A wave equation with discontinuous time delay

VOL. 15 NO. 4 (1992) 781-788

A WAVE

EQUATION

WITH

DISCONTINUOUS TIME

DELAY

JOSEPHWIENER

Department

of Mathematics TheUniversityof

Texas- Pan

American

Edinburg,

Texas

78539

and LOKENATH DEBNATH

Department

of Mathematics

UniversityofCentralFlorida

Orlando,Florida 32816

(Received

October 21,

1991)

ABSTRACT.

The influence of certain discontinuousdelaysonthe behaviorof the solutions of the wave

equationis studied.

KEY

WORDS

AND PHRASES:

Partial Differential Equation, Piecewise

Constant Delay,

Wave

Equation, BoundaryValueProblem,FourierMethod,Oscillation

1991

AMS

SUBJECT

CLASSIFICATION

CODES.

35L05, 35L20, 35L50, 35L99.

1.

INTRODUCTION.

Functional differentialequations

(FDE)

withdelayprovidea mathematicalmodel foraphysicalor

biological systemin which therateofchangeof thesystemdepends uponitspasthistory. The

theory

of

FDE

with continuousargumentiswelldevelopedand has numerousapplicationsin natural andengineering

sciences. Thispapercontinues our earlierwork

[1-5]

in anattempttoextend thistheorytodifferential

equationswith discontinuousargumentdeviations.

In

these

papers,

ordinarydifferentialequationswith

arguments havingintervalsofconstancyhavebeen studied.Such equations

re’present

ahybridof continuous and discretedynamical systemsand combinepropertiesofboth differential and differenceequations.

They

include asparticularcases loaded andimpulse equations,hencetheirimportanceincontroltheoryand in certainbiomedicalmodels. Continuityofasolutionatapoint joining anytwoconsecutiveintervalsimplies

recursion relationsfor the values of the solutionatsuchpoints. Therefore,differential equationswith

piecewiseconstantargument

(EPCA)

are

intrinsically

closertodifference rather than differentialequations.

In [6]

boundary

valueproblemsfor some linear

EPCA

in partial derivativeswere considered and the behaviorof their solutions studied. The results were also extended toequationswith positivedefinite

operatorsin Hilbert

spaces.

In [7]

initial valueproblemswere studiedfor

EPCA

inpartialderivatives.

A

class of loadedequationsthat arise insolvingcertain inverse

problems

was

explored

withinthe

general

frameworkof differentialequationswithpiecewiseconstantdelay. The

purpose

of thepresentnoteis to

investigatethe influenceof terms withpiecewiseconstant time onthe behavior ofthesolutions,especially

SEPARATION OF VARIABLES IN SYSTEMS OF PDE.

Consider theboundaryvalueproblem

(BVP)

consistingof the equation

U,(x, t)

A

U=(x,t)

+

BU=(x,[t]),

theboundaryconditions

andthe initial condition

(2.1)

U(O,t)-U(1,t)-O,

(2.2)

U(x,

O)

Uo(x

).

(2.3)

Here U(x,t)

and

Uo(x)

arereal m m matrices,

A

andB are real constant m xm matrices, and

[.]

designates

thegreatest integerfunction. Accordingto

[6],

a matrix

U(x,t)

iscalledasolutionof

BUT

(2.1)-(2.3)

if itsatisfies the conditions:

(i) U(x,t)

is continuous in

G-

[0,1] [0,

oo); (ii)

U,

and

U=

existandare continuous in

G,

with the pogsible exception of the points

(x,n),

where one-sided derivatives exist

(n

0,1,2

);

(iii) U(x,t)

satisfies

Eq. (2.1)

in

G,

with thepossible exception ofthepoints

(x,n),

and conditions

(2.2),

(2.3).

Lookingfor the solutionof

(2.1)

(2.3)

in theform

U(x,t)-

T(t)X(x)

(2.4)

gives

whence

r’(t)

X(x) A T(t) X"(x)

+

BT([t])

X"(x),

(A T(t

+

BT([t

]))-’

T’(t

X"(x

X-t(x

_p2,

whichgeneratesthe

BVP

x"(x)

+

pX(x)

O,

x(o)-x()-o

and theequationwithpiecewiseconstantargument

r’(t

-A

T(t

p2

BT([t])

e

2.

X(x)

cos

(xP)

ct

+ sin

(xP) c2,

(2.6)

The

general

solutionof

Eq. (2.5)

is

where

(-I

y’x

2"

P,’

0

(-l()’x2"P2n)

sin

(xP)

-,,0

(-"

i

cos

(xP

and

Ct,

C2

are

aitra

constant matrices.

