (2.4)
D(BL(t)) = Z
f o r each t ,
(2.5)
I I T ~ - ~ B ~ ( S4) Z 2 U I I Z I I, (t-s)"
t > s
ZETI(R~(S)):
o < a
< 1.
C o n d i t i o n s o n ?4,N,w,a w i l l b e o b t a i n e d s o t h a t t h e e v o l u t i o n o p e r a t o r I T ( t , s ) has the property (2.6)
llu(t,s)l
< ~ ~ e - ~ ' ( ~ , I- ~> )o
W e n o t e f i r s t t h a t ( 2 . 5 ) implies t h a t f o r any t > s t h e o n e r a t o r T
extension T
B (s) has
t-s L
R ( s ) t o a l l of Z with t-s L
and
E q u a t i o n ( 2 . 2 ) is now t a k e n a s t (2.9) I J ( ~ , S ) Z= T ~ - ~- z / T ~ - ~ B ~ ( P ) u ( o , s ) z ~ oo,
s
s < t
4
E4 t h e r e e x i s t s a
g
T.
S
Theorem 2 . 1
W i t h I I T ~ - ~ B ~ ( s ) IgIi v e n b y ( 2 . 7 ) a n d llTtll
unique s o l u t i o n of ( 2 . 9 ) w i t h t h e following p r o p e r t i e s : U ( t , . ) i s s t r o n g l y c o n t i n u o u s on [ O , T ] and U ( . , s ) is s t r o n g l y c o n t i n u o u s on [ s , T l , U ( t , t ) = I , 0 < s < r S t < T. ( i i ) U(t,r)U(r,s) = U(t,s), (i)
P r c o f : - We c o n s t r u c t I J ( t , s ) b y s u c c e s s i v e a p p r o x i m a t i o n s s o t h a t Q1
U(t,s)z =
1 Un(t,s)z n=O
with
Stability of Abstract Evolution Equations
Taking norms we obtain
*
Writing IlU (t,s)zA = gn(t,s) n
(n 2 0)
we have
then following Micklin [I], it can be shown that g (t,s) is hounded hv hn(t,s) n where M[ ~ ~ ( l - a ) ] ~ do hn(t,~) = s (t-o)a-(n-l)(l-a) r(n-na)
i
1-a lh (t,s) is a power series in (t-s) , uniformly converTent for all t < w O n m to hence lgn(t,s) converges uniformly on [O,T]. Hence 1 U n (t,s) is convergent n= 0 0 on A(T) in the uniform topology. It is also a solution of (2.9) and U(t,t)=I. To show that the solution is unique we suppose that there is another solution Ul(t,s) and let
Then which is of the same form as (2.12). ent number of times we obtain t II~(t,s)zll 9 C IIR(p,s)zlldp
1
If we iterate this inequality a suffici-
(c
> 0)
S
so R(t,s)z
=
0 for all zEZ by Gronwall's inequality.
The semi-group property can be proved in a similar way. t U(t,r)U(r,s)z = Tt-rTr-s~ + T t-r jTt-pRL(o)U(o,s)z dp S t + I T ~ - ~ , B ~ ( P ' ) U ( P ' , ~ ) T J (do' ~ , S.) ~ r Using the semi-group property for Tt along with (2.8) and setting lJ(t,r)U(r,s) - U(t,s) = R(t,r,s) we have t R(t,r,s)z = I T ~ - ~ , B ~ ( o ~ ) R ( P ~ , ~do' ,s)z r and so by Gronwall's inequality we have R(t,r,s)z U(t,r)U(r,s) - U(t,s) = 0.
=
0,zEX and s
r
t. Hence
For the proof of continuity we write
and note that Tt-s is strongly continuous in s and t. sider only $(t,s)z.
Therefore we need conTake h >-0,tlE[s,T), tzE
W. T. F . Blakeley
and
using (2.8). Then l l $ ( t l + h , s ) z - @ ( t l , s ) z l + 0 a s h 0 by t h e s t r o n c c o n t i n t u i t y o f Tt a n d t h e f a c t t h a t 0 S a < 1 and ~ U ( o , s ) z l l is b o u n d e d . +
llb(t2,s)z
-
@(t2-h,s)zl1
+
0 as h
+
0 f o r similar reasons.
