NONLINEAR SYSTEMS WITH CONTINUOUS MOTIONS

J. Math. & Math. Sci.

VOL. 17 NO. 3 (1994) 587-596 587

SOLUTIONS TO LYAPUNOV STABILITY PROBLEMS:

NONLINEAR SYSTEMS WITH CONTINUOUS MOTIONS

LJUBOMIR T.

GRUJIC Faculty

of MechanicalEngineering

Universityin

Belgrade P.O. Box

174, 11000 Belgrade

Serbia,

Yugoslavia

(Received January 23,

1991andinrevisedform

May 27, 1992)

Abstract. The

necessary

and sufficient conditionsforaccurate constructionofa

Lyapunov

functionand the

necessary

and sufficient conditionsforaset tobe theasymptotic stabilitydomain arealgorithmically solved fora nonlineardynamical systemwith continuous motions. Theconditions are establishedby utilizing

properties

of o-uniquely bounded sets,whichareexplainedinthe

paper. They

allow arbitrary selectionofano-uniquely boundedsettogeneratea

Lyapunov

function.

Simple examples

illustrate the

theory

anditsapplications.

Key

Words and Phrases:

Stability, Lyapunov

Method,

Lyapunov

Functions, Nonlinear

Systems,

Dynamical

Systems.

1991MathematicsSubjectClassificationCodes:

34A34, 34C35,

34D20,

70K15, 70K20,

73H10, 93D05, 93D20.

1.

INTRODUCTION

In

hisfundamentaldissertation

1] Lyapunov

^referredto

papers by

Poincar6

[2], [3]

^{as those}inspiring him to establish a methodthathas become fundamental for

qualitative

andstability

analysis

ofmotionsof a

very general

class ofnonlinearsystems.

The promising methodologicaleffectivenessofthe

Lyapunov

method has not been

fully

achieved due to the need to construct asystem

Lyapunov

function.

Significant

results on a

Lyapunov

function

generation

were initiated

by

Zubov

14].

^Theliteratureonthe

Lyapunov

^methodis too vast

[9]-[ 11],[ 13],[ 14]

to bereferred to herein.

The

problem

of the

necessary

andsufficientconditions for

constructing

a

Lyapunov

functionandthe

problem

ofthe

necessary

and sufficient conditionsforasetto bethe

asymptotic stability

domainhave not yetbeen solved.Solutionstotheseproblemswill be established

by

using properties^ofo-uniquelybounded sets.Theirfeatureswill beexplained

briefly by referring

to

[7],[8],

^where

they

werediscoveredand studied.

2.

NOTATION A,R"DA

B- {x-Ilxll

^<

},R" _Bo,

B--,- {: Ilxll ^},R" _,,

OB,- {x: Ilxll ^{},R" OB,,}

C(S)

an

open

connected neighborhood ofx

-0,

an

open hyperball,

the closure of

Ba,

the boundary ofboth

Ba

^and

^Ba,

thesetof allfunctionsof xcontinuouson

S,

D o,D,,D,R

D

Dt.,

^{the domain}of attraction,ofstability,ofasymptotic stability, respectively,ofx 0,

the Dini derivativeofv

along

thesystemmotion

(Yoshizawa 13]),

afamily offunctions determinedbyDefinition

5,

agivennonlinearvectorfunction,

thelargestsubintervalof

R/

over which a motion

x(t; Xo)

exists,

the dimension of thesystem,

an

open

connectedneighborhoodofx-0, the interior of

N (in

^fact

N),

the setofreal

numbers, [0,+oo[- {a: a eR,0 +o},

an

open

neighborhood of x

0,

ano-uniquelybounded set,

thegeneratingfunctionof theo-uniquelyboundedset

U,

asetgenerated bythe function u and apositivenumber

t;,

a tentative

Lyapunov

functionofthesystem, thesystemmotion

(solution), x(t;Xo),-x(t), x(O;xo)

Xo,

Euclidean norm on

R’,

theemptyset.

D’v(x) limsup{<v[x(O;x)]- v(x))/O:

^{0 0}

/}

f:R’--,R"

n

e{l,2 } N,R" D_N,

R R/

S,R"

^D_S

U,R"DU, u:R" R

u= ^{{x: u(x)}

^<

^}

3.

SYSTEM DESCRIFFION

Systems

tobe

analyzed

aredescribed

by

thefollowing equation

They

areassumed to

possess

eitherof the

following

^twofeatures:

Weak Smoothness

Property:

(i)

^{Thereis an}

open

neighborhood

S

ofx 0,

R" _.D S,

suchthat for

every Xo S (a)

^thesystem

(1)

^{hastheuniquesolution}

x(t;Xo) through

_x0at 0,and

(b)

^themotion

x(t;Xo)

is definedand continuous in

(t,xo) Io ^S.

