Volterra Integral and Differential Equations

24

2. LINEAR EQUATIONS

tion x(t) on an interval [to,T), such that x(t) = (p(t) for 0 < t < to- This yields x'(i) = A(t)x(t) + f C(t, s){s) ds + F(t) + f C(t, s)x(s) ds. Jo Jto As we saw before, a translation y(t) = x(t + to) results in

y'(t)=x'(t + to) = A(t + to)y(t) + / C(t0 + t, s)(f)(s) ds + F{t + t0) Jo + / C(t0 +t,s)x(s)ds Jtf) = A(t + to)y(t)+ / C(t0 + t, s + to)y(s) ds Jo + I C(t0 + t, s)(f)(s) ds + F(t + to), Jo which we write as

y'(t) = A(t)y(t) + f C(t, s)y(s) ds + F(t), Jo again of the form (2.1.2). The initial function is absorbed into the forcing function, and the last equation then has the initial condition y(0) = x(i 0 ) = (*o), so that an integration from 0 to t yields

y(t) =0(*o)+ / A(s)y(s)ds + f F(s)ds Jo

Jo

+ ( ( C(u,s)y(s)dsdu.

(2.1.3)

Jo Jo

Interchanging the order of integration in the last term yields an equation of the form of (2.1.1). Thus, the existence and uniqueness theorem that follows applies also to (2.1.2) with a given continuous initial function. The uniqueness part of the proof of the result is facilitated by the following relation.

2.1. EXISTENCE THEORY

25

L e m m a . Gronwall's Inequality Let f,g : [0,a] —> [0,oo) be continuous and let c be a nonnegative number. If f(t)
+ I g(s)f{s)ds, Jo

0
then f(t)
g(s)ds,

0
Jo

Proof. Suppose first that c > 0. Divide by c + jQ g(s)f(s) ds and multiply by g{t) to obtain

f(t)9(t)/\c+J

g(s)f(s)ds]
An integration from 0 to £ yields

In I c + J g(s)f(s) ds\ /c\ < f g(s) ds or f(t)
I g(s)f(s)ds 0 through positive values. This completes the proof. Theorem 2.1.1. Let 0 < a < oo and suppose that f : [0, a) —> i? n is continuous and that B(t, s) is an n x n matrix of functions continuous for 0 < s < t < a. IfO
(2.1.1)

Jo on [0,T]. Proof. Define a sequence of functions {x n (£)} on [0, T] by xi(t)=f(t) ,* Xn+i(t) = f ( * ) + / B ( t , s ) x n ( s ) d s ,

(2.1.4) 71 > 1 .

io

These are called Picard's successive approximations.

26

2. LINEAR EQUATIONS

One may show by mathematical induction that each xn(t) is defined on [0, T] and is continuous. Let M = maxo<s
x

l(*) + 5Z(xn+l(«)-Xn(*))

(2-1-5)

n=l

whose typical partial sum is xn(t). We now show by induction that xn +1 (t) - x n (i)| < [X(Mi)"]/n!.

(2.1.6)

It follows from (2.1.4) for n = 1 that x 2 ( i ) - x i ( i ) | = f(i)+ / B(i,s)f(s)ds - f ( i ) Jo < / |B(i,s)f(s)|ds

<MKt,

Jo

so that (2.1.6) is true for n = 1. Assume x f c + 1 (i)-x f c (i)| < [if(Mt)fe]/A;! and consider xfc +2 (i)-xfc + i(i)| = f(i)+ / JB(t,s)xfe+1(s)ds - f(i) Jo

- / B(t,s)xk(s)ds Jo < / |B(i,s)||x fc+ i(s)-x fc (s)|ds Jo <M

Jo

|x fe+ i(s) - x f e ( s ) | ds

M f e + 1 ATi f e + 1 =

as required.

(A + l ) !

