0387986995Basic TheoryC

	218		VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS
	and					_i pt)ti210
			Po(t)-'A(t)Po(t) _	[i	P(to 1/2
	The transformation y = Po (t) i changes system (VII.4.13) to
			- P0(t)-1 {A(t)Po(t) -			dP t)j z.
	Using the computations given above, we can write this system in the form
	(VII.4.14)		.ii = {t P(t) 1/ 2 [0				]
					01] - 49t) [ 11 11
	Suppose that
	(1)	a function p(t) is continuous on the interval Zo = {t : 0 < t < +oo},
	(2)	there exists a positive number c such that p(t)					c > 0 on the interval Zo,

(3) the derivative p'(t) of p(t) is absolutely integrable on Io,

Then, Theorem VII-4-3 applies to system (VII.4.14) and yields the following theorem (cf. IHN2]).

Theorem VII-4-9. If a function p(t) satisfies the conditions (1), (2), and (3) given above, every solution of equation (IV.4.12) and its derivative are bounded on the interval Zo.

The proof of this theorem is left to the reader as an exercise. Note that condition

(3) implies the boundedness of p(t) on the interval 1 . If p(t) is not absolutely integrable, set

(VII.4.15) z = [I2 + q(t)E]t,

where I2 is the 2 x 2 identity matrix, q(t) is a unknown complex-valued function, and E is a constant 2 x 2 unknown matrix. Then, the transformation (VII.4.15) changes system (VI I.4.14) to

(VII.4.16)

dt = [Iz + q(t)E]-1 { [ip(t)h12A0 -	4p(t) A11	[12 + q(t)E]	dtt) E} u,
	J
where
Ao =	Al =

Anticipating that

(i)q(t)I and Ip'(t)I are of the same size,

(ii)Jq'(t) I and Ip"(t)I are of the same size,

choose q(t) and E so that two off-diagonal entries of the matrix on the right-hand side of (VII.4.16) become as small as Ip'(t)I2 + [p"(t)I. In fact, choosing

q(t) =	-i p'(t)	,	E = [0	-11
	8 p(t)312		1	0 J

S. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

219

we obtain			= 1 + I9(t)2 (12 - q(t)E(
	(12 + q(t)E]'
and
(12 +	q(t)E]-1 { [1(t)1/2 Ao - 4p(t) A11 (12 + q(t)E1 -						dt) E}
1	jip(t)1/2A0		_ p'(t) EAo - p'(t)		Al
1 + q(t) 2			8p(t)	4p(t)
+g(t}AoE+E(p,p',p") - q(t)				dtt)12}
I		lp(t)1/2Ao- p,(t)12+E(p,p,p')-q(t)dq(t)12						,
1 -r q(t) 2 I			4p(t)				dt
where					0 Il
	E2 = -12,		EAo = -AoE =
					11	0

and E(p, p', p") is the sum of a finite number of terms of the form

a (P (t) 12 + $ p"(t) p(t)h/2

with some rational numbers or and /3 and some positive integers h. Applying Theorem VII-4-3 to system (VII.4.16), we can prove the following theorem.

Theorem VII-410. Suppose that

(1)a function p(t) is continuous on the interval Zo = {t : 0 < t < +oo},

(2)there exists a positive number c such that p(t) > c > 0 on the interval ID,

(3)

+00

(Ip(t)12 + Ip'(t)I } dt < +oo.

0

(4)limop'(t) = 0.

Then, every solution of equation (VII.4.12) is bounded on the interval 7o.

The proof of this theorem is left to the reader. Condition (3) of Theorem VII-4-10 does not imply the boundedness of p(t) on the interval Io. For example, Theorem

VII-4-10 applies to equation (VII.4.12) with p(t) = log(2 + t).

VII-5. Systems with asymptotically constant coefficients

In this section, we apply Theorem VII-4-3 to a system of the form

d9

(VII-5.1)

where A is a constant n x n matrix and V(t) is an n x n matrix whose entries and their derivatives are continuous in t on the interval Zo = It : 0 < t < +oo} under

the following assumption.