(ii) For every

_{Xo tE}

(R" -S) every

motion

x(t;xo)

^{of the}system

(1)

is continuous in

Io.

Strong

Smoothness

Property:

(i)

The system

(1)

^{has Weak}Smoothness

Property.

(ii)

(3.1)

If the

boundary

0S of

S

is

non-empty

then

every

motion of the system

(1)

passing

through Xo

G 0S at 0

obeys inf[I x(t;Xo)l[ I0]

^>0 for

every Xo

^/oS.

DEFINITIONS

ON THE DEFINITIONS OF STABILITY DOMAINS

For

thedefinitionsof the attraction domain

D,,

^see

[4]-[6],[9],[ 11],[ 14].

The stabilitydomain

D,

and

NONLINEAR SYSTEMS WITH CONTINUOUS

theasymptotic stabilitydomain

D

ofx 0are defined in

[5],[6]. We

shallrefertothose definitions in the

sequel.

For

the system

(1)

^{with WeakSmoothness}

Property,

the stabilitydomains are mutuallyrelated as follows:

LEMMA

1. Ifthestatex 0of thesystem

(1)

^{possessingWeakSmoothness}

Property

has both the domain of attraction

Do,

S

2)Do,

and thedomainofstability

D,,

thenthey and theasymptotic stability domain

D

are interrelatedby

D, 2)D, D-D,.

PROOF. Let

x 0 have

D, S

2)

D,

and

D,.

Thenithas also

D

because

D D, CID,

and both

Do

andD,

areneighborhoodsofx 0

[5],[6]. Letx0 D,.

^Then

^x(t;xo)

^{0ast +. This}and continuity

ofx(/;x0)

in

Io (Weak

^Smoothness

Property)

imply

max[ll x(/;Xo)[[ ^R/]

^ct<+oo.

^Let

^{e 2 a.}

^Hence,

x(t;x0)ll

^<e,

^{vt R/,}

^whichyields

^[5],[6]

Xo

D,

^so

thatD, D_Do

^andD

-D,, -D, tqD, [5],[6].

4.2

ON THE DEFINITION OF A POSITIVE DEFINITE FUNCTION

Thenotionofapositivedefinite function is used in abroader

Lyapunov

sense

[1 ].

DEFINITION

1.

A

function v:

R" R

^{is a}positivedefinite if andonlyif there is an

open

connected

neighborhood A

^{of x} 0,

R"

²⁾

A,

such that

1) v(x)

isuniquelydeterminedby x

CA

and v is continuous

onA" v(x) C(A), 2) v(0)-0,

and

3) v(x)

^{>0 for}

every (x 0)

6A.

4.3

DEFINITIONS AND PROPERTIES OF O-UNIQUELY BOUNDED SETS

O-uniquely boundedsetswere introduced,definedandstudied in

[7],[8].

DEFINITION

2.

A

set

U, R"

²⁾

U,

iso-uniquely boundedifand

only

ifit isboundedand for

every (x

^#

0) R"

there is

exactly

onepositivenumber

L,

k-

(x;U),

such that

(kx) OU.

DEFINITION

3.

A

functionu:

R" R

isradially increasingon an

open

neighborhood

N

ofx 0 if andonlyiffor

every (x ,, _{0) N}

_{andanyIx, 1, 2,obeying}both0 Ix1^<Ih^{and Ix.,x}

N

it satisfies u

Ctx

^<u

^{(o,:. ).}

PROPERTY U. Let N

be an

open

neighborhood of x 0and

U, N

²⁾

U,

be agivenboundedset.

Thereisafunctionu"

R"

^--,

R

that

obeys

the

following:

(a)

u is continuous

onN:u(x)C(N), Ca)

ifN-R" then

u(x)

^+ooas

Ilxll

^+,

(c) ^u(0)-

^0,

(d) u(x)

>

O

forall

(x , _{O) N,}

(e)

there ispositive number

, ^(U),

such that both 1.and2.hold:

1. u

(x)

^forx

N

if andonlyifx

U,

2.

u(x)

forx

N

ifandonlyifx

OU,

(f) u(kix) ,

1,2, holds for

any (x ,, _{0) N}

_ifand

_{only ifk L2 x; U) ]0, +oo],}

(g)

u isradially

increasing

on

N.

