V OF TO

VOL. 17 NO. (1994) 103-112

SOLUTIONS TO LYAPUNOV STABILITY PROBLEMS OF SETS:

NONLINEAR SYSTEMS WITH DIFFERENTIABLE MOTIONS

LJUBOMIR T.

GRUJI(

Department

of ElectricalEngineering UniversityofNatal

Rm.

1-05, Elec.

Eng.

Bldg.

King

George V Avenue,

Durban 4001, SouthAfrica

(Received January

23,1991and in revisedform April 28,

1993)

Abstract. Time-invariant nonlinearsystemswith differentiable motions are considered. The algorithmic necessaryand sufficient conditions are established in various forms for one-shot construction of a

Lyapunov

function, forasymptotic stabilityof acompactinvariant setandfor theexactdetermination of theasymptotic stabilitydomainofthe invariantset.

Theclassical conditionsareexpressedin terms of existenceof asystem

Lyapunov

functions. The conditions of theoremspresentedherein areexpressedviapropertiesof the solution vto

-p,

orof the solutionwto

-(1 w)p,

forarbitrarilyselectedp C.

P(S ;f)

orp C.

Pt(S ;f),

where families

P(S;f)

and

Pt(S;f)

arewelldefined. The equation -p,or itsequivalent^/,

-(1 w)p,

should be solvedonlyfor oneselection of the function

p.

Key

Words and Phrases: Nonlinear

Systems, Lyapunov

Functions,andasymptotic stability.

1991MathematicsSubjectClassificationCodes: 34A34, 93D05, 93D20.

1.

INTRODUCTION

The fundamentalclassicalproblemof the

Lyapunov

stability theory

[5]

^{has beenthatofthe exact}

one-shot construction of asystem

Lyapunov

function. This is a

consequence

of the conditions forasymptotic stabilitybecausetheyare

expressed

fornonlinearsystemsvia existence

6f

a

Lyapunov

function. Such classicalcriteriafor asymptotic stability ofasetwere

proved

by Zubov

[7,

p.

204],

^Bhatiaand

Sztige [1,

p.

207],

^and

^La

^Salle

[4, p. 32].

The

open

problemsare thefollowing:

Whatare the

necessary

andsufficient conditionsfor

asymptotic

stability ofa

compact

^invariantset

J,

which are^notexpressedvia existenceof a

Lyapunov

function?

What are the necessary and sufficient conditions for one-shot algorithmic construction ofa

Lyapunov

function?

Whatarethe

necessary

and sufficient conditionsforexactone-shot determinationoftheasymptotic stabilitydomain ofthe set

J?

The notion of the asymptotic stabilitydomain is defined inthe Appendix by following

[2], [3].

All threeproblemsare solved in various forms in what follows for alargeclassof time-invariant nonlinearsystems.

2.

NOTATION

Capital

Roman

letterswilldenotesetsandspaces.

J

will be an invariantsetof asystem,

J CR’. Its

neighborhoodwillbe denoted

by A(J),N(J)

or

S(J),

and its6-neighborhood^willbedesignated by

B6(J), Bt(J)- {x "p(x,J) <. 6},

^where

p(x,J)-inf{llx -yil _"Y J}

^with

Ilxll ^-(:x) .

^{Notice thatJ}

^CA(J)

and

J

C

Bn(J).

The closure, interior andboundaryofa

t J

aredenotedby

J’, ^j

and&/. Theemptysetis

O.

Ds(J),D,(J)

^and

D(J)

^willbe usedfor the domain ofstabilityof

J,

the domain of attraction of

J

and the domainof asymptotic stabilityof

J,

respectively.Their definitions aregivenin the

Appendix

by referring to

[21, 13].

A

^motion.of thesystem tobe considered isdenotedby

x_..(t;x0)

with

x_..(0;x0)- Xo.

^Ifv

:R" R

is differentiable then

dv(x)/dt O(x)

isthe

Eulefian

derivativeofvalongsystem motions.

Other notation will be explained in thesequel.

SYSTEM DESCRIPTION

A

systemtobe studied is describedby

--- ^{f(x), (3.1a)}

tR, xR’, f:R’--,R’. (3.1b)

Itisacceptedthat the systempossessesoneof thenext twosmoothnessproperties.

WeakSmoothness

Property:

(i) There is an

open

neighborhood

S(J)

of acompactinvariant

setJ

of thesystem

(3. lab)

such that for

every

_x0

IS ^{(J) -) ]:}

(a)

^thesystemhas theuniquesolution

x_.(t;x0),

and

(b)

the motion

x_(t;xo)

is defined, continuous and differentiable in

(t,x6)Iox[S(J)-)],

1o ^lo(xo)

^_C

R+

^and

1o .

