On linear differential equations with exponential coefficients and stationary delays of the argument. Regular case

ON LINEAR DIFFERENTIAL EQUATIONS WITD EXPONENTIAL COEFFICIENTS AND STATIONARY DELAYS OF THE ARGUMENT. REGULAR CASE (0 LINEINYKH DIFFERENTSIAL'NYKH URAVNENIIAKH S EKSPONENTSIAL'NYMI KOEFFITSIENTAMI I STATSIONARNYMI ZAPAZDYVANIIAMI ARGUMENTA REGULIARNYI SLUCHAI) PMM

Vo1.26,

No.3, K. G.

PP.

1962,

449-454

VALEEV

(Leningrad) (Received

The equations

with

[lags]

argument

in the

engineering [I].

is

solution

tained

that

coefficients

The present

is

of

reduced

a system

form for

The method presented

makes it

satisfying

tion

of

the

system

singular

point

in which

Yuler’s

tions

with

equations 1.

certain of

linear

constant by the

the

use of

large

system

coefficients

of

of

to

met in

carried

by the

construct

of linear from the

by a previously

transform

This the

out

author

[21 of

solution

is

ob-

argument. a particular

The construction

transform

linear

delays

frequently

of

in the neighborhood of method in a way similar

solution differs

and stationary

stationary

the Laplace

values

possible

the Laplace

is

equations.

equations

for

of

conditions.

coefficients,

are

one used

study

from Frobenius’

method,

The following

ponential

initial

differs

the

differential

in an asymptotic

tion

and with here,

investigation of

to the

of

1962)

5,

are considered

a generalization

The problem

the

exponential

problems.

method which

February

solu-

the

solu-

a regular to the one

differential solution of

equathese

[z].

differential delays

equations,

in the

with

argument,

is

excon-

sidered

(1.1) *=o

Here, satisfying

Y(t) the

is

an m-dimensional

conditions

vector,

A

9”

are complex

m x m rtlatrices

Linear

differential

with

equations

A,,

E

exponential

coefficients

669

(1.2)

E,

where E is the identity matrix. 'Ihesymbol 1.11denotes the norm of the matrix

.qkw

The elements aqk(fi) of the matrix A k(8) = II a II m are functions of bounded variatSllon on [-h, O], (h >Q 0) \[3] . ‘I% number'I in (1.1) is usually assumed to he finite. The case when 1 = m will be considered separately. !lerewe shall assume that (1.4) , n-l;

(k=O,l,...

s, i-l,..

bn)

The differential in front of the matrices ‘4qk (6) in (1.1) are Stieltjes integrals [3, p.2771. 'Ihecomplex numbers a9 satisfy the following conditions a, Go=0,

(q = 1, 2, .

Re as),0

, I)

(1.5)

Among the numbers 11~a there can be rationally noncomnensurate numbers. Suppose that the'transform of the vector 0(t) (t > 0) is a meromorphic vector Q(p) whose components are regular and bounded when Re p > b = const. In a particular case we shall assume that

0((t)= i; CjtYjFjt, .;=I

Q (p) = i CjYj!(p-Uo,)-"j--l j=l

(1.6)

where Cj are constant complex vectors, vi are non-negative integers, and the o. are complex numbers. With t > 0, we shall seek the system's (1.1) solut!on Y(t) satisfying the initial conditions

Y(t) = Yf’ (l), . . . ,

s

=Yy)

(t),

t E L--h, 01

(1.7)

where the vectors YO(j) (j = 0, 1, . . . . n - 1) satisfy Dirichlet's condition when t E [-h, O]. We assume that the vectors Y(t), . . . . cl”-‘Y(t)/&“l are continuous from the right at the point t = 0. I.’