Definition 2

implies

the next resultduetoDefinition

2, Corollary

1and

Proposition

4in

[8].

LEMMA

2.

For

abounded subset

U

ofan

open

neighborhood

N

ofx 0 to beo-uniquely bounded it isboth

necessary

and sufficient that itpossesses

Property U.

DEFINITION

4.

(i) A

^functionuisthe generating function on

N

of an

o-uniquely

boundedset

U

if andonlyif

they

have

Property U.

(ii)

The function uisthegeneratingfunction of theuniquelyboundedset

U

if andonlyiftheyobey

(i)

^for

N R".

Lemma

2and

Definition

4implythefollowing

corollary [8].

COROLLARY

1. Ifa function u is thegeneratingfunction on

N

of ano-uniquelyboundedset

U

thenforany ^>0forwhich

N __. _N;

the subset

U;

^of

N

^{is a}connected

open neighborhood

^{of x 0 that is} also ano-uniquely boundedsetwiththegeneratingfunction u on

N.

5. SOLUTIONS VIA O-UNIQUELY BOUNDED SEI We

shall make use of thefamily

E(S;f)

^{defined asfollows.}

DEFINITION

5.

A

^functionu"

R" R

belongstothefamily

E(S;.f)

if andonlyif

1)

^uiscontinuous on

S;

u

(x)

E

C(S),

and

2)

^thefollowing equations alongthe motions ofthesystem

(3.1),

D/v(x) -u(x), (5. la)

v(0)-0, (5.b)

have a solution v that is well defined in

R

and continuous for

every

x

-,

^{for some}

t el0, +, t t(u,/3.

TItEOREM

1.

In

orderfor the state x -0 of thesystem

(1)

^with

Strong

Smoothness

Property

^to

have the domain

D

ofasymptotic stability andfora set

N, R" __.N,

tobe the domain of itsasymptotic stability,

N -D,

^itisbothnecessaryand sufficient that

1)

^theset

^N

^isan

^open

^connectedneighborhood of x 0and

S _D N, 2) f(x)

^{0 for x}

E N

if and

only

if x 0,and

3)

^forarbitrarily selectedo-uniquelybounded set

U, S

D

U,

withthegeneratingfunction u on

S

obeyingu

E(S ;.f),

^theequations

(5.1)

^{have aunique}solution function v on

N

with thefollowing

properties:

(i)

^{v is}positivedefiniteon

N,

and

(ii)

^{if the}boundary

ON

of

N

isnon-emptythen

v(x)

^+ooas x

ON,

x

tEN.

PROOF.

_Necessity.

Let

x-0 of the system

(3.1)

^with

^Strong

^Smoothness

^Property

^{have the}

asymptotic stabilitydomain

D.

Definitionsof

Do

^and

^D [5],[6]

^showthat it has also the attraction domain

Do, Do

^D_D.

^It

^isaneighborhoodofx-0 duetoDefinitionof

D,

^and

S is

aneighborhood^{ofx -0 in} viewofthesmoothness

property. Hence, D,,

^fqS

,, _{O. Let}

us

prove S D D.

^If

^{OS O,}

^then

^{S R"}

^and

S

_D

D

^dueto

^R"

^:3

Do.

^If

^OS

^{0,then we}shall consider both x0tE

OS

and

x

^ft.

^{(R" ).}

^Ifx

. ^OS,

^then

x0

D,,

dueto

(ii)

^of

Strong

^Smoothness

Property.

Therefore,

OS

f’lD-0. ^If

x ^{e (R"-),}

^{then for}

x(t:x)

^{0as +ooit}is

necessary

that there is

t" tER/

such that

x(t*;x) tE

0S, because

D

and

S

are neighborhoodsofx

0,x

and the motion

x(t; x0)

is continuous in

R/

^dueto

(ii)

^{of WeakSmoothness}

Property

ensuredby

(i)

^of

Strong

Smoothness

Property. However, x(t’;x)

^tE

OS

impliesthat

x(t;Xo)

does not

converge

to x-0 because of

(ii)

^of

Strong

Smoothness

Property.

This yields

x ^D

^and

(R" -S)

^fqD

O. By

connectingthe aboveresults,thatis

Do

^tqS

,

_0,

_D

_{f30S 0and}

_Do t"I(R" -S) ^O,

weconclude that

S _D Do.

^Therefore,

^D D ^{(Lemma 1)}

^and

^{S D. Let N D}

^{so that}

^S _ ^{N. Hence,}

N

is

open

connectedneighborhoodofx 0 due to

(i-b)

^{of WeakSmoothness}

Property, N D Do,

and invariance

olD,,

withrespect^tosystemmotions

(Theorem

^1.5.14

by

Bhatiaand

Szegti [4],

Theorem 33.3 byHahn

[9]).

