0387986995Basic TheoryC

208	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Step 8. In this final step, we prove that the bounded solution Tjt of (VII.3.7) satisfies condition (2) of Theorem VII-3-1. It easily follows from Steps 6 and 4 that

		1
t	lim o	exp [(A3 - at)(t - s)1 ir4jt(t - s, s,T(s))ds = 0
		1+ 00
if	> X1.

For (j, f) such that ), < at, write the right-hand side of (VII.3.7') in the following form:

Jtot exp 12 (A	s)] IV1t(t - s, s,T(s))ds
Q	11	- s, s,T(s))ds
fexp	L(A - a)(t - s)]
or
r		1

+ t exp 12 (At - at)(t - s)] bt%tt(t - s, s, T(s) )ds

Jo l

for any a such that to < or < t. Observe that

fexp - At)(t - s)] Wjt(t - s,s,T(s))dso

< exp L2 (.1j - '\t)(t -a)]	12 (a) - at)(a - s)J 14 t(t -- s, s, T(s))ds
	I 1'. aexp `

< 2KO(t°) {l + C2} exp 1(aI - AE)(t - a)
At - A)		1:5	J ,
if JT(t)[ < C on I (cf. (VII.3.10)). Observe also that
f exp		j		A i3(a) {1 +C2}
	L2(,\j - at)(t - s)]ji)t(t - s,s,T(s))dsl
if T(t)j < C on I (cf. (VII.3.10)). Hence,
jexpt	1(A3 - a)(t - s)]		- s, s, T(s))dso
< 2K1\1 211-C2} {(to)exP {(A, - At)(t -				)1 +
				J
i f T(t) I < C on T. Therefore, letting a and t tend to +oo, we obtain
	rt
lim	I exp 12 (A, - At)(t - s)] Yt,jt(t - s, s, T(s))ds = 0.
t-+3o	to	L

This completes the proof of Theorem VII-3-1.

In order to find Liapounoff's type numbers of system (VII.3.4), it is necessary to establish the following lemma.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

209

Lemma VII-3-3. If N is a nilpotent matrix, then for any positive number e such that 0 < e < 1, there exists an invertible matrix P(e) such that

P(e)-INP(e)I < Xe

for some positive number IC independent of e.

Proof.

Assume without any loss of generality that N is in an upper-triangular form with zeros on the main diagonal (cf. Lemma IV-1-8). Set A(e) = diag[1, e, ... , e"-1].

Then, A(e)-'NA(e) = [ ek-Jz,k ] (a shearing transformation). Hence, as 0 < e < 1 and j < k, we obtain IA(e)-1NA(e)I < e(N(. 0

Let us find Liapounoff's type numbers of system (VII.3.4), i.e.,

dzj	= IA + B,,(t) +	z,	(7 = 1,2,... ,m).
a.

h*j

Theorem VII-3-4. A system of the form

dz = (AIn+E+N+B(t)Ja

dt

has only one Liapounoff's type number ,\ at t = +oo if

(i)A is a real number,

(ii)In is the n x n identity matrix, E is an n x n constant diagonal matrix whose entries on the main diagonal are all purely imaginary, N is an n x n constant nilpotent matrix, and EN = NE,

(iii)the entries of the n x n matrix B(t) are continuous on the interval I = {t :

to < t < +oo},

(iv) t lim B(t) = 0.

+00

Proof.

Set i = ea`e`EU. Then,

(VII.3.13) dt = [N + C(t)) u, C(t) = e-`EB(t)etE

Let p be any Liapounoff's type number of (VII.3.13) at t = +oo. Then, (µ( <

sup(N + C(t)( (cf. Theorem VII-2-1). This is true even if to -+ +oo. Since tez

lim C(t) = 0, we conclude that 1µ1 < (N(. Using Lemma VII-3-3, it can be t+00

shown that any Liapounoff's type number p of (VII.3.13) at t = +oo is zero. This, in turn, completes the proof of Theorem VII-3-4. 0

Remark VII-3-5. For a nilpotent matrix N, the two matrices N and eN are similar to each other for any nonzero number e. To prove this, use the Jordan canonical form of N.

