0387986995Basic TheoryC

98	IV. GENERAL THEORY OF LINEAR SYSTEMS

The function g(t, y-) is the nonlinear term which satisfies some suitable condition(s).

In the case when the matrix A is independent of t, formulas (N.6.6) and (IV.6.7) become

(IV.6.8)	g" = exp[(t - r)A]it + / t exp[(t - s)A]b(s)ds
			r
and
(IV.6.9)	exp[(t - r)A]>; +			exp[(t - s)A]9(s,y1s))ds,
			Jr
respectively.
Example IV-6-1. Let us solve the initial-value problem
(IV.6.10)	dt	= Ay + 6(t),		y(0)
where	-2			[2]	#p=
		1	0			1
	A= 0	-2 0 , b(t)=		0	,	1
	3	2	1	t		0

The matrix A was given in Example IV-3-3. As computed in §IV-3, a fundamental matrix solution of the associated homogeneous equation is

	e-2t	to-21		0
0) = exp(tA) =		e-2t
	0			0
	et - e-2t	et - (1 +	t)e-2t	et

Therefore, the solution of (IV.6.10) is

r	t	1	1 + to-2t
exp[tA] j i0 +		exp[-sA]b(s)ds }	e-2t
111	fo	111	-9-t+5et-(l+t)e

IV-7. Higher-order scalar equations

In this section, we explain how to solve the initial-value problem of an n-th order linear ordinary differential equation

ao(t)ucn) + al(t)u(n-1) + ... + an-1(t)u + an(t)u = b(t),

(IV.7.1)		U(n-1)(,r) =
u(r) = 'h, u'(r) = Q2, ...,
where the coefficients a°(t), a1(t), ... ,	and the nonhomogeneous term b(t) are

continuous on an interval I = {t : a < t < b}. In order to reduce (IV.7.1) to

7. HIGHER-ORDER SCALAR EQUATIONS

99

a system, setting yj = u and 112 = u', ...I yn = u(n-1), we introduce a vector yl

I. Then, problem (IV.7.1) is equivalent to

tin

(IV.7.2)	d11	= A(t)f + b(t),
	dt
where
0		1	0	0
0		0	1	0

A(t) =

... 0 1

... 0

		an0	_a 2(t)	al(t)
an(t)	an-1(t)	-3(t)
ao(t)	ao(t)	do(t)	ao(t)	ao(t)

0

0

b(t) =

0t)

b(t)

ao(

Assuming ao(t) 54 0 on Z, let 4?(t) = [ 1(t) 2(t) ... hn(t)J be a fundamental matrix solution of the associated homogeneous equation

(IV.7.3)	d	= A(t)y
	at

of (IV.7.2). Using 4i(t), we can solve problem (IV.7.2) (cf. §IV-6). The first component of the solution of (IV.7.2) is the solution of problem (IV.7.1). Also, the first components of n column vectors of the matrix $(t) give the n linearly independent solutions of the associated homogeneous equation

(IV.7.4)	ao(t)u(n) + al	(t)u(n-1)	+... +		an(t)u = 0
of (IV.7.1).
Example IV-7-1. Let us solve the initial-value problem
(IV.7.5)	u"' - 2u" - 5u' + 6u = 3t,			u(0) = 1, u'(0) = 2, u"(0) = 0.
To start with, set
				and y=	111
	111=u, 112=u',		y2=u",
					1122

IY31

100	IV. GENERAL THEORY OF LINEAR SYSTEMS
Then, problem (IV.7.5) is equivalent to

(IV-7.6)	dy"	= Ay" + b'(t),		y(0) =
	dt				#I,
where
10	1	0		0		1
(IV.7.7) A = 0	0 1,		b(t) =	0 I,	and	[2].
-6	5	2		3t		0