^This

proves

necessityof the condition

1). ^{From N -D} -D,,, D,

_D

D,,,

and Definitions of

Do

^and

^D

^{it resultsthat x 0isthe}unique equilibriumstateofthesystem

(1)

ⁱⁿ

N,

whichimplies

f(x)

⁰ for x

E N D

if andonlyifx 0

(Proposition 7

in

[6])

^and

proves

necessityof the condition

2).

From N D

itfollows that the interval

I0

^ofexistence of

x(t;Xo) equals R/, I0 ^R/,

^{foreveryx C}

^N,

duetoDefinitionsof

D,,, Ds

^and

^{D [5],[6]. Let U}

^bearbitrarilyselected

open

o-uniquelyboundedsetsuch that

N U

and itsgeneratingfunction uon

S

obeysu

6/E(S;f).

^Sucha set

U

exists becauseSisopen neighborhoodofx 0

CLemma ^2).

^Definition3,

Property U,

and

Lemma

2show that the function u is also positive^definiteon

S.

Since

S

_2)

N D

then the function u is thepositivedefinitegenerating^functionon

N, too.

Thepropertyof u

E(S;f)

ensuresexistenceof

Ix

^>0 such that there exists a solution functionv totheequations

(5.1),

which is well defined in

R

and continuousfor

every

x

6/B,,

that is that

vCx)l

^{+oo for}

^every x’,

^and

v(x)C(B’-. ^(5.2)

Let ]0, +oo[

be such that

B,IqU

^D

U. ^(5.3)

The existenceofsuch isassuredby

Corollary

1.

Let

x

[0, +oo[,

x-

X(Xo;f;u; ),

be such that forany Xo

N

thefollowingconditionholds,

x(t;xo) U _s

for

every Ix, +oo[. (5.4)

Such :exists in view ofDefinitionsof

D,,

^and

D, D,, ^D, N D

and

xo ^N.

Notice that

xo ^N

^implies

also

xC+o;xo) o. (5.5)

After integrating

(5.1a)

^from

6R/

to +oo wederive

v[x(+oO;Xo) ]- v[x(t;xo) f, u[x(o;xo)]do

^for

every (t,xo) U.R/xN. (5.6)

Since u 6

E(S;f)

^{then the}

following

holds,

v(0)-0. (5.7)

Now, (5.5)-(5.7)

^yield

v[x(t;Xo) f, u[x(o;x0)]do

^for

^every (t,Xo) R/xN (5.8)

Thiscanbe written in the

following form,

v[x(t;Xo)] f. ^utx(o;x

^for

^every (t,X0) R/xN. (5.9)

Positive invarianceof

D

withrespect tosystem motions,

N -D,

continuity ofthe motions

x

due tothe smoothnessproperty, continuity ofu on

N,

the definitionof’

(5.4)

^and

(5.2),

^andcompactness^of

[;, t]

for

any R/ prove

u[x(O;Xo)]do <+oo for

every (t,xo) _R.xN. (5.10) From (5.2)-(5.4)

^weobtain

u[x(O;Xo)]do

^{< +oo for}

^every xo ^N.

(5.9)-(5.11) together prove

boundedness of

v[x(t;xo)] expressed

as

v[x(t;Xo)]l

^{<+oo for}

^every (t,Xo)R/xN.

Hence,

by setting -0 and x -x in

(5.12)

^wederive

v(x)l

^{< +oo for}

^every

^x

^N.

(5.11)

(5.12)

(5.13)

Continuity of themotion x inXo

e N,

continuityofuinx

e N,

andofvinx

e ff,, -,

^_D

U--;,

^positive

invarianceofN

-D wiih

respecttosystemmotions,

(5.4), (5.9)

and

(5.12)

prove continuity^ofvinx

N

v(x) eC(N). (5.14)

Positiveinvariance of

N

withrespecttosystem motions,positivedefinitenessof uon

N

and

(5.8)

imply

v(x)>O

for all

(x ,0)N. (5.15)

Now, (5.7), (5.14)

^and

(5.15)

prove necessityof thepositivedefinitenessofvon

N.

