86

Chapter IІI. Study of stability of solutions of dynamic systems with cylindrical phase space

On the other hand

y

(t) y

(t) a y

(t) ... a

y

(t)

d ( (t))

,

t I ,

n 2

n 1

2

n 1

n 1

0

dt

where

d ( (t))

d ( (t))

d (t)

, |

d ( (t))

| |

d ( )

| | (t) | c

. From here it

dt

d

dt

dt

d

0

follows

that

the

function

yn 2

(t),

t I

is

limited. Then

the

function

y

n 1

(t),

t I

continuously differentiable,

| y

n 1

(t) | m

,

|

y

n 1

(t) | m

, m

const

.

n 1,1

n 1,2

n 1,1

Consequently, the function

z(t) ( y(t),..., yn 1(t)), t I

is limited: | z(t) | a,

t I ,

and continuously differentiable, moreover

| z(t) | c, t I ,

0 a ,

0 c ,

we have evaluation (3.17). By integrating the identity (3.16) after replacement

(t)

by z(t) , we get an equality (3.18). The lemma is proved.

Lemma 5. Let the conditions of Lemma 4 be satisfied, and let besides:

1) function

2	1	1
( ) C	(R	, R );

d ( )

2

2)

|

d

2

|

3

,

0

3

.

Then for any constant matrix

Q of order

(n 2) (n 2) the improper integral

*

(t)Q1 (t)dt

2

2

(t)]dt l0 ,

[q1 y1 (t)

... qn 2 yn 2

(3.19)

0

0

d

l0

[

*

(t)F (t)]dt

*

| l0

| ,

dt

(t)F (t) |0 ;

(3.20)

0

where

(t) ( y (t),..., y

n

2

(t)),

t I.

1

Proof. By integrating identity (3.16), we get ratios (3.19), (3.20). It is required to

prove that

| l

|

. The derivative

0

y

(t) a y (t) ... a

y

(t)

d 2 ( (t))

,

t I,

n 2

0 3

n 1

n 2

dt2

2

2

d ( (t))

d ( (t))

d

d (

(t))

d ( (t))

d ( (t))

where

(t)

(t), t I ,

|

| ,

2

2

dt

dt

dt

d

d

d

0 ,

|

d2 ( (t))

| ,

, | (t) | c , t I . As functions

y (t),..., y

(t), t I

n 2

d 2

3

3

0

1

, t,

t I . Then

are limited, then from (3.15) it follows that | (t) | c3, 0 c3

the

function

yn 2 (t),

t I satisfies

the

conditions

| yn 2 (t) | mn 2,1,

171

Lectures on the stability of the solution of an equation with differential inclusions

| yn 2 (t) | mn 2,2 , t, t I, 0		mn 2,1 , 0 mn 2,2 . Consequently, the vector
(t), t I	is limited:	| (t) | a, t I and is continuously differentiable,
	I , 0 a , 0 c and we have evaluation (3.20). The
| (t) | c, t

proved.

Lemma 3. Let the conditions of Lemma 4 be satisfied, and let besides:

function moreover lemma is

1) scalar function

V (0) 0;

V ( x)

be positive, continuous for all x Rn 1, x 0 and

2)		V
	0

(z(t))dt .

Then lim z(t) 0,	z(t) ( y1 (t),..., yn 1 (t)),	t I.
t

The proof of a similar lemma is given in Chapter I.

Lemma 7. Let the conditions of Lemma 5 be satisfied, and let besides:

1)	scalar function				V (x) be positive, continuous for all	x R	n 2	,	x 0
V (0) 0;
2)
		V ( (t))dt ,			(t) ( y1(t),..., yn 2 (t)), t I.
	0

and

Then

lim	(t) 0.
t

The lemma’s proof is similar to the proof of Lemma 3.

Lecture 27.

Improper integrals

Based on identities (3.12)-(3.15), assessments (3.9)-(3.11), and also Lemmas 4, 5 we can obtain evaluations of improper integrals along with the solution of the system (3.8).

