Chapter 2 Stability Domain

7/29/2019 Chapter 2 Stability Domain

1/22

C h a p t e r 2

S t a b i l i t y d o m a i n c o n c e p t s

2 . 1 I n t r o d u c t o r y c o m m e n t s

I n o r d e r t o g e t c o m p l e t e i n f o r m a t i o n a b o u t t h e c a u s a l i t y b e t w e e n i n i t i a l s t a t e s a n d

s y s t e m s m o t i o n s , c o n c e p t s o f d o m a i n s o f v a r i o u s s t a b i l i t y p r o p e r t i e s w e r e i n t r o -

d u c e d . I n t h e f r a m e w o r k o f t h e L y a p u n o v s t a b i l i t y t h e n o t i o n o f a t t r a c t i o n d o m a i n

o f t h e o r i g i n w a s d e n e d b y Z u b o v 2 1 8 ] a n d H a h n 9 4 ] , a n d n o t i o n s o f t h e s t a b i l -

i t y d o m a i n a n d t h e a s y m p t o t i c s t a b i l i t y d o m a i n w e r e d e n e d b y G r u j i c 6 4 ] , 6 5 ] ,

6 8 ] { 7 0 ] , 7 2 ] , a n d u s e d b y G r u j i c e t a l . 8 8 ] { 9 0 ] . T h e c o n c e p t o f p r a c t i c a l s t a b i l i t y

d o m a i n s w a s i n t r o d u c e d b y G r u j i c 7 0 ] .

I n t h e l i t e r a t u r e ( e . g . L a S a l l e a n d L e f s c h e t z 1 2 1 ] a n d Z u b o v 2 1 8 ] ) t h e n o t i o n o f

\ r e g i o n o f a s y m p t o t i c s t a b i l i t y " h a s b e e n u s e d i n t h e s e n s e o f t h e a t t r a c t i o n d o m a i n .

I n w h a t f o l l o w s t h e d i e r e n c e b e t w e e n t h e m w i l l b e c l a r i e d .

2 . 2 D o m a i n s o f L y a p u n o v s t a b i l i t y p r o p e r t i e s

2 . 2 . 1 T h e n o t i o n o f d o m a i n

T h e t e r m \ d o m a i n " d e n o t e s a s e t t h a t c a n b e , b u t n e e d n o t b e , o p e n o r c l o s e d .

D o m a i n s o f L y a p u n o v s t a b i l i t y p r o p e r t i e s w i l l b e c a l l e d f o r s h o r t \ L y a p u n o v

s t a b i l i t y d o m a i n s " i n a g e n e r a l s e n s e i n c o r p o r a t i n g d o m a i n s o f s t a b i l i t y , o f a t t r a c t i o n

a n d o f a s y m p t o t i c s t a b i l i t y . I n t h e c l o s e r s e n s e t h e n o t i o n \ L y a p u n o v s t a b i l i t y

d o m a i n " w i l l b e u s e d f o r t h e d o m a i n o f s t a b i l i t y ( f o r s h o r t , t h e s t a b i l i t y d o m a i n ) .

L y a p u n o v s t a b i l i t y d o m a i n s w i l l b e s t u d i e d h e r e i n i n t h e f r a m e w o r k o f t i m e -

i n v a r i a n t c o n t i n u o u s - t i m e n o n l i n e a r s y s t e m s g o v e r n e d b y

d X

d t

= f ( X ) ( 2 . 1 )

w i t h p o s s i b l y c e r t a i n s p e c i c f e a t u r e s t h a t w i l l b e d e s c r i b e d w h e n t h e y a r e n e e d e d .

I n t h e l i t e r a t u r e ( e . g . L a S a l l e a n d L e f s c h e t z 1 2 1 ] ) t h e n o t i o n \ r e g i o n " h a s b e e n

u s e d f o r a n o p e n c o n n e c t e d s e t . W e d o n o t w i s h a p r i o r i t o i m p o s e s u c h a r e s t r i c t i o n

2004 by Chapman & Hall/CRC


2/22

o n t h e l a r g e s t s e t o f i n i t i a l s t a t e s a p p r o p r i a t e f o r a c o r r e s p o n d i n g s t a b i l i t y p r o p e r t y .

T h e r e f o r e , w e s h a l l u s e t h e t e r m \ d o m a i n " i n g e n e r a l r a t h e r t h a n \ r e g i o n " . I n c a s e

a d o m a i n i s o p e n a n d c o n n e c t e d t h e n w e c a n a l s o c a l l i t a \ r e g i o n " .

