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<<< < Предыдущая 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 2223 / 2323

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

Lecture 35.

Global asymptotic stability of multidimensional dynamic systems II

А. Consider the case when

let,

	2

		i
i		i
			( )d		0,		, R		, i = 1, m.
								1
		i	i	i		i	i

	i

Theorem 6. Let the conditions of the lemmas 1-5 and Theorem 1 be satisfied, and

moreover, the		function	( ) 0	and	satisfy	the condition (4.29), matrix
*						1 = (11, . . . , 1 ) > 0,
= 2 .	Then	for any	diagonal		matrices	1 = (11, . . . , 1 ) > 0,

= (21, . . . , 2 ) along the solution of the system (4.13) the improper integral

(t)S1

y(t) y

(t)S2 y(t) y

(t)S3 y

(t) S4 y(t)]dt

= [ y

( )

( i

) 2 d

[ y

(t) 1 y

(t)]dt ,

(0)

(4.39)

where

S		=		,	S		= P		T		,	S		=		P	*	T	,
																	*
	1		5			2	1	4	3	1			3		3	2		1

S	4	=	6	P	T .
	4		6	3	2

Proof. As conditions of Lemmas 1-5 and (4.29), then we get evaluations (4.23), (4.28), integrals

Theorem 1 are satisfied, conditions (4.30). Then the sum of improper


y	*	(t)
y		(t)

I	1	I

2	y(t)

2I3

y* (t)



0 (

(t)

y(t) y

(t)(P T

) y(t) y

(t)

y(t)

T ) y(t) y

(t)(

P T

( )

d .

) y(t)]dt

(0)

(4.40)

As follows from Lemma 5 identities hold

(t)

y(t)

1 d

[ y

(t)

y(t)] y

(t)

y(t),

2 dt

t, t I. Substituting values y

* (t)

y(t),

y* (t)

y(t), t

from (4.40) we get an

assessment (4.39). From here taking into account (4.15). (4.16) we get I5

. The

theorem is proved.

Theorem 7. Let the conditions of Theorem 6 be satisfied, and let besides:

1) Matrix

S S*

(S4

S4 )

2) matrix

(S2

S2 )

221

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

Then a stationary set	of the	system (4.1),
stable.
Proof. Let the function	z(t) ( y(t), y(t)), t
following estimates are true	z(t) a,	z(t) c, t,

proper integral (4.39) can be represented as

(4.2) is globally asymptotically

I. As follows from Lemma 6 the t I , 0 a , 0 c . Im-


I5	*	(t)S1 y(t) z	*
I5	= [ y	(t)S1 y(t) z		(t)Wz(t)]dt .
	0

(4.41)

As the quadratic form

)]y

0, then from (4.41) we have

y [

W W

(t)Wz(t) ,

where z(t)

Consequently, all conditions of Lemma 7 are satisfied.

a, z(t) c, t I.

Then lim z(t) 0.

From here we have lim

y(t) 0,

lim y(t) 0. As x(t) K

y(t),

t I , then

lim x(t) 0.

follows from

identities

(4.17),

the function

( (t)) H 0 y(t) A11 y(t),

t I ,

then

lim (

(t)) H0

lim y(t) A11 lim y(t)

___

Notice

that

lim (t) ( C

R A11 ) lim y(t) RH

lim y(t) 0.

Consequently,

lim (t)	*	,	lim ( (t)) (lim (t
t			t	t

(t)	*	when	t .			It is	easy
(t)		when				It is	easy

theorem is proved.
5. Consider the case, when
			i i				___
				i	( i	)d i	i
				i	( i	)d i	i

)) (	*	) 0.
	*
to verify		that

i , i

So,

the

x(t) x	0
*

pair	(x	0,
	*

, i = 1, m.

when

	)
*

t ,. The

Theorem 8. Let the conditions of the lemmas 1-5 and Theorem 1 be satisfied, and

function ( ) 0

let, moreover, the

satisfy

the

condition

(4.33), matrix

= 2 .