(ii)

For every

_x0

JR" ^{-S(J)] every}

^motion

^x(t;Xo)

^{of thesystem}

(3. lab)

is continuousin

Strong

^Smoothness

Property:

(i)

The system

(3.1ab) possesses

the Weak Smoothness

Property.

(ii) Iftheboundary

dS(J) orS(J)

isnon-emptythen

every

motionofthesystem

(3. lab)

passing

through

xo ^dS(J)

^obeys

inf{p[x_(t;xo),S(J)]:t R/}

>0 for

every xo #S(J).

4.

LYAPUNOV FUNCTION GENERATION AND DE’rERMINATION OF THE ASYMPTOTIC STABILITY DOMAIN FOR THE SYSTEM WITH STRONG SMOOTHNESS PROPERTY

A

function v"

R" R

will be calledpositive

definite

^{with respectto}

^J

^{if and only ifthere is a}

neighborhood

A (J)

^of

J

such that (i)

v(x)

is continuous inx

E A (J), (ii) v(x

>

O

for

every

x

[A (J J ],

(iii)

v(x)

0 foreveryx

.

A

functionvispositive

definite

^on

^N(J)

with respectto

J

if and

only

if theprecedingconditions

(i)-(iii)

holdfor

A (J) N(J).

We

shall write

A, B.), D(.), ^N

^{andSinsteadof}

^A(J), B(.)(J), D+)(J), ^N(J)

^and

^S(J),

respectively, as soon as

J

isknown and fixed.

In

thesequeltheset

J

isassumedknown and fixed.

In

ordertogenerateav-function

(Lyapunov function)

^{for the}system

(3.lab)

^thefollowingdefinition isintroduced.

Definition

^{1. (i)}

^P(S;f)

^isthefamilyof all functionsp

:R" R

obeying

1)-3), 1)

p^isdifferentiable on

S

andpositivedefinite on

S

withrespectto

J,

2)

^for

^any

^{ct>Osuch that}

B

C

S

there is

[A

^>

O, [A l?,(ct;p ;S ),

satisfying

inf[p(x

x

(S B,)] [%

3)

thereis

Ix ]0, +oo[,Ix. Ix(o;/],

such thatthere existsasolutionv tothefollowing systemdetermined alongmotions of thesystem

(3.lab),

d

v(x) [grad v(x)f(x) -p(x) (4.1a)

dt

v(x)-O, Yx

^j

(4.t,)

which isdefinedand continuous in x

E B,.

(ii)

Pt(S;.D

^isthefamilyof all functionsp

P(S;D

for which the solution function v to

(4.lab)

^{is also}differentiable on

B,.

Notice thatp

P(S;f)

^{doesnot imply by itself}that the solution functionv to

(4.lab)

^ispositive

definite with respectto

J. In

fact, p

P(S;f)

guaranteesonly ^existenceof a continuous solution vto

(4.12ab)

^on

^any

^smallneighborhood

B,

^of

^J.

^Therefore,aselection of

p

^toobey

p P(S;f)

israther a

pure problem

of solving

(4. lab)

than astability problem. Methods forsolving

(4. lab)

will notbeconsidered herein.

The condition

2)

of Definition means that

p(x)

does not

converge

to zero as ^x---,0S or

Ilxll ^+o,xS. For example, p(x)-O

if

xJ

and

t,f)-Cllxll’-l)f2+llxll )

if

xqJ,

and

S-Bt-{x’p(x,J)<l}

with

J-{x’llxll <}

obey the condition

2). But, p(x)-O

if

xJ

and

p(x)-(xll - )(2-Ilxll )

^ifx

^J,

^and

^S --BI

^donotfulfillthe condition

2).

Notice that

p

ispositive definite on

BI

^{withrespectto}

^J

and differentiable on

B

in both cases.

Theorem1.

In

order

for

^{acompactinvariantset}

^J of

^thesystem

^(3.lab)

^{with the}

^Strong

^Smoothness

Property

tohave the domain

D o.f

asymptotic stability and

for

^{a set}

^{N, N}

^C_

^R’,

^{tobethedomain}

^{D N D,}

it isboth

necessary

and

sufficient

^that

1)

^theset

N

isan

open

connected neighborhood

o.f ^J

^and

^N

^C_

^S,

^and

2) (a) for

arbitrarilyselected

function

^p

^{P(S ;f),}

^theequations

^{(4. lab)}

^{havethe unique}solution v on

N

withthefollowing properties:

(i) ^{v is}positive

definite

^on

^N

^{withrespectto}

^J,

(ii) if

^the

boundary ON of ^N

^{isnon-emptythen}

^v(x

^+ooasx

^ON,

^x

^N,

^or,

(b)

for

arbitrarilyselectedp

Pt(S

;j’)theequations

(4.lab)

have theuniquesolution v on

N

with the.following properties:

(0

vis

differentiable

^on

^N

and positive

definite

^on

^N

^{withrespectto}

^J,

^and

(ii) if

^the

boundary ON of ^N

^{isnon-emptythen}

v(x)

^+ooasx

ON,

x

N.