In this section there is constructed a system of linear difference

670

K.G.

l’al

eev

equations for the image F(p) of the solution Y(t). The solution of this system of linear difference equations is obtained in the form of a series of matrices. 'Iherelationship between the image F(p) and the original function Y(t) will be denoted by a small arrow as follows

1'(t)+F

F (p) = f Y (t) e--pt dt n

(P),

(2.1)

Multiplying the system (1.1) by e-pt and integrating with respect to t from 0 to m, we obtain for F(p) the system of linear difference equation [l]

22L, (P -I- 4

F (P + aq)= R (p)

(2.2)

4=0 Here

n-1 J&(P)=

Aqnpn

0

(q =o,

. , 1)

(2.3)

(C is given in (1.6))

(2.4)

f-

R (P>= Q (P>+ 2 yq (P + 4

1,.

q=o y,(y) = A4"%s

Y$'(())pn-j-l+ ni2 ni1

j=o

{

ePs dAqk

(6) y$'(())pk-j-l-

j=O k=j+l -h n--l

0 0 ep (e--t)dAqk

23

(I?) Yhk’ (t) dt

(2.5)

ss

B=n -h 1 The elements of the known matrices L (p) are entire functions of p which satisfy the following condition ur&formly in Tm p

lim jj I L2 W Lq(P + aq)I<

P+w

P<

1f

Rep>b

= cons1

(2.6)

q-=1

'Ihisis implied by (1.2). llereR(p) is a known vector, and we have, in view of (2.~1), lim 1LF1 (p) R (p) 1 = 0,

Rep -

+ oc

P-7)

Making use of (2.3) and (2.1) we introduce the notation K,(p) = - L?(P)-%(P

+ xq)t

Q(P) = L?(P) H(P)

(2.8)

If the system of ciifferenceequations (2.3) is multiplied by :J0-'(p)

Linear

and solved

differential

for F(p)

equations

with

exponential

coefficients

671

one obtains W)=i

(2.9)

WP)WP+%)+Q(P) q=1

@hen Re p > b,, where b, is sufficiently large, the relation (2.9) can be considered as a contraction mapping E3, p.441 of the space of bounded regular (when Re p > b,) vector functions. The metric in this space is defined as

where the f,.(p) are the components of the vector Fs(p) (s = 1, 2). The system’s (2.4) so 1ution F(p), which is bounded for sufficiently large k p > b,, is unique and can be obtained by the method of successive approximations t3, p.45) . !Ye have (2.12) Fo (P) = 0,

F~+I (P) s

(i=O,i,2,

$J Kq (p) Fj (P Jr 6) + Q (P)

...)

(I=1

The sequence of functions Fj(p) converges uniformly F(p), which is regular when Re p > 6,) because

From (2.11)

we obtain

F(P) = Q(P) -I- $j 0=1

the following

expression

to the vector

for F(p)

(2.13)

&I (P) K,, (P -t h7,) &a (P + %, + %J * * *

x qj=1, 2, . . . , I

. . . Kq, (P -I- Rq, + an* + * * * + %,_J

Q (P -I- Q, + %2 $- * . * + a,“)

3. flegular case a0 z 0, Re a > 0 (“I = 1, . . . . I). In &is case the system (1.1) is especially simp 9e. Let us consider the equation which we shall call the generating one of (3.3) Det L, (p) = 0 We denote consideration

(3.1,!

its roots by pO, pl, pz, . . . . pk, . . . and introduce where the numbers pk kl, . . . . k,'

our

0'

PI;,, [s,,....I;~=PI;,- k,a,-h_2x?-.

Yext we designate

into

..

-

/
(1~~z 0% I, 2, ..a; g=(3, I,... ,1) (3.2)

by Z, the multiply-connected

region

of the coinpl.ex

672

Val eev

K.C.

plane

defined

p

by the

inequality

If

p E

and Re

t,

the

series

(2.13)

(2.8)

it

follows

that

pk

consider

the

k

0’ 1’ vector

= const,

b,

p >

that

the points

converges the poles

of

the

(3.2)

of

‘**I kl function

then

r is

a sufficiently

large

When E > 0 is smell Cs 1P - Pk boundary

converge ing

..

from

of

(2.13)

and must be of

ho, L . . . , kl

+

finite

C, uniformly.

finite

only

order.

and at

Let

)’

is

Rj(p) sequence

us

(3.4)

regular

theorem,

Ry Keierstrass’

\‘Je have

values.

thus

in the circle

converges

Hj(p)

converges uniformly inside the circle aylor’s expansion’ of Hi(p) at the point,

to definite

and (2.6)

From (3.1)

can lie

C,,

Il.(p)

of

(2.12)

(3.3)

integer.

