THE CAUCHY MATRIX

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VOL. 17 NO. 4 (1994), 753-758

ESTIMATES FOR

THE CAUCHY MATRIX

OF

PERTURBED LINEAR IMPULSIVE

^EQUATION

R S. SIMEONOVandD. D. BAINOV

P.O. Box

45 1504Sofia, Bulgaria (ReceivedMarch 24,

1992)

ABSTRACT. Estimates for the Cauchy matrix of a perturbed linear impulsive equation are obtained for given estimates for the Cauchy matrix of the corresponding unperturbed ^linear impulsive equation.

KEYWORDS AND PHRASES. Cauchy^matrixandperturbedlinearimpulsive equations.

1991AMS SUBJECT

CLASSIFICATION

CODE. 34A39.

INTRODUCTION.

Consider the linearimpulsiveequation

,’=A(t)r, #^rk,

(1.1)

A z

AkZ,

rk,

where belongs to the interval

JCR:rk <rk+l

^(kl); ^the ^sequence

^{rk}

^has ^no ^finite

accumulation point; z

Rn., Ak ^Rn

^x

n. Suppose

that A(t) belongstothe _spcePC(J,Rⁿ^x

n),

i.e.

A(t) is nxn matrix-valued function which is continuous for J, rk, d at the points

rk J it hs discontinuities of thefirst kind d is continuousfrom the left. We recM1

[1]

^that

thesolutionz(t) of

(1.1)

^for J, r

ksatisfiesthe equation

z’=

A(t)z d for ^r_k theconditions

z(rF de..=J

_t--rlim z(t)

Z(rk) z(r ^de.=J

^lira z(t)

Z(rk) ⁺

^A

C(rk) (rk) ⁺ AkX(rk).

k ⁰ ^t--*rk+⁰

Let z be a norm of thevector

R"

and A sup{ Az

I:

^z 1} be the corresponding normof the matrix A R

nxn.

Let theCauchy matrix w(t,s) of

(1.1)

^satisfy^an ^estimate^{of the}

form

W(t,8) _<(t)(8) (,,t J, _<t),

(1.2)

where thefunctions

, _:J--.R+

continuousand positive.

Based onestimate

(1.2),

weshall seek forvarious estimates for the Cauchy matrix Q(t,s)of theperturbedlinearequation

y’=_[A(t)

+

B(t)]y, :/:rk,

(1.3)

Ay [A_k

+ Bk]Y

rk,

(2)

where_B(t)_{E PC(J,R}

nxn)

andBk

.

^R

^nxn.

Weshallusethefollowinglemma:

LEMMA

1.1

[2].

^Letthe functionu PC(J,I

+

satisfy the inequality

s<_rk^<

wherec>0andPk^>^{0 are}^constants ^and^p(r)

_

^PC(J,I

₊

^).

Then

s<rk<t MAIN

RESULTS.

Recall

[1]

^that^if

Uk(t,s

^is^{the Cauchy}^matrixfor the equation

(s,t^e_J,s< t).

x’=

A(t)x

(rk-

^< ^<

rk),

then theCauchymatrixfor equation

(1.1)

^is

W(t,s

(s,t-(rk_l,rk]),

Uk+l(t,r)(E+Ak)Uk(rk

^,s)

(rk_l<S<rk<t<rk+l),

,+1

Uk+l(t,r)H(E+Aj)Uj(rj, r_ll(E+Ai)Ui(ri,

^s)

(ri_X<S<ri<rt<t<_rk+l).

3-k

Thenan arbitrary solutiony(t)of

(1.3)

^satisfiesthe integro-summary equation

(0 (.,(.+ (..’l(.’l(.’le."

+ (. (

r<

From

(2.1)

^and

(1.2)it

follows that

ly(t) (t)V(s)ly(s)l

+ fts(t)V(r)lB(r)l

^ly(r)

^ldr+ (t)V(rk)lBkl y(rk) srk<t

The the function u(t)= ly(t)l/(t)^satisfiestheinequality

f(s)ly(s)l

+ (r)f(r)lB(r)lu(r)dr ⁺ ^(rk)(rk)lB

^k

^u(rk).

srk<t

We apply Lemma1.1andobtn the estimate

ly(t) 5 y(s) M(t,s), where

(2.1)

M(t,s) (t)b(s) (1

s<rk<t

From

(2.2)

^{and the} equality y(t)=Q(t,s)y(s) there follow immediately the subsequent assertions:

THEOREM

2.1. LettheCauchymatrixW(t,s) of equation

(1.1)

^satisfyestimate

(1.2).

