An Invariance Principle in the Theory of Stability

8/20/2019 An Invariance Principle in the Theory of Stability

1/12

An Invariance Principle in the Theory

of Stability

JOSEPH P.LASALLE

THE stabili ty theorems of Lyapunov have been among

the oldest and strongest pi llars

of

control theory. The cen

tenary

of

Lyapunov's celebrated 1892 memoir was recently

marked with its English translation [17], while the 1907 French

translation was reprinted in 1949 by Princeton University

Press [16].

Applications

of

Lyapunov stability concepts to control prob

lems began to appear in Russia in the 1930s-1940s. Significant

theoreticalresults from the 1950s were summarizedin the books

by Chetayev [3], Lurie [15], Malkin [18], Letov [14], Zubov

[22], and Krasovskii [7]. In the post-1957 Sputnik era, English

translations of these books, and

of

other Russian works, further

increasedthe interest already stimulated by Contributions to the

Theory of Nonlinear Oscillations, a five volume series (1950,

1952, 1956, 1958, and 1960) edited by Lefschetz, and published

by Princeton University Press. The fourth volume contained a

detailed and rigorous survey of Lyapunov stability theory by

Antosiewicz [1]. The 1960 survey by Kalman and Bertram [6]

was more accessible and had a stronger impact on engineering

audiences, as did the books by Lefschetz [12], Lefschetz and

LaSalle [13], and a paper by LaSalle [10]. This activity contin

ued in the early 1960s when several volumes of Contributions

to Differential Equations were published, including the impor

tant results by Yoshizawa [20], whose 1966 book [21] presented

a collection of advanced stability results. The status of stability

theory in the pre-1965 per iod is summarized in the scholarly

work by Hahn [4], which is the most comprehens ive source

covering that period.

Among the innovations from that period are the results which

settle the issue of existence of Lyapunov functions. Particularly

important among the results

of

this type are the converse theo

rems of Massera [19], Krasovskii [14] and Kurzweil [8], which

found applications in diverse areas of control theory. Radially

unbounded Lyapunov functions were introduced in 1952 by

Barbashin and Krasovskii [2] to guarantee stability in the large,

that is, global stability. The same Barbashin-Krasovskii paper

and the bookby Krasovskii [14] initiatedanother line of research

which culminatedin this paper byLaSalle, and its extended 1968

version [11].

This line

of

research was aimed at extracting from Lyapunov's

stability theorem more information about the asymptotic behav

ior

of

the solutions

x t)

E IR

n

of

x

=

f x t) , f O, t)

=

0 for all

t

(1)

With a positive definite function Vex, t) , a theorem of Lya

punov establishes asymptotic stability of the equilibrium x = 0,

if V x, t) is negative definite along the solutions

x t)

of (1). IfV

is only nonpositive, the same theorem guarantees stability, but

does not reveal whether

x t)

converges to zero or to a set in IR

n

.

For autonomous systems,

x

=

f x),

a theorem ofBarbashin and

Krasovskii [2] (see also Theorem 14.1

onp.

67 of[14]) examines

the set

E

in which V =

O

If this set does not contain any positive

semi-trajectory (solution

x t)

for all t ::: 0) other than

x t) ==

0,

then x t) --- 0 as t ---

00.

This result gave a practical test for

asymptotic stability when

V

is only nonpositive, instead of neg

ative definite. In his 1960 paper [10], LaSalle extracted further

information on the asymptotic behavior of x t). Using the limit

sets and the largest invariant set

M

of x = f (x) contained in the

set E where V is zero, he showed that

x t)

converges to the set

M.

This set need not be an equil ibr ium, but can be a compact

limit set like a limit cycle.

This result of LaSalle, which he later termed

I

nvariance

Prin

ciple, had a significant connection with the then new concept

of

observability. For the system x = Ax, a positive definite

Lyapunov function V =

x

P

x

has the derivative

V

= A P +

PA =

-x Qx.