'

2.2. LINEAR PROPERTIES

27

But K(Mt)k/kl is the typical term of a Taylor series of KeMt that converges uniformly and absolutely on [0, T]. Thus (2.1.5) also converges uniformly on [0, T] to a continuous limit function, say x(i). We may, therefore, take the limit as n —> oo in (2.1.4) and pass it through the integral, obtaining x(t) = f(t)+ / B(t,s)x(s)ds, Jo so that the limit function x(i) is a solution of (2.1.1). To see that x(i) is the only solution, suppose there are two solutions, say x(£) and y(t), on an interval [0, T]. Then, from (2.1.1), x(«)-y(*)= / B(t,s)[x(s)-y(s)]ds, Jo so that X(t)-y(t)\<M

f |x(S)-y(S)|ds. Jo

This is of the form \z(t)\
2.2

Linear Properties

Discussion of the linear properties of (2.1.1) tends to be cumbersome, whereas the linear properties of (2.1.2) are very straightforward and analogous to properties of ordinary differential equations. In fact, in the convolution case for (2.1.2) with A constant, the entire theory is almost identical to that for ordinary differential equations.

28

2. LINEAR EQUATIONS

[0, a) —> Rn be continuous and B(t, s) be an Theorem 2.2.1. Let fi, f2 n x n matrix of functions continuous for 0 < s < t < a. If x(t) and y(t) are solutions of x(i) = f i ( t ) + / Jo

B(t,s)x(s)ds

y(i)=f2(i)+ / Jo

B(t,s)y(s)ds,

and

respectively, and if c\ and c-2 are real numbers, then cix(i) + C2y(i) is a solution of z(t) = [Cifi(*) +c 2 f 2 (t)] + /

B{t,s)z{s)ds.

Jo

Proof. We have cix(t) + c2y(t) = cifi(t) +c 2 f 2 (i) + / B(t,s)cix(s)ds Jo + /" B(t,s)c2y(s)ds, Jo and the proof is complete. We turn now to equations /"* x' = A(t)x+

C(t,s)x(s)ds +F(t)

(2.2.1)

Jo and /"* x' = A{t)x+

C{t,s)x{s)ds, (2.2.2) Jo with F : [0, a) —> Rn being continuous, A(t) an n x n matrix of functions continuous on [0,a), and C(t,s) an n x n matrix of functions continuous for 0 < s < t < a. Notice that if the initial condition for (2.2.1) is x(0) = xo, then an integration from 0 to t yields

2.2. LINEAR PROPERTIES

pt

pt

pt

x ( i ) = x o + / F(s)ds + A(s)x(s)ds Jo Jo

+

29

pU

/ C(u,s)x(s)dsdu, Jo Jo

(2.2.3)

and upon change of order of integration, we have an equation of the form (2.1.1) with f(i) = x o + / Jo

F(s)ds,

so that when F(t) = 0, then f (t) = x 0 . Theorem 2.2.2.

Consider (2.2.1) and (2.2.2) on [0,a).

(a) For each xo there is a solution x(t) of (2.2.1) for 0 < t < a with x(0) = x 0 . (b) If xi(t) and X2(t) are two solutions of (2.2.1), then xi(t) — X2(i) is a solution of (2.2.2). (c) If xi(t) and X2(i) are two solutions of (2.2.2) and if ci and C2 are real numbers, then cixi(t) + C2X2(t) is a solution of (2.2.2). (d) There are n linearly independent solutions of (2.2.2) on [0,a) and any solution on [0, a) may be expressed as a linear combination of them. Proof. In view of the remarks preceding the theorem, (a) was established by Theorem 2.1.1. Parts (b) and (c) follow by direct substitution into the equations. To prove (d), consider the n constant vectors e i , . . . , e n , where ej = ( 0 , . . . , 0, lj, 0 , . . . , 0) T and let Xj(i) be the solution with Xj(0) = e^. For a given x 0 = (xOi,. ,xOn)T, we have x 0 = zoiei H \-xOnen, so the unique solution x(i) with x(0) = xo may be expressed as x(i) = xoixi(t) H

h xOnx-n(t)

Now, the x i ( i ) , . . . ,xn(t)

are linearly independent on [0, a); for if

n

^cIxl(t)=0

on

[0,a)

8=1

is a nontrivial linear relation, then n

n

CX

0 = ^2 i i(°) = X! Ciei is a nontrivial linear relation among the e^, which is impossible. completes the proof.

This

30

2. LINEAR EQUATIONS

Corollary. If x i ( i ) , . . . ,xn(t) are n linearly independent solutions of (2.2.2) and ifxp(t) is any solution of (2.2.1), then every solution of (2.2.1) can be expressed as x(t) = xp(t) + cixi(i) +

+ cnxn(t)

for appropriate constants c\,..., cn. Proof. If x(t) and xp(t) are solutions of (2.2.1), then x(t) — xp(t) is a solution of (2.2.2) and, hence, may be expressed as CiXi(i) -I

\-CnXn(t) .