220	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Assumption 4. The matrix A has n mutually distinct eigenvalues p 1, u2, ... , An, and the matrix V(t) satisfies the conditions

lim V(t) = 0		+00
	and	+ [V'(t)[dt < +oo.
t + Cc		o

Let A1(t), 1\2 (t).... , an(t) be the eigenvalues of the matrix A+V(t). Then, these are continuous on the interval I. Furthermore, it can be assumed that

lim ),(t) = µj	(j = 1,2,... ,n).
t-+,

Choose to > 0 so that A) (t),.\2(t), ... , an(t) are mutually distinct on the interval

I={t:to<t<+oo}. Set

F(t, A) = det[AII - A - V(t)].

Then, F(t, \j(t)) = 0 on II for 1 = 1,2,... ,n, and

	Also,	j(t, A, (t)) + as (t, \,(t)) A (t) = 0 (j =1,2,... n)		on I.
		OF (t, \j (t)) # 0	(j = 1,2,... ,n),
		8A
				8F (t,
	since III, µ2i ... , p, are mutually distinct. Observe that .X (t)			A, (t ))	is
				aF

8a (t, af(t)) linear homogeneous in the entries of the matrix V'(t). In this way, we obtain the following lemma.

Lemma VII-5-1. Under Assumption 4, the derivatives of the eigenvalues of the matrix A + V (t) are absolutely integrable over the interval I = {t : to < t < +oo}, i. e.,

JA(t)dtl < +oo (j = 1,2,... ,n).

w

An eigenvector p, (t) of the matrix A + V(t) associated with the eigenvalue as(t) can be constructed in the following manner. Observe that the characteristic polynomial of A + V(t) can be factored as

F(t, \) = (A - \1(t))(A -1\2(t)) ... (A - \.(t))

Hence, by virtue of Theorem IV-1-5 (Cayley-Hamilton), we obtain

(VII.5.2) (A+V(t)-.11(t)In)(A+V(t)-.\2(t)In)...(A+V(t)-an(t)IJ) = 0

5. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

221

on Zo. Furthermore, if t > to,

J1 (A+V(t) - Ah(t)In) # 0,

h 0j

lim fl(A + V(t) - Ah(t)In) = H(A - phln) 36 0,
t-+oo h#l	h#j
for j = 1, 2,... , n. From (VII.5.2), it follows that
[A + V(t)] 1(A+ V(t) - Ah(t)In) = A,(t) [I(A + V(t) - Ah(t)In)
h#J	h#)
Hence, choosing a suitable column vector j,(t) of	H(A+V(t)-Ah(t)In), we obtain
	h#)

p3(t) 1-1 0 and [A+V(t)]p,(t) = \j(t)jYj(t)

on the interval I = It : to < t < +oo} if to > 0 is sufficiently large. Furthermore,

t+oo j5, (t) = 4,	(.7 = 1,2,... n)
lim

are eigenvectors of the matrix A associated with the eigenvalues p. (j = 1, 2,... , n), respectively. Observe that the entries of the vectors pF,(t) (j = 1,2,... , n) are polynomials in the entries of V(t) and A1(t),... ,An(t) with constant coefficients.

Hence,

+00

Ip"j'(t)I dt < +oc (j = 1,2,... ,n).

Jta

Thus, we proved the following lemma.

Lemma VII-5-2. Under Assumption 4, them exists a non-negative number to such that

(1)the matrix A + V (t) has n mutually distinct eigenvalues A1(t)...... n (t) on the interval I = It : to < t < +oo},

(2)the eigenvalues Al(t), ... , An(t) are continuously differentiable on I and

lim Aj(t) = p, (j = 1,2,... ,n),

(3) the matrix A + V(t) has n eigenvectors p1(t),... ,#n(t) associated with the

eigenvalues \I(t) .... ,An(t), respectively, such that	lim 73 (t) = 4j (j =
	t-+oo

1, 2, ... , n) are eigenvectors of the matrix A associated with the eigenvalues

p)(j = 1,2,... ,n),

(4)the derivatives of the eigenvalues A,(t) (j = 1,2,... , n) and the derivatives

o f the eigenvectors p ' , (t) (j = 1, 2, ... , n) with respect to t are absolutely integrable on the interval Z.