Then

for any

diagonal

matrices 1

0, 3 , 4 0,

0 such

that

4 4 5

( 3 )( 3 )

___

and

along

the

solution

the

system (4.13)

the

improper integral

222

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

where


						*	(t) 1 y(t) y						*	(t) 2 y
I		6	= [ y				(t) 1 y(t) y							(t) 2 y			(t)
			0
						m			i	( )								d
						m			i
										i ( i )			3i d i							[
										i ( i )			3i d i							[
						i=1				(0)							0	dt
									i
	1	=		5	,			2		=	4					,			3	=
	1			5				2			4	1		3	1				3

y	*	(t

1	y		*

2

	3

)		y	(t) y	*	(t)E		y(t)]dt
	3					4

2 y y* 1 y]dt < ,

P2* 1 , 4 = 6 P3 2 .

(4.42)

Proof. For the given case the following estimates are true (4.23), (4.28), (4.34). Then the sum of improper integrals


		I		I			I				[ y			(t)			y(t) y		*	(t)( P ) y						(t) y		*	(t)			y(t)
			1			2		3					*						*									*
			1			2		3								5							4	1	3						1
																5							4	1	3						1
											0
																											m			( )
y		(t)			y(t) y					(t)( P								) y(t) y			(t)(		P ) y(t)]dt				m		i			(	)		d .
	*								*						*		*		*
				2								3		2			1					6		3	2						i	i		3i	i
																											i 1			(0)
																													i

From here considering identities

y	*		y(t)	1 d	[ y	*	(t)		y	(t)],

		1		2 dt				2

y(t)

( y

(t)

y(t)] y

(t)

(t),

t I ,

we get an assessment (4.42). The theorem is proved.

Theorem 9. Let the conditions of Theorem 8 be satisfied, let besides:

	E1	*		0;
1)	E1	E1		0;
			1		(E		E	*	)		1	E
2)			2		(E	4	E	4	)		2	E	3
2)	W		2								2
	1				1		*			1			*
	1
					2		E3			2	(E2	E2
					2					2



	0.

)

Then a stationary set

of the system (4.1), (4.2) is globally asymptotically

stable.

The proof of the Theorem is similar to the proof of Theorem 7. Б. Consider the case when

i i

i ( i )d i 0,

___ i = 1, s;

i i			___
	i	( i )d i	i 0,	i = s 1, m	(4.43)

where (1)			( (1) ) (				(	),...,				S	(	S	)), (2) ( (2) ) (			S 1			(	S 1	),...,			m	(	m	)), (1)	(	1	),...,	S	),
						1	1					S		S				S 1				S 1				m		m			1		S
(2)	(	S 1		,...,	m	),
		(1) (			(1) ) H					y						___ (1)	(2) (								y	___ (2)
		(1) (			(1) ) H				01	y	(t)					A 11 y(t),	(2) (		2	) H				02	y		A 11		y(t), t I ,
									01										2					02

223

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

(1)

(1) (t) = (C

y(t),

R A11

) y(t) R H

(2)

(2) (t) = (C

) y(t) R

t I.

A11

Improper integrals

[ y* 1 y y* 2 y y* 3 y y* 4 y y* 5 y y* ]dt 0,

[ y

P y y

y y

P y]dt 0,

y y

y]dt

i ( )

(

[ y

)

i 1

(0)

y y

y]dt

i ( )

(

d .

[ y

)

i S 1

(0)

Theorem 10. Let the conditions of the lemmas 1-5 and Theorem 1 be satisfied,

( ) 0

and let, moreover, the function

satisfy the condition (4.43), matrix

= 2 .