Suppose that

Q

is positive semi-definite, so that

it can be expressed as Q = C'C for some matrix C. It then fol

lows that

V

= -

y

y, where y

=

Cx can be treated as an output

ofx =Ax. Clearly, the set E where

V

= 0 is y t)

==

O If the pair

A, C) is completelyobservable, then y t) == 0 implies x t) = O

Hence, no nonzero positive semi-trajectory

x t)

is contained in

E, which proves, via Barbashin-Krasovskii [2], that x =

Ax

is asymptotically stable. However, if the system is only stable

with a pair of purely imaginary eigenvalues unobservable from

y

= Cx, then LaSalle's Principle shows that x t) converges to

the periodic solution in the unobservable subspace.

The Invariance Principle was subsequently extended to peri

odic and almost periodic systems, but it does not hold for more

309


2/12

general nonautonomous systems (1). Toobtain similar informa

tion on the asymptotic behavior of x t), Yoshizawa [20] derived

a set of conditions under which

V

x, t) :s

W x) ,

where

W x)

is

positive definite with respect to a set M, implies that x t) con

verges to M. The main theorem in this paper by LaSalle mod

ifies and improves this result of Yoshizawa, and provides the

strongest convergence result for nonautonomous systems. Togo

beyondthis result, further restrictions on f x, t) are needed. One

of them is the so-called persistency of excitation condition in

adaptive identification and control. Typically, an adaptive algo

rithm guarantees that V :s -e

2

, where e is the scalar tracking

error. The Yoshizawa-LaSalle theorem provides the conditions

under which e(t) converges to zero. It has thus become an in

dispensable tool in adaptive control design. It has also become

instrumental in deducing stability from passivity properties as

in feedback passivation and backstepping designs of nonlinear

systems.

Other settings where LaSalle's

Invariance Principle

has been

studied are infinite dimensional systems, as in the work of

Hale [5], and stochastic systems governed by continuous-time

Markov processes, as discussed by Kushner in [9].

REFERENCES

[1]

H. ANTOSIEWICZ, A survey of Lyapunov's second method, in

Contr.

to Nonlinear Oscillations, S. Lefschetz, Ed., 4:141-166 (Princeton Univ.

Press, Princeton NJ), 1958.

[2] E.A. BARBASHIN AND N.N. KRASOVSKII, On the stabilityof motion in the

large,

Dokl.Akad. Nauk USSR,

86:453-456,1952.

[3] N.G. CHETAYEV,

TheStability

of

Motion

, PergamonPress (Oxford), 1961.

(Russian original, 1946.)

[4] W.

HAHN,

Stability

of

Motion

(Springer-Verlag, New York), 1967.

[5] J.K. HALE, Dynamical systemsand stability, J.

Math.Anal. Appl., 26:39

59,1969.

[6] R.E. KALMAN

AND

J.E.

BERTRAM,

Control system analysis and design

via the 'secondmethod' of Lyapunov, I Continuous-time systems, 1.Basic

Engineering (Trans.ASME), 82D:371-393, 1960.

[7] N.N. KRASOVSKII,

Stability

of

Motion

,StanfordUniv. Press (Stanford,CA),

1963. (Russian original, 1959.)

[8] J.

KURZWEIL, The

converse second Liapunov's theorem concerning the

stability of motion, Czechoslovak Math.

1.,6(81):217-259

&

455-473,

1956.

[9] H,J.

KUSHNER,

Stochastic stability, in Stability of StochasticDynamical

Systems,R. Curtain, Ed., Lect. Notes inMath., Springer-Verlag (New York),

294:97-124,1972.

[10] J.P.

LASALLE,

Some extensions ofLiapunov's secondmethod, IRETrans.

Circuit Theory,

CT·7:52G-527, 1960.

[11] J.P.LASALLE, Stability theory for ordinary differential equations,

1.

Dif

ferential Equations,

4:57-65,1968.

[12] S. LEFSCHETZ,

Differential Equations: Geometric Theory,

Interscience,

Wiley (New York), 1957.

[13] S.

LEFSCHETZ

AND

J.P. LASALLE,

Stability by Liapunov s Direct Method,

with Applications,Academic Press (New York), 1961.

