0387986995Basic TheoryC

198	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS
numbers a such that exp[-at] f (t) is bounded on the interval Z. Set
	+oo	ifA =0,
(VII.1.1)	1 (f) = inf{o : a E A}	i f A j4R and A 3 40 ,
	-00	ifA=R.

The quantity A (f) is called Liapounoff's type number off at t = +oo.

Note that if a E A and a <,6, then a E A.

Example VII-1-2.

J+ (0) = -00, A (exp[-t2j) _ -00, A (tm) = 0 (for all constants m),

{ A (exp[ort]) = a, A (exp[t2j) _ +00.

Here, it was assumed implicitly that to > 0 if m < 0.

The following lemma can be proved easily.

Lemma VII-1-3. Let A (f) be Liapounoff's type number of a C' -valued function

f (t) whose entries are continuous on an interval Z = It: to < t < +oo}. Then,

(i) exp I - (. (f) + e) t] f(t) is bounded on I if e > 0, and unbounded if e < 0,

whenever A (f) # -oo,

(ii) A fi < max{A (f,) : j = 1,2,... m

(tii) A	f, = A(ft), tfA(f1) > A(fi)forj=2....,m,
	1<f<m

(iv) f1, f2, ... , f, are linearly independent on the interval if A if 1) , ... , A (fm)

are mutually distinct,

m

(V) A(f1f2...fm) <- FA(f,) in the case when f1i... , fm are C-valued junc-

tions.

(vi)A (P(t)f) = A (f), if the entries of an n x n matrix P(t) and the entries of its inverse P(t)-1 are bounded on the interval Z.

The following lemma characterizes Liapounoff's type number.

Lemma VII-1-4. If A(f) is Liapounoff 's type number of a function f(t) at t =

+oo, then

	a(f)	= lim	sup	log If (s) I
				s
		t-.+oo lt<s<oo

2. LIAPOUNOFF'S TYPE NUMBERS OF A LINEAR SYSTEM

199

Proof.

Since there exists a positive constant K such that I f (s)I < Keia(f)+`i' for e > 0 and large values of s, it follows that

logIf(s)I <A(f)+e + logK 5.1U) +e+logtK				for t < s.
Hence,
lim	sup	log If(3)1	< a(f)
t-+m	It<5<+m	s

Also, for a fixed positive number a and any positive integer in, there exists a large

value of sm such that		If (sm)1 and		lim sm = +oo. Hence, log m +
				sm
e < log If (sm)1	Therefore, we obtain
Sm
	lim	sup	log If(s)I	> (f )
			s
	t-+oo<a<+ao

Thus, Lemma VII-1-4 is proved.

VII-2. Liapounoff's type numbers of a homogeneous linear system

In this section, we explain Liapounoff's type numbers of solutions of a homogeneous linear system

(VII.2.1)	dy = A(t)9
	dt

under the assumption that the entries of the n x n matrix A(t) are continuous and bounded on an interval T = It : to < t < +oc}. Let us start with the following fundamental result.

Theorem VII-2-1. If y" = fi(t) is a nontrivial solution of system (VII. 2. 1) and if f A(t)e < K on the interval I = It : to < t < +oo} for some non-negative number

K, then

(VII.2.2)

Proof.

Let the n x n invertible matrix 4P(t) be the unique solution of the initial-value

dY

problem 7 = A(t)Y, Y(to) = In, where In is the n x n identity matrix. Then,

200 VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

4i(t)-1 is the unique solution of the initial-value problem		dZ	= -ZA(t), Z(to) _
I (cf. Lemma IV-2-4). Since	1, we obtain	dt

1 + KJ I4(t)Idt,

	to	for t r- T.
	+ KI/ t
1		, I dt
Therefore,
)4i(t)I < exp[K(t - to))	and	I < exp[K(t - to)]	for	t E T
(cf. Lemma 1-1-5). Observe that fi(t) = 4D(t)j(to) and b(to)				From
the definition of the norm of a matrix, it follows that
exp[-K(t - to)] < 1 (t)I < 1		(to)j exp[K(t - to)]	for	t E Z.