Proof

^Necessity.

^Let

^{thecompactinvariant}

^setJ

^{of thesystem}

^{(3. lab)}

^withthe

^Strong

^Smoothness

^Property

have theasymptotic stabilitydomain

D. Hence,

italsohas

Do (Definitions

^{A-2andA-3 of the}

Appendix),

which is a connected

open

neighborhoodof

J.

Evidently,

D,

^fqS

, . ^{Let D C._ S}

^{be first}

^proved.

^If0S

thenS

R" andDo

^_C

^S

^dueto

D

^C_

^{R n.}

^If0S

, _{t thenxo E}

0S

andx ^{E (R" ’)}

^{will be}analyzed separately.

In

case x0tE 0Sthen

Xo Do

^duetothe

^Strong

^Smoothness

Property. Hence,

0S

tqDo

and

0S t"ID

dueto

D D,,

implied bytheWeak Smoothness

Property CLemma

A-l, Appendix).

In

case

x

⁶⁵

^{(R" -’)}

then

x_(t;x)

does not

converge

to

J

dueto the

Strong

Smoothness

Property.

Therefore,

x ^Do

^and

Dfq(R

-S)-9

^{so that}

D CI(R" -S)- 9. ^{From D CIS}

^"9,

^D

^f3c

-

^and

D f3O(R" -S)-

itresults that

D

^C_

S. Let N D. Hence, N

C_

S

and

N

isopenconnectedneighborhoodof

J,

whichproves necessity ofthe condition

1).

N -D

implies

(Definitions

^A-1to

A-3)

that

10(Xo)-R., fXo

^65N.

^Let

^p65

^{P (S ;f), [p} 65Pt(S;[)],

be arbitrarilyselected.

Hence,

there is a solution v to

(3.lab),

which is defined and continuous on

B,, [and

differentiable on

B],

respectively,

v(x)

65

C(-ff,) ^[v(x)

⁶⁵

C"(-ff _,)] (4.2) Let

"r

C]0, +oo[,

^"t

"t(x0;]’;la’,J),

be such that foranyx0 65

N

the condition

(4.3)

^holds,

x_(t;x0)

65

B,, ^Vt

⁶⁵

^{[x, +o]. (4.3)}

Existence of such x isimplied by

N D Do (Definition A-2).

^Besides,

x_(+oO;Xo) 65, Vx

65N.

(4.4)

Afterintegrating

(4.1.a)

^andusing

(4.1.b)

together^with

(4.4)

^wederive

(4.5),

,,[x_(t;x0)] , p[x_(O;Xo)]ao, v(,Xo) R.xV. ^(4.5)

Invariance of

Do,D Do,N D,

continuity ofmotions

x_ (the

^{weakSmoothness}

^Property),

continuity of

p

^on

N,

the definitionofx

(4.3)

andcompactnessof

It, ;]

for

any

65

R/ prove

V(t,Xo)

⁶⁵

R/xN (4.6)

+oo,

(4.2), (4.3)

^{and x}

(o;xo) x_[o;x;x_ (x;x0)]

65

B

foro65

Ix, +oo]

and thecondition

3)

of Definition 1 yield

p[x(o;x0)]do <+oo,

x065N. ^(4.7)

Now, (4.5)-(4.7)

gives

or, for 0 and x

xo,

v[x_.(t;xo)]J

^<+oo,

V(t,Xo) 65R+xN, (4.8)

v(x)l

^<+oo,

^Vx

^65N.

^(4.9)

Differentiability of

x_

ⁱⁿXo⁶⁵

N,p

⁶⁵

P(S;./’), [p

65

pI(S;f)],

invariance ofN

D D,,, (4.5)

^and

(4.8) prove

continuityof v on

N,

[differentiability ofvon

N],

vCx) C(N) [vCx) C")(N)] ^(4.10)

respectively. Invarianceof

N,

positivedefiniteness

ofp

on

S

withrespect

toJ,N

^C.