Now t a k e h > 0 , s l E [ O , T ) a n d s z ~ ( O , T l t h e n
and
Since U(o,sl)
=
U ( p , s ~ + h ) U ( s l + h , s l )a s h > 0 we h a v e
IIU(p,sl+h)z
-
U(p,sl)zll
s ~ ~ U ( o , s l + h ) lII l I - U ( s l + h , s i ) z l l
h e n c e I l @ ( t , s l + h ) z - @ ( t , s l ) z l l -+ 0 a s h + 0 b y t h e s t r o n g c o n t i n u i t y o f r J ( . , s ) o n [ s , T ] , t h e b o u n d e d n e s s o f IIU(t , s ) l l a n d t h e p r o p e r t y o f IITt-oBL(0)II . Il@(t,s2-h)z - @(t,s2)zll+ 0 a s h + 0 f o r s i m i l a r reasons. W e now h a v e t o c o n s i d e r t h e p r o b l e m o f o b t a i n i n g a n e s t i m a t e o f t h e f o r m ( 3 . 6 ) f o r I I U ( t , s ) l when RTtR < Mexp(-wt), o > 0 .
F o r s i m p l i c i t y i n t h e f o l l o w i n g we a s s u m e t h a t B i s i n d e n e n d e n t o f t s o t h a t t h e o p e r a t o r U ( t , s ) i s i n f a c t a s e m i - g r o u p U ( t , s ) = U t-s w h e r e tU z = T z - I T t q o B U o zdp t t o An e s t i m a t e o f lLJtll c o u l d b e f o u n d by summing t h e s e r i e s
b u t i n a p p l i c a t i o n s t h i s could-be d i f f i c u l t . An a l t e r n a t i v e a ~ p r o a c hi s t o i s d e t e r m i n e d by i t e r a t e t h e i n e q u a l i t y ( 2 . 1 2 ) N t i m e s o n l y where (2.14)
n(1-a) n(1-a)
-
a 2 0 a < o
for for
n 2 n < N.
Stability of Abstract Evolution Equations
T h e n we h a v e
w i t h g o ( t ) = 9e-wt, a n d we m y u s e G r o n w a l l ' s lemma t o o b t a i n
-N
-
0
-
where b ( t ) = I g n ( t ) and C = T 0
N(1-a)-a
[ Nr(1-a)]
T( ( g + 1 ) ( 1 - a )
S i n c e C > O t h i s r e s u l t h a s no u s e f o r o u r p u r p o s e s .
W e c a n f i n d a b e t t e r e s t i m a t e i f we u s e t h e s e m i - g r o u p p r o p e r t y o f T i f y t h e e s t i m a t e o f 1g i v e n bv ( 2 . 7 ) and u s e d above. s = y t a n d u s i n g ( ~ - 3 ) we ~ find that
t o modFrom ( 2 . 8 1 w i t h
where
T h e o r e m 2 . 1 i s s t i l l v a l i d as t h e a d d i t i o n a l term i n t h e k e r n e l K ( t , o ) i s
e- B ( t - p )
w h i c h is b o u n d e d b y u n i t y .
T h e c o r r e s p o n d i n g gn ( t ) a r e q i v e n by
a n d ( 2 . 1 5 ) is r e p l a c e d by
W e now u s e G r o n w a l l ' s lemma t o e s t i m a t e e e t g ( t )
-
and have
g ( t ) e R t G ?leCt + b ( t ) e C t
N where b ( t ) = Ign(t)eBt 1
and
C
=
-
F(l-a)-a T
[ Nlr(l-a)] N+l
I'[ ( N + l ) ( 1 - a ) ]
T h u s i n p l a c e o f ( 2 . 1 6 ) we h a v e
-
C l e a r l y t h i s e s t i m a t e w i l l depend upon T u n l e s s a = 0 ,
w i l l b e o f n o u s e a s s u c h when we c o n s i d e r T NUtN
<
g ( t ) S File - w ~ t
t h e n u s i n g t h e semi-group 1 unT1
If y = log M1 then
s
~
m.