From

X(0)-

O

we conclude

at

Ct-

O,

and

e

condition

X(1)-

O

enables toch sinP

-O (although

is isnot

e

necesry conquen

of

e

equation

(sinP)

C2

O).

is canbewritten asew-e

O

or e

I.

uming

at

all

eigenvalues

pt,

p ...,p=

of

P

aredistinct and

S-PS

dgt,p

p,),

wehave e

-

I,

whence

SeS

-

I

and e

I.

erefore,

-g(j,nA,...,nj,),

where the

A

are integer, and

P-S

-t,

P2-SS-l-Sd&g(j,jf

,j)S

-,

sin

(xP)

S

sin

(x)S

-

-Sdg(sinjx, ...,sinxj)S

-.

Furthermore,

wecanput

P-diag(a(mO

1)+

1),...,amj),

1,2,...)

(2.7)

in

(2.5)

and obtain thefollowing result.

(2.8)

THEOREM

2.1. Thereexists an infinite

sequence

of matrixeigenfunctionsfor

BVP

(2.5)

X(x)-Vdiag(sint(m(j’-1)+

1)x

sinnmjx),

(]-

1,2

fo

’X(x)X,(xlax-

{O,

j#k

t,

where

!

istheidentitymatrix.

REMARK.

Thematrices

SXj(x)S

-

satisfyTheorem2.1 forany nonsingular

S.

THEOREM

2.2. Let

E(t)

bethe solution of theproblem

T’(t)

-AT(t)P

,

T(O)

t

(2.9)

andlet

n(t)

E(t)

+

(E(t)

l)a-lB.

(2.10)

Ifthematrix

A

isnonsingular,then

Eq. (2.6)

withthe initial condition

T(0)-

Co

has on

[0, oo)

aunique

solution

T(t)

M(t

It])

Mr’l(1)

Co.

(2.11)

PROOF.

On

the interval n s <n +1,where n 0isaninteger,

Eq.

(2.6)

turnsinto

T’(t)--AT(t)p2-BC.P2,

C.

T(n)

withthegeneralsolution

T(t)-E(t-n)C

-A-IBC..

At

nwehave

C.

C

A

-IBC.,

whence

C

(I

+

A

-tB

Cn

and

T(t

(E (t

n +

(E (t

n

I) A

-ZB

C.,

that is,

Att-n

+

wehaveC.+t=M(1)C,

and

Hence,

which isequivalentto

(2.11).

r(t)-M(t

-n)C,.

(2.12)

C,

-M’(1)

C0.

(2.13)

T(t)

M(t

n

)M’(1) Co,

THEOREM

2.3. If

M(1)[[

<1,then

r(t)ll

exponentiallytendstozero as

EXAMPLE. For

the scalarparabolic equation

ut(x,t

a2u=(x,t)

+

bu=(x,[t])

wehave m and

Pj

nj, accordingto

(2.7).

For

Eq. (2.9),

withA a and

P

Pj,

wehave

E(t)

e and

M(t)

e

-’2’2’

(1

e-"2’).

Hence,

theinequality

M,(1)[

< isequivalentto

-1<e 1- <1,

whence

a(1

+

1-Since the function

(1

+

e-’)/(1

-e-’)

isdecreasing,all functions

T(t)

exponentiallytendtozero as if andonlyif

-a<b

a

.

areunbounded andoscillatory,forj>j0- Indeed, lettingb

a:’

+ andsolvingtheinequality

a +

>u(1

+

e-"’)/(l

gives

e <

/(2a’-

+

),

whichholds for any >0 and sufficiently large jand implies

Ms(1

<-1. Ifb -a

2,

then

Ms(t)=

1,

T(t)- Ts(0

and

u(x,t)--Uo(X),

for allt. Therefore,the condition

Ibl

a isnecessaryand sufficientfor the series

u(x,t)

i.

Ti(t)xi(x)

(2.14)

tobe a solution of the scalar

BVP

(2.

I) (2.3),

with

A

a and

B

b,if

Uo(X)

isthree timescontinuously

differentiable. The coefficients

T(0)

aregiven by

r(0)

I

’Uo(X)x(x),

where

X(x)

vsin

(=jx)and

uo(x)

C[O,

1]

satisfies

uo(0)

uo(1)

0.

THEOREM

2.4. The solution

T

O

of

Eq.

(2.6)

isglobally asymptoticallystable as +ooif and

onlyif theeigenvalues of the matrix

M(1)

satisfytheinequalities

I,1

<

,

r-1 m.