Therefore, (VII.2.2) follows, since I¢(to)I 0 0.

Set A = {a Q is a nontrivial solution of (VII.2.1)}. Then, (iv) of Lemma

VII-1-3 implies that A is a nonempty subset of R which contains at most n numbers.

Let Al > A2 > ... > Am (1 < m < n) be all of the distinct numbers in A. Set

(VI1.2.3) Vj = {j : is a solution of (VII.2.1) such that A (p) < A-}

for j = 1,2.... Tn. Then, (ii) and (vi) of Lemma VII-1-3 imply that V, is a vector space over C. Set

(VII.2.4) 1i = dim Vj (j = 1,2,... ,m).

The following lemma states that there exists a particular basis for each space V, which consists of y, solutions whose type numbers are equal to A,.

Lemma VII-2-2. For system (VI1.2.1) and y, given by (VII. 2-4), it holds that

(_) yi = n,

(ii) ym < -Ym-1 < ... < y,

(ii:) for each j, there exists a basis for V, that consists of y, linearly independent

solutions d,,- (v = 1,2,... , y,) of (VIL2.1) such that A (p,,°) = A3.

Proof

It is easy to derive (i) and (ii) from the definition of V, and the definition of the numbers at, ... , am. To prove (iii), let y,,,, (v = 1, 2, ... . y,) be a basis for Vj.

	Assume that	A,	for	v = 1,...t,
	(
			for	v=f+1...... ).

It follows from (ii) of Lemma VII-1-3 and definitions of V, and A, that t > 1. Set

for

+for

v = 1....	, t,
v = t + l	.... , y2.

Then, , , (v = 1, ... , y,) satisfy all the requirements of (iii).

2. LIAPOUNOFF'S TYPE NUMBERS OF A LINEAR SYSTEM

201

Observation VII-2-3. The maximum number of linearly independent solutions of (VII.2.1) having Liapounoff's type number A,, is rye.

Definition VII-2-4. The numbers Al, A2, ... , Am are called Liapounoff's type numbers of system (VIL2.1) at t = +oc. For every j = 1, 2,..., m, the multiplicity of Liapounoff's type number A, is defined by

	(VII.2.5)	hj _	for j=1,2,... in- 1,
			for j = in.
		7m

The structure of solutions of (VI1.2.1) according with their type numbers is given in the following result.

Theorem VII-2-5. Let {(1, ¢2r ... , iin} be a fundamental set of n linearly in- dependent solutions of system (VII.2.1). Then, the following four conditions are mutually equivalent:

(1) for every j, the total number of those 4t such that A (y5t) = A, is h, (cf.

(VII. 2.5)),

(2) for every j, the subset {¢t : A (Qt) < A, } is a basis for V,,

(3) A	ctt	> A) if the constants ct are not all zero,
(4) A	= max {A (dt) :		ct 0} for every nontrwzal linear combi-
	n
nation FC, it of		e	Wn}

t=1

Proof.

Assume that (1) is satisfied. Then, the total number of those ¢t such that

m

A (3t) < A, is Fht = ryr = dims Vj. Hence, (2) is also satisfied. Conversely, t=j

assume that (2) is satisfied. Then, the total number of those t such that A (fit) < Aj is equal to dime V) = -,. Hence, (1) is also satisfied (cf. (VII.2.5)).

Assume that (2) is satisfied. Then, if A		ct4	A, for some ct, it
follows that	ctc E V,, and hence

clot =

tt

Since V. C Vm-1 c

202	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

for some c e. Since {¢j : t = 1,2,... , n} is linearly independent, all constants cc

must be zero. Thus, (3) is satisfied.

n

Assume that (3) is satisfied. Write a linear combination I:ct4t in the form

		t=1
m
cj¢t. Then, A	CWI = A), if	ci4r 96 0. Hence, (4)
j=1 a(ft)=A,	a(mt)=a1	a(Ot}=a,

follows from (iii) of Lemma VII-1-3.