This completes the proof.

2.3

Convolution and the Laplace Transform

When A is constant and C is of convolution type, then the variation of parameters formula for (2.2.1) becomes identical to that for ordinary differential equations. Consider the systems x' = P x +

ft

Jo

D(t - s)x(s) ds + F(t)

(2.3.1)

and *

x' = P x + / D(t - s)x(s) ds , (2.3.2) Jo in which P an n x n constant matrix, D(t) is n x n matrix of functions continuous on [0, oo), and F : [0, oo) —> Rn continuous. We suppose also that |F(i)| and \D(t)\ may be bounded by a function Meat for M > 0 and a > 0. That is, F and D are said to be of exponential order. Laplace transforms are particularly well suited to the study of convolution problems. A good (and elementary) discussion of transforms may be found in Churchill (1958). Our use here will be primarily symbolic and the necessary rudiments may be found in many elementary texts on ordinary differential equations. The following is a list of some of the essential properties of Laplace transforms of continuous functions of exponential order [(v) requires differentiability]. The first property is a definition from which all the others may be derived very easily with the exception of (vii).

2.3. CONVOLUTION AND THE LAPLACE TRANSFORM

31

(i) If h : [0, oo) —> Rn, then the Laplace transform of h is />oo

e-8th{t)dt.

L(h) = H{s)=

(2.3.3)

Jo (ii) If D(t) = (dij(t)) is a matrix (or vector), then L(D(t)) = (£(%(*))). (This is merely notation.) (iii) If c is a constant and hi, hi functions, then L(ch\ + h2) = cL{h\) + L(h2). (iv) If D(t) is an n x n matrix and h(i) a vector function, then L[

[ D(t - s)h(s)ds)

\Jo

=L(D)L(h).

(2.3.4)

J

(v) L(h'(t)) = sL(h)-h(0). (vi) i " 1 is linear. (vii) If h\(t) and /i2(i) a r e continuous functions of exponential order and L(/U(t)) = L(h2(t)), then hx(t) = h2(t). Theorem 2.3.1. Let Z{t) be the nxn matrix whose columns are solutions of (2.3.2) with Z(0) = I. The solution of (2.3.1) satisfying x(0) = x 0 is x(t) = Z(t)x0 + f Z(t- s)F(s) ds . Jo

(2.3.5)

Proof. Notice that Z(t) satisfies (2.3.2):

r* Z'(t) = PZ(t)+

D(t-s)Z(s)ds. Jo

We first suppose that F and D are in L1[0, 00). If we convert (2.3.1) into an integral equation, we have ft ft r ft 1 x(t) = x(0) + / F(s) ds + / \P+ D(u - s)du\ x(s) ds , io Jo I Js 1 and as D and F are in L1, we have x(t)\<\x(0)\

+K +K

\x(s)\ds, Jo

some

AT > 0 and 0 < t < 00.

32

2. LINEAR EQUATIONS

By Gronwall's inequality \x(t)\<[\x(0)\+K]eKt. Thus both x(£) and Z(t) are of exponential order, so we can take their Laplace transforms. We have /"* Z'(t) = PZ(t) + D{t-s)Z{s)ds, Jo and upon transforming both sides, we obtain sL{Z) - Z(0) = PL{Z) + L(D)L(Z), using (i)-(v). Thus [sI-P-

L{D)]L(Z) = Z(0) = I,

and because the right side is nonsingular, so is [si — P — L(D)~\ for appropriate s. (Actually, L(Z) is an analytic function of s in the half-plane Re s > a, where \Z(t)\ < Keat.) [See Churchill (1958, p. 171).] We then have L(Z) = [si - P -

HD)]'1.

Now, transform both sides of (2.3.1): sL(x) - x(0) = PL(x) + L(D)L(x) + L(F) or [sI-P-

L{D)]L{x) = x(0) + L(F),

so that L(x) = [si -P- L(D)] -1 [x(0) + L(F)] = L(Z)x(0) + L(Z)L(F)

ft1

\

= L(Zx(0))+L[ /

Z(t-s)F(s)ds)

(

\Jo /"*

J \

= L Z(t)x(0) + Z{t- s)F(s) ds . \ Jo J Because x, Z, and F are of exponential order and continuous, by (vii) we have the required formula. Thus, the proof is complete for D and F being in L^Coo).