< &, and the

$1

follows

and uniformly.

terms

enough, the vector

the’circie

o!’

sequence cients

kl

it

absolutely

Hj (PI = J’i (P)(Pwhere

1, . . . ( I)

(“Q = 0, 1, 2, . . . , q = 0,

(P--k.,k,,...,klI),&>O

p

on the the

and the coeffi-

= pk

establish:;

kl,

. . . .

the

follow-

kl

theorem. Theorem

3.2.

Suppose

that

in the

system

(1.1)

a0 E 0, Se a

> 0

transform F(p) (‘I = 1, . . . . 11, and 0,(t) z 0. In this case the Laplace is representable in the form of the series of Y(t), given by (2.11, (2.13). The meromorphic vector *v(p) can have poles, of finite multidetermined by (3.2). The coplicity, Only at the points pk efficients converge

to

kl,

0’

of Laurent’s

expansion

the corresponding

of

...I

kl

F(p)

at the points

coefficients

of

I: = pk

the expansion

k 0’ I#“‘., kl of the vector

F(p). ‘Ihe last of

Y(t)

assertion

of 7lleorem

as an asymptotic

series

5

Y(t)k,, k,, From the properties following

asymptotic

Re pk,, ‘IIeorem

(3.1)

of

permits large

one to obtain

values

of

the expansion

t

rescF(P)ep’)IP=Pko, k,,

,,,

, kl

(3.5j

, kl =CI

the Laplace

transform

Dl ) we now obtain

the

result

Ix

Y(t)-

3.1 for

T’s

cF (p) ep’)IP’Pk,,

h’,,

k,, . . . , kl >t~

and property

(3.5)

imply

the next

,,,

,

ki

e--bt + ’ 1+co

theorem.

(3.6)

Linear

Theorem

differential

3.2.

equations

with

exponential

coefficients

673

Suppose that in the system (1.1) a0 E 0 ,

(q=1,

Rea,>O

. . . . I)

Then we have the following results: (1) The solutions of the system (1.1) are asymptotically stable if Re pk < 0 (k = 0, 1, ...).

(2) The solutions of the system (1.1) are not stable if Re pk > 0 0

for at least one pk

0

(3) Suppose that Re pk < 0 (k = 0, 1, 2, ...). ?he solutions of the system (1.1) will be stable if and only if the elements of the matrix L,'(p) (2.3) have simple poles for all roots pk lying on the imaginary axis. lhe conclusions of Theorem 3.2 can be reformulated in the following way. Theorem 3.3. Suppose that in the system of Equations (1.1) a0 E 0, Re a9 > 0 (q = 1, .... 1), and Q(t) e 0. In order that all solutions of the system (1.1) may be stable, it is necessary and sufficient that all solutions of the abbrev ated system of (l.l), that is, of

A

d”Y (1)

on -

(3-V

dt”

be stable. Note 3.1. If one considers the solutions of the system (1.1) and (3.7) with the initial conditions (1.7), then one can distinguish between their behaviors at infinity, when t + + 0~. For example, the solution of will tend to zero while the solution of the system of the system (1.1) Equations (3.7), with the same initial conditions, will be unbounded. Note 3.2. In the 'Iheorems3.2 and 3.3 the main assertions follow already from known results (see, for example, [4]). If 0(t) in (1.1) has the form (1.6), then it follows from (2.13), (2.8), (2.4) and (1.6) that the vector F(,o)will have additional poles at the points ll'j, k,. . . .

, k[

=

mj

-

k,a,- &.a,-

..-

klal

(j=i,..., L k, = 0, 1,2,. ..) (3.8)

Example

3.1.