Three eigenvalues of the matrix A are 1, -2, and 3. The corresponding projections

are	1 3 -4 1
-6 -1 1

	P1(A) _ -6 -6	-1	1 I , P2(A)	15	-6	8	-2
	-6	-1	1		12	-16	4

		1-2	1	1
P3 (A)	= 10	-6	3	3
		-18 9		9

and N = 0. Therefore,

4)(t; 0) = exp(tAJ = etP1(A) + e-2tP2(A) + e3tP3(A)

et -6 -1 1				2t 3		-4	1	3t -2 1			1
=-6 -6 -1			1	e -6		8	-2 + e -6			3 3.
	-6 -1		1	15	12	-16	4	10	-18 9		9
Thus,
t
4D(s;0)-1 6(s) ds
1 1 - (t + 1)e-t			1	1 + (2t - 1)e2t				1 1 - (3t + 1)e '3t
1-(t+1)e-` +- -2(1+(2t-1)e2tJ +-											3(1-(3t+1)e-39	,
1 - (t +	1)e-t		20					911 - (3t +			1)e-3t1
				4[1 + (2t - 1)e2t]
and, consequently,
	1	-	1	[1_(t+1)e_tl		1		1 + (2t - 1)e2t
				1 - (t + 1)e' + - -2(1 + (2t - 1)e2CJ
	0			1 - (t +	1)e-t	20		4[1 + (2t - 1)e2L]

1	1 - (3t +
+ -	3[1 - (3t +
	9(1 - (3t +

1)e-3t

1)e-3t]

1)e-3t]

=	30t + 25 - 17e'2t + 50et + 2e3L
	2(15 + 17e-2t + 25et + 3e3t)

60	1	2(-34e-2t + 25et + 9e3t)

The first component of g(t) gives the solution of (IV.7.5).

7. HIGHER-ORDER SCALAR EQUATIONS

101

Remark IV-7-2. In the case of a second-order linear differential equation

(IV.7.8) ao(t)u" + al (t)u' + a2(t)u = b(t),

a fundamental matrix solution of (IV.7.3) has	the form 4P(t) = 0' (t)	¢2(t) ,
	.01(t)	02(t)

where 01 (t) and ¢2 (t) are two linearly independent solutions of the associated ho-

mogeneous equation (IV.7.4). Since 4i(t)-1 =	1	.02(t)	-02(t) where
	W(t)
		-011(t) 01(t)

W(t) = det(4i(t)), the first component of the formula

y"(t) _ (t}it + -t(t) J t 4>(s)-16(s)ds

gives the general solution of (IV.7.8) in the form

u(t) = r)1451(t) + 77202(t)

(IV.7.9)	42(s)	r	t	01(s)
	- 41(t) It ao(s)W(s)b(s)ds + 02(t)			ao(s)W(s)b{s)ds,

where ill and q2 are two arbitrary constants. This is known as the formula of variation of parameters (see, for example, [Rab, pp. 241-2461). Moreover,

	(IV.7.10)	W(t) =W(r)exp	f do (s) dsJ
	as in (4) of Remark IV-2-7.
	Remark N-7-3. Write (IV.7.10) in the form
	(IV.7.11)			[	-		t s)ds
		01(t)02(t) - dl (t)t2(t) = c2 exp				ft ao(s) I		,
				L

where C2 is a constant. Regarding (IV.7.11) as a differential equation for 02(t), w

	obtain				al(s)
	02(t) = C101(t) +C201(t)	1	[ - J
		t 01(T)2		t	ao(s)ds }dry
	J
	where cris anoth1er constant.	Therefore,	1(t) and ¢1(t) f 1(r)2

x exp [ - J t _(s)ds] d7form a fundamental set of solutions of ao(t)u"+al(t)u'+ s

a2(t)u = 0.

Remark IV-7-4. Let 01 (t) and 02(t) be linearly independent solutions of a thirdorder linear homogeneous differential equation

(IV.7.12)

ao(t)u"' + a1(t)u" + a2(t)u' + a3(t)u = 0.