Finally, assume that (4) is satisfied. Then, every solution d of (VII.2.1) with A (j) < A, must be a linear combination of the subset {¢t : A ((rt) S Aj }. Hence,

(2) is satisfied. 0

Definition VII-2-6. A fundamental set (01,02, ... , On } of n linearly independent solutions of system (VIL2.1) is said to be normal if one of four conditions (1) - (4) of Theorem VII-2-5 is satisfied.

. C V2 C V1, it is easy to construct a fundamental set of (VII.2.1) that satisfies condition (1) of Theorem VII-2-5. Thus, we obtain the following theorem.

Theorem VII-2-7. If the entries of the matrix A(t) are continuous and bounded on an interval Z = It : to < t < +oo}, system (VIL2.1) has a normal fundamental set of n linearly independent solutions on the interval Z.

Example VII-2-8. For a system = Ay" with a constant matrix A, Lia

pounoff's type numbers A1, A2, ... , Am at t = +oo and their respective multiplicities h1, h2, ... , hm are determined in the following way.

Let IL I , p2, ... , Pk be the distinct eigenvalues of A and M1, m2, ... , Mk be their

respective multiplicities. Set v, = R(pj) (j = 1, 2,... , k). Let Al > A2 >	> Am
be the distinct real numbers in the set {v1i v2, ... , vk }. Set h, _	mr for
t=at

j = 1, 2, ... , m. Then, A1, A2, .... Am are Liapounoff's type numbers of dy = Ay"

at t = +oc and h1, h2, ... , hm are their respective multiplicities. The prom of this result is left to the reader as an exercise.

Example VII-2-9. For a system

dy
(VII.2.6)	= A(t)y
dt

with a matrix A(t) whose entries are continuous and periodic of a positive period c..' on the entire real line R, Liapounoff's type numbers )k1, A2, ... , Am at t = +oo and their respective multiplicities h1, h2, ... , hm are determined in the following way:

There exists an n x n matrix P(t) such that

(i) the entries of P(t) are continuous and periodic of period w, (ii) P(t) is invertible for all t E R.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

203

(iii) the transformation y = P(t)z changes system (VII.2.6) to

(VII.2.7)

with a constant matrix B.

Therefore, (vi) of Lemma VII-1-3 implies that systems (VII.2.6) and (VII.2.7) have the same Liapounoff's type numbers at t = +oo with the same respective multiplic- ities. Liapounoff's type numbers of system (VII.2.7) can be determined by using

Example VII-2-8. Note that if pl, p2,. .. , p" are the multipliers of system (VII.2.6),

then p, log[p,] (j = 1, 2,... , n) are the characteristic exponents of system

(VII.2.6), i.e., the eigenvalues of B, if we choose log[p3] in a suitable way. Hence,

R(µ1) log[[p2I] (j = 1, 2,... , n).

Those numbers are independent of the choice of branches of log[p,].

Example VHI-2-10. For the system	= I	the fundamental set {et
	J	[b],

I	{[]		-e
eu [0] I is normal, but the fundamental set	2t	,	[J} is	not normal.

VII-3. Calculation of Liapunoff's type numbers of solutions

The main concern of this section is to show that Liapounoff's type numbers of

a system J = B(t)y" at t = +oc and their respective multiplicities are exactly the

same as those of the system	= Ay" with a constant matrix A if	iimoB(t) = A.
dt		t-+0

It is known that any constant matrix A is similar to a block-diagonal form

diag[pllm, + W.421,., + Rf2i ... , µkl,,, + Mk],

where µl, ... , Pk are distinct eigenvalues of A whose respective multiplicities are m1, ... , mk, is the mj x mj identity matrix, and M1 is an mj x m, nilpotent matrix (cf. (IV.1.10)).