Set

Po(t) _ [ 91 (t) #2 (t) ...	( t ) [[ ,	Q = [ 91 92 ... 4n [ ,
A(t) = diag[A1(t), A2(t), ...	, An (t)[,	M = diag[pl, p2, ... , pn]-

222 VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Then,	lim Po(t) = Q,		lim A(t) = M,
			t-+oo
	Po(t)-'[A + V(t)]Po(t) = A(t),			Q-'AQ = M,
	f+oo I Po(t)_ 1 dP°(t) I		dt < +oo.
	to	dt
Observe that the transformation y"= P°(t)z changes system (V11.5. 1) to
dt =	POW-1 [[A + V(t)]P0(t) - dl t)1 i= [A(t)				- Po(t)-'d t)] Y.

Applying Theorem VII-4-3 to this system, we obtain the following theorem.

Theorem V11-5-3. Under Assumption 4, if the eigenvalues A1(t)......

n (t) of

the matrix A + V (t) satisfy all requirements given in Assumption 3 (cf. § VII--4) on the interval .7 = {t : to < t < +oo}, a fundamental matrix solution of (VII.5.1) will be given by

1(t) = P°(t)[In + H(t)]exp I / ( A(s)ds],

where H(t) is an n x n matrix whose entries are continuously differentiable on the

interval I and lim H(t) = O. t+00

Remark VII-5-4.

r:1

(a)As t -. +oo, the matrix exp [ - J A(s)ds] 4i(t) approaches the matrix Q,

(b)we can prove a result similar to Theorem VII-5-3 even if system (VII.5.1) is replaced by

d = (A+V(t)+R(t)]y,

where the matrix A + V (t) satisfies all the requirements of Assumption 4 and the entries of the matrix R(t) are absolutely integrable for t > 0, i.e.,

f+00 JR(t)jdt < +oo.

0

Observation VII-5-5. Let us look into the case when the matrix A has multi- ple eigenvalues. To do this, consider system (VII.3.1) under Assumption 1 given in §VII-3. By virtue of Theorem VII-3-1, system (VII.3.1) is changed to system (VII.3.4) by transformation (VII.3.3). Furthermore, (VII.3.4) is changed to

(VII.5.3)	dt	[ N 3 + R 3 (t)) u 3	(j = 1, 2, ... m),
where
R3(t) = e-'E, B33(t)+E B3h(t)Ttj(t) etE,				(j=1,2,...,m)
		h #)
by the transformation
	z3 = explt(A3I., + E, )]u3		(.) = 1, 2, ... , m).

Therefore, we look into the following two cases.

5. SYSTEMS WITH ALMOST CONSTANT COEFFICIENTS

223

Case VII-5-5-1. If N is an n x n nilpotent matrix such that N' = 0 and R(t) is an n x n matrix whose entries are continuous on the interval 0 < t < +oo and satisfy the condition

(VII.5.4)		+ jt2(T_1)IR(t)Idt < +oo,
we can construct a fundamental matrix solution 45(t) of the system
(VII.5.5)		dy	= (N + R(t))y'
		dt
such that	lim	e-' N4(t) = I, ,	where I is the n x n identity matrix. In fact, the
	t -.+00

transformation y" = etNii changes system (VII.5.5) to dii = e-tNR(t)e1NU. Since leftNI < Kjtjr-1 for some positive number K, the integral equation X(t) = In

j	e-R(r)erNX(r)dT can be solved easily.

Case VII-5-5-2. Let N be an n x n nilpotent matrix such that Nr = 0 and let

R(t) be an n x n matrix whose entries are continuous on the interval 0 < t < +oo and satisfy the condition

	+00
(VII.5.6)	tr-1IR(t)jdt < +oo.

10

Exercise V-4 shows that this case is different from Case VII-5-5-1. As a matter of fact, in this case, we can construct a fundamental matrix solution 'P(t) of system (VII.5.5) such that

(VII.5.7)	lira t-(k-t) (41(t)	- etN) c" = 0',	whenever Nkc_ = 0,
	t-+00

where k < r. We prove this result in three steps.