Then for any diagonal matrices

__	__	__		__
3	diag (	31 ,...,	3m S ),		4

diag

= (1, . . . , 1

__	__
(	41 ,...,	4m S ) 0,

		__		__	__
),		2	diag (		21,..., 2S ),
__			__		__
	5	diag (		51 ,..., 5m S )		0,

__	__	__	__
4	4	5 (	3 )( 3 ) 0,	diag( 1 ,..., m S ),

__	__
0,

__	__	__
diag (		1 ,..., m S ),

__	__		__
diag (		1	,...,	m S	),
		1		m S

I	7	I	1	I	2	I	30

along the solution of the system (4.13)the improper integral

D y y

D y

D y]dt

[ y

S i

( )

y* 2 y y* 1 y]dt < ,

i ( i ) 2i d i

i ( i ) 3i d i

[

i=1

(0)

i=S 1

(0)

dt 2

where D =	5	,	D	2	=	4				,	D =	3	P* ,
1	5			2		4	1	31	1		3	3	2	11

(4.44)

D	=	6	P	.
4		6	3	21

The proof of the Theorem is similar to the proof of Theorem 8.

Theorem 11. Let the conditions of Theorem 10 be satisfied, and let besides:

1) D D* 0;
	1	1
			1			*		1
				(D4		D4 )			D3
2)			2	(D4		D4 )		2	D3
2)	W		2	1			1	2		0.
	2			1			1			*
	2
					*			(D2
					D3					D2 )
				2	D3		2			D2 )
				2			2

224

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

Then a stationary set of the system (4.1), (4.2) is globally asymptotically stable.

The proof of the Theorem follows from evaluation (4.44).

Lecture 36.

The solution of the model problem of the phase system

Equations of the phase system have a form
̇ =			+					−			(			), ̇ = −2									− 1.03 − 0.03									− 0.75 ( )
1		1				2			2			2				2					1						2				3								1
̇ = −0.01					− 1.01							− 0.25							( )			+ ( ),
3			2								3							1			1				2		2
		̇= + +														(		), ̇= − + − +																(			),
			1					2			3				1		1				2				1		2			3		2			2
where
( )			= { ( ) = ( ( ),															( )) C							(R	2	1, R	2	) /			d ( )									,
																								1		2		2						1			1
		0										1		1		2				2											11			d						21
																																		d		1
																																				1
	d			(				)				,		( ) = (										),					(		) =		(			),
			2				2					,		( ) = (										),					(		) =		(			),
			2				2
12			d							22				1		1					1		1		1			2		2		2			2			2
			d			2
						2							, R ,											,			R }.
																		1										1
												1											2			2

(4.45)

(4.46)

In particular In this case

(		) = sin	,
1	1	1

	(	2	) = sin	2	,
2		2		2

(0,1),

=	2	= 2 .
1	2

1	2
		1	( 1 )d 1	= 0,
		1	( 1 )d 1	= 0,
	1

2 2

2 ( 1 )d = 0.

In the vector form equation (4.45) can be written as

			x = Ax B ( ), = Cx R ( ),						(4.47)
					1
where
1	1		0		0	−1		0	−1
= (−2	−1.03	−0.03), = ( 1, 2) = (−0.75				0 ), 1 = (−0.75), 2 = ( 0 ),
0	−0.01	−1.01			−0.25	1	−0.25		1
0,	1, 1		1,	0	C = (0,1,1),	C = ( 1,1, 1),		1	( 1 )
C =		,	R =	,	C = (0,1,1),	C = ( 1,1, 1),		( ) =	.
					1	2
1	1 1		0	1				2	( 2 )

	Nonsingular transformation. We choose vector * = (														, , )			so that				*B = 1,
													1	11	12	13						1	1
*B = 0.		Vector *						= (1; 5/3;1).		Similarly, we define vector *					= (	21	,	22	,	23	) from the
1	2					1								2		21		22		23
condition			*B = 0,					*B =1.		Vector		* = (0; 1/3;1).		Finally,		we		choose				vector
				2	1			2	2				2
	* = (	31	,	32	,	33	)	so that B = 0, B					= 0. Vector *	= (1; 1/3;1). The determinant
3		31		32		33			3	1	3	2	3