S,

and

(4.5)

^imply

v(x)

>

O, Vx

65

(N-J). (4.11)

Now, (4. la), (4.1 0)

^and

(4.1 1)

verify positivedefiniteness ofv on

N

withrespect

toJ [and

^itsdifferentiability on

N],

respectively. Positivedefiniteness ofp, uniquenessof the motions

x_(t;Xo)

^forevery Xo⁶⁵

S,N

^C_.

S,

invariance of

N

and

(4.5) prove

uniquenessof the solutionv to

(4. lab).

This

completes

the

proof

of necessity ofthe conditions

2-a-i)

^and

^2-b-i),

respectively.

Let x,

beasequence,

x, .

^{ask +o,.f,65}

^{ON, ON}

^#9,and

^x,

⁶⁵

^{N. Let 65]0, +oo[}

^bearbitrarily selectedso that

B; ^{CN. Let} T, T ^T,(x,;)

⁶⁵

^{[0, +oo[,}

be the first momentsatisfying

(4.12),

x(t’,x)B, ^{Vt E[T,+oo[. (4.12)}

Such

T

exists due to

x ^N

^and

^{N D} Do (Definition A-2). Continuity

of

x_(t;Xo)

ⁱⁿ

(t,Xo) N D, D Do Do

^and

^D

^_C

^S

^imply

T

^{+oo ask +o=.}

^Let

m be such a natural number that

x .(N-BOfor

all k -m,m +

I,....

Suchm exists becauseNis

open, B ^CN andxk ^ON

^ask

Let

cxbe introducedby

ct-

min[p(x)’x _(N-BO]. ^(4.13)

p P(S;f),[p P(S;f)],

guarantees

(due

^tothe condition

2)

of Definition

1)

ct

tel0, +oo[. (4.14)

From (4.6)

^and

(.4.13)

^wederive

(4.15) by

setting 0and

x_(0;x,) x,,

t,

v(x)a foculo+ r,p[x_(o.:t)]do, ^{x (N-J),}

^{k -m,m+}

^(4.15)

lnvarianceofN

-D -Do,

positive definiteness

ofp

on

S

withrespectto

J (the

condition

1)

of Definition

1), N

_C

S

and

(4.16)

imply

v(x) T

^x

^(N-J),

^{k m,m+}

^(4.16)

Thisresult,

T

^{+ as k + and}

^(4.14)

^yield

^v(x)

^+oo

asx ^ON, x ^N,

^which

^proves

^necessity

of the conditions

2-a-ii)

^and

2-b-ii),

respectively.

Sufficiency.

Let

allconditionsofTheorem be valid. Then,theset

J

isasymptoticallystable

[1, p. 208], [7,

p.

204].

^Thesystem

(3.lab)

has the domain

D

ofasymptotic stabilityof

J (Definitions

A-1to

A-3).

The condition

1)

^implies

N S. Two

possiblecases willbe considered:

a)

^theboundary

ON

of

N

isempty, and

b) ^ON

^isnon-empty.

a) ^{Let ON} -.

^Then

^{N -R"}

^thatobviously implies

N -D -R"

duetothe conditions 2-a-i and

2-b-i).

b) ^{Let ON} .

^If

^{OS O#}

^then

^{S R"}

^{so that}

^{D C.S.}

^If

^{OS ,,}

^{thenc3SfqD due to}

⁽ⁱⁱ⁾

^{of the}

^Strong

Smoothness

Property.

This fact aswell as that both

D

and

S

areneighborhoodsof

J prove D

^_CS.

In

bothcases

D

^C_.

S. Let

now

OD ,

and

OD

be treated

separately.

If

OD ,

thenthe definition of v as the solutionto

(4.lab), D

^_C

^S

^{and the}

proof

of thenecessity partshowthat v is continuouson

D

and

v(x)

+oasx 0D,x

D.

Thesefacts, continuity of^{v on}

N, N C._ S,

thefactthat

D

and

N

are connectedneighborhoodsof

J

and

v(x)

^+ooasx

ON,

x

N,

imply

N-D (4.17)

Let

now

OD . ^{Hence, D R".}

The solution vto

(4.lab)

is continuous on

D R"

asshown in the necessity part.

Hence, v(x)[

^<+ooforeveryx

ER".

Since

v(x)---*

^+ooasx

ON,

x

.N,

thenONtqR"

and

N -R".

Finally,

(4.17)

holds alsoin case

OD ,

^which

^proves (4.17)

in all casesand

completes

the

proof.

From

thecomputational pointof view theformof the condition

"v(x)

^+ooasx

^{ON, x}

U.

N"

is notsuitable.