(2.15)

PROOF.

There exists anonsingularmatrix

Q

such

thatM(1)

QJQ-,

where

J

isadiagonalor Jordan matrix with thediagonalelements

,.,.

Hence,

from

(2.13)

itfollowsthat

C,,

(QJQ-’)"Co

-Q

)J"

Q-Co

or

C,-

E

Q,(n)’,,

km

(2.16)

where the entries of the matrices

Q,(n)

arepolynomialsofdegreenotexceeding k 1.Thisimplies

C,

O

as n ooif andonlyif

(2.15)

holds,and the conclusion of the theorem follows from

(2.12).

LEMMA.

Allentriesofeverysolution of theequation

T’(t)

-AtT(t)A

(2.17)

withconstant m xm matrices

A andA2,

are linear combinationsofterms

exp()ct.2)t),

where

)and

ct?

areeigenvalues

ofAt

andAz

andkisinteger.

PROOF.

Assume,

forsimplicity,that theeigenvalues of

A

and

A

aredistinct,and let

SIAIS!

DI

diag(ct),

t

),

ct,

SA2

2

diag(ctt-),

ct),

ct,

Then the substitution

T

-S1V

changes

(2.17)

to

V’(t)

DV(t)A

and the substitution

V

WS

transforms

(2.18)

into

W’(t)

D

W(t)

D_2.

The entriesof

W(t)

are c₀

exp(ctl)?)t),

witharbitraryconstants c0,andthecompletionof the

proof

follows

WAVE EQUATION WITH DISCONTINUOUS TIME DELAY 785

fromthe formula

T

StWS

.

Ifsomeof theeigenvaluesaremultiple, polynomial factors will

replace

the

constants

c#.

THEOREM

2.5.

If

alleigenvaluesof

A

havepositiverealparts,

Uo(x)IE

C3[0,1],

and

IIA-’II

<

,

then

BVP

(2.1)

(2.3)

has a solution

(2.14).

This seriesand all its

term-by-term

derivatives

converge

uniformly.

PROOF.

Thefunctions

T(t)

satisfy

Eq. (2.6),

with

P;

given by

(2.7).

Therefore, settingin the above

lemmaA

-a,

A

Pf

andnotingthat

Re

ctl

)

<

0,

a}

2)>0,we concludethat

E;(t)

0asj ooand all

>0,where

E;(t)is

the solution of

(2.9). Hence,

(2.10)implies

g;(t)

-A-B

asj ooand

g(t)ll

<

L

for

sufficiently

large

j,

by

virtueof the condition

IIA-BII

<1.

From

theformula

Ti(t

M;(t

-[t])Mid(1)

wenotethat

T(t)ll

0exponentiallyas o,forsufficiently

large

j. Therefore,for

any

>0,ther exist

to

0 andj

N

such that

,,(t)ll

<

,

for >to,where

Rc(t)

isthe remainder of series

(2.14)

after

theNthterm.

For

0<

to

and

large

j,wehave

TAt)If

<

CoAl,

and in this case the uniform

convergence

of series

(2.14),

togetherwith itsrespectivederivatives in andx,follows from the smoothnessofthe initial function

Uo(x)

and theformula

r(o)

Co

]o

UoCx)X(xWx.

THEOREM

2.6. Ifall eigenvalues

k,

of thematrix

M(1)

arenegative,where

M(t)

is defined

by

(2.10),

then eachentryofeverymatrix solutionof

Eq.

(2.6)

isoscillatory and,moreprecisely,has a zero in each interval ns n +1,forsufficiently largen.

PROOF.

Let

)(t)

denote anentryof a solution

T(t)

of

Eq.

(2.6)

and let

c,’)-xc’)(n).

Then from

(2.16)

weget

t,-n+

r-I r-I

where

q,’)

aretheentriesof the matrices

Q,.

If

c,

’)

0,

for infinitely

many

n,then

’)(t)

is

oscillatory.

Assuming

c,

’)

,,

O,

forall sufficiently

large

n,weconcludethat

lim-(’) "-(’)

(since

the coefficients

q,)(n)

arepolynomials of

n),

where

h

isone of the eigenvalues of

M(I). Hence,

c,./c,

b_{<0, startingwithsomen,whichimpliesthat}

t’Xt

_{has a zero in each intervaln < <n +}

1,

_for

sufficientlylarget.

THEOREM

2.7. Ifthe matrix

M(1)

has anegative

eigenvalue,

thenthereexistsaninitial matrix

Co

such that thecorrespondingsolution

T(t)

of

Eq.