Consider a system of the form

(VII.3.1)	10 = A2y1 + EBjt(t)Ut (j = 1,2,...
	t=1

where y2 E C'-,, A. is an n, x n) constant matrix, and B2t(t) is an n, x nt matrix whose entries are continuous on the interval Za = {t : 0 < t < +oo}, under the

following assumption.

204 VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Assumption 1.

(i) For each j, the matrix A. has the form
(VII-3.2)	Aj = )1In, + E. + N.,	(j = 1,2,... ,m),

where Al is a real number, In, is the nJ x nl identity matrix, E. is an n. x nj constant diagonal matrix whose entries on the main diagonal are all purely imaginary, and Nj is an nj x n, nilpotent matrix,

(ii) Am < )1m-1 < ... < 1\2 < Al,

(iii) N,E, = E?NJ (j = 1,2,... ,m),

(iv) t limp B,t(t) = 0 (j,t = 1,2,... ,m).

The following result is a basic block-diagonalization theorem.

Theorem VII-3-1. Under Assumption 1, there exist a non-negative number to and a linear transformation

(VII-3.3)	y , = z ' , + I: T , (t)zt		(j = 1,2,... ,m)
	tj
with ni x nt matrices T,t(t) such that
(1) for every pair (j, t) such that j		f, the derivative dt tt (t) exists and the

entries of Tat and dj tt are continuous on the intervalI = (t : to < t < +oo)

(2)	lim T;t(t) = O (j # e),
	t-+00
(3) transformation (VII.3.3) changes system (VI1.3.1) to
	(VII.3.4) t Lzj	A,+B,,(t)+EBjh(t)Th,(t)I	(j=1,2,...,m).

Proof.

We prove this theorem in eight steps.

Step 1. Differentiating both sides of (VII.3.3), we obtain

dzl	+	T,tdz"t	dTit5t
dt		dt	t#? df
		t0r		m
				t=1#t
(VII.3.5) = A) % + E Tj tat				Bjt +	Tt V
		Aj 4' Bjj + h* BjhThj		i, + t AjTjt + Bjt + hit BjhTht zt,

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

205

where j = 1, 2,... , m. Note that

		in		n+	r
					1
			2, = F		BjnTnv zv
		e=1	Vol	v=1	h#v
Rewrite (VII.3.5) in the form
{	-[At+Fj		}+ET,t{ Lz' -[At+Fe]zt}
			t#i	l	1
	_	[AJTJt + Bit +		BlhThe - Tlt (At + Ft)		dtt	Ztf
		t#,		h#t
where j = 1, 2,... , m, and
		F t = B» +		B)hTh)	M )
			h#j

Define the Tit by the following system of differential equations:

(VII.3.6)	d t t	T	(.l r e)
		h#t

Then,

{-[A,+F,]x",}+Tj dt -[At+Ft]zt}=Q (j=1,2,...,m).

This implies that we can derive (VII.3.4) on the interval I if the Tit satisfy (VII.3.6) and condition (2) of Theorem V1l-3-1, and to is sufficiently large.

Step 2. Let us find a solution T of system (VII.3.6) that satisfies condition (2) of Theorem VII-3-1. To do this, change (VII.3.6) to a system of nonlinear integral equations

(VII.3.7)	T,t(t) = J exp[(t - s)A,]U,t(s,T(s))exp[-(t - s)A1]ds,
for j 0 e, where the initial points rat are to be specified and
U,t(t,T) = B11(t) + E Bjh(t)Tht - Tjt		Btt(t) + E Bth(t)Tht
	h961		h#t
Using conditions given in Assumption 1, rewrite (V11.3.7) in the form
	rt	1
(VII.3.7')	Tit(t) = J exp 2(a1 - t)(t - s)]		s,s,T(s})ds,

206 VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

where j t and

Wjt(r,s,T)

(VII.3.8)

= exp [(A, - At)r] exp[rEj j exp[rNj[UJt(s, T) exp[-rEtj exp[-rN,].

On the right-hand side of (VII.3.7'), choose the initial points rat in the following

way				A1t< A ..(ie j ,> .t)
		( +00
(VII.3.9)	ra t		if Aj > At		(i e., j < t),
		j`	if
		to

Step 3. Let us prove the following lemma.