Houlever s i n c e we h a v e
O < t S T
p r o p e r t y o f lJt we c a n show :
e
-
~
l
~
~
. . . 4 , $, m+N i . . .
and so
W. T. F. Blakeley
S i n c e w l = 6-C, w l d e p e n d s o n T s o t h a t i n d e t e r m i n i n g c o n d i t i o n s o n t h e o p e r a t o r B , t h r o u g h t h e p a r a m e t e r N 1 , f o r t h e o r i g i n o f t h e ~ e r t u r b e ds y s t e m t o b e a s y k p t o t i c a l l y s t a b l e we m u s t o p t i r n i s e o n T . I f N i s s m a l l e n o u g h t h e n we c a n f i n d T such t h a t
s i n c e C a n d b ( T ) i n c r e a s e c o n t i n u o u s l y w i t h N 1 a n d T a n d a r e z e r o when N 1 = 0 . I n c o n t r a s t t o t h e a b o v e a n a l y s i s we w i l l now f i n d a stability c r i t e r i o n b y estimating d i r e c t l y t h e s o l u t i o n of t h e i n t e g r a l equation t
< ~ e - + ~N l ' I- ~h ( ~ >d o
(2.21)
h(t)
(2.22) where
h ( t ) = eBtl1u 11 t
0
and
( t - ~ ) ~ U ' = w-6
Now s u p p o s e we c a n f i n d a f u n c t i o n H ( t ) w h i c h s a t i s f i e s
then
w h e r e h o ( t ) is t h e s o l u t i o n o f
>
Moreover h o ( t ) is g r e a t e r t h a n any h ( t ) which s a t i s f i e s ( 2 . 2 1 ) h e n c e H ( t ) h ( t ) f o r a l l t > 0.
W e l o o k f o r E ( t ) i n t h e f o r m o f a n e x o n e n t i a l b o u n d iM1eQt a n d n o t e t h a t R m u s t h e p o s i t i v e s i n c e i f P ( t ) = b l l e - G t ( 0 > 0 ) , when r e s u b s t i t u t e i n t o ( 2 . 2 3 ) we r e q u i r e ~ 4 l e - ' ~> Ile - w ' t
e-or, + N F4 1 t -----do
1
0
>
f o r which t h e r e . s no s o l u t i o n i f a W i t h H ( t ) = Mlent (R > 0 ) we s o l v e
(t-0)
a
0 since the integral diverges a t t
1
t
Since
10
e R ~
0-d
do =
L
Rt
l
% e-'y
1-CY o
dy
ut > 0
< -e
Rt
1 m
je-'y 1-a o
1-a
dy
Rt
=
we c a n see t h a t ( 2 . 2 4 ) i s s a t i s f i e d i f R i s c h o s e n s o t h a t
If t h e r e e x i s t s 6 > 0 such t h a t
t h e n ( 2 . 2 5 ) is s a t i s f i e d f o r a l l t 2 T s u c h t h a t
e ol-a'r ( l - a ) ,
-
m.
Stability of Abstract Evolution Equations
a n d i f M16 Z such t h a t
hfl
then ( 2 . 2 5 ) is v a l i d f o r a l l t
> 0.
Hence we c a n f i n d Vl,P(>c))
< n.l,eRt
h(t)
a n e s t i m a t e f o r t h i s v a l u e o f 9 b e i n g ~ i v e nby (2.28)
1 > 52
1-a T(1-a).
U s i n g ( 2 . 2 2 ) a n d ( 2 . 2 8 ) we now h a v e a n e s t i m a t e o f
w h e r e R1-a
kJtl
i n t h e form
F o r s t a h i l i t y we r e q u i r e R < P s o o n u s i n g ( 2 . 1 8 ) we
> NIT(l-a).
n e e d R < -!X l+Y
(2.30) i.e. Rut
N1
NlT(1-a)
<
= ~ r ~ ( l + y ) s" ol ( 2 . 3 0 ) becomes
The p a r a m e t e r y i s s t i l l a t o u r d i s p o s a l . when y = - s o t h a t t h e b e s t e s t i m a t e i s
The maximum v a l u e o f
occurs 1-Y
CL
(2.31)
MNr(1-a)
< w
1-a 1-a a a (1-a)
.