(2.6)

has anoscillatory entrywithazeroineachinterval n< <n+ 1,forsufficientlylargen.

PROOF. Assume,

for simplicity, that in the equation

M(1)-

QJQ-i

the matrix

J

is diagonal,

J-

d/ag(Xa,

k2,

k,,),

andthat

.

<0.

In

theequation

C,-QJ’Q-Co

put

Co-Q;

then

C,-QJ"

and

c,

)-

(qll,

q21

q,,,:X7),

_where

c

l)

isthe firstcolumn of

C,,

and

(qll,q21

q,t)

is the firstcolumnof

,.i)/r.0).

k

_{<0 asn}

--

,

_which

Q.

Since

Q

isnonsingular,there exists an elementqil O.

Hence,

lim

,.,,

1,,--,

proves

that theentry

)(t)

of the solution

T(t)

has a zero in each interval

(n,n+ 1),

forsufficiently

large

n.

Following

[8],

wecan

prove

that ifthe matrix

M(1)

has nopositive eigenvalues and no eigenvalues

Separationof variables in the matrixequationwith constant coefficients

U(x,

t)

AoU

(x,

t)

+

A2U=(x,

t)

+

BoU(x,

It

])

+

B2U=(x

It

])

leadsto

(2.5)

andtothe

EPCA

T’(t)

-AoT(t

-A2T(t)

p2

+

BoT([t])-B2T([t])

P2,

whichcan be alsoinvestigated bytheabovemethod.

3.

A

SCALAR

WAVE EPCA.

Separation

of variables in theequationwithconstantcoefficients

u,,(x,

t)

a-u=(x,

t)

bu=(x, [t])

(3.

I)

andboundaryconditions

(2.2)

yields

X/(x) vsin

(n ix)

and leads to the

EPCA

T"j(t

+ a

=nj=T(t

bnj=T([t]).

(3.2)

For

brevity,weomitthe subindexjand makethe substitution

T’(t)

V(t),

whichchanges

(3.2)

toa vector

EPCA

where w col

(T,

V)

and

w’(t)

-A

w(t)

+

Bw([t]),

0

A=

Eq. (3.3)

onthe interval n< _{<n + becomes}

withthe solution

where

w’(t)-Aw(t)+Bc.,

c.

w(n)

w(t)-M(t

-n)c.,

(3.3)

M(t)

eA’+

(e

’

-I)A qB.

(3.4)

Therefore,

Eq.

(3.3)

withthe initial condition

w(0)

co

hason

[0, oo)

auniquesolutiongiven by the right-hand

sideof

(2.11),

where

M(t)

is defined in

(3.4).

THEOREM

3.1.

For

b<0,the solution w 0 of

Eq.

(3.3)

isunstable.

PROOF.

Computationsshow that

e

cos(cot)/+

co-I

sin(cot)A

and

whereco anj.

Also,

Hence,

and

e

’’

-I

(coscot

1

co-

sincot

/

,-cosincot

coscot-

1’

eA,

I

A

qB

(

b

(1

coscot)/a

bcosincot/a

M(t)-

(coscot

+

ha-2(1

coscot)

(ha

--

1)co

sincot

co-

sincot coscot /

b b

The condition b<0implies

detM(1)

> and shows thatatleast one of theeigenvalues

.

of

M(I)

satisfies

I

>1. Therefore

w(t)ll

as /,fosomeinitial vector c

,,

0.

THEOREM

3.2.

For

b> a

2,

the solutionw 0 of

Eq. (3.3)

isunstable.

PROOF.

Calculationsyield

b

2ta

b b

det(M(1)-l)-),.2-2

costa+s,n

-)k+

l-+cosco

andthe expressions

.--s

+d,

L-s-d

for theeigenvalues

),.,g

of

M(1),

wheres-

costa+mn

,

()

d

-

sin:’ta

+

sm

-.

The condition b>a shows that

d:’

>0 and

.

>1. The latterinequality implies

w(t)]l

ooas +oo, for some initialvector_Co 0.

THEOREM

3.3. Thesolutionw 0of

Eq.

(3.3)

isasymptoticallystable as +if anonlyif

0<b<a

2,

(3.5)

andta,2.nn, n -0,1,2

PROOF.

Theconditiond <0,which meansthat theeigenvaluesof

M(1)

arecomplex,leads to

ta

cos2

-

b

2/(2a

b)2,

whence

Since

kll

and

detM(1)

3qgz,

theinequality

X,

< isequivalentto

detM(1)

<

I,

thatis,to b>

O.