Lemma VII-3-2. Let a be a positive number and let f (t, s) be continuous in (t, s) on the region Z x Z, where Z = {t : to < t < +oo}. Then,

(VII.3.10)		t exp[-a(t - s)] f (t, s)dsl	< - max	l	f (t, s)to	-	s < t y
	III	ft"					J
for each fixed t E T. Also, if I f (t, s)' is bounded for to < t < s < +oo, then
(VII.3.11)	rt exp[a(t - s)j f (t, s)dsl <_		1 sup { If (t, $)I : to < t < s < +oo}
	I J+00		a

for each fixed t E Z.

Proof

In fact, estimates (VII.3.10) and (VII.3.11) follow respectively from

rt

Jto exp[-a(t - s)]ds = a {1 - exp[-a(t - to)]} <

t	exp[a(t - s)]ds = - 1 .
+oo		O
	a

Step 4. Let us estimate s, T) fort < s < +oo if A j > at, and for to < s < t if A_, < at. If A, > At, r < 0, and s > t, it follows that

i1',t(r,s,T)I < Ko(l +		Irlm'+mi-2) exp	12(A, - A,)7-1 IVit(s,T)I
for some positive number 1Co.		It can be shown	`easily	that there exist a positive
number X and a non-negative valued function p(t) such that
I + Ir1m'*m`_2 exp [(A1 - At)r]			< X,	for r < 0,
B, (s)	Q(t)	for t < s < +oo and all pairs (p, q),

lim p(t) = 0.

t-+00

Hence, if Aj > At, r < 0, and s > t, we obtain IW,t(r,a,T)I < /C2,3(t) {1 + ITI2}

for some positive number 1C2, where ITI = maxIT,t[. Similarly, the estimate t#j

I Wjt(r, s, T) I < 1C219(to) { 1 + ITI2) is obtained by choosing a positive number 1C2

sufficiently large, if A., < at, r > 0, and to < s < t.

3. CALCULATION OF LIAPOUNOFF'S TYPE NUMBERS

207

Step 5. In a manner similar to Step 4, we can derive the following estimates:

IWj1(T,s,T) - Wj1(r,s,T)I
K3$(t) {1 +ITI + ITI} IT - TI	if A, > A1, T < 0, s > t,
1 K3$(to){1+IT{+ITI}IT-T{	if A<A1, T>0, to<s<t

for some positive number K.

Step 6. Define successive approximations on the interval I as follows:

TO ye(t) = 0,

TP1t;)t(t) =		fi..	exp [(A,		-	A,)(t - s)j 9,1(t - s, s, Tp(s))ds,
		[
where j 54 a and p = 1, 2, .... Suppose that
(VU.3.12)	I Tp;j,(t)	j < C		on the interval I = ft: to < t < +oo}
for some positive number C. Then, Lemma VII-3-2 and Step 4 imply that
				2	K2i3(t){1 + C2}		if	A) > A,,
I TP .iat(t)		I <_	A' - Al
				2			if	A, < '\1
			Al	Aj K21300){I +C2}
on the interval Z. Hence, choosing to so large that
		2	K2/3(t){1 + C2} < C				on Z,
	IA, - A,I
we obtain						on the interval I
	ITp+1,je(t)j < C
from (VII.3.12).
Step 7. Suppose that (VII.3.12) holds for p = 0, 1, 2,.... Then,
lTp+i.t(t) - Tp,1(t)I S 2				aup ITP(s) - TP-1(s)I <_ WPC on Z,
where ITpI =	i ITpael if to is so large that K30(t){1 + 2C} <								on Z. Since
P									lim Tpe(t) _
Tp.e(t) _	{TQae(t) - TQ_lae(t)}, it can be shown easily that

Tje(t) exist for all (j, 1) such that j # f uniformly on the interval T. The limit

Tj1(t) satisfies integral equation (VII.3.7).