Then the Cauchymatrix Q(t,s)of equation

(1.3)

^satisfies ^theestimate

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PERTURBED LINEAR IMPULSIVE EQUATION 755

Q(t,s) _< M(t,s) (s,t ^J,s<t), where M(t,s) isgiven by

(2.3)

COROLLARY

2.1. If

w(t,s) _<Ke(t s) (s,t J,s_<t), whereK > anda areconstants, then

s<rk<t COROLLARY

2.2.

constant8>0such that

(2.4)

(2.5)

If in the interval J

R+

^estimate

^(2.4)

îs ^valid ^{and there} êxists â

sup IB(r) <6, sup

Bkl

^<^6,

r+ ^rkER

⁺

then

O(t,s) <^h’ea(t-s) K(t-s)+tn(1

+ K6)i[s,t),

wherei[s,t)^is^thenumberofpointsrklyingin the interval[s,t).

Moreover,

if thereexistconstants q>0ande>0such that

then

i[,,t)<_(t- ,)

+ , (2.8)

IO(t,s)l < K(I+K6) exp{[a+K$+qtn(l+K)l(t-s)} (c<s<t).

(2.9)

Taking^intoaccount that

1-I

⁽¹

+KIBkl)

^<_exp ^K

IBkl,

^we^obtain

s<_rk < ^S<rk <

COROLLARY

2.3.

In

the interval J’

=it+

^let ^estimate

^(2.4)

^be ^valid ^{and let} ^a ^constant

M>0exist such that

Then

0 IB(r)

ldr+ _, Bkl

^<_^M.

^(2.10)

Q(t,s)I

<_KeKM’e

^a(t-s) (O<s<t).

(2.11) REMARK

1. If equation

(1.1)

^is^uniformlyasymptoticallystable,i.e.,estimate

(2.4)

isvalid with a<0, then under perturbations for which

(2.6)

^is satisfied with small enough equation

(1.3)

^isalso uniformly asymptotically stable.

If equation

(1.1)

îs ûniformly ^stable, i.e., â=0 in

(2.4)

^and ^condition

(2.10)

^is valid, then equation

(1.3)

^isalso uniformly stable.

The goalof thefollowing considerations isto obtain estimatesfor O(t,s) inwhich instead of theintegral ^{and the} sum of thenorms of B(r) and B_k thenormofthefollowingfunction should enter

D(s)

[ ^tB(r)dr +

^B_k ^(s,t^e^J,s^<^t).

s<rk<t

Weshallnotethat O(s)is continuousfors

:

^rk, O(t- ⁰and

O(r- O(rk) O(r +

^B_k.

Let y(t)beanarbitrarysolutionof

(1.3).

From

(2.1),

takingintoaccountthat

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and

W(t,t- )- W(t,s) (t,r)D(r)y(r)dr

+

W(t,s)D’(r)y(r)dr

+

W(t,s)D(r)y’(r)dr

sSrk<t

W(t,

r ^)D(r ^)y(r

^{)- W(t,}

^rf ^)n(rff

^)y(rf⁾⁺^W(t,

r ^)BkY(rk)

W(t,^r

t

^)[D(r

t

^)(E

⁺

^Ak

+ Bk)-

^(E

⁺ Ak)D(r f ⁺ Bk]Y(rk)

w(t,

- ^)[o( ^)(A ⁺ ^)- ^AO(f

weobtainthat

y(t) w(t,s)[E

+

D(s)]y(s)

+

W(t,s)[D(r)(A(r)

+

B(r)) A(r)D(r)]y(r)dr

and

+

W(t,

r ^)[D(r

^)(A^k

⁺ ^{Bk)- AkD(r} ^f ^)]y(rk). ^(2.12)

S<rk<t

If W(t,s)satisfies estimate

(1.2)

^and^there^exist^constantsM_>O, m>⁰andr/>⁰such that IA(t)l <M,IB(t)l <M,

IAkl

^<m,

IBkl

^<m

S<rk<t

(t,

rkeJ ^(2.13)

<__rt (s <t),

(2.14)

then from

(2.12)

^we^obtain^that

<o(t)(s)(1 /0) y(s) /

ItsSO(t)(r)’3Moly(r)ldr+

Y(t)

and by Lemma1.1 weobtainthat

(t)(rk)" 3moly(rk) s<rk<t

y(t) < ly(s)lN(t,s) (s,t J,s<_t),

(2.15)

where

(2.16) S<rk<t

Fromtheestimate

(2.15)

^obtainedthere follows immediately.