[14] A.M.

LETOV,

Stability inNonlinear ControlSystems (English translation),

PrincetonUniv. Press (Princeton, NJ), 1961. (Russian original, 1955.)

[15] A.E. LURIE,

SomeNon-linearProblemsin theTheory

of

AutomaticControl

(English translation), H.M.S.O., London, 1957. (Russian original, 1951.)

[16] A.M. LY

APUNOV,

Probleme general de la stabilite du mouvement (in

French),

Ann.Fac.Sci. Toulouse,

9:203-474, 1907.Reprinted

inAnn. Math.

Study,

No.

17,1949,

Princeton Univ. Press (Princeton, NJ).

[17] A.M. LYAPUNOV, The general problem of the stability of motion (trans

lated into English by A.T. Fuller),

Int.

J.

Control,

55:531-773,1992.

[18] I.G. MALKIN,

Theory

of

Stability

of

Motion

,AEC (AtomicEnergy Commis

sion) Translation 3352, Dept. of Commerce, United States, 1958. (Russian

original, 1952.)

[19] J.L.MASSERA , Contributionsto stability theory, Ann.Math.,64:182-206,

1956.

[20] T. YOSHIZAWA, Asymptotic behavior of solutions of a system of differ

ential equations, in

Contributions to Differential Equations, 1:371-387,

1963.

[21] T.YOSHIZAWA, StabilityTheory byLiapunov s SecondMethod, The Math

ematical Society of Japan, Publication No.9 (Tokyo), 1966.

[22] V.1. ZUBOV, Mathematical Methods for the Study of Automatic Control

Systems,

Pergamon Press (Oxford), 1962. (Russian original, 1957.)

P.V.K.

J.B.

3


3/12

An

Invariance Principle in the

Theory

of Stability

JOSEPH P. LASALLEl

Center for Dynamical Systems

Brown University, Providence, Rhode Island

1.

Introduction

The purpose of this paper is to give a unified

presentation of Liapunov's

theory

of

stability

that

includes

the

classical Liapunov

theorems

on

stability

and instability as well as

their

more recent extensions. The idea being ex

ploited here had its

beginnings

some time ago. It was, however, the use made

of this

idea by Yoshizawa in [7J in his

study of

nonautonomous differential

equations and by Hale in [1]in his study of autonomous functional-differen

tial equations that caused the

author

to return to this subject and to adopt

the general approach and point of view

of

this

paper.

This

produces some

new

results fo r

dynamical

systems defined

by

ordinary differential equations

which demonstrate the essential nature of a Liapunov function and which

may be useful in applications.

Of

greater

importance,

however, is the possi

bility,

as

already indicated by

Hale's results for

functional-differential

equa

tions, that these ideas can be

extended

to more general classes

of

dynamical

systems. It is hoped, for instance, that it may be possible to

do

this for some

special types

of

dynamical

systems defined by

partial

differential equations.

In Section 2 we present some basic results for ordinary differential equa

tions. Theorem I is a

fundamental

stability theorem for nonautonomous

systems and is a modified version

of Yoshizawa's

Theorem 6 in [7]. A simple

example

shows that

the conclusion of

this theorem is the best possible.

However, whenever

the

limit sets

of solutions

are

known

to have an invar

iance

property,

then

sharper

results

can

be

obtained. This invar iance

principle explains

the

title

of

this paper.

It

had its origin for autonomous

and periodic systems in [2]

and

[4],. although we present here improved ver

sions

of those results. Miller in [5] has estab li shed an invariance property

1

This

research was supported in

part

by the National Aeronautics

and

Space Adrnini

stration under

Grant No.

NGR-40-002-015

and under Contrac t No. NAS8-11264, in part

by

the

United States Air Force through the Air Force Office of Scientific Research under

Grant

No.

AF-AFOSR-693-65, and in part by the United Sta tes Army Research Office,

Durham,

under Contract

No.

DA-31-124-ARO-D-270.

Reprinted

with permission from Differential Equations and Dynamical Systems

(New

York: Academic Press, 1967),1. Hale and 1.P.LaSalle, eds.,

Joseph

LaSalle,

An

Invariance Principle in the

Theory

of Stability, pp. 277-286.