2 . 2 . 2 D e n i t i o n s o f s t a b i l i t y d o m a i n s

T h e d e n i t i o n s o f s t a b i l i t y d o m a i n s w e r e i n t r o d u c e d t o c o m p l y w i t h t h e d e n i t i o n s

o f s t a b i l i t y o f a s t a t e a n d o f a s e t 6 4 ] , 6 5 ] , 6 8 ] , 6 9 ] , 7 0 ] , 7 2 ] .

D e n i t i o n 2 . 1 ( a ) T h e s t a t e X = 0 o f t h e s y s t e m ( 2 . 1 ) h a s t h e s t a b i l i t y d o m a i n

d e n o t e d b y D

s

i f a n d o n l y i f b o t h

( i ) f o r e v e r y " 2


3/22

W e a r e n o w i n t e r e s t e d o n l y i n s t a b i l i t y o f X = 0 . L e t "

=

p

2 = 2 . F o r a n y

" 2 0 "

] , t h e m a x i m a l ( " ) d e n o t e d b y

M

( " ) ( D e n i t i o n 1 . 2 , S e c t i o n 1 . 1 . 1 ) o b e y s

M

( " ) = " . F o r a n y " 2 "

+ 1 t h e m a x i m a l

M

( " ) = "

H o w e v e r

k X ( t X

0

) k < " 8 t 2

>

>

>

>

:

B

"

" 2 0 "

B

"

\ S

" 2 "

S

" 2 + 1

w h e r e

S

=

X X 2

>

>

>

>

:

B

"

" 2 0 "

B

"

\ S

" 2 "

S

" 2 + 1

s o t h a t

D

s

=

D

s

( " ) " 2


4/22

x 2

x 1

S

SeO

Ds() = B

x 2

x 1

Ds() = B

a ) b )

" 2 0 "

) D

s

( " ) = B

"

" = "

=

p

2

2

) D

s

( "

) = B

"

x 2

x 1

Ds( )

B

=

x 2

x1

Ds()

B=B

c ) d )

" 2 "

) D

s

( " ) = B

"

\ S

" = ) D

s

( ) = B

"

\ S

x2

x 1

Ds()=Ds

B

x 2

x 1

Ds=S

e ) f )

" 2 + 1 ) D

s

( " ) = S

D

s

= D

s

( " ) " 2


5/22

x2

x1

A

2

11 4-1-4

-1

-2

F i g u r e 2 . 2 : T h e s t a t e p o r t r a i t o f t h e s y s t e m X = ( 1 ; x

1

; x

2

) ( 4 ; x

1

; 2 x

2

) X . T h e

s h a d e d a r e a t o g e t h e r w i t h i t s b o u n d a r y r e p r e s e n t s t h e s e t A

E x a m p l e 2 . 2 L e t t h e s e c o n d o r d e r s y s t e m ( 2 . 1 ) b e s p e c i e d b y

d X

d t

= ( 1 ; x

1

; x

2

) ( 4 ; x

1

; 2 x

2

) X

W e a r e i n t e r e s t e d i n s t a b i l i t y o f t h e s e t A

A =

X X 2


6/22

x 2

x 1

A2

1

1 4-1-4

-1

-2

0

S4

N(,A)=Ds(,A)

]0,25]5

a)

x2

x1

2

1

1 4-1-4

-1

-2

0=

= 255

b)

N(25,A)=Ds(25,

A)5 5

=

25

5

x2

x1

2

1

1 4-1-4

-1

-2

0

N(,A)

c)

N(,A)S4=Ds(,A)

x 2

x 1

2

1

1 4-1-4

-1

-2

0

d)

S4=Ds(,A)=Ds(A)

N(,A)