THEOREM 2.2. Let theCauchy matrix W(t,s) of equation

(1.1)

^satisfy ^estimate

(1.2)

^and

letconditions

(2.13)

^and

(2.14)

^hold.

Then the CauchymatrixQ(t,s) of equation

(1.3)

^satisfies ^theestimate Q(t,s) <N(t,s) (s,t S,s<t), where N(t,s)is givenby

(2.16).

COROLLARY

2.4. If W(t,s) <^Ke^{a(t- s)} (s,t J,s<_t), then

Q(t,s) < (1

+

)Kea(t-

s).e3KMo(t-

^s)

⁺

^en(1

⁺

3Kmo)i[s,t)

(2.17)

fors,t J,s<t.

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Moreover,

if condition

(2.8)

holds,then

Q(t,s) < (1

+

r/)(1

+

^3Kmo)^e^K ^[a

+

3KMl+qtn(1

+

3Krnrl)](t- s)

(2.18)

fors,t J,s<^t.

COROLLARY

2.5.

In

the assumptions of Theorem 2.2 let condition

(2.14)

^bereplaced by themoregeneral condition

tB(r)

^dr

+

^B_k <_1 (s,teJ,s< <^s

+

h),

(2.19)

$

s<rk<t

whereh>0is aconstant. Then Q(t,s) satisfiestheestimate

Q(t,s) _<K(1

+

t/) ezp{[a

+

3KMo

+ en(K ⁺

^Kt/)](t ^s)

⁺

^en(1

⁺

3Kml)i[s,t)}

fors,t J,s<^t.

Indeed,estimate

(2.20)

^followsimmediatelyfrom

(2.17)

and the fact that theestimate y(t) <_ ly(s)

Lez’r(t

^s)+

^ri[s,t)]

_(s< <s

+

h)

implies

Then

ly(t)l _< lu(s)l^Lezp[v

+nL)(t-

^s)

⁺

^ri[s,t] ^(s

<

^t).

REMARK

2.

In

somecasesestimate

(2.17)

^isbetter thanestimate

(2.7).

EXAMPLE

1. Letequations

(1.1)

^and

(1.3)

be scalar and A(t)= 1,

B(t)=sinwt, Ak=^l,

Bk=(-1)kb,

0<b<l,

rk=S=0,1,2,--.,t51+.

IW(t,s)

e-(t-

^s)

⁺

n2i[s,t) <

Kea(t-

^s)

whereK 2, a

+

^en2.

In

thenotationintroduced

6=1, M=I, re=l,

I

^B(T)dT

⁺ ^Z

ThenQ(t,s) is estimated:

(i)

byestimate

(2.7)

(ii)

Q(t,s) <Kea(t

S)’e2(t-

^{s) +/n(1}

⁺

^2)i[s,t)

byestimate

(2.17)

(O<_s<_t),

B_k <

2

^+b ^r/.

(2.20)

(2.21)

Q(t,s) <_ (1

+

l)Kea(t

s).e6rl(t-

^{s) +/n(1}

⁺

^6o)i[s,t)

_(2.22)

Estimate

(2.22)

^is better than estimate

(2.21)

^if 60<2, i.e., if

2+b

^<

1/2

which is fulfilled for largewand smallb.

ACKNOWLEDGEMENT.

The present investigationis supported by the Bulgarian Ministry of Science andHigherEducationunderGrantMM-7.

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REFERENCES

1.

BAINOV,

D.D.

& SIMEONOV, P.S.,

Systems with ImpulseEffect: Stability,Theoryand Applications,Ellis Horwood Limited, (1989).

2.

SAMOILENKO,

A.M.

& PERESTYUK, N.A.,

Stability of the solutions of differential equationswithimpulseeffect, Differ.

Eqns.

13

(11) (1981-1992)(in Russia.

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THE CAUCHY MATRIX