Theorem 3. Let the conditions 5 of the lemmas 1, 2 be satisfied, the matrix

А be

a Hurwitz matrix, the function	( ) 0 . Then for any value	1 0	, along with the
solution of the system (3.8) the	improper integral

	I					[ (t) (t)]				[ (t) (t)]dt									[M		y	2		(t) M		y	2	(t)
		1																		0		n	2		1		n 1
								1	1			2
				0														0
														d
	M				y		2	(t) ... M		y	2	(t)]dt			[	*	(t)F (t)]dt							0,
				2					n 1		n
						1								dt					1
													0

(3.21)

d

[ * (t)F1 (t)]dt * ( )F1 ( ) * (0)F1 (0)

l1,

| l1 | ,

(3.20)

dt

0

where (t) ( y (t),..., y

(t)), t I, | | , |

| , ( ) C2

(R1

, R1 ), |

d2

| , 0

.

n 2

d 2

1

1

2

3

3

172

Chapter IІI. Study of stability of solutions of dynamic systems with cylindrical phase space

Proof.

where

From inclusion

d ( (t))

d

( ) 0

it follows that

							0,	(t), (t),	t I ,
					2
	1
	d ( (t))

					(t)	d ( (t))
		dt
					(t) , (t)		, t I.
		d				dt
		dt

(3.23)

Multiplying identity (3.23) by

2

(t)

, we get

[ (t) (t)] [ (t) (t)] 0, t, t I,

1

1

2

where 1

0

is some number. As follows from (3.13), (3.14) following identities hold

a place

(t) (t)

1

y

(t) (

1

) y

(t) y (t) ...

n

2

a

n 1

1

0 1

n 1

1 1 1

(

1

) y

(t),

t I ,

n

a

1

n 2

n

(t) (t)

1

y

(t) (

1

) y

(t) y (t) ...

n 2

a

n 1

2

n 1

2 0

2 1 1

(

1

) y

(t),

t I.

a

2

n

2 n

n

It is easy to verify that product (3.23) is a quadratic form with respect to variables

y

, y ,..., y

n

, y

n 1

, y

n 2 . Then the improper integral

1

1

I1

*

(t)Q (t)dt 0,

0

where

*

. As all

(t)Q (t) [ 1 (t) (t)] 1[ (t) 2 (t)], t, t I, (t) ( y1(t),..., yn 2 (t))

conditions of product of Lemma 5 are satisfied, then

I					*	(t)Q (t)dt		[M		y	2		(t) M		y	2		(t) M		y	2	...
	1								0		n	2		1		n	1		2
																				1
			0				0

		2
	M n 1 yn ]dt
				0
	l	d		[ * (t
		dt
	1
		0

where F1 is a constant matrix of order

d	[	*	(t)F (t)]dt 0,
dt			1

)F1 (t)]dt, | l1 | ,

(n 2) (n 2) . The theorem is proved.

173

Lectures on the stability of the solution of an equation with differential inclusions

Theorem 4. Let the conditions of Theorem 3 be satisfied. Then for any quantities0 , 1,..., n , n 1 , along with the solution of the system (3.8) the improper integral

I

[

y

(t)

y

(t)

y (t) ...

y

(t)

y

2

dt

2

0

n 2

1

n 1

2

n

n 1

n 1

n

(t)]

1

0

(3.24)

[ y

2

(t)

y

2

(t) y

2

(t) ...

y

2

(t)]dt

l

0,

n 2

n 1

n

0

1

2 1

n 1

2

0

d

l

[

*

(t)F

(t)]dt,

| l

| ,

2

2

dt

2

0

(3.25)

where

F

is a constant matrix of order

(n 2) (n 2),

( y

,..., y

n 2

) .

2

1

The proof of the Theorem is similar to the proof of Theorem 3.

Theorem 5. Let the conditions of Theorem 3 be satisfied. Then for any quantities

h

, h , h

, along the solution of the system (3.8) the improper integral

0

1

2

I													2	dt			[H		y	2		(t) H	y	2	(t)
	2				[h (t) h (t) h ( (t))]													0		n	2			n 1
							0	1				2										1
				0												0
H				y		2	(t) ... H			y	2	(t)]dt l	0,
			2						n 1		n
					1							3

(3.26)

					d
			l			[	*	(t)F (t)]dt, | l		| ,
			3		dt			3	3
				0
where	F	is a constant matrix of order						(n 2) (n 2),		( y1,..., yn 2 ) .
	3
									are denoted by identities
Proof. As			(t), (t), ( (t)), t I
(3.12), respectively, then
			* (t)Q (t) [h (t) h						(t) h ( (t))], t I.
					0			1	2

(3.27)

(3.15), (3.13),

Further, repeating the proofs of Theorem 3, we get ratios (3.26), (3.27). The theorem is proved.