We w i l l d e m o n s t r a t e t h e a p p l i c a b i l i t y o f t h e s e r e s u l t s b y c o n s i d e r i n g two e x a m p l e s . The f i r s t i s t h e o n e - d i m e n s i o n a l d i f f u s i o n p r o c e s s , w h e r e t h e u n n e r t u r b e d s y s t e m is z(x,O) = z ( x ) . 0
z ( 0 , t ) = z ( 1 , t ) = 0 a n d Z = L2 [ 0 , 1 1 . A b s t r a c t i n g t h i s e q u a t i o n i t is e a s y t o show t h a t t h e s o l u t i o n is g i v e n i n terms o f a s e m i - g r o u p Tt w h e r e ca
TtzO
=
1 2 e - n 2 m 2 ts i n nmx n=l
1 I s i n n.rryzo(y)dy 0
< e - ~ 2. t Hence t h e c l a s s o f p e r t u r b a t i o n o p e r a t o r s we c a n a l l o w and II T II t L(Z) m u s t s a t i s f y ( 2 . 5 ) a n d i f t h e p e r t u r b e d s e m i - g r o u n is t o h e n e g a t i v e l y e x p o n e n t i a l l y bounded w e r e q u i r e ( 2 . 3 1 ) . For examnle i f Bz = a z X t h e n s i n c e i t i s e a s y t o show t h a t
we r e q u i r e la1 <
me -- 2 . 0 6 f o r a s y m p t o t i c s t a b i l i t y o f t h e n e r t u r b e d s y s t e m . 2
F o r o u r s e c o n d e x a m p l e we c o n s i d e r t h e h e a t c o n d u c t i o n p r o b l e m i n R"
w i t h Z = L 2 ( R ) w h e r e R is a n o p e n b o u n d e d s e t i n Rn w i t h b o u n d a r y
az
r.
On
r
we and t h e
r l i s i n s u l a t e d s o t h a t - -- = 0 on r 1 i s k e p t a t a m b i e n t t e m p e r a t u r e w h i c h we t a k e t o b e z e r o . U n d e r c e r t a i n s m o o t h n e s s c o n d i t i o n s o n R a n d T i t is p o s s i b l e t o show t h a t t h e s o l u t i o n is g i v e n i n terms o f a s e m i - g r o u p Tt s u c h t h a t assume t h a t p a r t o f t h e boundary
rest o f t h e b o u n d a r y
IITtII
r/rl
< ~le-~'
f o r some
w > O.
W. T . F. B l a k e l e y
Now l e t u s a s s u m e t h a t t h e b o u n d a r y n a r t T I i s n o t p e r f e c t l y i n s u l a t e d a n d i n fact
t h e n we r e g a r d t h i s new p r o b l e m a s a p e r t u r b a t i o n o f t h e o r i g i n a l p r o b l e m . U s i n v t h e m e t h o d s o f C u r t a i n a n d P r i t c h a r d [ 21 we c a n r e f o r m u l a t e t h e ~ r o b l e m s o t h a t t h e a n a l y s i s o f t h i s s e c t i o n c a n b e a p p l i e d . Vow B w i l l map ~ ' ( r l ) f u n c t i o n s i n t o a l a r g e r s p a c e t h a n L ' ( P ) , however t h e semi-groun Tt is s m o o t h i n g i n t h a t we a r e a b l e t o s h o w t h a t
T h e r e f o r e we w i l l h a v e e x i s t e n c e a n d u n i q u e n e s s o f t h e m i l d s o l u t i o n o f t h e p e r t u r b e d problem and s t a b i l i t y w i l l f o l l o w i f
3
PERTURBATION THEOREMS FOR NON-LINEAR OPERATORS
I n t h i s s e c t i o n w e assume t h a t A and B a r e n o n - l i n e a r accretive. Definition
A is m-accretive
o p e r a t o r s w i t h A rn-
i f A is a c c r e t i v e i . e . f o r e a c h y,zED(A),X > 0 .