From the argument given above, we conclude that Liapounoff's type numbers of system (VII.3.4) are at, ... ,.1,,, and their respective multiplicities are n1, ... , r4,,.

Thus, the following theorem was proved.

210	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS
Theorem VII-3-6. Liapounof's type numbers of a system		= B(t)y at t =

+oo and their respective multiplicities are exactly the same as those of the system dy dt= Ay with a constant matrix A if the entries of the matrix B(t) are continuous

on the interval Io = {t : 0 < t < +oo} and lim B(t) = A.

t-.+ao

Applying Lemma VII-1-4 to the solutions of the system

= B(t)y', we obtain

the following corollary of Theorem VII-3-6.

Corollary VII-3-7. If an n x n matrix B(t) satisfies the conditions

(i)the entries of B(t) are continuous on the interval To = {t : 0 < t < +oo},

(ii)lim B(t) = A exists,

t-.

then, for every nontrivial solution ¢(t) of the system	dy	= B(t)y,	lim	log 1-0(t)I
	dt		t--+oo	t

p exists and p is the real part of an eigenvalue of the matrix A.

Remark VII-3-8. The conclusion of Theorem VII-3-1 still holds even if condition

(iv) of Assumption 1 of this section is replaced by IB)k(t)j < f (t) (j, k = 1, 2,... , m) on the interval Zo, where f (t) satisfies

		rP
(VII.3.14)	sup(1 +p- t)-1 J f(s)ds - 0 as t +oo.
	p>t	t

To see this, let us assume that a positive-valued function f (t) satisfies condition tP

(VII.3.14). Set h(t) = sup ((1 + p - t)-1 ( f (s)ds) for a fixed positive number
p>t \\	ft

t. Also, set E(t) = suph(r). Then, lim E(t) = 0, and E(t2) < E(t1) if t1 < t2.

r> t

t--+oo

Now, it is sufficient to prove the following lemma.

Lemma VII-3-9. For any positive numbers t, to, and c, it holds that

(a) rt e-c(t-a) f(s)ds < (I + -) E(to), to

f	+ ec(t-s) f(s)ds <	1 +

lim ft e-`(t-`) f (s)ds = 0. t-.+oo

o

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

211

Proof of (a).

Set p(t) = f et -`(t-°)f (s)ds. Then, 0'(t) = -cb(t) + f (t) and, hence,

I.

0(r) - 0(o) = -c / T th(s)ds +	/ T f (s)ds
0	0

for to < a < r. Suppose that there exists a positive number 6 such that

b(r) = C1 + - + 6 E(to),	0(a) = C- + 6 E(to),
and
0(s) > (!± b E(to)	for a < s < r.

Then,

E(to) + c l! + 6 E(to)(r - a) < E(a)(1 + (r - o)).

This is a contradiction. \

Proof of (b).

Set ,L(t) = f et-f (s)ds for 0 t T for a fixed T > 0. Then, "(t) _

r

cO(t) - f (t) and, hence, w(r) - ?P(t) = cJ ;(s)ds J f (s)ds for t < r < T.

	r
Suppose that there exists a positive number 6 such that t
><i(r) = I ! + 6) E(t),	y(t) = (1 + 1 +6) E(t),
and
1l'(s) > (!+o)E(t)	for	t < s < r.

Then,

E(t) + c (1+ 6) E(t)(r - t) < E(t)(1 + (r - t)).

This is a contradiction. This, in turn, proves that

		T	C1 + c j E(t)				T > t.
	J	f (s)ds <				for
	lim E(t) = 0, the integral		+430
Since			+ e'('-")f (s)ds exists and
	t-+o		ft
		j+00 ec(t-e)f(s)ds <		\	1+ 1 f E(t).
		e				c

212	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

The proof of (c) is left to the reader as an exercise. 0

For the argument given above, see [Harl).