Step 1. First of all, if an n x n matrix T(t) satisfies condition (VII-5.7), then

c' = 0 if 41(t)c = 0 for large t. In fact, noice first that		We	N'5 if
	lim	e tr
			f!
NPe = 0 for p = F + 1, ... , r - 1. Also, note that lim	etNt:	= 0 if a(t)e = 0
t-+oo tktkl

and Nkcc = 0. Therefore, N'lE = 0 if k = r. Hence, Nr-2c' = 0. In this way, we obtain c" = 0. Thus, we showed that if the matrix %P(t) satisfies the differential

equation (N + R(t))* and condition (VII.5.7), then %P(t) is a fundamental matrix solution of (VII.5.5).

Step 2. To prove the existence of such a matrix W(t), we estimate integrals of ske(t-")NR(s) with respect to s, where 0 < k < r - 1. Observe that

ske(t_e)N =	r1 Sk(t - S)hNh
	h1
	h=0

Let us look at the quantity sk(t - s)h. Note that 0 5 k 5 r - 1 and 0 5 h < r - 1.

224	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Case 1. k + h <-r - 1. In this case, since sk(t - s)h = sk+h C t _ 11, define s

	Sk(th' 8)h N°R(s)ds = / t	8k(th! S)h NhR(s)ds.
	Jt
	Case 2. k + h > r. In this case, look at sk(t - 8)h = D-1)p (hp) Sk+pth-p
		p_Q
	Subcase 2(i). k + p < r - 1. In this case,	sk+pth-p = Sr-I	1	t", where
		\ s
	r-1-u=k+p,u+v=h-p. Since

v = h - p - p = (h-p) + (k+p) - (r-1) = (k+h) - (r-1) = k - (r-1-h),

we obtain 0 < v < k. Now, in this case, define

h

j	Irt	t
	Sk+pth-PNhR(S)dS
	= J+ao p) sk+Pth-PNhR(s)ds.
Subcase 2(ii). k + p > r. In this case,
Sk+pth-p = Sr-1(tlµt", where		r-l+p=k+p, v -p=h -p.

Since

v = h - p + p = (h-p) + (k+p) - (r-1) = (k+h) - (r-1) = k - (r-i-h),

we obtain 0 < v < k. Also, note that h-p<(r-1)+(k-r)=k-1. Now, in

this case, define

=	k;h )	k+P h PNhR		t	-kh	$k+hPt	-	P Nh R(s)d$,	where	t > to.
	-3	t -	{s)d s = ro
From the definition given above, it follows that
		t Ske(t-s)NR(S)d8 = etN / t Ske sNR(s)ds,
		f',				v
			I	rt
		lim		/ ske(`'")NR(s)ds = O (k = 0,..., r - 1),
		t+00 F.		q
Setting					r
				U(t) =
					f t ske(`-e)NR(s)ds,

t, and

t112, l01 when h(t) is a small function such as i ,

sin(tt/2) since the

6. AN APPLICATION OF THE FLOQUET THEOREM

225

we obtain

dU(t) = NU(t) + tkR(t).

dt

Step 3. Let us construct n x n matrices Uk (t) (k = 0, 1, 2, ... , r - 1) by the integral

equations

Then,

lira t-++00

ft

Uk(t) = Nk + 1 ske(t-")NR(s)Ui(s)ds

Uk(t) = Nk

Observe that

tkUk(t)	tkNk	rt e(t_s)NR(s)skUk(s)ds.
k!	k!		n		k!
Hence, setting
				tkUk(t)
				k!
		k=0
we obtain		r-1
1(t) = eIN +			t		k r
				e(t-,)NR(s) s		ds.
	Y' r				k!
	k=o n
This implies that
do(t)		_ (N + R(t))W(t).
It can be easily shown that	dt
				etNc' -
lim	L(t)c	=			Nk"
	tk		Urn
t-+oo		t-+00 tic

if Nk+16= 0. Thus, the construction of W(t) is completed.

Case VII-5-5-2 was given as Exercise 35 in [CL, p. 1061.

VII-6. An application of the Floquet theorem

The method discussed in §VII-5 does not apply directly to the scalar differential equation

d-t2 + {1 + h(t)sin(at)} ri = 0

g log t

derivative of h(t) sin(at) is not absolutely integrable over the interval 0 < t < +oo.