2.3. CONVOLUTION AND THE LAPLACE TRANSFORM

33

In the general case (i.e., D and F not in L1), for each T > 0 define continuous i 1 [0,c«) functions FT and DT by

= lF{t)

FT(t)

\F(T){l/[(t-T)2

T[)

ifO
iit>T

and = T

(l>(t) \ D ( T ) { l / [ ( t - T ) 2 + l]}

ifOT.

Consider (2.3.1) and x'(t) = P x ( t ) + F T ( t ) + / £>r(*-s)x(s)ds, (2.3.1)r Jo with x(0) = xo for both. Because the equations are identical on [0,T], so are their solutions; this is true for each T > 0. Thus because (2.3.5) holds for (2.3.1)T for each T > 0, it holds for (2.3.1) on each interval [0,T]. This completes the proof. Exercise 2.3.1. This exercise if not trivial. Substitute (2.3.5) into (2.3.1), interchange the order of integration, and show that (2.3.5) is a solution of (2.3.1). Although one can seldom find Z(i), we shall discover certain properties that make the variation of parameters formula very useful. For example, by a change of variable / Z(t - s)F(s) ds = / Z(s)F(t Jo Jo

-s)ds,

so if we can show that O

/ \Z(t)\dt
34

2. LINEAR EQUATIONS We turn now to the integral equation ft x(t) = f ( i ) + /

B(t-s)x(s)ds,

(2.3.6)

Jo with f : [0, oo) —> Rn being continuous, B(t) an n x n matrix continuous on [0, oo), and both f and B of exponential order. The goal is to obtain a variation of parameters formula for (2.3.6). Naturally if B and f are both differentiate, we could convert (2.3.6) to an equation of the form (2.3.1) and apply Theorem 2.3.1. But that seems too indirect. Such a formula should exist independently of B' and f. We shall see, however, that the derivative of f will enter, in a natural way, even when the transform of (2.3.6) is taken directly. Theorem 2.3.2. Let H{t) be the n x n matrix satisfying H(t) = I+

ft

Jo

B(t-s)H(s)ds

(2.3.7)

and let f (£) and B be continuous and of exponential order. The unique solution x(t) of (2.3.6) is given by ft x(i) = H(t)f(0) + / H(t-s)f'(s)ds. Jo

(2.3.8)

Proof. The Laplace transform of (2.3.7) is

L(H) = L(I) + L(B)L(H) and, as 1,(1) = s " \ L(I) = s" 1 /. Thus [I - L(B)]L(H) = L(I), and, because L(I) is nonsingular, so is [/ — L(B)]. This implies that L(H) =

[I-L(B)]-1s-1.

Now the transform of (2.3.6) is L(x) = L(f) + i ( B ) i ( x ) so that [/ - L(B)]L(X) = L(f) or L(X) = [I-L(B)]-1L({).

2.3. CONVOLUTION AND THE LAPLACE TRANSFORM

35

Multiply and divide by s on the right side and recall that i(f') = s i ( f ) - f ( O ) . This yields L(X) =

{s[I-L(B)]}-1sL(i)

= L(H)[L(i')+i(0)] = L(H)L(?) + L(H)t(0)

= L( I H(t-s)t'(s)ds*)+L(H(t)t(O)) = L(ff(i)f(0) + f H(t- s)f'(s) ds "\ , so that (2.3.8) follows from (vii). This completes the proof. Notice that (2.3.7) represents the n integral equations x(t)=el+

Jo

B(t - s)x(s)ds.