VII-4. A diagonalization theorem

Liapounoff's type number of a solution is useful information since it provides some idea about the behavior of the solution as t --. +oo. However, it is not quite enough when we look for a more specific information. For example, Liapounoff's type number of the second-order differential equation

x

dt2 + {1 + R(t)}r, = 0

is 0 and its multiplicity is 2 if the function R(t) is continuous on the interval 0 < t <

+oc and	lim R(t) = 0 (cf. Theorem VII-3-6). However, this does not imply the
	t-+"0

boundedness of solutions as t --+ oc. 0. Perron [Per3[ constructed a function R(t) satisfying the two conditions given above in such a way that solutions of (VII.4.1) are not bounded as t +oc. Looking for better information concerning equation

(VII.4.1), M. Hukuhara and M. Nagumo [HNI[ proved the following theorem.

Theorem VII-4-1. Every solution of differentud equation (VII.4.1) is bounded as

t +x, if 1	+C0
	+ jR(t)ldt < +oc.
0
Proof.
First, fix to > 0 in such a way that a		r+x
		jR(t)ldt < 1. Write a solution 0(t)

SU

of (VII.4.1) in the form

b(t) = q(to) cost - to) + 0'(to)sin(t - to) - J t R(s)¢(s) sin(t - s)ds.

to

Choose a positive number K so that 4(to)j + kd'(to)j < K and choose another

positive number At so that M >

K + At J

1

t to

KQ > K. Then,

JR(s)jds < At

for to < t < tt

if jb(t)j < At for to < t < t1. Hence, 14(t)I < M for to < t < +oc. 0

In this section, we explain the behavior of solutions of a system of linear dif- ferential equations under a condition similar to the Hukuhara-Nagumo condition.

Precisely speaking, we consider a system of the form

(VII.9.2)	d	[A(t) + R(t))1i

under the following assumptions.

4. A DIAGONALIZATION THEOREM

213

Assumption 2. Assume that A(t) is an n x n diagonal matrix

(VII.4.3) A(t) = diag(A1(t), A2(t), ... , An (t)],

R(t) is an n x n matrix whose entries are continuous on the interval Zo = {t : 0 < t < +00)1 and

(VII.4.4)	J	IR(t)Idt < +oo.
Set
(VII.4.5)	A,k(t) = al(t) -)tk(t) and D-,k(t) = R(AJk(t)) (j,k = 1,2,... ,n).

Concerning the functions A. (t) (j = 1, 2,... , n), the following is the main assumption.

Assumption 3. The functions Al (t), A2(t), ... , A,, (t) are continuous on the inter- val 4. Furthermore, for each fixed j, the set of all positive integers not greater

than n is the union of two disjoint subsets P.t and P32, where

(i) kEP) I if

lim I Djk(r)dr = -oo and	Dik(r)dr < K	for 0 < s < t
t-+oo 0	Li

for some positive number K,

(ii) k E Pj2 if

`t

f/a Dik(r)dr < K for s > t > 0

for some positive number K.

Remark VII-4-2. Assume that the functions A1(t), ... , An(t) are continuous on

the interval Z o and that lim A, (t) = pj (j = 1, 2, ... , n) exist. Then, the functions

t +oo

A1(t),... ,An(t) satisfy Assumption 3 if the real parts of pl,... p, are mutually distinct. The proof of this fact is left to the reader as an exercise.

The main concern in this section is to prove the following theorem due to N.

Levinson.

Theorem VII-4-3 ([Levil]). Under Assumptions 2 and 3, there exists an n x n matrix Q(t) such that

(1) the derivative dQ(t) exists and the entries of Q and Q are continuous on
d
the interval Z0,
(2) t1 1 Q(t)
(3) the transformation
(VII.4.6)	y' = V. + Q(t)] E
changes system (VII.4.2) to
(VII.4.7)	d" dt= A(t)i

on the interval Z0, where I is the n x n identity matrix.

Prool

We prove this theorem in six steps.

214	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Step 1. Differentiating both sides of (VII.4.6), derive

a + (In + Qj dr = [A(t) + R(t)J (I + Q) z.