Consider the case when

( )d 0, , R1.

(3.28)

Theorem 3. Let the conditions 4 of Hurwitz matrix, the function ( ) 0

lemmas 1, 2 be satisfied, the matrix А be a and an equality holds (3.28). Then for any

value 2 , along the solution of the system (3.8) the improper integral

174

Chapter IІI. Study of stability of solutions of dynamic systems with cylindrical phase space

		I					( (t)) (t)dt						[N			y	2		(t) N		y	2	(t) ... N		y	2
			4														n	1		2				n 1		n
										2					1						1
						0					0
					d					( )
								*		4								2
					dt		[z		(t)F z(t)]dt			( ) d ,
				0						(0)
						d
l									*	(t)F z(t)]dt,		| l			| ,
	4							[z						4
						dt				4
					0

(t)]dt

(3.29)

where F4 is a constant matrix of order				(n 1) (n 1),	z(t) ( y1(t),..., y
Proof. For solutions of the system (3.8) the following identities
hold. The product	( (t)) (t) z	*	(t)Q z(t), z(t) ( y (t),..., y				(t)),	t
						n 1
	2			1	1

applying Lemma 4, we get ratios (3.29), (3.30). As

		(3.30)
n 1	(t)),	t I.
(3.12),		(3.13)
I
	. Further,

for any

	2

	2	( )		2				2
					d ,				d 0,	, R	,
	( (t)) (t)dt		( )				( )
										1
0		(0)

, then

( )

( ) 2d .

(0)

The theorem is proved. Consider the case when

Let the values

,

( )d

suchthat

	0,		, R .
				1
		| ( ) | d ,			.

(3.31)

Theorem 7. Let the conditions a Hurwitz matrix, the function ( )

4 of thelemmas 1, 2 be satisfied, the matrix0 and an equality hold (3.31), the values

,

be

be such that

4 4 5 2 32 0 , along the

0

. Then for any quantities 3 , 4

0, 5

0 such that

solution of the system (3.8) the improper integral

I

5

[ ( (t)) (t) 2

( (t))

2 (t)]dt

[P y2

(t) P y2

(t)

3

4

5

1 n 1

2 1

0

0

(3.32)

dt

... P

y2

(t)]dt

d

[z*

(t)F z(t)]dt

n 1

n

5

0

( )

[ ( (t))

| ( (t)) |]

3 (t)dt

[ ( ) | ( ) |] 3d ,

0

(0)

175

Lectures on the stability of the solution of an equation with differential inclusions

d

l

*

(t)F z(t)]dt,

| l

| ,

[z

5

dt

5

5

0

where

F

is a constant matrix of order (n 1) (n 1),

z ( y1,..., yn 1 ).

5

Proof. Denote by

( ) ( ) | ( ) | . The function ( )

perty

( )d

( )d

| ( ) | d 0,

, R

for any

1

. From here it follows that

(3.33)

has the pro-

[ (
0
for any numbers		3

3	( )		3
(t)) | ( (t)) |] (t)dt		( )		d ,
	(0)

. Consider the function

(3.34)

S(t) ( (t)) (t)																2	( (t))									2	(t) ( (t)) (t)
									3						4		| |]						5										3	|
				2							2		[																2			2	|
3	4								5											3						4				5				3
[															| |]	(		2			2				)				0, t,				t I ,
												3			2						3							2
			5						2									4						4
													5							5
due to the fact that 4 4 5									2	2		0			. Then
										3
( (t)) (t)														2	( (t))				2	(t) ( (t)) (t), t, t I.
					3						4						5													3
Consequently, the improper integral
	I	5						[ ( (t)) (t) 2															( (t)) 2								(t)]dt
															3				4											5
							0																											(3.35)
																		( )
						( (t)) 3 (t)dt															( ) 3d ,
					0													(0)

by virtue of inequalities (3.34). As along the solution of the system (3.8), the following identities hold (3.12), (3.13), then

* *						2	2		2	2	t I.I.
z z(t()tF)Fz(zt()t) ( ( (t())t)) (t()t)					( ( (t())t))			(t(),t),t
5	5	3	3		4	4		5	5

Further, applying Lemma 4, we get ratios (3.32), (3.33). The theorem is proved.