a n d i f t h e r a n g e o f I + X A is t h e w h o l e o f 7 f o r some X > 0 . K a t o [ 3 ] h a s shown t h a t i n t h i s c a s e
h a s a u n i q u e s o l u t i o n f o r e a c h z ED(A) i f Z i s u n i f o r m l y c o n v e x a n d h a s a l s o d e r i v e d s e v e r a l c o n d i t i o n s o n t h 8 o p e r a t o r R s u c h t h a t A+B i s m - a c c r e t i v e . Unf o r t u n a t e l y a l l o f t h e s e c o n d i t i o n s r e q u i r e t h a t B is a c c r e t i v e and t h i s is r e s t r i c t i v e i n a n v l i c a t i o n s s i n c e i t i m p l i e s t h a t t h e p e r t u r b a t i o n is s t a b i l i z i n g . Our a i m is t o r e l a x t h i s c o n d i t i o n . T h e r e s u l t o f K a t o t h a t we u s e i s t h e following:Let A be a single-valued m-accretive operator, B be a singleTheorem 3.1. v a l u e d a c c r e t i v e o p e r a t o r on a r e f l e x i v e Banach s p a c e Z w i t h n ( B ) 3 D(A). If A 0 = BO = O a n d i f f o r e a c h zo€D(A) t h e r e a r e a n e i g h b o u r h o o d U o f zo a n d c o n s t a n t s a ' , b' such t h a t (3.3 )
II Bx-Byll
< a'll x-yll
t h e n A+B i s m - a c c r e t i v e
+ blllAx-Ayll
f o r x , ycD(A)nU
if b' < 1.
W e d e r i v e t h e f o l l o w i n g s e q u e n c e o f lemmas.
Lemma 3 . 1 .
I? b e as i n T h e o r e m 3 . 1 . b u t w i t h ( 3 . 3 ) r e p l a c e d b y
Let A,
IIBx-Byll
G a'llx-yll
t h e n A+B is m - a c c r e t i v e I n t h e l i m i t as n (3.5 )
+
IIBx-Byll
if b'
+ b'Il(2-
2
( 2 n - 1 ) )(Ax-Ay)
+ (1-
1
2(2n-1)
) ( Ex-By ) ll
< 1
t h i s c o n d i t i o n becomes G a'llx-yll
+ b'll2(Ax-Ay)
+ (Bx-Fy)ll.
Proof S p l i t t h e o p e r a t o r A+B b y w r i t i n g i t a s P + P B + ( l - S ) B , w h e r e 0 < ? < 1. A p p l y i n g K a t o ' s r e s u l t , T h e o r e m 3 . 1 . , r e g a r d i n g PR as a n e r t u r b a t i o n o f A t h e n (1-B)B a s a p e r t u r b a t i o n o f A+RR we f i n d t h a t A+B i s m - a c c r e t i v e i f t h e r e e x i s t c o n s t a n t s a , a l l b , b l w i t h e a c h b < 1 s u c h t h a t f o r x,y€D(A)nU
II R(Bx-Rv)ll
G all x-vll
+
bll Ax-Ayll
Stability of Abstract Evolution Equations If (3.7) holds true then we have
1 IVe require h < 1 which gives (3.6) if we set a = l-?-b,c a n d h = l - Fb c-hl i.e. Pbl < 1-3-bl? which gives - . - 1 / ( 1 + _ ? I , ) Since b l < 1 11.e can choose 6 = 13 ' Thus (3.6) and (3.7) can be replaced by the single condition II Rx-Byll 1I-$3a X-yll + bill $(Ax-Av) + ~(Bx-Ry)ll ~,yFQ(8)'7~1.
We note that (3.4) with n
=
1 is identical to (3.8).
To prove (3.4) with n = 2 we let A+yR+(l-y)B where 0 < Y < 1 and derive two conditions similar to (3.6) and (3.7) using (3.8) instead of Kato's condition (3.3) These conditions are
(1-v)ll Rx-Byll < +l3a
x-yll + hill {(AX-by)
1+2v
+ T(R~-Ey)ll
u
If (3.10) is valid then
'fa 1 vh 1 , b = . e which is identical to (3.9) if a = b b 1--f- -'(l+y) 2 Y 2-b1 b require b < 1 i.e. 1-y --'(l+~) > y which glves y < ;i~-i;;. Since hl < 1 we can 2 With this value of v (3.10) becomes (3.4) with n = 2. choose y = :. The general result can be obtained by an induction argument. The accretiveness condition on R can be relaxed to obtain the following result. Lemma 3.2. Let A be single-valued and m-accretive and A+B single-valued and If An = BO = 0 and there exists non-negative conaccretive with D(B)>D(A). stants a,b with b < 1 and an a , 0 < n < 1 such that
then A+aB is m-accretive Proof:- Set A+aB = (1-n)A + a(A+B) for 0 S a < 1 then by Lemma 1 , accretive if there exists an a > 0 , b < 1 such that
A+%R
is m-
which is (3.11). We now consider these results when Z is a Hilhert snace H. condition for accretiveness (3.1) can be replaced by Re
(3.12)
In this case the
for all x,y€D(A)
where<.,. > denotes the inner product on H. Lemma 3.3 Let A be single-valued and m-accretive, A+B be single valued and accretive on a Hilbert space H with D(B)>D(A)>O. If A 0 = BO = 0 and there exist a, b > 1 such that
II Bx-Byll
(3.13) then A +
2 l+b2 B
S
a211x-y11
is m-accretive.