Then, it follows from (VII.4.7) that Q should satisfy the linear differential equation

(VII.4.8)	dt _ (A(t) + R(t)1[I,, + Q] - (In + Q] A(t)
or, equivalently,
(VII.4.9)	dQ = A(t)Q - Q A(t) + R(t) (In + Qf .
The general solution Q(t) of (VII.4.9) can be written in the form
(VII.4.9')	Q(t)	='Nt)C41(t)-1 + f0(t)O(s)-1R(s)`I,(s),I,(t)-1ds,

where C is an arbitrary constant matrix, 44(t) is an n x n fundamental matrix of

dt = [A(t) + R(t)]4>, and 'F(t) is an n x n fundamental matrix of Al =

(cf.

Exercise IV-13). Thus, the general solution Q(t) of (VII.4.9) exists and satisfies condition (1) of Theorem VII-4-3 on 10. Therefore, the proof of Theorem VII-4-3 will be completed if we prove the existence of a solution of (V1I.4.9) which satisfies condition (2) of Theorem VII-4-3 on an interval I = {t : to < t < +oo} for a large to.

Step 2. Now, construct Q(t) by using equation (VII.4.9) and condition (2) of

Theorem VII-4-3. To do this, let 4?(t. s) be the unique solution of the initial-value problem

dY = A(t)Y,	Y(s) = In.
dt

Then, (VII.4.9) is equivalent to the following linear integral equation:

(VII.4.10)	Q(t) = J1 (t, s)R(s) f 1 + Q(s)145(t, s)'1ds,
where	4?(t, s) = diag(Fl (t, s), F2(t,1s), ... , Fn(t, s)],
	F, (t, s) = exp [ I A,(r)d; {	(j = 1,2,. .. ,n).

JJJ

Step 3. Letting q,k(t) and rik(t) be the entries on the j-th row and the k-th column of Q(t) and R(t), respectively, write the integral equation (VII.4.10) in the form

[ft

(VII.4.10') ggk(t) =	J	exp	Aik(r)drr)k(s) + F		ds,
			11	A=1

4. A DIAGONALIZATION THEOREM			215
where j, k = 1, 2,... , n and the .1jk(t) are defined by (VII.4.5). Note that
t		t
( J(Ajk(r)dr}	I = exp I J Djk(T)dr] <		eK
lexp I
	ll	a
if
0< s< t for k E Pj1	and	t< s< +oo for	k E Pj2,

where j, k = 1, 2,... , n and the Djk(t) are defined by (VII.4.5). The initial points rjk are chosen as follows:

to	if k E P1I,
rjk = { +00	if k E Pj 2

for some to > 0.

Step 4. Define successive approximations by

gojk(t) = 0,

	gPJk(t) = j t	exp	l	rk(S) +	rja(S)gp-I;hk(s) ds,
			t Ajk(r)dTJ
		LJ			h=1
	where p = 1, 2, .... Then, we obtain I gpuk(t) I < eK (1 + nC]J					r(s)ds on the
	interval I={t:to <t<+oc} if				to
	(VII.4.11)	Igp_I,jk(t)I < C		on the interval 2,

where r(t) = max(j,k) Irjk(t)I. Using assumption (VII.4.4), choose to so large that

r+oo	r(s)ds < C.
e'11 + nCJ Jto

Then, from (VII.4.11), it follows that Igpjk(t)I < C on the interval Z.

Step 5. Similarly, if (VII.4.11) holds for p = 1, 2, ... , we obtain

P

Igp+Iuk(t) - gpjk(t)l < (2} C

on the interval I = {t : to < t < +oo} if to is chosen sufficiently large. Hence, plim gpUk(t) = gjk(t) exists uniformly on the interval Z, and the limit satisfies

integral equation (VII.4.10').

216	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Step 6. Let us prove that the bounded solution qjk(t) satisfies condition (2) of

Theorem VII-4-3. If k E Pj2, then Tjk = +oo. Hence, lim q)k(t)

1im f exp f f

AJk(T)d-rl [rik(s)+rih(s)chk(s)](is

= 0.