F i g u r e 2 . 3 : T h e d e p e n d e n c e o f t h e s u b s e t D

s

( A ) o f D

s

( A ) o n 2 0 + 1 ( E x a m p l e 2 . 2 ) :

a ) D

s

( A ) = N ( A ) = N ( A ) \ S

4

2 0

2

p

5

5

b ) D

s

(

2

p

5

5

A ) = N (

2

p

5

5

A ) = N (

2

p

5

5

A ) \ S

4

=

2

p

5

5

c ) D

s

( A ) = N ( A ) \ S

4

2

2

p

5

5

3

d ) D

s

( A ) = S

4

2 3 + 1

H e n c e D

s

( A ) = D

s

( A ) 2 0 + 1 = S

4

w h e r e

S

4

=

X X 2


7/22

3

2

1

01 2 3

-1

-2

-3

-1-2-3 x1

x2

F i g u r e 2 . 4 : T h e s t a t e p o r t r a i t o f t h e s y s t e m ( E x a m p l e 2 . 3 ) d e s c r i b e d v i a t h e p o l a r c o o r -

d i n a t e s = k X k = a r c t a n

x

2

x

1

b y = ; ( 1 ;

2

) ( 4 ;

2

) ( 9 ;

2

) = ; 1

s o t h a t t h e s t a b i l i t y d o m a i n D

s

( A ) o f t h e s e t A i s f o u n d a s D

s

( A ) = S

4

D

s

( A ) =

X X 2


8/22

I t i s n o w o b v i o u s i n v i e w o f F i g . 2 . 4 t h a t

k X ( t X

0

) k < " i

8

>

>

>

>

>

>

>

>

>

:

k X

0

k < " " 2 0 1

k X

0

k 1 " 2 1 2

k X

0

k < " " 2 2 3

k X

0

k 3 " 2 3 + 1

T h i s m e a n s

D

s

( " ) =

8

>

>

>

>

>

>

>

>

>

:

B

"

" 2 0 1

B

1

" 2 1 2

B

"

" 2 2 3

B

3

" 2 3 + 1

w h i c h y i e l d s

D

s

= D

s

( " ) " 2 0 + 1 ] = ( B

"

" 2 0 1 ] ) ]

B

1

B

"

" 2 2 3

B

3

= B

1

B

1

B

3

B

3

=

B

3

H e n c e , t h e s t a b i l i t y d o m a i n D

s

o f X = 0 o f t h e s y s t e m e q u a l s c o m p a c t c i r c l e

B

3

D

s

=

B

3

= f X X 2


9/22

x2

x10

F i g u r e 2 . 5 : T h e s t a t e p o r t r a i t o f t h e s y s t e m : X = ( ; + x

1

+ x

2

) X ( E x a m p l e 2 . 4 ) .

E x a m p l e 2 . 4 L e t t h e s y s t e m c o n s i d e r e d i n E x a m p l e 2 . 1 b e r e a n a l y z e d ,

d X

d t

= ( ; + x

1

+ x

2

) X 2


10/22

x2

0 x1

Da

F i g u r e 2 . 6 : T h e a t t r a c t i o n d o m a i n D

a

o f X = 0 o f t h e s y s t e m : X = ( ; + x

1

+ x

2

) X

T h e y a r e i n t e r r e l a t e d s o t h a t D

a

i s t h e i n t e r i o r o f D

s

a n d D

s

i s t h e c l o s u r e o f D

a

h e n c e , D

a

i s s u b s e t o f D

s

D

a

=

D

s

D

s

=

D

a

D

s

D

a

E x a m p l e 2 . 5 L e t

( k X

0

k ) =

8

>

:

0 i k X

0

k 1

k X

0

k + 1

k X

0

k ; 1

i k X

0

k > 1

a n d t h e s y s t e m m o t i o n s b e d e n e d b y

X ( t X

0

) = e x p ( ; t ) 1 + ( X

0

) t X

0

T h e y s a t i s f y t h e i n i t i a l c o n d i t i o n , X ( 0 X

0

) X

0

, a n d a r e c o n t i n u o u s i n t 2


11/22

||X(

t;X

0)||

1

1

1< ||X0

||^

2||X

0||+1

0t

||X0||+1

||X0||-1

||X0|| exp(- )2

||X0||+1

||X(t;X0)||

^

F i g u r e 2 . 7 : T h e n o r m o f m o t i o n s d e n e d b y X ( t X

0

) = 1 + ( X

0

) t X

0

e x p ( ; t ) ( X

0

) = 0

k X

0

k 1 a n d ( X

0

) =

k X

0

k + 1

k X

0

k ; 1

k X

0

k > 1 ( E x a m p l e 2 . 5 ) .