102 IV. GENERAL THEORY OF LINEAR. SYSTEMS

Also, let 0(t) be any solution of (IV.7.12). Then, (4) of Remark IV-2-7 implies that

		0	01	02		[ft
	(IV.7.13)	10'	401	452	=c3exp	ao(s)ds]
		'P	4i
				oz

where c3 is a constant. Write (IV.7.13) in the form
(IV.7.14)	Au(t)o" + A1(t)O' + A2(t)4 = c3exp					[-I	c al(s)ds ,
							ao(s)	]
where
	461	02	01	02	As(t)		COI	0'2
	A1(t) = 101	¢2 4	A2(t) - 1,i	'02" '			40	oz

A fundamental set of solutions of the homogeneous part of (IV.7.14) is given by

{01,02}. Therefore, using (IV.7.9), we obtain

		I-01(t) I	02		eXp I -	T ao(s)	I dT
0(t) = C101 (t) + c202(t) + C3			Ao
				(T3
					J
		- J ao(s)ds]dr}
+ 02 (t) I ` Ao(r)2 e		L
where cl and c2 are constants. This implies that 01(t), 02(t), and
02(r)	i)				1(r)	[ - f	)
Aa(T)2 exp	[f' ao(s)dsj dr - -02(t) I				Ao(r)2 exp		ao(s) ds] dr

form a fundamental set of solutions of (IV.7.12).

EXERCISES IV

IV-1. Let A bean n x n matrix, and let A1, A2, ... , Ak be all the distinct eigenvalues of A. On the complex A-plane, consider k small closed disks Of = {A : IA - A2 I < r)} (j = 1,... , k}. Assume that Aj n At = 0 for j # 1. Set

f P]	A)-1dA	(j = 1,... ,k),
	ail=rj(AI,, -
	2'rfd

and N=A-S,

where the integrals are taken in the counterclockwise orientation. Show that

(i)Pl + ... + Pk = In,

(ii)Pi Pt = O if U'41),

(iii) A = S + N is the S-N decomposition of A.

EXERCISES IV

103

Hint. Note that

1 {(u1n - A) -1 - (7-In - A)-1 } = (oln - A) -1 (rln - A)-1 . r-o

Also, if p(A) is a polynomial in A with constant coefficients, then

	k
p(A) =	1 j	p(A)(Aln - A)'1dA.
	J=12rri

For general information, see [Ka, pp. 38-43).

IV-2. Under the same assumptions as in Exercise IV-1, show that if f (A) is analytic in a neighborhood of each eigenvalue AJ, and an n x n matrix B is in a sufficiently small neighborhood of A, then f (B) is well defined and Biim f (B) = f (A).

Hint. If f (A) is analytic in a neighborhood of each eigenvalue A,, then

f (A) _	1
	J=12rri

IV-3.	Let A = S + N be the S-N decomposition of an n x n matrix A. Show that,
as functions of A, the matrices S and N are not continuous.
Hint. Use Lemma IV-1-4.
IV-4.	Let A be an n x n matrix with n eigenvalues Al, ... , An. Fix a real number
r satisfying the condition r > R[AJ]		(j = 1,... , n). Show that
	j+00
(F)	exp[tA] =	io)In - A)-'do
	2T	00

for t E R.

Remark. This result means that exp[tA] is the inverse Laplace transform of (sln -

	A) 1. For example, if A = [ U1	], then (s12 - A) -1	=	s2 + 1 [_i		1 ]. Hence,
	taking the inverse Laplace transform, we obtain exp(tA)			=	cos t	sin t	For-
					I-sint	cost

I.

mula (F) is useful in the study of exp[tA] of an unbounded operator A defined on a Banach space (cf. [HUP, Chapter XII, pp. 356-386]).

IV-5. Show that J(A2 +T 21" )-'dr = A-1 arctan(tA-1) if every eigenvalue of

an n x n matrix A is a nonzero real number.

m

IV-6. For an n x n matrix A, show that lim (In + A ) = eA.