311


4/12

for almost periodic systems

and obtains

thereby a similar stability

theorem

for almost periodic systems. Since little attention has been paid to theorems

which make possible estimates of regions of attraction (regions of asymptot

ic stability) for nonautonomous systems results of this type are included.

Section 3 is devoted to a brief discussion of some

of

Hale's recent results [I]

for

autonomous

functional-differential equations.

2.

Ordinary

Differential

Equations

Consider

the

system

i =f(t,

x) (1)

where x is an n-vector,fis a continuous function on Rn,+l to

R

and satisfies

anyone

of

the

conditions

guaranteeing

uniqueness

of

solutions.

Fo r each x

in Rn, we define I x I

== (Xl

+

... + X

n

2 )1 I2 ,

and

for E a closed set in Rn

we define d(x, E) =

Min

{ Ix - y I; y in E}. Since we do

no t

wish to

confine ourselves to bounded solutions, we introduce the point

at

CX

and

defined(x,

00) ==

IX 1-1. Thus, when wewrite E* == E U

{oo},

we shall mean

d(x, E*) =

Min

{d(x, E), d(x,

oo)}. If x t) is a solution of (1), we say

that

x t)

approaches

E

as t

0 0 ,

if

d x t), E) --.0 as

t

- ) 0 0 0 0 .

If

we

can

find such

a set E, we have obtained information

about

the asymptotic behavior of

x t)

as

t

CXJ.

The

best

that

we

could hope

to

do

is to find the smal lest

closed set

J

that x(t) approaches as t

0 0 . This set Q is called the positive

limit set of x(t)

and

the points

p

in Q are called the

positive

limit points

of

x t). In exactly the same way, one defines

x t)

E as t --+- - 0 0 , negative

limit sets,

and

negative l imit points. This is exact ly Birkhoff's concept of

limit sets. A

point

p is a positive limit

point of x t), if and

only

if there

is a

sequence of times t

n

approaching

CX ) as n 0 0

and

such

that x t

n

) - p as

n ----.

CXJ. In

the

above, it may be

t ha t the

maximal interval

of

definition of

x t)

is [0, r ),

This

causes no difficulty since, in

the

resul ts to be presented

here, we need only, with respect to t ime

t,

replace

00

by

T.

We usually ignore

this possibility

and speak

as

though ou r

solutions

are

defined on [0, X or

0 0 ,

(0).

Let

V(t,

x)

be

a Cl function on [0,

oo ]

x

R

to

R,

and

let G be

any

set

in R .

We shall say

that V

is a

Liapunov function

on G for Eq. (1),

if V(t, x)

0

and

V(t, x) -

W x)

0 for all t > 0

and

all x in G, where W is

continuous

on

Rn

to

R, and

. av

n

av

r i:

+

;lfi·

U i I ox;

312

(2)


5/12

We define (G is the closure

of

G)

E = {x; W(x) ==

0,

x

in

G}.

The following result is

then

a modified but closely related version of

Yoshizawa's Theorem 6 in [7].

Theorem 1. If V is a Liapunov function on G for Eq. (I), then each

solution x(t)

of (1) that remains in G for all

t > to

0 approaches

E*

=

E u {oo} as

t oo,

provided one

of the

following conditions is

satisfied:

(i) Fo r each pinG the re is a

neighborhood N of

p such that

If t,

x)

I

is bounded

fo r

all t

>

0 and all

x

in

N.

(ii)

W is Cl and W is bounded from above

or

below

along

each

solu

t ion which remains in

G

for all t > to o

If

E

is bounded,

then each

solution

of

(1)

that

remains in

G

for

t >

to

0

either

approaches E or

0 0

as

t

0 0 .

Thus

this

theorem explains precisely the nature

of

the

information

given

by a Liapunov funct ion. A Liapunov functi on relative to a set G defines a

set E, which under

the conditions

of

the

theorem

contains

(locates) all

the

positive limit sets of

solutions

which for positive time remain in G.The prob

lem in applying

the

result is to find good Liapunov functions. For in

stance, the

zero function

V

=

0 is a Liapunov

function

for the whole space

Rn

and

condition (ii) is satisfied bu t gives no

information

since

E = R .