LJ

h=1

In the case when k E PJ1, note that

t

q)k(t) = f exp [f>tik(r)dr]

[r)k(s)+r)h(s)hk(s)} ds

h=1

and

ftv

IrJ//

l1

A=1

r)h(S)hk(S)

L

s

exp IJ

\Jk(r)dr] rik(S) +

o

[ft

ll

= f exp

.\)k(T)dr] [r)k(S) +> rJh(S)hk(s)J ds

h=1

+

\.k(T)dr] {r)k(S) +

n

rJh(S)ghk(S)ds

s

h-1

dsl

for any a such that to <l1a < t. Observe that

exp [ft ak(T)dr] fIrk(s)+

rJh(s)hk(s)

1

1

= eXp Lf t Dk(r)dTJ

f exP [fC.Jk(r)dr] [rJk(s) +

r)h(s)hk(s)l1 ds

h=1

JI

(

t

+oo

< exp J f D)k(r)dr] eK (1 + nCj ft)

r(s)ds

l

o

if (q)k(t)(< C on Z. Observe also that

ft

exp I ft

AJk(T)dr]

[r)k(s) +

h=t

rJh(s)hk(s) ds

I

< eK(1 + tnC} f+co r(s)ds

0

fix a positive number a so large

if lq)k(t)E < C on I. For a given positive number

ft-

r(s)ds <

that eK (1 + nCj

Since

lim f t D)k(r)dT

= -oo

if

k E P)

t-»+0

for a fixed a, we obtain

+

t

1

r(s)ds <

e

exp f DJk(r)dr] eK [1 + nC] f

L

o

to

for large t. Therfore, for any positive number e, there exists

such that Iq,k(t)( <

e for t > t(e). This complete the proof of Theorem IV-4-3.

4. A DIAGONALIZATION THEOREM

217

Remark VII-4-4. The n x n matrix W(t, s) = [In + Q(t)j$(t, s) is the unique solution of the initial-value problem

dY = [A(t) + R(t)]Y,	Y(s) = In + Q(s),
dt

where di(t, s) is the diagonal matrix defined in Step 2 of the proof of Theorem

VII-4-3 (cf. (VII.4.10)). Since det[%P(t,s)] i4 0 for large t, the matrix WY(t,s) is invertible for all (t, s) in Yo x Yo (cf. Exercises IV-8). This, in turn, implies that the matrix In + Q(t) is invertible for all t in Io.

Remark VII-4-5. Theorem VII-4-3 has been shown to be the basis for many results concerning asymptotic integration (cf. [El, E2] and [HarLl, HarL2, HarL3]).

Remark VII-4-6. Using the results of globally analytic simplifications of matrix functions in [GH[, H. Gingold, et al. [GHS] shows some results similar to Theorem

VII-4-3 with Q(t) analytic on the entire interval Io under suitable conditions.

Remark VII-4-7. Instead of (VII.4.4), [HX1] and [HX2] assume only the inte- grability at t = oc for above (or below) the diagonal entries of R(t) and obtain the results similar to Theorem VII-4-3. More results were obtained in [HX3] applying a result of [Si9j.

The following example illustrates applications of Theorem VII-4-3.

Example VII-4-8. Let us look at a second-order linear differential equation

	(VII.4.12)		d 22 + p(t)rt = 0.
	If we set
	1 =	do	,	A(t) =	0	1l
					-P(t)	0J
		dt
	equation (VII.4.12) becomes the system
	(VII.4.13)		dy	= A(t)y
			dt

The two eigenvalues and corresponding eigenvectors of the matrix A(t) are

	'\t(t)	= i p(t)112,		911 _ [i p(t)1/2]
					1
	A2(t) = -t p(t)'/			pct) _ [-i pt)1/2J
Set Po(t) _ [i P(t)'/2				Then,
dPo(t) -	ip'(t)	[0	0 1	-i	[ip(t)'/2 1 1
dt	2p(t)'/2			POW-1 =	1 ,
		1	1 '	2p(t) '/2	jP(t)1/2