+ b211Ax-Ay11
\d.
bf(2-a)/a = 1
and
T. F . Blakeley
then = ll A X - ~ ~ 1 1
II Ax-Ay+Rx-Byll
+ < Ax-Ay ,Rx-Rp > + < Bx-Ry ,Ax-4y > + b:lI Bx-Byll
+ ( 1 - h : )ll Bx-Byll 9
11 A X - ~ y l l + < Ax-Ay ,Bx-Ry > +
'Fx-Ry ll lu-Ay11
= a:ll x-yll =
1 + Ilb(Ax-Ay)
~ : I I X - ~ I I+
,Ax-4y > + b:ll3x-r3yll
+ h , (Bx-By)ll
~ : I I ~ - " ( A x - A+ ~ ()R X - F ~ ) I I
a
Now ( 3 . 1 4 ) i s v a l i d i f b 2 = l / b : , a 2 = a : / ( l - b : ) , u s i n g ( 3 . 1 3 ) , a n d s o we h a v e S i n c e b > 1 t h e n b l < 1 and s o a l l t h e c o n d i t i o n s o f ( 3 . 1 1 ) o f Lemma 3 . 2 . R is m - a c c r e t i v e . Lemma 3 . 2 . a r e s a t i s f i e d a n d we h a v e A+aR = A + l+b2
-
C o r o l l a r y W i t h t h e s a m e c o n d i t i o n s a s i n Lemma 3 . 3 . b u t w i t h ( I - € ) A + R a c c r e t i v e i n s t e a d o f A+B a c c r e t i v e a n d ( 3 . 1 3 ) r e p l a c e d h y
-
t h e n A+B i s m - a c c r e t i v e . Set B
Proof:-
=(%)B
2 a n d ----i= 1 - r . l+b
If
( l - ~ ) i l + gi s a c c r e t i v e t h e n s o i s
( 1 - E ) ( A + B ) a n d a l s o A+B. I l ~ x - % y l l< ~ a211x-y112 + ( 1 - ~ ~ ) 1 l A x - A y 1 1 ~
which is ( 3 . 1 3 ) .
A+(1-e)B i . e .
H e n c e b y Lemma 3 . 3 .
A+E
then
is r n - a c c r e t i v e .
Lemma 3 . 4 . L e t A b e s i n g l e - v a l u e d and m - a c c r e t i v e , A+R h e s i n g l e - v a l u e d and I f A 0 = BO = 0 a n ? a c c r e t i v e o n a r e a l H i l b e r t s p a c e H, w i t h n ( B ) > D ( A ) > O . t h e r e e x i s t s non-nogative c o n s t a n t s a ' , b ' > 1 such t h a t
1 ( 1 + 6 ) b , - R B is m - a c c r e t i v e . 1 b'-1 Proof:C h o o s e a s . t . 0 9 r < --, a n d l e t b = t h e n b < 1 and b 2-T -q- b' s i n c e b ' > 1. S e t a = a ' ( l + b ) t h e n s q u a r i n g ( 3 . 1 5 ) g i v e s
+
t h e n f o r a n y 6 > 0, A
o g
[ ( ) b
2
- 11 II AX-AyII
5
[a'll n-yll + b t ll A X - ~ y l l ]+ a
2abll x-711
+
< 1
- 2abll x-yll
21 h.-2 2 - 3 - 1 1 I\ An-Ayl 1 a'il x-rH + b 9 l l.4x-PgII 1
-
I
l l ~ ( ~ x - ~ y )+ l (lb 2 - 1 )ll Rx-!3yll
r
'3
which i m p l i e s
11 A X - ~ y l l + 2
< Ax-Ay,
+ b211*(~x-~y) a
a211x-y11
>
+ 11 R X - B ~ ~ I I + Rx-Byll
II Ax-Ay + Bx-Byll
i.e. s o b y Lemma 3 . 2 . Since b'
Bx-By
=
'
+ 2abll x-3-11. l l2--1 -(4~-rl~.)