It is trivial

bu t

useful for applications to note that if VI

and

V

2

are Liapunov

functions on G, then

V

= VI

+ V

2

is

also

a

Liapunov

function

and

E = £1 £2 .

If

E is smaller than

either

£1 or £2' then V is a better

Liapunov function

than either

E.

or £2 and is always at least as good

as e ither of the two.

Condition

(i )

of

Theorem 1 is essentially

the

one used by Yoshizawa.

We

now

look at a simple example,

where condition (ii)

is satisfied and

condi

tion

(i)

is not.

The

example also shows

that

the conclusion

of

the theorem is

the best possible. Consider

x

+ p t) i + x = :: 0, where p(t) (j > o Define

2V

= x

2

+ y2,

where

Y ==

i . Then

V

===

- p t)y2 -

b

y

2

and

V

is a Lia

punov function

on

R2. Now

W ===

l5y2

and W ==

2t5YJi

=== -

2c5 xy

+

p(t)y

2

)

- 2f5xy. Since all solut ions are evident ly bounded for all

t

> 0,

condi

tion (ii) is satisfied. Here E is the x-axis (y:=:

0 ) and

for each

solution

x t),

yet) =

X(/) --+- 0 as

t 0 0 .

Noting that

the equat ion

.t (2

+

exp[t))

x

+ x == 0 has a solution x t) = 1

.t-

exp( - t) , we see that this is the best

possible result without further restrictions on p.

313


6/12

In

order

to use

Theorem

1, there

must

be some means

of

determining

which solut ions remain in G.

The

following corollary, which is an obvious

consequence of

Theorem

1, gives one way of

doing

this

and

also provides,

for

nonautonomous

systems, a

method

for estimating regions

of

attraction.

Corollary 1.

Assume

that

there exist

continuous

functions

u x) and

II(X) on Rn

to

R

such

that u x)

V(t,

x)

v x)

for all

t

O. Define

Q,/+ = {x; u x) < 1J}

and

let

G+

be a

component

of Q 1+.

Let G denote

the

component

of

Q 7

=

{x ;

v(x) <

YJ}

containing

G+.

If V

is a

Liapunov

function

on

G for (1)

and

the

conditions

of Theorem 1

are satisfied,

then

each solution

of

(1) s ta rt ing in

G+

at any time

to

0

remains in

G for

all t > to

and

approaches

E*

as

t

0 0 . If G is

bounded and

EO =

E

G

C

G+, then

EO is an

attractor and G+

is in its region of at

traction.

In general we

know

that

if

x t)

is a solut ion

of I ) - i n

fact,

if x t)

is any

continuous

function on R to Rtl-then its posit ive l imit set is closed

and

connected. If x t) is

bounded,

then its positive limit set iscompact. There are,

however, special classes of differential

equations

where

the

limit sets of solu

t ions have an addit ional invariance property which makes possible a refi

nement

of

Theorem

1. The first of these are the autonomous systems

x

=f(x).

(3)

The

limit sets

of

solutions

of

(3) are invariant sets.

If

x t)

is defined on [0, 00)

and

ifp is a positive limit

point

of

x{t),

then

points on the

solution

through

p

on

its

maximal interval of definition are positive limit points

of x t).

If

x t)

is

bounded

for

t >

0, then it is defined on [0,

00),

its positive limit set

Q

is

compact,

nonempty

and

solutions

through points p of

Q

are defined on

-00, 00)

(i.e.,

(J

is invariant). If the maximal

domain

of definition

of

x t) for t

> 0 is finite, then x t) has no finite posit ive limit

points: That

is,

if

the maximal interval of definition of x t) for t

>

0 is [0,

fJ),

then

x t)

--+ 0 0

as

t

fJ

As we have said before, we will always

speak

as

though our

so

lutions

are

defined on

-0 0 , 0 0 )

and

it

should

be remembered

that

finite

escape time is always a possibility unless there is, as for example in

Corol

lary 2

below,

some condition

that

rules it

out.