It is necessary that functions be defined on [0, oo) for the Laplace transforms to be applied. Also, F and D need to be of exponential order. The following exercise suggests that one may try to circumvent both problems. Exercise 2.3.2. Suppose that F and D in (2.3.1) are continuous on [0, T] but not defined for t > T. Define F and D on [0, oo) by asking that F(t) = F(T) and D(t) = D(T) if t > 1. Check the details to see if the variation of parameters formula will work on [0, T]. Exercise 2.3.3. Continue the reasoning of Exercise 2.3.2 and suppose that it is known that Jo \Z(t)\ dt < oo. If D is not of exponential order but F is bounded, can one still conclude that solutions of (2.3.1) are bounded? We return to x(i) = f (t) + [ B(t- s)x(s) ds

(2.3.6)

Jo with f : [0, oo) —> Rn being continuous and B a continuous n x n matrix, both f and B of exponential order. Theorem 2.3.3. If H is defined by (2.3.7) and if H is differentiable, then the unique solution of (2.3.6) is given by x(t) = f(t) + / H'(t-s)f(s)ds.

Jo

(2.3.9)

36

2. LINEAR EQUATIONS

Proof. We have L(H) = {s[I -

L(B)}}-1

and [I - L(B)]L(x) = L(t), whose product yields s" 1 L(x) = L(H)L({) or L(l)L(x) = L(H)L(f), so that L

/ x ( i - s ) d s ) = i ( / H(t - s)i(s) ds ,

V^o

J

\Jo

)

which implies /* x(s)ds= /* H(t-s)f(s)ds.

Jo

Jo

We differentiate this to obtain (2.3.9), because H(0) = I. This completes the proof. The matrices Z and H are also called resolvents, which will be discussed in Chapter 7 in some detail.

2.4

Stability

Consider the system x' = ^(t)x+ / C(t,s)x(s)ds

(2.4.1)

Jo with A an n x n matrix of functions continuous for 0 < t < oo and C(t, s) an n x n matrix of functions continuous for 0 < s < £ < oo. If : [O,£o] —> -R" is a continuous initial function, then x(t, 0) will denote the solution on [to, oo). If the information is needed, we may denote the solution by x(t,to,(f>). Frequently, it suffices to write x(t). Notice that x(£) = 0 is a solution of (2.4.1), and it is called the zero solution.

2.4. STABILITY

37

Definition 2.4.1. The zero solution of (2.4.1) is (Liapunov) stable if, for each s > 0 and each to > 0, there exists S > 0 such that \(f)(t)\ <S

on

[0,t0]

and

t > t0

imply |x(i, 0, there exists S > 0 such that to>O,

\(f>(t)\<6

on

[0,t 0 ],

and

t > t0

imply |x(t, (p)\ < e. Definition 2.4.3. The zero solution of (2.4.1) is asymptotically stable if it is stable and if for each to > 0 there exists 5 > 0 such that \)\ —> 0 as t —> oo. Definition 2.4.4. The zero solution of (2.4.1) is uniformly asymptotically stable (U.A.S.) if it is uniformly stable and if there exists r] > 0 such that, for each s > 0, there is a T > 0 such that io>O,

|0(i)|<7?

on

[0,£ 0 ],

and

£ > t0 + T

imply |x(t, (p)\ < e. We begin with a brief reminder of Liapunov theory for ordinary differential equations. The basis idea is particularly simple. Consider a system of ordinary differential equations x' = G ( t , x ) ,

(2.4.2)

with G : [0, oo) xRn -> Rn being continuous and G(t, 0) = 0, so that x = 0 is a solution. The stability definitions apply to (2.4.2) with (t) = x(to) on [0,t 0 ]. Suppose first that there is a scalar function V : [0,oo) x Rn ^ [0,oo) having continuous first partial derivatives with respect to all variables. Suppose also that V(£, x) —> co as |x| —> co uniformly for 0 < t < CXD; for example, suppose there is a continuous function W : Rn —> [0, oo) with

W(x) -^ oo as |x| -> oo and V{t,x) > W(x).

38

2. LINEAR EQUATIONS

Notice that if x(t) is any solution of (2.4.2) on [0, oo), then V(t,x(t)) is a scalar function of t, and even if x(t) is not explicitly known, using the chain rule and (2.4.2) it is possible to compute V'(t,x(t)). We have . NN

,, y ( i

dV dxx ^ ^ ^

x ( f ) )

'

But G(t,x) = [dxi/dt,... V'(t,x(t))=gradV-G

+

dV dxn - - - + ^ ^

,dxn/dtj

+

dV ^ -

(a)

and so (a) is actually

+ dV/dt.