In this section, using the Floquet theorem (cf. Theorem IV-4-1), we eliminate periodic parts of coefficients so that Theorem VII-4-3 applies. Keeping this scheme in mind, consider a system of the form

(VII-6-1)

djj = A(t,h(t))y

under the following assumption.

226	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Assumption VII-6-1.

(1)The entries of an n x n matrix A(t, ej are continuous in (t, e) E R x A(r) and analytic in f E A(r) for each fixed t E LR, where FE C'" unth the entries et, ... , A(r) = IF: Ie < r}, and r is a positive number.

(2)The entries of A(t, e) are periodic in t of a positive period w.

(3)The entrees hl,... , h,,, of a C"t -valued function h(t) are continuous on the interval T(to) = {t : to < t < +oo} for some non-negative number to and

lim h(t) = 6. t+00

Let us consider the following two cases.

Case 1. The function h(t) is supposed to be continuously differentiable on T(to) and satisfy the condition

(VII.6.2)	+00	Ih(t)Idt = +oo	and	f + Ih'(t)Idt < +oo.
	J
	I.

Case 2. The function h(t) is supposed to be twice continuously differentiable on T(to) and satisfy the condition

lim t -+00

	+00	/	+00
(VII.6.3)	r= +00,		= +oo,
	L00I;(t)ldt	0
	Ih"(t)Idt < +oo,	J + Ih'(t)I2dt< +oo
	J0		0

For example, the function h(t) _ I satisfies (VI1.6.2), whereas h(t) =

satisfies (VII.6.3).

Observation in Case 1. In Case 1, we use the following lemma.

sin(f)

f

Lemma VII-6-2. If a matrix A(x, ej satisfies conditions (1) and (2) of Assumption VII-6-1, them exist n x n matrices P(t,e) and H(t") such that

(i) the entries of P(t, ) are continuous in (t, e) E R x A(f) and analytic in

FE A(f) for each fixed t E R, where f is a suitable positive number.

(ii) P(t + w, ej = P(t, t-) for (t, F) E R x ©(f),

(iii)P(t, fj is invertible for every (t, t) E R x A(f),

(iv)the entries of H(t) are analytic in FE A(f),

2ai

(v) any two distinct eigenvalues of H(0) do not differ by integral multiples of -, w

(vi) a P(t, a exists for (t, e) E R x L(f) and given by

(VII.6.4)

P(t, eM = A(t, t)P(t, e) - P(t, e)H(e)

for (t, e) E R x L(f).

6. AN APPLICATION OF THE FLOQUET THEOREM

227

Proof

In order to prove this lemma, construct a fundamental matrix solution $(t, e) of the differential equation d = A(t, e)y by solving the initial-value problem

dX = A(t, e-) X, X (O) = I, where In is the n x n identity matrix.		The en-
dt	bog((w ,
tries of 4!(w, ej are analytic in A(r). Define H(e) by H(ej =		and

P(t,') = 4i(t, cl exp(-tH(E)]. Then, (VII.6.4) follows.

The most delicate part of this proof is the definition of H(). Details are left for the reader as an exercise (cf. [Sill).

Changing system (VI I.6.1) by the transformation

(VII-6-5) W = P(t, i(t))u

we obtain the following theorem.

Theorem VII-6-3. Transformation (VII.6.5) changes system (VII.6.1) to

(VII.6.6)	di =H(i(t)) -	P(t,h(t))-1	hj(t)aP(t,h(t)) }u.
			1<j<m
Proof
In fact,
	diliiT = P(t, h(t))-1 {A(t(t))P(t&(t)) - dt [P(t, h(t))] } u"
(VII.6.7)	= P(t, i(t))-1( A(t, h(t))P(t, h(t)) -			(t, K (t))

1 < <m

Since H(e) is given by (VII.6.4), equation (VII.6.6) follows from (VII.6.7).

Observe that

	r+oo		h(t)) dt < +oo
(IV.6.8)	P(t, h(t))-'
	fro	1<j<m	j

under assumption (VII-6.2). Also, observe that H(i(t)) does not contain any pe-

riodic quantities and	dH(i(t))	is absolutely integrable. Therefore, if eigenvalues
	dt

of H(6) satisfy suitable conditions, the argument given in §VII-5 applies to system

(VII.6.6).