To prove

uniquenessof the solutionv to

(5.1.ab)

^weshallsupposethat there aretwosolutions

v

^and

v to

(5.1). Hence,

Vl(X)-v2(x)" fo {U[Xl(;x)]-u[x2(;x)]}d

for every

x0eN. ^(5.16)

Since

u(x)

^isuniquelydeterminedby x @N,due to

(a)

^of

Property U

and Definition4, and themotion of thesystemisuniquethroughx0,

xl(o;xo) xz(o;x0)

and

u[xl(o;xo)] u[x2(o;x0)]

^{so that}

vl(x0) v:,(xo)

⁰ for

every

_x0

N.

This

proves

uniquenessof the solutionvto

(5.1)

^andcompletesthe

proof

of

3(i).

Let ON

be non-empty,

x,x

xk, bea

sequence

convergingto

x’,

_xk

x’

^ask +=, where

x, N,

for allk- 1,2 and

x’ ON. Let ]0, +oo[

bearbitrarilychosen so that

U {x" u(x)< }, U _D U s.

^{Such exists}because theset

U

iso-uniquelybounded and the functionu isits

generating

^function on

N (Definitions

^{2 and3,}

Property U, Lemma

2and Definition

4).

Theset

U

is a connected open neighborhoodofx-0

(Corollary 1). Let T, T- T(x,)[0, +oo[,

be the first instantobeyingthefol- lowing

x(t;xk)U

^forall

^{[T,+oo[. (5.17)}

Theexistence of such

T

^isguaranteed by

x, N

andN

-D (Definitions

^of

Do

^and

D [5], [6]).

^Continuity ofthe motions

x

in

(t,x0)_RN

due to

Strong

^Smoothness

Property

and

-N-D (Theorem

^{33.1 by}

Hahn

[9])

^and

^S

^_.D

^D

^imply

T

^{+ooask +oo}

(Theorem

^33.2byHahn

[9]). ^Let

m be a natural number such that

x,

G

(N Us)

^{for allk m,m+ Such}rn exists because

N

isopen,

N D U

and

x, ON

as k +=.

Let a’

be definedby

(18),

ct’ min[u(x):x (N Us) ^(5.18)

The o-uniqueboundednessofthe set

U

s, the fact that the function u is itsgeneratingfunction on

N(corollary 1), N D U,

and

U _D U

guarantee

(Property U

and

Lemma 2)

^that

^ct’

^defined

by (5.18)

^satisfies

ct’- e]0, +oo[. (5.19)

From (5.9)

^weget,^afterreplacing^x

by T,,

v[,,t;x.)]- j, ,.[,,o;x)]do

+

,[,,o;x)]do ^eor

^vry

^t.x.)

and for k- m,m+1,....

(5.20)

Setting 0in

(5.20)

^andusing

(5.18)

^and

(5.19)

^wederive

Tk

v(x,,a f,do+ ,u[x(o;x,,]do

^for

^x,_N

^andall

^{k-m,m+a (5.21,}

Positive invarianceof

N- D

withrespect^tosystemmotions,positivedefinitenessofuon

N,

and

(5.21)

imply

v(x,) T,

^for xk

N

and all k m,m +1

(5.22)

Since

T,

+ ask +oo, the lastinequality,the definitions of

T,, T, T(x, ,),

^and

ofxk,

and^ct>0imply

v(x,)-+oo

^as

x,--ON

^{dueto k+oo,}

x,N,

which

proves

necessityof the condition

(3-ii).

Sufficiency. Let

all the conditions ofTheorem hold. Then,

S _D N. Two

_possiblecaseswillbe considered

separately: a) N

isabounded set,

b) ^N

^isanunboundedset.

a) ^{Let N}

be a boundedset. Then,underthe conditions of thetheoremtobeprovedall theconditions of Theorem byVanelli andVidyasagar

[12]

are satisfied, which

proves N

D.,. Since

Do D (in

^view of WeakSmoothness

Property

implied by

Strong

^Smoothness

Property

and

Lemma 1), ^{N -D.}

b) ^Let N

be an unbounded set. Underthe conditionsofthetheoremtobeprovedthezero statex 0 of thesystem

(1)

isasymptoticallystable

(cf.

^Yoshizawa

[13]). Hence,

ithasthedomain ofasymptotic stability

D.

Sinceboth

N

and

D

are

open

connectedneighborhoods of x 0,

N ND , _{O. (5.23)}

Since

S __. N, S

isalso unbounded. If

OS

isempty,then

S R",

whichimplies

S

:)

D.

If 0Sis non-empty, then

OS ND

0 dueto

(ii)

^of

Strong

Smoothness

Property

andDefinitions

ofDo, D,

^and

D [5],[6].

^This

result

impliesS _D D

becauseboth

D

and

S

areneighborhoodsofx 0and

D

isalsoconnected.