B ) k X

0

k > 1 y i e l d s k X ( t X

0

) k = e x p ( ; t )

1 +

k X

0

k + 1

k X

0

k ; 1

t

k X

0

k . I n t h i s c a s e ,

m a x k X ( t X

0

) k t 2


12/22

D

s c

( " ) =

8


13/22

x2

2

A

4x1

- 4

- 2

1

- 1

- 1 10

D(A)

F i g u r e 2 . 8 : T h e a t t r a c t i o n d o m a i n D

a

( A ) o f t h e s e t A = f X X 2


14/22

x2

x1

A

Da(A)

2

-2

-2

1

-1-1 1

0

-3 2 3

-3

3

a)

x2

x1

A

Ds(A)

2

-2

-2

1

-1-1 1

0

-3 2 3

-3

3

b)

F i g u r e 2 . 9 : a ) T h e d o m a i n D

a

( A ) o f a t t r a c t i o n o f t h e s e t A = f x x 2


15/22

I n v i e w o f D e n i t i o n 2 . 5 , t h e s t a t e X = 0 o f t h e s y s t e m h a s a l s o t h e a s y m p t o t i c

s t a b i l i t y d o m a i n D

D = D

s

\ D

a

=

X X 2


16/22

D e n i t i o n 2 . 6 ( a ) A s e t A


17/22

o b e y t h e e s t i m a t e

k X ( t X

0

) k k X

0

k e x p ( ; t ) 8 t 2


18/22

C o m m e n t 2 . 4 T h e s t a t e X = 0 o f t h e s y s t e m

d X

d t

= ( ; + x

1

+ x

2

) X 2


19/22

2 . 2 . 6 D e n i t i o n s o f a s y m p t o t i c s t a b i l i t y d o m a i n s o n N

( )

( )

L e t t h e s y s t e m ( 2 . 1 ) b e o f t h e L u r i e f o r m ( S e c t i o n 1 . 1 . 5 ) ,

d X

d t

= A X + B f ( w ) ( 2 . 2 a )

w = C X + D f ( w ) ( 2 . 2 b )

I n o r d e r t o e x p l a i n t h e n e e d f o r t h e s t u d y o f d o m a i n s o f a s y m p t o t i c s t a b i l i t y

o n N

i

( ) w e p r e s e n t t h e f o l l o w i n g s i m p l e e x a m p l e .

E x a m p l e 2 . 1 3 L e t n = 1 a n d

d X

d t

= ; s i n X :

T h i s s y s t e m h a s i n n i t e l y m a n y e q u i l i b r i u m p o i n t s l o c a t e d a t X = k , w h e r e k

i s a n y i n t e g e r . H e n c e , X = 0 o b v i o u s l y i s n o t a s y m p t o t i c a l l y s t a b l e i n t h e l a r g e .

H o w e v e r , i t i s a s y m p t o t i c a l l y s t a b l e w i t h t h e d o m a i n o f a s y m p t o t i c s t a b i l i t y D =

; . O v e r S = D t h e n o n l i n e a r i t y f f ( X ) = ; s i n X , b e l o n g s t o t h e f a m i l y

N

1

( L M S ) f o r L = 0 1 ] a n d M = ; 1 1 . I f t h e s y s t e m i s e m b e d d e d i n t o t h e c l a s s

o f L u r i e s y s t e m s ( 2 . 2 ) , t h e n w e c a n s p e a k o n l y a b o u t a s y m p t o t i c s t a b i l i t y o f X = 0

f o r a p a r t i c u l a r f ( ) o r f o r a n y f 2 N

1

( L M S ) , o r f o r a n y f 2 N

0

( L S ) . T h i s

m e a n s t h a t w e c a n l o o k o n l y f o r t h e a s y m p t o t i c s t a b i l i t y d o m a i n f o r a p a r t i c u l a r f

e . g . f ( X ) = ; s i n X , o r f o r e v e r y f 2 N

1

( L M S ) , o r f o r e v e r y f 2 N

0

( L S )

L e t D

f

d e n o t e t h e a s y m p t o t i c s t a b i l i t y d o m a i n o f X = 0 o f t h e ( L u r i e )

s y s t e m ( 2 . 2 ) f o r a p a r t i c u l a r n o n l i n e a r i t y f

D e n i t i o n 2 . 9 T h e s t a t e X = 0 o f t h e s y s t e m ( 2 . 2 ) h a s t h e s t r i c t ] a s y m p t o t i c

s t a b i l i t y d o m a i n o n N

i

( L M S ) , w h i c h i s d e n o t e d b y D

i

( L M S ) D

i c

( L M S ) ]

i f a n d o n l y i f

a ) i t h a s t h e s t r i c t ] a s y m p t o t i c s t a b i l i t y d o m a i n D

f

D

f

c

] f o r e v e r y f 2

N

i

( L M S )

a n d

b ) D

i

( L M S ) = \ D

f

f 2 N

i

( L M S ) i s a n e i g h b o u r h o o d o f X = 0

D

i c

( L M S ) = \ D

f

c

f 2 N

i

( L M S ) i s a c o n n e c t e d n e i g h b o u r h o o d

o f X = 0 ] , r e s p e c t i v e l y .