225

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

					<	,		>	<		,	2	>	<		,	3	>		43/9	14/9	23/9
					<	,		>	<		,		>	<		,		>		43/9	14/9	23/9
					1		1			1					1									16
(	,		,	) =	<			>	<		,		>	<		,		>	=	14/9	10/9	10/9	=	16	> 0.
(	,	2	,	) =	<			>	<	2	,	2	>	<	2	,	3	>	=	14/9	10/9	10/9	=		> 0.
1		2	3		2		1			2		2			2		3							9
					<	,		>	<		,		>	<		,		>		23/9	10/9	19/9		9
					<	,		>	<		,	2	>	<		,	3	>		23/9	10/9	19/9
					3		1			3		2			3		3

Consequently, vectors		,	2
	1

1 = (13/3; 8.12/3, −0.96), 2

,	3	are linearly

= (2/3; 1/3, −1),

independent. As vectors

3 = (5/3; 4/3, −1), then

1 = −

	*
where	y1 = 1

5.37

x,	y	*
		2

15.881 − 3

=	x,	y	3	=
2			3

̇ = −

+	18.37	; = −		5.37		−	15.88		+	18.37	,

2	3	3	1	3	1		3	2		3	3

	*		x.	Consequently,

	3
5.37				1	−	15.88	2 +	18.37	3 + 1( 1).
						3		3
		3

In a similar way, we find

y		=	5	y		2	y			(		),	y		=	3	y	8	y		29	y	.
	2				2			3	2		2			3						2
			3			3										4	1	3			13	3

As 1 = (0,1,1) = −1

C x = y		y	2	y	,
1	1		2	3

1 + 1 2 + 1 3, then

C		= ( 1,1, 1) =	1	*	1	*	,

	2		2	1	2	3

C	x =	1	y	1	y	.

2		2	1	2	3

From the above, it follows that equation (4.47) with nonsingular transformation is reduced to (4.see (4.46)).

	̇ = −		5.37					−			15.88				+		18.37					+ ( ),
	̇ = −							−							+							+ ( ),
	1			3		1						3				2	3			3					1	1
̇ = − 5				3	2							3					3		3					8		+ 29
̇ = − 5				+	2			+				2	(			), ̇ = −			3			−		8		+ 29				,		(4.48)
2	3	2			3	3						2		2		3		1	4	1			1	3	2	12				3
̇= − +				+			+					(		), ̇= −				1	−				1	+					(		),
̇= − +				+			+					(		), ̇= −					−					+					(		),
1	1	2				3				1			1			2	2		1			2			3			2		2
			( ) = ( 1( 1), 2( 2)) Φ0.
Matrix
										1						0	1
																											4
	P = 1 , 2 , 3						=			5/3						1/3	1/3			,		\| P \|=					4	0.
	P = 1 , 2 , 3						=			5/3						1/3	1/3			,		\| P \|=						0.
										1						1	1									3
										1						1	1
			*						1		5/3					1						0			1					1
				1

K = P*		= 2*				=			0		1/3					1 ,	K 1		=		3/4					0		3/4			.
				*
			3						1		1/3					1					1/4					1		1/4

226

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

−1.79	−15.88/3	18.37/3		1	0
= −1 = ( 0	−5/3	2/3	) , = = (0		1),
−3/4	−8/3	29/12	0		0

	1		1	1	1
C = CK
		=
			1/2	0	1/2

the equation (4.48) can be written in the vector form

y = Ay B ( ), = Cy R ( ), ( )	.
0

6. Solution properties. The characteristic equation of the matrix

( ) = ( 3 − ) = | 3 − | = 3 + 1.04 2 + + 0.98 = 0.