It

canbesetinanother form

by utilizing

w as usedbyVanelli andVidyasagar

[6],

w(x)=

^1-

exp[-v(x)]. (4.18)

Evidently,thefollowingaretrue:

a)

^{w is}defined and continuous

[and differentiable]

on

S

ifandonlyifvisdefined and continuous

[and differentiable],

respectively,on

S,

b)

^positivedefiniteness ofvon

S

withrespect^to

J

implies positivedefiniteness of w on

S

withrespect to

J,

and viceversa,

c) d)

v +ooimpliesw +1 andviceversa,

theequations

(4.lab)

are equivalenttothefollowingsystem

dw(x) -[1 w(x)]p(x)

dt

(4.19a)

w(x)-

0,

Vx J. (4.19b)

The factslistedabove under

a)

^to

d)

^andTheorem directlyyieldthenextresult:

Theorem2.

Let

the

function

^vbereplaced by^wand theequations

(4.lab) by

theequations

(4.19ab)

in

Definition

^1.

In

order

for

a compact invariant^set

J of

^{the system}

^(3.lab)

^withthe

Strong

^Smoothness

Property

to have the domain

D of

asymptotic stability and

for

^{a set}

^{N, N}

^C_

^R’,

^tobe the domain

D N -D,

it isboth

necessary

and

sufficient

^that

theset

N

isanopenconnectedneighborhood

of ^J

^and

^N

^C_

^S,

(a)

for

arbitrarilyselected

function

^p

P(S;f),

theequations

(4.19ab)

have theuniquesolution w on

N

withthefollowing properties:

(i) ^wispositive

definite

^on

^N

^{withrespectto}

^J,

and

(ii)

if

^theboundary ON

of ^N

is non-empty then

w(x)

+1 as x

ON,

x

N,

.for

arbitrarilyselectedp

PI(S ;f)

^theequations

(4.19ab)

have theuniquesolution won

N

with thefollowing properties:

(i)

^wis

&fferentiable

^on

^N

^andpositive

definite

^on

^N

^{withrespectto}

^J,

and

(iO if

^the

boundary ON of ^N

^{isnon-emptythen}

^w(x

^{+1 asx}

^ON,

^x

^N.

1)

and

9.)

5.

GENERATION OF A LYAPUNOV FUNCTION AND DETERMINATION OF THE ASYMPTOTIC STABILITY DOMAIN FOR THE SYSTEMS WITH THE WEAK SMOOTHNESS PROPERTY

The class of the systems describedby

(3.lab)

with the WeakSmoothness

Property

^is

larger

thanthat with the

Strong

Smoothness

Property. It

is notsurprising^thatthe conditionsof the precedingtheorems slightly change^{for the}systemswith the WeakSmoothness propertyasfollows.

Theorem3.

In

order

for

^{acompactinvariantset}

^J of

^{the system}

^(3.lab)

^{with the}WeakSmoothness

Property

tohavethe domain

D of

asymptotic stabilityand that a subset

N orS ^{equals D N D,}

^{it isboth}

^necessary

and

sufficient

^that

1)

theset

N

isopen^connectedneighborhood

of ^J,

2) (a) for

arbitrarily selected

function ^{p P(R’;f),}

^theequations

^(4.lab)

^{have theuniquesolution}

function

^von

^N

^withthefollowing properties:

(i)

^{v is}positive

definite

^on

^N

^{withrespectto}

^J,

and

(ii) if

^the

boundary ON of ^N

^{isnon-emptythen}

^v(x

^+asx

^ON,

^x

^N,

or,

(b)

for

arbitrarilyselected

function

^p

P(R’;f),

^theequations

(4.lab)

have theuniquesolution

function

^von

^N

^withthefollowing properties:

(i) vis

differentiable

^on

^N

^andpositive

definite

^on

^N

with respectto

J,

and (ii)

if

^theboundary

^ON of ^N

^{isnon-emptythen}

^v(x)

^+asx

^ON,

^x

^N.

Proof.

^Necessity.

^Let

^thesystem

(3.lab) possess

theWeakSmoothness

Property. Let

thesystem

(3.lab)

have theasymptotic stabilitydomain

D

and let

N-D,

for

N

^C_S.

Let

p

P(R";f), [p PI(R";f)],

be arbitrarilyselected.

From

thispoint^on weshouldsimply repeatthecorresponding partof the

proof

of necessity of the conditions of Theorem inordertoshownecessityof all conditions of Theorem3.

Sufficiency.