T h i s d e n i t i o n w a s i n t r o d u c e d i n 6 8 ] , 6 9 ] . I t c a n b e e x t e n d e d t o s e t s a s f o l l o w s :

D e n i t i o n 2 . 1 0 A s e t A


20/22

a n d

b ) D

i

( L M A S ) = \ D

f

f 2 N

i

( L M A S ) i s a n e i g h b o u r h o o d o f t h e

s e t A D

i c

= \ D

f

c

f 2 N

i

( L M A S ) i s a c o n n e c t e d n e i g h b o u r h o o d o f

t h e s e t A ] , r e s p e c t i v e l y .

2 . 3 D o m a i n s o f p r a c t i c a l s t a b i l i t y p r o p e r t i e s

2 . 3 . 1 D e n i t i o n s o f d o m a i n s o f p r a c t i c a l s t a b i l i t y

B y f o l l o w i n g 7 0 ] a n d S e c t i o n 1 . 2 . 2 w e a c c e p t t h e f o l l o w i n g d e n i t i o n f o r t h e

s y s t e m ( 2 . 3 ) :

d X

d t

= f ( X i ) f


21/22

2 . 3 . 2 D e n i t i o n s o f d o m a i n s o f p r a c t i c a l c o n t r a c t i o n

w i t h s e t t l i n g t i m e

A s f o r p r a c t i c a l s t a b i l i t y d o m a i n s , w e r s t i n t r o d u c e t h e n o t i o n o f t h e d o m a i n o f

p r a c t i c a l c o n t r a c t i o n w i t h s e t t l i n g t i m e f o r t h e s y s t e m ( 2 . 3 ) ( s e e S e c t i o n 1 . 2 . 3 ) .

D e n i t i o n 2 . 1 3 T h e s y s t e m ( 2 . 3 ) h a s t h e d o m a i n o f p r a c t i c a l c o n t r a c t i o n w i t h t h e

s e t t l i n g t i m e

s

w i t h r e s p e c t t o f X

F

I g , w h i c h i s d e n o t e d b y D

p c

(

s

X

F

I ) i f

a n d o n l y i f b o t h

a ) i t s m o t i o n s o b e y

X ( t X

0

i ) 2 X

F

f o r e v e r y ( t i ) 2 T

s

I

p r o v i d e d o n l y t h a t X

0

2 D

p c

(

s

X

F

I )

a n d

b ) t h e i n t e r i o r

D

p c

(

s

X

F

I ) o f D

p c

(

s

X

F

I ) i s n o n - e m p t y .

W h e n

s

X

F

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c o n t r a c t i o n w i t h t h e s e t t l i n g t i m e

s


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f o r e v e r y ( t i ) 2 T

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2 D

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s

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a n d

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s

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s

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s t a t e X

t h e n w e m a y u s e D e n i t i o n 2 . 1 4 w i t h A = f X

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2 . 3 . 3 D e n i t i o n s o f d o m a i n s o f p r a c t i c a l s t a b i l i t y

w i t h s e t t l i n g t i m e

I n v i e w o f t h e p r e c e d i n g d e n i t i o n a n d t h e n o t i o n o f p r a c t i c a l s t a b i l i t y w i t h s e t t l i n g

t i m e ( S e c t i o n 1 . 2 . 4 ) w e a c c e p t t h e f o l l o w i n g d e n i t i o n :

2004 by Chapman & Hall/CRC


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D e n i t i o n 2 . 1 5 T h e s y s t e m ( 2 . 3 ) h a s t h e d o m a i n o f p r a c t i c a l ( c o n t r a c t i v e ) s t a -

b i l i t y w i t h t h e s e t t l i n g t i m e

s


A

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1 ) X ( t X

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f o r e v e r y ( t i ) 2

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Chapter 2 Stability Domain