In +00

M

104	IV. GENERAL THEORY OF LINEAR SYSTEMS
IV-7. Find the solution of the following initial-value problem
	2 - 2A + A21r = etAe	b7(0) = n,	V(0) = S,

where A is a constant n x n matrix, {6, ,j,S are three constant vectors in C", and y E C' is the unknown quantity.

Hint. If we set y = e`AU, the given problem is changed to

Al =E,	u"(0) =r1',	u(0)
dt2

IV-8. Find the solution of the following initial-value problem

2 +A21 =D, 9(0) = 40 , !Ly (0) = 41

where E C" is an unknown quantity, A is a constant n x n matrix such that det A = 0, and {rjo, >7j } are two constant vectors in C".

Hint. Note that cos(xA) and sin(xA) are not linearly independent when det A = 0.

If we define F(u) =	sin u	then the general solution of the given differential equation
is	u

j(x) = cos(xA)c"1 + xF(xA)c2.

IV-9. Find explicitly a fundamental matrix solution of the system dy = A(t)y

if there exists a constant matrix P E GL(n,C) such that P-"A(t)P is in Jordan canonical form, i.e.,

P-1A(t)P = diag[A, (t)I1 + N1iA2(t)12 + N2,... ,.1k(t)Ik +Nkj,

where for each j, Ap(t) is a C-valued continuous function on the interval a S t S 6,

I1 is the n, x n j identity matrix, and N, is an nl x n, matrix whose entries are 1 on the superdiagonal and zero everywhere else.

Hint. See [GH].

	/r		3	0	-1
IV-10. Find log (I 11		]).cos([ 1		2	-1 j).anii
	\` `		0 1		1
	252	498	4134	698 1).
arctan	-234	-465	-3885	-656
	15	30	252	42
	-10	-20	-166	-25

IV-11. In the case when two invertible matrices A and B commute, find

log(AB) - log(A) - log B

if these three logarithms are defined as in Example IV-3-6.

EXERCISES IV

105

IV-12. Given that

		0	0	1	0	yi
	A =	0	0	0	1	I/2
		2	3	0	0 '	1/3
		4	-2 0		0	1/4

find a nonzero constant vector ii E C4 in such a way that the solution l(t) of the

initial-value problem

d9

A9, 9(0)

satisfies the condition	lim y"(t) = 0.
	t -+oo

IV-13. Assume that A(t), B(t), and F(t) are n x n, m x m, and n x m matrices, respectively, whose entries are continuous on the interval a < t < b. Let 4i(t) be

an n x n fundamental matrix of - = A(t)4 and W(t) be an in x in fundamental

matrix of Z = B(t)u. Show that the general solution of the differential equation on an ii x m matrix Y

(E)	dY = A(t)Y - YB(t) F(t),
	_WT
is given by
	Y(t) = 4;(t) C 4,(t)-1 r`

where C is an arbitrary constant n x m matrix.

IV-14. Given the n x n linear system

dYdt = A(t)Y - YB(t),

where A(t) and B(t) are n x n matrices continuous in the interval a < t < b,

(1)show that, if Y(to) -I exists at some point to in the interval a < t < b, then

Y(t)-i exists for all points of the interval a < t < b,

(2)show that Z = Y-1 satisfies the differential equation

dZdt = B(t)Z - ZA(t).

IV-15. Find the multipliers of the periodic system dt = A(t)f, where A(t) _

[cost simt

-sint cost

106	IV. GENERAL THEORY OF LINEAR SYSTEMS
Hint. A(t) = exp It 101		0]]
	l`

IV-16. Let A(t) and B(t) be n x n and n x m matrices whose entries are realvalued and continuous on the interval 0 < t < +oo. Denote by U the set of all R'-valued measurable functions u(t) such that ,u(t)j < 1 for 0 < t < +oo. Fix a

vector { E R". Denote by ¢(t, u) the unique R"-valued function which satisfies the

initial-value problem at = A(t)i + B(t)u(t), i(0)

where u E U. Also, set

R = {(t, ¢(t, u)) : 0:5 t < +oo, u E R}. Show that R is closed in

Hint. See ILM2, Theorem 1 of Chapter 2 on pp. 69-72 and Lemma 3A of Appendix of Chapter 2 on pp. 161-163j.