In Corol la ry 3 below,

the

solutions

might

welt go to infinity in finite time.

The invariance property of the limit sets of solutions of

autonomous

systems (3)

now

enables us to refine

Theorem I.

Let V be a

Cl

function

on

R

to

R. If G

is

any

arbit rary set in

R ,

we say

that V

is a

Liapunou

function on G

for

Eq. (3)

if V

= :

(grad

V) • f

does

not

change sign on G.

Define E = {x; V(x) = 0, x in G}, where G is

the

closure of G. Let M be

314


7/12

the largest invariant set in E. M will be a closed set.

The

fundamental sta

bility theorem for

autonomous

systems is then the following:

Theorem 2. If V is a Liapunov function on G for (3), then each so

lution

x t)

of (3) that remains in G for all

I

> 0

(I

< 0) approaches

M* = M

u

{oo} as t -- . 0 0 (I -- +

- 0 0 ) .

If M

is

bounded, then e ithe r

x t)

M or x t) 0 0

as

1---.

0 0

(I

-- +

- 0 0 ) .

This one theorem contains all of

the

usual Liapunov like theorems on

stability

and

instability

of autonomous

systems. Here, however, there

are

no conditions of definiteness for

V

or V,

and

it is often possible to obtain

stability

information about

a sys tem with these

more

general types

of

Lia

pUDOV functions.

The

first corol la ry below is a stabil ity resul t which for

applications

has

been quite useful, and the second illustrates how

one

obtains

information on instability. Cetaev's instability

theorem

is similarly

an

im

mediate consequence of

Theorem

2 (see Section 3).

Corollary

2. Let G be a

component

of

Q

=

{x ; V(x)

<

1]}.

Assume

that G

is

bounded,

V;;£ 0

on G, and MO = M n

G

c:

G.

Then

MO is an

attractor

and

G is in its region of attraction. If, in addition, V is constant

on

the boundary of

MO ,

then MO is a stable attractor.

Note

that if

MO

consists of a single

point p, then p

is asymptotically stable

and

G prov ides an est imate of its region of asymptotic stability.

Corollary 3. Assume

that

relative to (3)

that

V V> 0 on G

and

on

the boundary

of G

that V

= O. Then each solution

of

(3) s tart ing in G a p

proaches 0 0 as t 0 0

(or

possibly in finite time).

There are

also some special classes of

nonautonomous

systems where the

limit sets of solut ions have an invariance property.

The

simplest

of

these

are

periodic systems (see [2])

x = ((t, x) , f(1

+ T, x =

.f(t)

for all

t

and

.r. (4)

Here, in

order

to avoid introducing the concept

of

a periodic

approach

of a

solution of (4) to a set

and

the concept of a periodic limit point, let us con

fine ourselves to solutions

x t) of

(4) which are bounded for t > O Let J be

the

positive limit set

of

such a

solution x t), and

let p be a

point

in

Q.

Then there is a solut ion

of

(4) starting

at

p which remains in

Q

for all

t

in

-

00, 00);

that

is, if

one starts at p

at

the proper

time,

the solution

remains

in Q for all time. This is the sense now in which Q is an invarian t set. Let

V(t, x) be Cl on R x R and periodic in t of period T. For an arbitrary set

G

of RIJ. we say

that

V is a

Liapunou

function on G for the periodic system (4)

315


8/12

if V does not change sign for all t and all

x

in G. Define E· = {(t, x); V(t, x)

=

0,

x

in

G}

and let

M

be

the

union

of

all solutions

x t) of

(4) with the

property

that (t,

x(t» is in

E

for all

t. M

could be called

the

largest invar

iant set relative to E. One then obtain the following version

of

Theorem 2

for periodic systems:

Theorem

3. If

V

is a Liapunov function on G for the periodicsystem (4),

then each solution

of

(4)

that

is bounded and remains in G for all t

>

0

(t

<

0)

approaches

M

as

t

--+ 00

(t

- 00).