Altogether,

in bothcases

S

_D

D.

Weshalltreat

separately

the cases ofnon-empty

OD

and ofempty

OD.

The definition of the function

v, S D ^D,

^{and the}

proof

^ofthenecessity part

prove continuity

ofvon

D

and

v(x)

^+ooas

x

OD,

whichtogetherwithcontinuity of^valso on

N, S _D N

and

v(x)

+asx

ON [the

^condition

3(ii)]

^implyboth

OD

fN and

D ON

Theseequationsand

(5.23)

prove both

OD ON

and

D N

duetothefact that both

D

and

N

are

open

connectedneighborhoods ofx 0.

Let

now

OD

beempty. Then

D R’. Hence,

vis

positive

definiteon

R" (see

^the

proof

of thenecessity

part).

Thus,it is continuous on

R",

whichimplies

v(x)

< +oofor

every

x

R". Therefore, ON R"

-0 duetotheconditions

3(ii),

whichyields

N- R"

sothat

N- D. Finally, N -D

in all thecases,which

completes

the

proof.

|

Theconditions

slightly change

ifthe

system (3.1) possesses

Weak Smoothness

Property

rather than

Strong

Smoothness

Property.

THEOREM

2.

For

the statex 0 ofthesystem

(1)

possessing Weak Smoothness

Property

tohave the domain

D

ofasymptotic stabilityand that a subset

N

of

S, S _D N,

equals

D" N D,

it is both

necessary

and sufficient that

1)

^theset

N

is an

open

connectedneighborhoodof x 0,

2) .f(x)

0 for x

N

ifand onlyifx 0, and

3)

for arbitrarilyselectedo-uniquelybounded set

U, S D U,

withthegeneratingfunction u on

obeying

u

E(S; .f),

^theequations

(5.1)

^{have aunique}solution function v on Nwith thefollowing properties:

(i)

^{v is}positivedefiniteon

N,

and

(ii)

^{if the}

boundary ON

^of

N

isnon-empty^then

v(x)

^+ooasx

ON,

x

N.

PROOF.

Necessity.

Let

thesystem

(3.1) possess

Weak Smoothness

Property. Let

x -0have the asymptotic stabilitydomain

D, S _.D D,

and let

N, S

_D

N,

beequal^to

D. Let

ano-uniquelyboundedset

U, S

D

U,

with thegeneratingfunction uobeyingu

E(S;f),

bearbitrarilyselected.

From

thispoint^{on we} havetorepeattheproofof thenecessity partof Theorem 1toshowthat the conditions

1)-3)

^ofTheorem

2hold.

In

that

way

^we

complete

the

proof

ofthenecessity part.

Sufficiency. Let

thesystem

(3.1) ^possess

Weak Smoothness

Property

and the conditions

1)-3)

^be

valid. Thenx- 0of the system

(3.1)

^isasymptotically^stable

[1].

^{Therefore,x-0}has the domainof asymptotic stability

(Definitions

^of

D,,, D,

^and

D [5],[6]). Let

_x0C

(R" -N).

Since

x(t;x0)

is continuous in C

I0,

then it canenter

N

iff it

passes

through

ON. But v(x)

^+ooasx

ON,

x

N [the

^condition

3(ii)].

^Thisand

D /v(x)

<0for x C

(R -N)

inviewof positivedefinitenessofu on

R"

and

(5.1a),

show that

x(t;x0)

cannot reach

ON. Hence, x(t;x0)(R"-N)

^forall

Cl0.

^Therefore,

^{N DD.}

Furthermore,

(5.1 a)

^andpositivedefiniteness ofu on

R"

imply

(see

^the

proof

of thenecessity partofTheorem

1) ^v(x

asx

OD,

x

D,

whichtogether^withthe condition

3(i) proves

dD("IN

O.

Thisresult,

N

D

D,

and thefact that

D

and

N

arenon-empty

open

connectedneighborhoodsof x 0 imply

D N

andcomplete the

proof.

Thepropertiesof thegeneratingfunctionuof ano-uniquelyboundedset

U

are essential for theaccurate one-shot determinationof theasymptotic stabilitydomain.

However,

suchproperties^{are notneeded for} asymptotic stabilityof x 0only. This is clarifiedbythe next result.

THEOREM

3.

For

thestatex 0of thesystem

(3.1)

^possessingWeak Smoothness

Property

^tobe asymptoticallystableit isbothnecessaryand sufficient that forany positive^definitefunctionp

E(S;f)

there exists auniquesolutionfunction vto

(5.24)

with

(5.24a)

determinedalong systemmotions,

O /v(x -p (x (5.24a)

v(0)

0,

(5.24b)

which is alsopositivedefinite.