(b)

The right-hand side of (b) consists of known functions of t and x. If V is shrewdly chosen, many conclusions may be drawn from the properties of V. For example, i(V'{t,x(t)) < 0, then t > t0 implies V{t,x.(t)) < V(to,x(to)), and because V(t,x) —> oo as |x| —> oo uniformly for 0 < t < oo, x(t) is bounded. The object is to find a suitable V function. We now illustrate how V may be constructed in the linear constant coefficient case. Let A be an n x n constant matrix all of whose characteristic roots have negative real parts, and consider the system x' = ,4x.

(2.4.3)

All solutions tend to zero exponentially, so that the matrix /"CO

B=

[exp Atf [e-xp At] dt

(2.4.4)

Jo is well defined, symmetric, and positive definite. Furthermore, ATB + BA=-I

(2.4.5)

because - / = [exp Atf [exp At]\™ O

= / (d/dt) [exp At]T [exp At] dt Jo O

= / Jo

( A T [exp At]T [exp At] + [exp At]T [exp At] A) dt

= ATB + BA. Thus, if we select V as a function of x alone, say V{x) =xTBx,

(2.4.6)

2.4. STABILITY

39

then for x(i) a solution of (2.4.3) we have

V'(x(t)) = (xT)'Bx + xTBx' = (x') T Bx + x T Bx' = xTATBx + xTBAx = xT(ATB + BA)x T = —X X .

The matrix B will be used extensively throughout the following discussions. In some of the most elementary problems asking V(t,x) to have continuous first partial derivatives is too severe. Instead, it suffices to ask that V : [0, oo) x Rn —> [0, oo) is continuous and

(2.4.7) V satisfies a local Lipschitz condition in x.

Definition 2.4.5. A function V(i, x) satisfies a local Lipschitz condition in x on a subset D of [0, oo) x Rn if, for each compact subset L of D, there is a constant K = K(L) such that (i,xi) and (£, X2) in L imply that V(t,x1)-V(t,x2)\
V/ 242) (i,x) = limsup [V(t + h,x + hG(t,x)) -V(t,x)]/h.

(2.4.8)

Because V satisfies a local Lipschitz condition in x, when V is independent of t [so that V = V(x)], we see that l^(2.4.2)(x)|<^|G(t,x)|. Next, define V'(t,x(t)) = limsup [V(t + h,x{t + h)) -V(t,x(t))]/h.

(2.4.9)

It can be shown [see T. Yoshizawa, (1966; p. 3)] that V'(t,x(t)) = V(2A.2)(t,x).

(2.4.10)

Moreover, from integration theory it is known that V'(t,x(t)) < 0 implies that V(t,x(t)) is nonincreasing.

40

2. LINEAR EQUATIONS

The next problem will be encountered frequently in the following, and it is best taken care of here. Refer to (2.4.3) and select B as in (2.4.4). Then form V(x) = [ x ^ x ] 1 / 2 and compute the derivative along a solution of (2.4.3). If x ^ 0, then V has continuous first partial derivatives and F'(x) = {xTBx)'/2[xTBx\1/2 = -xTx/2[xT5x]1/2. Now there is a positive constant k with |x| > 2fc[xTBx]1/2, so for x / f l , V'(x) < -k\x\. But we noted after (2.4.8) that \V'\
=K\Ax\,

so when x = 0 we have V'(x) < 0. Hence, for all x we see that V'(x) < -fc|x|. The theory is almost identical for integro-differential equations, although the function V(t,x) is generally replaced by a functional V(t,x(-)) = V(t,x(s); 0 < s < t). We develop this idea more fully later when we consider general functional differential equations; however, we now have sufficient material for some general results.

2.5

Liapunov Functionals and Small Kernels

We consider the system /"* x' = Ax+

C(t,s)x(s)ds,

(2.5.1)

Jo in which A is an n x n matrix all of whose characteristic roots have negative real parts, C(i, s) an n x n matrix of functions continuous for 0 < s < t < oo, and / Jt

\C(u, s)\ du is continuous for 0 < s < t < oo .

2.5. LIAPUNOV FUNCTIONALS AND SMALL KERNELS

41

Find a symmetric positive definite matrix B with ATB + BA = -I.

(2.5.2)

There are positive constants r, k, and K (not unique) with |x| >2fc[x T Bx] 1/2 ,

(2.5.3)

\Bx\
(2.5.4)

r\x\<[xTBx}1/2.