S all x-yll
3
2-9 + bll -(Ax-4y)
+ Rx-PyII
+ Rx-By11
A+aB i s m - a c c r e t i v e .
1 12-0 l+b,Tb+l)
then
-i
=
( l + "1 ) b l - ' irith
=
a n d s i n c e h < 1,
"
W e now h a v e o u r f i n a l r e s u l t w h i c h is e s s e n t i a l l y a c o r o l l a r y t o Lemma 3 . 4 .
> O
Stability of Abstract Evolution Equations
-
Theorem 3 . 2 . W i t h t h e same c o n d i t i o n s a s i n Lemma 3 . 4 . b u t w i t h ( 1 - E ) A + R a c c r e t i v e i n s t e a d o f A+B a c c r e t i v e a n d w i t h ( 3 . 1 5 ) r e p l a c e d h y
- -
(3.16)
<
IIBx-Byll
allx-yll
+ b'(1-E)IIAx-Ayll
t h e n A + B is m - a c c r e t i v e . Proof :-
-
1 (b,
Set B =
B = ( l - € ) R i n Lemma 3 . 4
W e i l l u s t r a t e t h i s r e s u l t by considering t h e equation (3.17)
z
t t + 2 S z t + z XXXX + u s i n h a z = 0 ,
x
0
<
1,
t
>
0,
cr > 0
w h i c h d e s c r i b e s t h e b e n d i n g o f a heam o n a n o n - l i n e a r f o u n d a t i o n ( S h a r m a a n d Dasgupta [ 4 ] ) . T h i s e q u a t i o n c a n b e w r i t t e n i n t h e a b s t r a c t form
by s e t t i n g w = [ z , v ] with inner product.
'1'
with v = z
t
and i n t r o d u c i n g t h e r e a l H i l h e r t s p a c e
1
(3.19)
>
I [z
=
~
~
+
V~I V Z z +
EP( V I~~ Z ~+ V ~ +Y 2I i) 2 z 1 z 2 1 d x
0
T h e o p e r a t o r A is f o r m a l l y d e f i n e d b y Aw = [ - v , 2 < v + r a n d D(A) = {*H t i v e because
v = z = v
: AtfiH,
=
<
XXXX
1"
=
XX
zxx = 0 a t x = 0 , 1 1 .
{ ( ~ ~ + z : ~ ) d >x 0
Then A is m-accre-
f o r a l l *o(A).
0
T h e p e r t u r b i n g t e r m is f o r m a l l y d e f i n e d by Bw = [ 0 , p s i n h a z ] T
>
hence
(yv s i n h a z +
=
E sinh az)dx 1
0,
o
I n o r d e r t o show t h a t t h e c o n d i t i o n s o f T h e o r e m 3 . 2 . a p p l y j n a n e i g h h o u r h o o d o f t h e e q u i l i b r i u m p o i n t z = 0 we o b s e r v e f i r s t t h a t i f y l > y z t h e n
cr a [ + v ( ~ T + ~ ~ ~ ~f g2( ~ + t y; +~y ?) v 2 + . ) + I s i n h a y ~- s i n h ay2 = Q ( Y I - Y Z ) 1 a a a" 4 < & ~ 1 - ~ 2 ) [ +2 5 ( ~ : + ~ ; ) + F ( Y I + Y Z ) I ,
+
...
L e t llw112 G k 2 t h e n u s i n g t h e m e t h o d s o f F r e u n d a n d P l a u t 151 m e c a n e s t a b l i s h the inequality k2
a
~ ' ( z ~ ~ + < z r2 ~ )c a2 xr 2
where 8
0
and
X + cos2L s i n h 22 c 2 = l E X 3 sinh - sin with
2r2 =
E
O Smax x G lIzl
5.