PROOF.

Necessity.

Let

thesystem

(3.1) possess

the Weak Smoothness

Property. ^Let

^{x-0 be} asymptoticallystable. Thenithas

Do, D,

^and

D,

and

D,, S ,,, _{O, D,}

_f"lS

,, _O

_and

_D

_f’lS

, O,

because

D,,, D,, D

and

S

areneighborhoods ofx

O. Let p E(S;f)

^{be an}arbitrarilyselectedpositivedefinite function

(Definition 1).

Such propertiesof

p

and itsmembershipto

E(S;f)

guaranteeexistenceofa solution vto theequations

(5.24),

^{which is}well defined in

R

and continuous

(see

the

proof

^ofthenecessity partof Theorem

1)

^onthe

^setA

determined in Definition 1. The

setL -A OD,D D_ L,

is also an

open

connected neighborhood of x^,-0

(see

^the

proof

of Theorem 1 forsuch aproperty of

D). Let

e satisfying

L D B,

be arbitrarilyselected.ThenD

D_ B,. Let p ]0, e[ obeyingD,(e) _ ^B,

^bealso

arbitrarily selected,

^where

D,(e)

isdefined

[5],[6]

^astheneighborhood ofx 0 suchthat

x(t;xo)ll

^<eforall

R/

^holds

iffxo ^{D,(e). By}

followingthe

proofs

of

(5.13)

^and

(5.14),

^we

prove

thatv,definedby

(5.24),

has thefollowing properties since

A _ ^L _ ^B, ___ ^D,(e) _ ^B,,

v(x)l

^{<+ for}

every

x

B,, ^(5.25a)

v(x) C(B,) ^(5.25b)

Notice that

D,(e)D_Bp

and the definitions of

D,(e)and D guarantee [5],[6] x(t;x0)B,

^for

every (t,xo) R,xB

^Thisresult,

^A

^I’qD

^D_B,,

^positivedefiniteness of thefunctionp

onA, x(+;xo)

0forevery

xo B, ^{(because D}

²⁾

B,)

^and

^(5.24a),

^{integratedfrom 0to +,togetherwith}

^{(5.24b) prove (5.26),} V(Xo)

^{>0 for}

^{every (x}

o^#

(0)) B,. ^(5.26)

Now, (5.24b)

through

(5.26)

prove positivedefinitness of the solutionvto

(5.24)

^on

B,. ^Its

^uniquenessis

proved

in the samewayasin the

proof

ofTheorem1,whichcompletesthe

proof

of thenecessity part.

Sufficiency. Sufficiency ofthe conditionsof Theorem 3 forasymptotic stability ofx 0 ofthesystem

(3.1)

^withWeak Smoothness

Property

iswell known

[13].

^Thiscompletesthe

proof

of Theorem 3. |

6.

EXAMPL

Example 1.

Let

n 1,

dx

{x ^[x {I)x

^{a for}

^{]xl[0,1], (6.1)}

--

^{-x +}

^{h(x) h(x)}

^{x for}

^{Ix I[1,+}

The system

possesses Strong

Smoothness

Property

because

f(x)--x +h(x)

^is Lipschitzian on

R x.

The equilibrium ^{states are}

x,x---1, x,2-0

^and

x,3-+l.

They suggest

S-]-1, +1[

^and

U-{x:x GRX, lx I< ^a}-]-ct,+,

^for

^aG]0,1[.

^The

^generating

function u on

N, u(x)-Ix l,

^{of the}

NONLINEAR SYSTEMS WITH CONTINUOUS MOTIONS 595

o-uniquelyboundedset

U

and

(5. lab)

yield

D/v(x)’-I xl, ^xS.

Thesolution vtothisequationis

v(x) -In(1 Ix ^I),

^x

^S. (6.2)

The function

v(27)

and theset

N S -]

1,

+1[

satisfyall the requirements ofTheorem 1,that isthat,

1) ^N -]

1,

+1[

isan

open

connectedneighborhoodof x 0and

N S,

2) f(x

-x +h

(x

⁰for x

N

iffx 0,

3) (i) v(x)-Oforx

F_N iffx--0,

v(x)CF.C(N),andv(x)>Oforevery(x #0)N, whichprove

positivedefiniteness of v on

N,

(ii) v(x)

+asx

ON {-1,

⁺¹

},x N.

Hence N -] -?

^{1,+1 isthe domain}

^D

of asymptoticstabilityofx 0,

D --]- 1,+1[.