(4.49)

is equal to

1,04 >

all coefficients of the

0,98,	then the matrix	A

characteristic polynomial are greater than zero and is a Hurwitz matrix. Then, as follows from Theorem 1


[16] the following estimates are true		\|	x(t) \| c		,
				0
\| (t) \| c4 ,
	t I = [0, ). As matrices		A,	A

1,2,3. Consequently, the matrix From (4.48), (4.49) we have

is a Hurwitz matrix.

where

(4.50)


		(		1	(t))			1	0	0					H	0
( (t)) =		1		1	,	H		=		,	H		= (0,0,1),			0	= I	,
( (t)) =					,	H	0	=		,	H	1	= (0,0,1),				= I	,
			(				0		1			1					3
	2		(	2 (t))				0	1	0					H1

= (	11) ,		= (−1.79			−15.88/3			18.37/3) ,
= (	11) ,		= (−1.79			−15.88/3			18.37/3) ,					= (−3/4, −8/3,29/12).
12		11			0		−5/3			2/3			12
					0		−5/3			2/3

	From (4.50) it follows that
	( ( )) = 5.37			− 15.88				+ 18.37			+ ̇,			= ( ), = 1,2,3, ;
1	1		3	1	3	2		3		3		1
			3		3			3
		2	( ( )) = 5		− 2		+ ̇, , ̇= ̇( ), = 1,2,3;
		2	2	3	2	3		3	2
			̇( ) = 2.37				− 12.88				+ 21.37			+ ̇, ;
			1		3	1		3		2		3	3	1
					3			3				3
				̇( ) = − 1				+	5	− 7		+ ̇, .
				2		2		1	3	2	6	3		2

	Based on identities (4.51) improper integrals												I10 , I20 , I30 , I40 , I 30 , I 40 ,

calculated and conditions of global asymptotic stability can be established.

(4.51)

can be

227

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

Comments

A general theory of global asymptotic stability of multidimensional phase systems with countable equilibrium state, based on a priori estimation of improper integrals along the solution of the system has been developed.

The main results obtained in this chapter are:

–boundedness of solutions of multidimensional phase systems and their derivatives of the first and second-order was established;

–conditions under which еру solution of dynamic systems with countable equilibrium state and its derivative of the first order have asymptotic properties were determined;

–conditions of global asymptotic stability of multidimensional phase systems with equal to zero in period values of integrals from components of periodic nonlinearities were determined;

–conditions of global asymptotic stability of phase systems with not equal to zero in period values of integrals from components of nonlinear periodic functions were determined;

–asymptotic properties of solutions of dynamic systems with countable equilibrium state in the general case, when part of components of nonlinear periodic functions have values of integrals in a period equal to zero, and for the other part of the values of the components of integrals in a period not equal to zero, were studied.

Literature

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2.Andronov A.A., Vitt A.A., Hajkin S.Je. Teorija kolebanij. – M.: Fizmatgiz, 1959. – 600 s.

3.Barbashin E.A., Tabueva V.A. Dinamicheskie sistemy s cilindricheskimi fazovym prostranstvom. – M.: Nauka, 1969. – 305 s.

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5.Bakaev Ju.N., Guzh A.A. Optimal'nyj priem signalov chastotnoj moduljacii v uslovijah jeffekta Dopplera // Radiotehnika i jelektronika, 1965. – T. 10. – № 1. – S. 35–46.

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–S. 7–15.

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228

Chapter IV. Global asymptotic stability of multidimensional dynamic systems with cylindrical …

Еducational issue

Aisagaliev Serikbai Abdigalievich

LECTURES ON THE STABILITY OF THE SOLUTION OF AN EQUATION

WITH DIFFERENTIAL INCLUSIONS

Educational manual

Editor L. Strautman

Typesetting G. Kaliyeva

Cover design B. Malayeva

Cover photo is from www.freepik.com.

IB No.13799

Signed for publishing 05.10.2020. Format 70x100 1/12. Offset paper.