In [5] Miller showed that the limit sets of solutions of almost periodic

systems have a similar invariance property

and

from this he obtains a resul t

quite like Theorem 3 for almost periodic systems. This then yields, for pe

riodic and almost periodic systems, a whole chain

of

theorems on stability

and instability quite similar to that for autonomous systems. Fo r example,

one has

Corollary

4. Let Q

FI

+

=

{x ;

V(t, x) < Y}, all t in [0,

T]}, and

let G+

be

a component

of

Q,,+. Let G be the component

of

Q

FI

= {x ; V{t, x) < T

for some r in [0, T]} containing

G+. If G

is bounded, V 0 for all t and

all

x in G, and if MO = M n G

c

G+, then MO is an att ractor and G+ is an

its region of attraction. If V(t, x) = , t for all t and all

x

on the boundary

of

MO, then MO is a stable att ractor .

OUf

last example

of

an invariance principle for ordinary differential equa

tions is that due to Yoshizawa in [7] for asymptotically autonomous

systems. It is a consequence of Theorem 1 and results by Markus

and

Opial

(see [7] for references) on the limit sets

of

such systems. A system

of

the form

x =

F x)

+ g(t, x) +

h(t, x)

(5)

is said to be asymptoticallyautonomousif (i) g(t, x) 0 as t

--.

00 uniformly

for x in

an

arbitrary compact set

of

Rn, (ii)

f:

Ih(t,

9 (t»

Idt

< 0 0

for all

q; bounded and continuous on [0, 00) to R ,. The combined results of Markus

and Opial then state that the positive limit sets

of

solutions

of

(5) are invar

iant sets

of

x = F(x). Using this, Yoshizawa then improved Theorem I

for asymptotically

autonomous systems.

It turns out to be useful, as we shall jIlustrate in a moment on the sim

plest possible example, in studying systems

I

which are

not

necessarily

asymptotically autonomous to state the theorem in the following manner:

Theorem 4. If, in addition to the conditions

of

Theorem 1,

it

is known

that a solution x t)

of

(1) remains in G for t > 0 and is also a solution

of

an

316


9/12

asymptotically

autonomous system (5), then

x t)

approaches M* == M

u

{oo} as

t

0 0 , where M is

the largest invariant

set of

x

==

F x) in

E.

It can happen that the sys tem (1) is itself asymptotically autonomous, in

which case the above theorem

can

be

applied.

However, as the following

example

illustrates, the original system may

no t

itself be asymptotically au

tonomous, bu t it still may be possible to

construct

for

each solution of

(I)

an

asymptotically autonomous system (5) which it also satisfies.

Consider again

the

example

x==y

y

== -

x

-

p t)y,

o

<

D

pet)

m

for all t > o

(6)

Now we have the additional assumption

that pet)

is bounded

from

above.

Let x t), be any solution

of

(6). As was argued previously below

Theorem 1, all

solutions are

bounded

and yet)

0 as t 0 0 . Now

(X(/),

satisfies x ==

Y,

rV = -

X

- pet) yet), and this system is asymptotically

autonomous to

(*)

.¥

y, y := :

- x.

With the same

Liapunov function as

before, E is

the

x axis and the largest

invariant

set of (*) in E is the origin.

Thus fo r (6)

the

origin is asymptotically

stable

in the large.

3.

Autonomous

Functional-Differential Equations

In

this

section

we

adopt

completely

the

notations

and assumptions in

troduced by Hale in his paper in these

proceedings and

present a few

of

the

stability results that he has

obtained

for autonomous differential equations

(7)

A more complete account with numerous examples

is

given in (1).

For

the

extension to per iodic and

almost

periodic functional-differential

equations

by Miller see [6].