176

Chapter IІI. Study of stability of solutions of dynamic systems with cylindrical phase space

Theorem 8. Let the conditions of Theorem 3 be satisfied. Then for any quantities1 0, 2 , 3 , along the solution of the system (3.8) the improper integral

	I		[ 2 22 2 32 2 2			2	2 2 3	]dt
	6		1	1	1	2	1
		0

		[L y2			(t) L y2				(t) L y2 (t) ... L							1		y2 (t)]dt l	6
			0 n 2			1 n 1						2 1			n			n
	0
(			2 3 ) | |dt							2 3			[ | |] dt ,
	0			1					0	1
									d
					l					[	*	(t)F (t)]dt, | l					| ,
					6			dt				6		6
							0
		|		( 2 3 ) | |dt | , |										2 3	[ | |] dt | ,
			0	1								0		1
where 0				, the equality (3.31)holds.

Proof. Along the solution of the system (3.8) the inequality holds

(3.36)

(3.37)

(3.38)

	[																| |]														2							2			2
											2			3					2							2				2				2		3							2
						1																		1
												1			1																1								1
	2 2										| |			2					|				|		2										2							0.
														2			3										2		3							2				3
		2						3
																1												1										1
From here it follows that
				2																																									2
																								2							2		2
2 | |					2		3 [						| |] 2											2 2						3				2		2									2		3 .
3																	1																									2
						1																		1								1														1
Then improper integral
	I				[ 2					22 2 32 2 2 2																							2 2 3				]dt
		6						1					1				1									2								1
					0
								( 2 )						| |dt						2 2 3					[ | |] dt.
													3									1
								0										0
Further, using					identities								(3.12)-(3.14)									we				get				inequality (3.36). Notice that
(t)	d ( (t))	,	t I , the function ( (t)),																					t I					is limited, so
	dt

177

Lectures on the stability of the solution of an equation with differential inclusions

	( 2 3 ) | | dt		( 2 3 ) dt		2 3 dt ,	I1 I2 I [0, ),
0		I1		I2

where ( (t As (

)) 0 )

when				t I ,
					1
		|			|, then
	2
					[
		2		3
0			1

	( (t)) 0		when
		( )
	| |] dt		2
			2		3
		(0)		1

t I	.
2
( )d .

From here evaluation (3.38) follows. Evaluation (3.37) follows from Lemma 4. The theorem is proved.

Lecture 28.

Global asymptotic stability

Based on estimates of improper integrals, we can formulate sufficient conditions

of global asymptotic stability of a stationary set	of the system (3.2), (3.3) for two
cases:

а)

( )d 0;

б)

( )d 0;

separately.

Theorem 9. Let the following conditions be satisfied:

1) matrix	А is a Hurwitz matrix,					( )	;
						0
2) there exists vector		*	R	n	such that

B

3) rank of the matrix

4)		( )d 0;

0, AB
P ||	*	,

		n 2	B
0,..., A
*	*	*n 1		*
A		,..., A

n 1	B
0, A
||

is equal to

0; n;

5) conditions 4 of the lemmas 1, 2 are satisfied, and let besides:

Ni i 0, i 1,n 1, 0 0.

Then the stationary set of the system (3.2), (3.3) is globally asymptotically stable.

Proof. Under conditions 1) - 5) of the Theorem, we get statements of Theorems 4, 6 for variables yn 1, y1,..., yn . In particular, from Theorem 4 when 0 0 , inequalities (3.24), (3.25) can be written:

178

Chapter IІI. Study of stability of solutions of dynamic systems with cylindrical phase space

			I					[ y							(t) y									(t) ...					y			(t)]dt l				0,
												2											2							2
				2						1		n 1								2		1				n 1				n					2
							0
																d
										l									*		(t)F z(t)]dt,				| l				| ,
											2							[z									2
																dt								2
														0
where	z(t) ( y1(t),..., yn 1(t)),											t I					. From here it follows that
		[ y							(t) y										(t) ...							y			(t)]dt l							.
						2												2										2
				1		n 1									2		1							n 1			n							2
	0
As follows from Theorem 6 we have the evaluation (v. (3.29), (3.30))
							[N			y	2		(t) N									y	2	(t) ... N							y		2	(t)]dt
											n 1									2								n	1				n
									1													1
					0
			( )																				( )
			( ) 2d l4															,						( ) 2d l4										,	| l4 | .
			(0)																				(0)