Notice that

f(x)[ -]

^x

[11 -Ix

^,x

,

^isnota

generating

functionon

N

of

any

o-uniquely bounded setbecauseit is notradially

increasing

on

N.

Example

2.

Let

thefunction h bedefined as in

Example

and

.-

^x

^{-h(x). (6.3)}

It

isclear that thesystem possesses

Strong

Smoothness

Property

on

R

andhas theequilibrium states

x,

--1,

x,:,-

0 and

x,-

+1

(see Example 1). Let,

again,

U {x’x R , [x ]< c} -]- +

^{so that}

u(x) "1 ^x I. ^{From (5.1a)}

^weget

D/v( x)’-I

^x

l,

^x

^.N.

Integrating

thisequation

along

motionsofthesystem

(6.3)

^wederive

v(x)-m(-Ixl), ^xN,

which isnegative definite on

N

and, thus, does not

satisfy

the

necessary

and sufficient conditionsfor asymptotic stabilityofx -0of thesystem

(6.3). Hence,

^{x -0}of thesystem

(6.3)

^{is not}asymptotically stableand does not have theasymptotic stabilitydomain.

Example 3.

Let

n 2and

2 -(1 -xl ^I)(1 -I ^l) "/’(x). (6.4)

The functionj’is

globally

Lipschitzcontinuous. Thesystemhas

Strong

Smoothness

Property

on

R 2.

The set

S,

ofitsequilibriumstates is determined

by

S,-{x’xR2,(x-O)

or

(Ix l- ^1,11-1)}.

Thissuggests

S {x:x R 2, Ixx I<

^1,

^[x l< ^1}.

^{The system}

^(6.4)

has Weak Smoothness

Property

on

S.

Let U {x:x ^R 2, xx

⁺

x [< ct},

ct

]0,1[,

^sothat

U

is

o-uniquely

boundedsetwiththegenerating function u on

R

definedby

u(x)"1 x l+ x ],

^whichtogetherwith

(5.1)

^and

(6.4)

yields

v(x)-- In[(1 -Ixa I)(1 -Ix,21)].

Thefunction vandthe

setN S obey

alltheconditionsofTheorem 2.

Therefore,

x 0 of the system

(6.4)

isasymptotically stablewiththe domain

D

ofits

asymptotic’stability

obtained as

D -N -S,

that is that

D {x:x S:’, Ix1 I<

^1,

Ix I< ^1}.

7.

CONCLUSION

Thenecessaryand sufficientconditionsforasymptotic stabilityof the zeroequilibriumstateandfor aset tobethe domain ofitsasymptotic stabilityare

proved

inanalgorithmicform that enablesaccurate constructionofasystem

Lyapunov

function. Ifafunctionvobtained from D/v--u foranarbitrarily chosenu,whichisageneratingfunctionofano-uniquelybounded set, isnotpositive^{definitethenthe zero} stateisnotasymptoticallystable. Thereis no sense totrywithanother function u.

However,

ifsoderived functionvispositivedefinitethen thezero stateisasymptoticallystable.

In

thiswaytheproblemofan algorithm^toconstructaccuratelyanddirectlyasystem

Lyapunov

functionhas been solved.

However,

it imposesothervery

complex

mathematical

problems:

the

problem

offindingconditions on uguaranteeing existence of well defined andcontinuous vsatisfying

(5.1)on

anyhowsmallneighborhood

’,

^{ofx 0,and}

the

problem

^ofsolving

(5.1).

Theseproblemshave notbeen solved.

Theorems of thepaper openand initiatenewdirectionsin the

Lyapunov

stability analysis.

ACKNOWLEDGEMENT.

Theauthor is

grateful

toall the members of the

Department

of Elec- trical and

Computer Engineering,

Louisiana

State University, Baton Rouge, LA, U.S.A.,

fortheir won- derful

hospitality

that was

helpful

^tohimforhisresearch.

The author also

expresses

^hisgratitudetoProfessor LokenathDebnath,Editor, and reviewers for their comments and thankstoEcoleCentraldeLille, 59651Villeneuve

d’Ascq-Cedex, ^France,

^{for the} opportunity^tobe with theSchool andtorevisethe

paper

effectively.

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The General Problem ofStabilityof Motion

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ofScience,Vol.

II, pp.

5-263, 1956.

(French

translation: Problme

gnral

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[2] POINCAR], H., "Sur

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[3] POINCAR, H.,

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[4] BHATIA, N. P.

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[5] GRUJI,

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Vanelli,

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Mathematical Problems in Engineering

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