Let

us

consider the differential equation

17.CT. Val eev

674

dy (t)

--Jj-

ay

+

By (2.3)

be-‘y(t -

(t) +

and (2.4)

image

Ez

0

(t < 0)

(3.9)

we have

L (PI = For the

y(t)

Y(0) = 1,

z) = 0,

f(p)

of

Lq (p) = be-PT,

a,

P +

the

solution

y(t)

(3.10)

R (P) = 1

of

the

Equation

(3. s),

ob-

we

tain f (p) = (p + a)+ The series

(2.13)

takes

on the

be-(P+l)

f(P)=*-

we obtain

e--at

1 -

i

~

I! b2,r

be+ (-O) x

-t

i\ I+Te

Example tion

y(t)

3.2. of

For the equation

+

Let

equation

dy/dt+

y (I -

f(p)

+

the

the

bneT (na--n @+I) / 2)

ePt

asymptotic

y(t)

solution

expansion

=

y(t)

sin

and (3.5)

we obtain

(IflZ)-

t,

p-‘e-’

O’+l)

+.‘.+(--l)“fl

-\-$-e-2r

+-$-

4

. ..+

n!

i

The series values

of

t,

(3.16), but

it

as well

of

as the

series

describes very well if we take the For example,

(3.13)

1

the

solu-

(3.24)

the

difference

f (p + 1)

(n-1)x

(3.15!

i 2

(1+n2)(n-11)!

e-3t

:-<

expansion

en(n-l)

l+ebL

+...

tE I---n, 01

(1 _t ,-7Xx) e-n

(1$-22)1!

X

(3.5)

we now have

the

‘i

f 2--nU)

nf

(1 + e-s%) e-”

1 + e+

y(t)-

(2.13)

+...

nl (n (n-1)

f(P) = (e-x (Pfl) + 1) [(p + 1)2 + I] pi-’ From Formulas

(3.12)

(3.5)

...+

TC)e-* = 0,

of

-I-.‘.

expansion

bne’

us find

the

image

asymptotic

e-2!

=

(p+a)(p+a+l)(P+a+2)

6~5 @Se) 21 -...+(--)n

2r

(3.11)

form

(1--%a)

L

f (p + 1)

&-@P-3)

the

her (a-1) N

T (p + a)+

T

rP+4w-a+~)+

From (3.12)

y (t)

be(++‘)

(3.13),

> :Y...

!

%

x/2

e--nt -f-

diverges

for

(3.16)

.. all

the asymptotic behavior of y(t) first five terms of the series when t - a. the limiting in the first set of parentheses in (3. 16). then we obtain -22 , By making use of another value y( +m) with an accuracy of up to 10 grouping in finding the original function, one can show that the

Linear

corresponding if

one

series

series the

in

6, then

one obtains

for

all

In order

ential

equations

usually

tion

explain

but without in Section of

the

neighborhood various

3 is

solution of

of

in control

of

of

t. Thus,

form of

a power

and the

expansion

will

we note

that

a system

a regular

argument

[Il.

linear

linear

singular to ones

the is

regular which

for

the

prob-

differ-

point

the

The methoa

the most cenvenient of

of

substitution,

system

was pro-

construc-

differential equations in the This is especially so for the

singular

point.

Let

the

fundamental

us find

a system

of

differential

normalized

equations

matrix

(which

is

of

the

frequently

met

problems) z dZ/dx== (.A - zB) Z (cc),

in the

values

in the

675

cases.

4.1.