From (3.39), (3.40), we have

(3.39)

(3.40)

V (z(t))dt

[( N ) y

2

(t) (N

) y

2

(t)

...

n 1

2

1

2

1

(3.41)

0

0

... (N

) y

2

(t)]dt

,

V (z) 0,

V (0) 0,

n

1

n

n 1

2

(N2

2

(Nn 1

2

where V (z) (N1 1 ) yn 1

2 ) y1 ...

n 1 ) yn .

Further, applying Lemma 6 from evaluation (3.41), we get

lim y

(t) 0,

i 1,n 1,

t

i

by virtue of conditions 5) of the theorem. Notice that function z(t) ( y1(t),..., yn 1(t)),

t I

satisfies the conditions:

| z(t) | a, | z(t) | c, 0

a ,

0

c .

As follows from Lemma 2, if lim yi (t) 0,

lim x(t) 0

i 1, n , then

. Accor-

t

t

ding to the statement of the theorem 2, function (t) 0 yn 1(t) 1 y1(t) ... n yn (t),

t I .

Then lim (t) 0 . Consequently, lim (t)

*

, (

(t)) (

*

) when

t

. So,

t

t

t . It

is easy to

verify that

the

pair

x(t)

x* 0 when

t , (t) * when

(x* 0, * ) . The theorem is proved.

Theorem 10. Let the conditions 1) – 4) of Theorem 9 be satisfied, and let besides:

1)conditions 5 of the lemmas 1, 2 be satisfied;

2)M0 0 0, Ni Mi i 0, i 1,n 1.

Then the stationary set of the system (3.2), (3.3) is globally asymptotically stable.

179

Lectures on the stability of the solution of an equation with differential inclusions

Proof. Under the conditions of the Theorem, we get statements of Theorems 3, 4, 6. As follows from formulas (3.21), (3.22), (3.24), (3.25), (3.29), (3.30) the following ratios are true

	I				[ M			y	2			(t) M			y		2			(t) ... M				y		2	(t)]dt l 0,
		1					0		n 2					1			n 1				n 1					n
																													1
				0
	I					[ y							(t) y						(t) ...			y			(t)]dt l				0,
										2						2							2
			2				0			n 2			1			n 1				n 1			n					2
					0
	I4								2							2						2		(t)]dt l4
				[N1 yn 1							(t) N 2 y1							(t) ... N n 1 yn											,
					0

where |l1| < ∞, |l2| < ∞, |l4| < ∞. From here it follows that

[( M

) y

2

(t) (N

M

) y

2

(t) ...

0

n 2

1

1

n 1

0

1

0

... (N

M

) y

2

(t)]dt l l

l

,

n 1

n 1

n

2

4

n 1

1

where M

0

0,

N

i

M

i

0,

i 1,n 1

. Scalar function

0

i

V ( (t)) ( M0

) y

2

(N

M

) y

2

... (N

M

) y

2

, (t) ( y (t),...,

n 2

1

1

n

1

n 1

n 1

n

0

1

n 1

1

Further, applying Lemma 7, taking into account that

V (x) 0, x,

and due

to

inequalities

(3.42)

V ( (t))dt , we

get

lim (t)

0

t

0

(3.42)

yn 2 (t)),		t I .
x R	n 2	, V (0) 0	,

. Consequently,

lim x (t) 0,		i 1, n,
t	i

lim (t) 0, t

(t)	,
*

( (t)) (	) 0
*

when t . The

theorem is proved.

Theorem 11. Let the conditions 1) – 4) of the Theorem 9 be satisfied, and let besides:

1) conditions of the lemmas 3, 4, 5, 6 be satisfied;

2) M

0

H

0

0, M

i

H

i

N

, i 1,n 1.

0

i

i

Then

h0 (t) h1 (t) h2 ( (t)) 0

when t .

(3.43)

If in addition the solution of the system of the second order

t I

(3.44)

h0 h1 h2 ( ) 0,

is globally asymptotically stable, then the stationary set of the system (3.2), (3.3) is globally asymptotically stable.

180