H e n c e f r o m ( 3 . 2 1 ) we h a v e sinh ayl
- s i n h nyp
C o n s i d e r now t h e o p e r a t o r ( 1 - E ) A + B
<
a(yl-y2)cosh IF
(yl 2 y 2 )
W. T . F . Blakeley
U s i n g ( 3 . 2 3 ) and t h e i n e q u a l i t y
we see t h a t i f ( 1 - c ) A + R is t o b e a c c r e t i v e t h e n we m u s t h e a b l e t o f i n d such t h a t
E
> 0
We a l s o h a v e
hence
Thus p r o v i d e d ( 3 . 2 5 ) c a n b e s a t i s f i e d a l l t h e c o n d i t i o n s o f Theorem 3 . 2 . a r e f u l f i l l e d s o t h a t A+B i s m - a c c r e t i v e i n a n e i g h b o u r h o o d o f z = 0 a n d h e n c e -(A+B) g e n e r a t e s a n o n - l i n e a r c o n t r a c t i o n s e m i - g r o u p . W e n o t e t h a t assuming 5 , ~l a n d a r e given then (3.25) e f f e c t i v e l y determines t h e size of t h e neinhb o u r h o o d f o r w h i c h t h e s y s t e m is s t a h l e t h r o u g h t h e v a l u e o f k (= c R ) . F u r t h e r m o r e i f k is s u c h t h a t t h e i n e q u a l i t y h o l d s i n ( 3 . 2 5 ) t h e n t h e r e is a c e r t a i n a l l o w a b l e c l a s s o f p e r t u r b a t i o n s of t h e n o n - l i n e a r s y s t e m which w i l l e n s u r e e x i s t e n c e , u n i q u e n e s s and s t a h i l i t y o f t h e p e r t u r b e d s v s t e m . T h i s class o f p e r t u r b a t i o n s w i l l h e d e t e r m i n e d hy a n w l i c a t l o n o f Theorew 3 . 2 . a g a i n w i t h A r e p l a c e d h y t h e o p e r a t o r A+F o f ( 3 . 1 8 ) . A f u r t h e r example i l l u s t r a t i n g Lenna 3 . 4 .
is g i v e n i n [ 61
References [ I ] I l i c k l i n S . G . " I n t e g r a l E q u a t i o n s " Pergamon P r e s s 1 9 6 4 . 121 C u r t a i n R . a n d P r i t c h a r d A.J. "An a b s t r a c t t h e o r y f o r u n b o u n d e d c o n t r o l a c t i o n f c r d i s t r i b u t e d p a r a m e t e r s y s t e m s " I i n i v e r s i t y o f lvarwick r o n t r o l C e n t r e R e p o r t No. 3 9 . [ 3 ] K a t o T. " A c c r e t i v e o p e r a t o r s a n d n o n - l i n e a r e v o l u t i o n e q u a t i o n s i n B a n a c h s p a c e s " P r o c . Symp. P u r e ? l a t h . A m e r . \ l a t h . S o c . 2 p t 1 1 3 8 - 1 6 1 ( 1 9 7 0 ) . [ 4 ] Sharma S . a n d D a s g u p t a S."The b e n d i n g p r o b l e m o f a x i a l l y - c o n s t r a i n e d beams o n n o n - l i n e a r e l a s t i c f o u n d a t i o n s " I n t . J . S o l i d s S t r u c t u r e s 11 8 5 3 - 8 5 9 (1975). I 5 1 F r e u n d L . B . a n d P l a u t R.H."An e n e r g y - d i s p l a c e m e n t i n e q u a l i t y a p r s l i c a h l e t o p r o b l e m s i n t h e d y n a m i c s t a b i l i t y o f s t r u c t u r e s " J . A p p l . Mech. 5 3 6 - 7 ( 1 9 7 1 ) [ 6 1 P r i t c h a r d A . J . a n d B l a k e l e y W.T. F . " P e r t u r b a t i o n r e s u l t s a n d t h e i r a p p l i c a t i o n s t o problems i n s t r u c t u r a l dynamics" L e c t u r e Notes i n nfathematics V o l . 503 4 3 8 - 4 4 9 S p r i n g e r - V e r l a g ( 1 9 7 6 ) .
.