IV-17. Let u(t) be a real-valued, continuous, and periodic of period w > 0 in t on R. Also, for every real r, let ¢(t, r) and t1'(t, r) be two solutions of the differential equation

(Eq)	d2y	- u(t) y = 0
	dt2
such that
o(r, r) = 1,	0'(r, r) = 0,	and	0(7-,T) = 0, t;J'(T,T) = 1,

where the prime denotes d Set

C(r) = 10'(00(0, r).'r)V(0, r) 1

i,I (0,r) J ,

M(r) _ 0(r+w,r) Vl(r+w,r)

0(r+w,r) tG'(r+w,r)

(I) Show that

M(r) = C(r)'1M(O)C(r).

(11) Let A+ and A_ be two eigenvalues of the matrix M(r). Let I K+(r)J and

K_(r)	be eigenvectors of M(r) corresponding to A+ and A_`, respectively.

how that

(i) K±(r) are two periodic solutions of period w of the differential equation

dK ± K2 + u(r) = 0,

dT

(ii) if we set

Qf{t,r) = exp [J rt Kt(() d(l

these two functions satisfy the differential equation (Eq) and the conditions

13f(t+w,r) = A*O*(t,r).

EXERCISES IV

107

Hint for (I). Set 4'(t,r) _	(t	(t'	I Then, ((t, r) is a fundamental
			)JJJ -
matrix solution of the system

and y = 4'(t, r)c is the solution satisfying the initial condition y(r) = c'. Note that

C(r) = 4'(0, r) and M(r) = 4'(r + w, r). Therefore, 4'(t, r) = 4'(t, 0)C(r) and

4'(t + r) = 4'(t, r)M(r). Now, it is not difficult to see 4'(w, r) = 4'(w, 0)C(r) _ 4'(0, r)M(r) = C(r)M(r) and 4'(w, 0) = M(O).

Hint for (II). Since eigenvalues of M(r) and M(0) are the same, the eigenvalues of

M(r) is independent of r. Solutions 77+ (t) of (Eq) satisfying the condition Y7+ (t +

w) = A+n+(t) are linearly dependent on each other. Hence, W+(t) = K+(t) is

independent of any particular choice of such solutions q+(t). In particular K+(t +

W) = 17'+ (t + w) = K+(t). Problem (II) claims that the quantity K+(r) can be

17+(t + w)

found by calculating eigenvectors of M(r) corresponding to A+. Furthermore,

A+ = exp I o K+(r)drJJ I I. The same remark applies to A_. The solutions 0±(x, r)

o

of (Eq) are called the Block solutions.

IV-18. Show that

(a) A real 2 x 2 matrix A is symplectic (i.e., A E Sp(2, R)) if and only if det A = 1;

(b) the matrix l

01 0 J N is symmetric for any 2 x 2 real nilpotent matrix N.

Hint for (b). Note that det(etNJ = I for any real 2 x 2 nilpotent matrix N.

IV-19. Let G, H, and J are real (21n ) x (2n) matrices such that G is symplectic,

H is symmetric, and J = I 0n . Show that G-1JHGJ is symmetric.

IV-20. Suppose that the (2n) x (2n) matrix 4'(t) is the unique solution of the

initial-value problem L'P = JH(t)4', 4'(0) = 12n, where J = I	0n n and H(t)
l	T	J
is symmetric. Set L(t) = 4'(t)-1JH(t)4'(t)J. Show that d4'dt)		= JL(t)4'(t)T,

where 4'(t)T is the transpose of 4'(t).