We continue where Hale left off in Sec tion 2

of

his paper, except that we

shall assume

that

the

open

set Q is

the whole

state space C

of

continuous

functions. We also confine ourselves to solutions

x of

(7) that

are

bounded

and

hence defined on [- r, 00). Except

that

we

are

in t he s ta te space

C,

the definition of

the

positive limit set

of

a

trajectory x

f

of

(7)

is essentially

the same

as for

ordinary

differential

equations,

and

the

notion of an invar

iant set is modified to take into account

the

fact that there is no longer

uniqueness to

the

left. A set M c C is invariant in the sense that if qJ E M,

317


10/12

then

x(tp)

is defined on

[ - , 0 0 ) ,

there is an extension on

-00, - r], and

x, fP)

remains in

M

for all

t

in

-

00, 00). With these extensions of these

geometr ic not ions to the state space C, Hale then showed that

the

positive

limit set of a

trajectory

of

(7)

bounded

in

the fu-ture

is a

nonempty,

compact,

connected,

and

invariant set in C. He was then able to

obtain

a

theory

of

stability

quite

similar to

that

for

autonomous ordinary

differential equations.

Let

V

be a

continuous

function on C to

R

and

define relative to (7)

· - 1

V(tp) ==

lim - [V xr fP» - V

0 on

U

when

cp :;t:

0

and V O =

0'

and at

the

end

...

in

tersect the

boundary

of C; ... . This is clear from his

proof and

is necessary,

since he wanted to generalize the usual statement of Cetaev's theorem to

include

the

possibility

that the

equilibrium

point

be

inside

U

as well as on

its boundary.

Corollary

6.

Let p E

C be an equil ibrium point of (7) contained in

the

closure of an

open

set

U and

let

N

be a

neighborhood

of p. Assume that:

(i)

V

is a

Liapunov

function on

G

=

U

n

N,

(ii)

M

n G is either

the

empty

set or

p,

(iii)

V qJ)

< 11

on G

when

cp

= =

p, and

(iv)

V(P) = Y and V(


11/12

its

boundary or approach p. Conditions

(i)

and

(iv) imply

that it cannot

reach

or approach tha t par t of

the

boundary

of Go

inside

No

nor

can

it

approach

p as

1 -+

0 0 .

Now

(iii) states

that

there

are

no points of

M

on

that

part of

the boundary

of

No

inside

G.

Hence each such trajectory

must

leave

No in finite time. Since

p

is either in the inter ior or on

the boundary

of G,

each neighborhood

of

p contains

such trajectories,

and p is

therefore un

stable.

In .[1]

it was shown

that

the

equilibrium

point

q; = 0 of

X(/) = ax

3

( / )

+ bx3 1 - r)

was unstable

if a

>

0 and Ib I < Ia I.

Using the same Liapunov function

and

Theorem

6 we

can

show a

bit

more.

With

V(q»

= -

q>4 O)

+.. - JO q;6(6)

dO,

4a

r

V(x,) =

-- + £ r x6(8)

dO,

and

which is nonposi tive

when

I

b J <

I

a

I (negative definite

with

respect to

, (0) and tp(- r»; that is, V is a Liapunov funct ion on C

and

E

= {

0,

the

region

G = {q;; V(qJ) < O}

is nonempty,

and

no trajectory starting

in G

can

have

lp

=

0 a s a

positive limit

point

nor can

it

leave

G.

Hence

by

Theo

rem 5,

each

trajectory starting in

G must be

unbounded. Since

qJ =

0 is a

boundary

point of

G,

it is unstable.

It

is also easily seen [J] that

if a < 0

and I

b

1< Ia I then cp =

0 is asymptotically stable in the large.

In [1]

Hale

has a lso extended this theory for systems with infinite lag

r =

00), and

in

that

same

paper

gives a

number

of significant examples

of the application of this theory.

REFERENCES

(I)

Hale,

I. ,

Sufficient conditions for stability and instability of autonomous functional

differential eauations,

J. Diff. Eqs.

1, 452-482 (1965).

[2] LaSalle,

J., Someextensionsof

Liapunov's second

method, IRE

Trans. Circuit Theory

CT-7,

520-527 (1960).

319


12/12

Date post:	07-Aug-2018
Category:	Documents
Upload:	castrojp
View:	214 times
Download:	0 times

An Invariance Principle in the Theory of Stability

Documents