Let

the system

(3.lab)

have theWeakSmoothness

Property. Let

the conditionsofTheorem 3hold.Then,the invariant

setJ

isasymptoticallystable.The system

(3. lab)

hasthe domain

D

of asymptotic stabilityof

J (Definition A-3),

^which

equalsDo (Lemma A-l). Letx0 (R" -N).

Continuity of

x_ (t ;x0)

in

I0

^{due to the Weak Smoothness}

Property,

positive definiteness ofp on

R"

due top

P(R";.f), [p PI(R ;.f)],

negativenessof

O(x)

on

(R" N)

guaranteed by positivedefiniteness

ofp onR"

and

(4. lab),

andthe condition

2-a-ii), [2-b-ii)],

respectively, imply

x_(t;x0) (R" N)

forall

I0. ^{Hence, D _C N.}

Furthermore,

(4.1a)

and positivedefiniteness ofp^on

R"

imply

(see

theproofof the necessitypart ^of Theorem

1) ^v(x)

^+oasx

^OD,

^x

D,

which

together

withthe condition

2-a-i), [2-b-i)],

respectively, implies

OD

^fqN

, _t.

_Thisresult,

_{D N}

_{and the}fact thatD is aneighborhoodof

J

imply

D N

and

complete

theproof.

ThecounterparttoTheorem2inthisframework is thenextresult thatfollowsdirectlyfrom Theorem 3and

(4.18).

Theorem4.

Let

the

function

^vbe

replaced by

wand theequations

(4.lab) by

theequations

(4.19ab)

in

Definition

^1.

In

order

for

^acompact invariantset

J of

^thesystem

^(3.lab)

^{withtheWeakSmoothness}

^Property

^to

have the domain

D of

asymptotic stabilityand thatasubset

N ors

^equals

^{D N D,}

^{it is bothnecessary}

and

sufficient

^that

1)

^theset

N

isopenconnectedneighborhood

of ^J,

and

2)

^(a)

for

arbitrarilyselected

function

p

P(R;f),

^theequations

(4.19ab)

have theuniquesolution w on

N

withthefollowingproperties:

(i) w ispositive

definite

^on

^N

^{withrespectto}

^J,

^and

(iO

if

^theboundary

^ON of ^N

^isnon-empty then

w(x

+ as x

ON,

x

N,

or

(b) for

arbitrarilyselected

function

p

P(R;f),

^theequations

(4.19,ab)

have theuniquesolution won

N

withthefollowing properties:

(i)

^{w is}

differentiable

^on

^N

^andpositive

definite

^on

^N

^{withrespectto}

^J,

^and

(ii)

if

^the

boundary ON of ^N

^{isnon-emptythen}

^w(x

^{+ asx}

^ON,

^x

^N.

6.

GENERATION OF A LYAPUNOV FUNCTION AND ASYMPTOTIC STABILITY

The classicalproblem^{of the}

Lyapunov

stability theory has been theproblem of^the

necessary

and sufficientconditionsforasymptoticstability

(without

determination of theasymptotic stability

domain).

It

generatedtheproblemof thenecessaryand sufficient conditionsfor an exact, direct and one-shot con- structionof asystem

Lyapunov

function. The solutiontothese

problems

resultsdirectly from^the

proof

ofTheorem andTheorem3 in thefollowingform.

Theorem5.

In

order

for

^{acompactinvariant}

^setJ of

^{the system(3.lab)withthe WeakSmoothness}

^Property

tobeasymptoticallystableitisboth

necessary

and

sufficient

^that

1) .for

arbitrarilyselected

function

p obeyingthe conditions

1)

and

3) of

⁽ⁱ⁾

ofDefinition

^{1,theequations}

(4.lab)

have theunique positive

definite

^solution

function

^{vwithrespectto}

^J,

0,

2) for

arbitrarilyselected

function

^{p obeying}the conditions1)and

3) of

^(i)and(ii)

of Defmition ^I,

^the

equations

(4.lab)

have theunique

eh’fferentiable

^positive

de.finite

^solution

function

^{v withrespectto}

J.

Thistheorem, Theorem andTheorem 3show that the condition

2)

^of

(i)

of Definition 1isneeded onlyfor the exact determinationoftheasymptotic stabilitydomain

D

of

J.

By

makinguseof

(4.18)

the solution can be stated in termsofthe solution w to

(4.19ab).

Theorem6.

Let

the

function

^vbe

replaced by

^wandtheequations

(4.lab) by

the

equations (4.19ab)

in

Definition

^1.