Example

case”

of

a simple

a system

a regular

critical

solutions

one of of

finite 3.1

solution,

solution

in the

coefficients

t.

neighborhood

a delay

all

term “regular

the

by means of

for

usual

of

the of

in the

reduced,

the

exponential

in Example

y(f)

values

construction

with

converges

original

finite to

the

posed

y{ t)

seeking

lem of

(1.1)

for

is

converge 4.

equations

differential

neighborhood

ables

with

leads

to

the

the

of

aid

system

of

a regular the

of

2 (1) = E,

singular

formulas

De obtain

of

solution

equations

in the

(2.2)

Let us denote by pl, pq, A) = 0, and let us consider

Be shall

+

we have the 4-l

denote

=

the

A change

p Y(f)

= Y( -In

Cl

(P

-Id-

takes

of

vari-

r),

on the

(4.2)

form

(E (P -I- 1) + 4-l +

. . . , p, the roots the

t E 10, 00)

(E (p -!- 2) + A)-’ + . . .

most simple

(i? II -7i,“.,...,?n;

P: -p/,+k

(Ep

(4.1)

form

+ (Ep + A)-‘B (E (p +- 1) + A)9

that

x = 0.

, Z(Z)

Y (0) = E,

F (PII = (EP + 4-’ -t WP 4- A)--

Suppose

11, 0)

equations

difference

its

point

x = e

dY,Mt + AY (t) = e-{ BY (t), The system

-t

xE

simplest -t-

c2

noncommutative

i#h;

k=O,

of

partial

tyep (P

-

of the equation case when

pz)-’

product

+

. . . +

of

*i,

*2,...1

fractions c,

iP

matrices

-

&J-l

(4.4) Det (Ep +

(4 LJJ expansion (4.1::

in the usual

K.C.

67G

Valeev

[I A, = n,, A% . . . , A,

(4.7)

k-1

We can find the

change

the

of

Z(x)

original

variables,

Y(t)

with

5 x-j k + 5 2 j=:

=

the

aid

of

and recalling

(3.5).

we obtain

k-l

{E+

x

fi

[(Q-

‘WI} cj x

k + 1 + s) +

S=l

sir

(4.5)

[B (E (Pj + s) -t WI}

k=1 s=l

Example

ential

Let

4.2.

equation

us find

which

the

solution,

has a regular

when x E [l,

singular

point

at

o),

of

a differ-

x = 0 z (1) = ll

By means of

the

change

of

variables

x = ,-*,

t E LO, m),

(4.9)

Z(X) S y(t),

we

obtain d2!/ (t) -__ dt2

The difference

,--1 y (t) = 0,

equation

Here we have for

the

expansion

original

the

critical

by the of

ing two constants series

f(P)=~+p”,p_tl)t+

image has a pole

the

and its

solution

(2.13)

(4.10) have the

1

P’f(P)=l+j(p+‘),

when the

(2.2)

(0) = 1

g

y (0) = 0,

f(p)

case of

usual we

the

differential

equation

(4.12) (4. lo),

order

in terms

a and b,

for

p’(pi_1~(171~+“’

forms

higher than the first. We are looking determining the principal part of rules, of the poles p = 0, -1, -2, . . . Introducobtain the solution in the form of the

Linear

differential

equations

with

ezponential

coefficients

677

BIBLIOGRAPHY 1.

Valeev,

K. G.,

shenii

0 resbenii

nekotorykh

periodicheskimi exponents tions 2.

Levrent’ev, nogo

of

with

Shilov,

solutions

of

Fizmatgiz,

0. R.,

Krasovskii,

the

the

differential

24,

teorii

Theory

of

s

and characteristic

linear

Metody

No.

4.

equa-

1960.

funktsii

Functions

Cpetsial

analit.

Fizmatgiz,

Course).

of

of

re-

uravnenii

kompleksof

a Complex.

1958.

N. N. , Nekotorye

(Some Problems

solution

PM4 Vol.

B.V., of

Matematicheskii Special

(On the

some systems

(Methods

pokazateliakh

differentsial’nykh

coefficients).

M.A. and Shabat,

peremennogo

Analysis. 4.

lineinykh

koeffitsientami

periodic

Variable).

3.

i kharakteristicheskikh

sistem

radachi

Theory

of

teorii Stability

‘nyi

kurs

(Mathematical

1960. ustoichivosti of

Motion).

Translated

dvizheniia GIFML, 1959.

by H. P. T.