Therefore, in the case of

complex

eigenvalues, a criterion for asymptotic stability is 0 <b<

max(a2(1

tan2),

a2(1-

cot27)).

The,nequahtyd >0 m the case of distinct realeigenvalues

2to

leads tob>

max(a2(1

-tan

7),

a

(1

-cotZ7)),

and thenequahaes <1,

g

>-1 yieldb<a

2.

Hence,

in this case a criterion ofasymptotic stabilityis

max(a2(1-

tan2-),

a2(1-

cot2-))<b

<a

2.

2,0

ct27))’

then d 0 and

:k

costa+

ba-2

sin

2ta/2

whence

Finally, ifb

max(a2(1

-tan

7),

a2(1

costa<

.

<

cos2ta/2

and

g[

<1. Accordingto

(2.15),

thisimplies

asymptdtic

stabilityandcompletesthe

proof

ofcriterion

(3.5).

REMARK

1. Ifb -a

2,

then

-

1,

-

costa, andthe solutions of

(3.3)

arebounded but not

asymptoticallystable.

REMARK

2.

In

Theorems 3.1 and3.2itwasimplicitlyassumed thatta

,,

2atn. Ifta 2nn,then

.

1,whichleadstothe existenceofunbounded solutions for

(3.3).

COROLLARY

1. Ifthe coefficient a is irrational, then

(3.5)

is a criterionofasymptotic stabilityof thesolutions to

(3.2)

for allj.

PROOF.

Recallingthatta-taj-aj,we note that theequality aj 2n isimpossiblefor

any

irrationala.

COROLLARY

2.

For

anyrational a, there existinfinitely

many

integersjsuch that thecorresponding

solutions

wj(t)

of

(3.3)

areunbounded.

PROOF.

Theequation

ta

2nimpliesj 2n/a, andjwill be anintegerforinfinitelymany integers

THEOREM

3.4. Eachcomponentofeverysolution of

Eq. (3.3)

oscillatesif andonlyifeitherof the following conditions holdstrue:

(i)

b<

max(a2(1 tan2), a2(1 cot2));

(ii)

max(a:’(1 -tan),

a(1-cot

-))

<b<

2s’"2

and costa<-I/2.

PROOF.

Hypothesis

(i)

meansthat theeigenvaluesof

M(1)

arecomplex,and inthiscaseall nontriviai solutions of

(3.3)

oscillate componentwise.

In

the case of real eigenvalues, we require

d2>

0 and

kt

s + d<0,whichalsoimplies

L2

s d<0,where s and d are defined in Theorem 3.2. The condition d >0leadstothe left-sideinequalityin

(ii),

and the condition

k

<0 yieldsthe

right-side

inequalityin

(ii).

Finally,the restrictioncosta<-1/2arisesfrom thecomparisonof theinequalitiess<0and

k

<0.

REMARK

3.

The restrictioncosta<-1/2 implies

a2/2sin2(ta/2)

<

2a2/3.

In

conclusion,it isworthnotingthattheasymptotic propertiesof

Eq. (3.2)

depend

onthealgebraic

natureof the coefficienta.

For

b<0,all solutionsof

Eq. (3.2)

are unstable and oscillatory; for b a all solutions of

Eq. (3.2)

are unstableandnonoscillatory. Thesetwocasesholdtruefor both rational and irrational values of a.

For

0<b<

max(a2(1-tan2),

a(1-cot2)),

all solutions of

(3.2)

areasymptoticallystable andoscillatory, providedthatta

,,

2n.

However,

asindicated in

Corollary

2,for

any

rational a, there existinfinitely

many

integers jsuch that

tar

"2 n,whichleadsto the existence of unbounded solutions for

(3.2).

Furthermore,sinceta t.o anj,theinequalitycosta<-1/2

breaks down forinfinitely

many

integers j. Therefore,underhypothesis

(ii)

of Theorem 3.4,there are

infinitely

many

solutionsof

Eq.

(3.2)

which areasymptoticallystable andoscillatory,aswell asinfinitely many solutions whichare asymptoticallystable and

nonosciilatory(ta

,

2n).

Also, forto

,

2n and

a/2 sin(ta/2)

<b<a

,

the solutionsof

(3.2)

areasymptoticallystable andnonoscillatory. Problems ofthis naturedeservefurtherinvestigation.

ACKNOWIDGEMENT.

Thisresearch waspartially

supported

by

U.S.

Army

Grant

DAAL03-89-G-0107,

NASA Grant NAG

9-553,andbytheUniversityofCentralFlorida.

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