Digital printing. Volume 24,3 printer’s sheet. 70 copies. Order No.11430.

Publishing house «Qazaq University» Al-Farabi Kazakh National University KazNU, 71 Al-Farabi, 050040, Almaty

Printed in the printing office of the «Qazaq University» Publishing House.

229

«ҚАЗАҚ УНИВЕРСИТЕТІ» баспа үйінің

жаңа кітаптары

Ақанбай Н. Ықтималдықтар теориясы және математикалық статистика. ІІ: оқулық / Н. Ақанбай. – Алматы: Қазақ универси теті , 2017. – 458 б.

ISBN 978-601-04-2292-6 (ортақ)

ISBN 978-601-04-2294-0 (2-кітап)

Оқулық механика-математика факультетінің математика және басқа да маман дықтар студент теріне , магистрлеріне, докторанттарына және ҚазҰУ-дің білім жетілдіру институтының тыңдаушыла рына оқылған дәрістер негі зіндежазылдыжәнеҚРБжҒМ-ніңшешімімен2011жылы«Оқулық»грифі беріліп, баспадан шыққан аттас оқулығының өңделген және бірқатар жаңа материалдармен толықты рылған І бөлімінің тікелей жалғасы болып табылады. Оқулықтың бұл ІІ бөлімінде І-бөлімге кітап көлемінің көтермеуіне байланысты енбей қалған, бірақ «Ықтималдықтар теориясы және математикалық статистика» пәнінің типтік оқу бағдарламасы бойынша оқытылуға тиісті келесідей тараулар қамтыл ған : Туындатқыш функциялар; Сипаттамалық функциялар; Тәуелсіз кездейсоқ шамалар тізбектері үшін шектік теоремалар; Математикалық статистикаға кіріс пе; Таңдамалық теорияның негізгі ұғымдары ; Үлестірімдердің белгісіз параметрлерін бағалау; Статистикалық гипотезаларды тексеру теория сы элементтері; Кездейсоқ процестер теориясының жалпы және корре ляциялық теориясы элементтері; Стационар процестер теориясына кіріс пе; Марков процестері: Негізгі ұғымдар.

Оқулықтың құрылымы және оқулықтағы материалдардың орналастырылуы бұл оқулықты математика ғана емес, математикаға жақын басқа да (информатика, механика, физика, математикалық және компьютерлік мо дельдеу , т.с.с.) мамандықтар студенттері, магистрлері, докторанттарымен қатар оқытушыларының да пайдалануына мүмкіндік береді.

Аканбай Н. Теория вероятностей и математическая статистика: в 2 частях. Часть I: учебник / перев. с каз. языка Н. Аканбай. – Алматы: Қазақ уни верситеті, 2019. – 376 с.

ISBN 978-601-04-4228-3

В учебнике математические основы теории вероятностей изложены на базе аксиоматики А.Н. Колмогорова. В 1 главе материалы о случайных со бытиях и их вероятностях рассматриваются в рамках дискретного ве роятностного пространства. 2 глава посвящена общему вероятностному пространству. В 3 главе случайная величина определена как изме ри мая функция . Рассматриваемое в 4 главе понятие математического ожи дания введено как интеграл Лебега по вероятностной мере на общем вероятностном пространстве , при этом от читателей не требуется знание каких-либо сведений об интеграле Лебега. В 5 главе рассматриваются предельные теоремы в схеме Бернулли. В 6 и 7 главах изложены мате риалы об различных видах сходимости последовательнос тей случайных величин и о законах больших чисел.

Учебник рекомендован студентам, магистрантам и докторантам PhD, обучаю щимся по специальности «Математика».

Кітаптарды сатып алу үшін «Қазақ университеті» баспа үйінің маркетинг және сату бөліміне хабарласу керек. Байланыс тел: 8(727) 377-34-11.

E-mail: baspa@kaznu.kz, cайт: www.read.kz, www.magkaznu.com

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