In order.for

^{acompactinvariantset}

^J of

^thesystem

^(3.lab)

^{withtheWeakSmoothness}

Property

^to

be

asymptotically

^stableit isboth

necessary

and

sufficient

^that

1) for

arbitrarily selected

function ^p

^obeyingthe conditions

1)

and

3) of

⁽ⁱ⁾

of Definition

^{1,theequations}

(4.19ab)

have theuniquepositive

definite

solution w withrespectto

J,

for

arbitrarily^selected

function

p obeying theconditions

1)

and

3) of

(i)^and

(ii) of Definition ^1,

^the

equations

(4.19ab)

have theunique

differentiable

^positive

definite

^solution

function

^{w withrespect}

to

J.

7. EXAMPLES

Example^1.

Let

asimplesecond order nonlinearsystem

(3.lab)

^{have the}following

specific

form:

dx

2)

d-7-(1-Ilxll ^{(10o- Ilxll )x (7.1)}

Thesystemhas theset

S,

of theequilibriumstates,

s,- (x:llxll

^{-0 or}

Ilxll

^or

Ilxll ^{10}. (7.2)}

Theset

J,

J {x: xll ^{1, (7.3)}

isacompactinvariantsetof thesystem.

From (7.1)

^and

(7.2)

^{itfollowsthat the}system

possesses

the

Strong

Smoothness

Property

withtheset

S

givenby

s {x: Ilxll

^<

^{10}. (7.4)}

Let

the function

p

be selected in the form

0

Ilxll -

^1,

pox)-

(llxil_)llxll Ilxll

^1.

^(7.5)

It

is differentiableon

R

andpositivedefinite on

R

^withrespecttothe set

J (7.3).

^Thesolution function v to

(4. lab)

^for

p

definedby

(7.5)

is obtained in theform

v(x) xll’-

¹

^(7.6)

[98(100_11x11), Ilxll

^1.

Thefunction v isdefined,continuous and differentiable on the set

S (7.4). Hence, p P(S;f).

^Besides,

the function vispositivedefinite on

S

withrespect

toJ (7.3)

and

v(x)

^+ooasx

OS,

x

S,

where 0s

(x: Ilxll ^10}.

Since theset

S

isopenconnectedneighborhoodof theset

J (7.3)

thenall the conditions ofTheorem 1are satisfiedfor theset

N -S (7.4).

Thismeans that thedomain

D

ofasymptotic stability of thecompact invariant

setJ (7.3)

^ofthesystem

(7.1) equals S,

D

-S

{x: Ilxll

^<

^10}.

Since the system ^issimplethisresult can beeasily^verified.

Exalnple^2. Letathirdordernonlinearsystem

(3. lab)

be describedby dx

2.10’{1 sin[n(2 lO’)-t(xrnx lO)2]}(xrnx- 10)

d-" _ncos[n(2 10’)-’ (xrHx 10) 2]

x

^-./(x),

-?61

10 4

H

-4 10

-H r.

-6 2

The matrix

H

ispositivedefinite. The closedinvariant set

J

of thesystemis

J {x :xrHx _10}.

Thesystem

possesses

the

Strong

Smoothness

Property

withtheset

S

given

by

S {x :xrnx

<

110}. (7.9)

Theset

S (7.9)

^is

open

connectedneighborhoodof

J (7.8). ^Let

the function

p

be

accepted

in the nextform:

0 x_J

p(x)=

4(xrHx 10)xrHx

^x

(R3-,]). ^(7.10)

Thefunctionpisdefined, continuousand differentiableon

R 3,

andpositivedefinite on

R

^withrespectto

J. It

leadstothefollowingsolutionfunctionv totheequations

(4.lab),

0 x^j

(7.t) v(x)-

-In{I-sin[n(2. 104)-l(xrHx 10)2]},

^x

^(R3-,]).

Thefunctionv

(7.11)

isdefined, continuousand differentiableon

S. Hence,

the function

p belongs

to

Pl(S;f).

Furthermore,the function v

(7.11)

^isalsopositivedefiniteon

S

withrespect

toJ.

Besides,

v(x)

+ as x

OS {x :xrHx 100}. (7.7)

AllconditionsofTheorem havebeenverified.

Hence,

thesystem

(7.7)

^{has thedomain}

D

of asymptotic stabilityof

J,

whichequals

N S,

D -S {x:xrHx

^<

110}.

(7.7a)

(7.8)

8.

CONCLUSION

Nonlinear time-invariantsystemscharacterizedbythe smoothnesspropertiesare considered. The problems^ofthenecessaryandsufficientconditionsfor anexactdirect construction of asystem

Lyapunov

function, forasymptotic stabilityofacompactinvariant

setJ

andfortheexactdeterminationofitsasymptotic stabilitydomainaresolvedalgorithmically. Thismeans that the invariantset

J

isasymptotically^{arable if} andonlyif thesolution v to -p

(4.1a)

^with

v(x)

0on

J (4.1b)

^ispositivedefinite

[and differentiable]

for any p

.P(S;f), [.p PI(S;f)],

respectively. Theequation

(4.1a)

^{is tobesolved}

^only

^foroneaueh

arbitrarily

selected

p.

Ifthe solution v ispositivedefinite with

respect

to

J

then

J

is

asymptotically

^stable.

However,

if the solution vis notpositivedefinitewithrespectto

J

then

J

isnot

asymptotically

stable.

In

the latter case thereis notsensetotrysolving9

-p

with

any

other function

p.

Thesestatementsresult from Theorem 5that

together

with Theorem 6

opens

new direction onto the

asymptotic

stability

analysis.

The completesetsofconditionsforthe exact determinationof the asymptotic

stability

domain of

J

aregiveninvariousforms of Theorem toTheorem 4.

They

establishessentiallynew

approach

tosolving the

Lyapunov

stabilityproblems.

Ifthe

setJ {0}

and the system

(3.lab)

^{is linear}thenTheorem5becomes a

generalization

^{of the} well known criterionforasymptotic stabilityof the zeroequilibriumstateoftime-invariantlinearsystems

[5,

p.

76].

ACKNOWLEDGEMENT.

Thisresearch wasfinancially supported bythe ScienceFundof

Ser-

bia,grant number 0401E,throughthe Instituteof Mathematics,Belgrade, Serbia,under thecontract 0401

E,

whiletheauthorwas attheFacultyof MechanicalEngineering,

Belgrade,

Serbia,Yugoslavia.

APPENDIX A-I. DEFINITIONS OF STABILITY DOMAINS

Definition

^A-1.

^{A setJ}

^{of thestatesof thesystem}

(3.lab)

^{hasthe domain}

D,(J) of

^{stabilityif andonlyif}

tbrevery

mepsilon ]0, +oo[

there isaneighborhood

D,(e;J)

^{of theset}

J

such that

x_(t;xo)

belongstothe e-neighborhood

B,(J)

of the set

J

for all

E[0,+oo[

provided only that Xo

D,(e;J)

and that

D,(J) U[D,(e’,J):

^e

]0, +=o[ ].

Definition

^A-2.

^A

^set

^J

^{of thestatesof thesystem}

(3.lab)

has the domain

Do(J) of

attraction if andonly if there is aneighborhood

Do(J)

^{of theset}

J

such that

lim{p[x(t;x0),J]: +*].

-0 provided onlythat

xo

^Do(g

^).

Definition

^A-3.

^A

^set

^J

^{of thestatesof thesystem}

(3.lab)

has the domain

D(J) of

asymptotic stabilityif and only if it has both the domain

D,(J)

of stability and the domain

Do(J)

of attraction and

D (J D, (J

f’I

D, (J ).

A-II. PROPERTIES OF STABILrrY DOMAINS

Lemma

A-1.

If

^{the system}

⁽¹⁾

^possessingthe Weak Smoothness

Property

has the domain

D (J of

^asymptotic

stability

of

^aset

^J

^then

Do(J)

^C_

D,(J)

^and

Do(J) D(J).

Proofi ^Let

^thesystem

^{(3.lab) possess}

^{theWeakSmoothness}

^Property

^andhave the domain

D(J).

Then ithas also the domains

D(J)

^and

D,,(J) (Definition A-3). Letx0 ^Do(J).

^{Then,DefinitionA-2and}

^(i-b)

and

(ii)

of the Weak Smoothness

Property

imply

max{p[x_(t’,Xo),J]:te[O,+oo[}-m(x0)

<+oo.

Hence,

for

e[m(Xo),+oo[,x(t’,Xo)B,(J),Vte[O,+oo[,

^which

proves XotED(e;J),

hence,

xoD,(J)

^and

Do(J)

^C_

D,(J).

The last result and

D(J)- D,(J)tqD,(J) prove Do(J)- D(J).

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1.

BHATIA,

N.

P.

and

SZEGO, G. P.,

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Systems:

Stability Theory andApplications, Springer-Verlag,Berlin, 1967.

GRUJI(,

Lj.

T., "Sets

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Systems," Systems

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No.

4, 1979, 327-338.

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Zagreb,

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No.

1-2, 1985,5-10.

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J.

P.,

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Press,

Princeton,1949.

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VANELLI, A.

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VIDYASAGAR, M.,

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1, 1985, 69-80.

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ZUBOV,

V.

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Methods

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