(2.5.5)

and

A basic tool in the investigation of (2.5.1) is the functional V{t,x{-)) = [xTBx\1'2 + K

/

\C(u,s)\du\x(s)\ds,

(2.5.6)

Jo Jt where K is a positive constant. This functional has continuous first partial derivatives with respect to all variables (when x ^ 0) and it satisfies a global Lipschitz condition in x(i). Let us compute the derivative of (2.5.6) along a solution x(t) of (2.5.1). For x ^ O we have

V'(t,xC)) =

{(xTBXy/2lxTBx}1/2} ft

OO

/

\C(u,t)\du\x\-K

/ Jo

\C(t,s)\\x(s)\ds,

and because (xTBx)' = (x'fBx + xTBx' r

i-t

= \xTAT+ /

I

Jo

r

+ xTB \Ax+

I

-i

xT(s)CT(t,s)ds\Bx

1

ft

Jo

i

C(t, s)x(s) ds

1

ft

= - x T x + 2 / xT(s)CT(t, s) ds Bx, Jo

by (2.5.3) and (2.5.4) we have ft

(x T Bx)72(x T Bx) 1 / 2 < -fc|x| + K / \C(t, s)\ |x(s)| ds . Jo

42

2. LINEAR EQUATIONS

This yields V < \-k+K

f

\C(u,t)\du\

\x\-(K-K)

I \C(t,s)\\x(s)\ds.

(2.5.7)

Our basic assumption is

_ There exists K >K and k > 0 with k < k-K /

\C(u,t)\ du . (2.5.8)

Theorem 2.5.1. Let B, k, and K be defined by Eqs. (2.5.2)-(2.5.4). (a) If (2.5.8) holds, the zero solution of (2.5.1) is stable. (b) If (2.5.8) holds with K > K and k > 0, then x = 0 is asymptotically stable. (c) If (2.5.8) holds and j0 / t °° \C(u, s)\ duds is bounded, then x = 0 is uniformly stable. (d) Suppose (c) holds and K > K and k > 0. If for each p > 0 there exists S > 0 such that P > S and t > 0 imply f* f™p \C(u, s)\ duds < p, then x = 0 is uniformly asymptotically stable. Proof of (a). Let e > 0 and to > 0 be given. We must find S > 0 such that \(f)(t)\<S

on

[0,i0]

and t > t0

imply |x(i, <j>)\ < e. Because V'(t,x(-)) < 0, if |0(t)| < 5 on [0,i 0 ], then we have

r|x(t)|
|C(u, s)| du |0(s)| ds

I-OO

t()

/

/

K\C(u,s)\duds

"'to

or |x(t)| < J{(l/2fc) + / " / Jo

K\C(u,s)\duds}/r,

Jtlt

which will be smaller than e if IV

5<er

rtn

,oo _

1

(l/2fe)+ / / if |C(u,s)| duds . / L ^0 Jtv, J This choice of 5 = S(e,to) fulfills the conditions for stability.

(2.5.9)

2.5. LIAPUNOV FUNCTIONALS AND SMALL KERNELS

43

Proof of (c). For uniform stability, 5 must be independent of to. If Jo It° I C '( U ' s)\duds < M ioi Q < t < oo and some M > 0, then (2.5.9) may be replaced by (2.5.10)

5 <sr/[(l/2k) + KM)], yielding uniform stability. Proof of (b). We have V'(t,X(-))<-k\x\-(K-K)

I

\C(t,s)\\x(s)\ds,

Jo

and hence, there is a \i > 0 with

y'(t,x(0)<-M[|x| + |x'|],

(2.5.11)

which is a fundamental relation. If x(i) is a solution of (2.5.1), then rt

r\x(t)\ < V ( t , x ( - ) ) < V ( t 0 ,

,-t

\x(s)\ds-n Jto

\x'(s)\ds. Jtn

If Euclidean length is used for |x|, then the second integral is arc length. Let x[a, b] denote arc length of x(i) on [a, b].

(2.5.12)

Then we have r\x(t)\

< V(t0

[ |x(s)|ds-//x[to,i].

(2.5.13)

Jto

Because |x(t)| > 0, we have Jt°° |x(s)| ds < oo, which implies that there is a sequence {£„} —> oo with |x(i n )| —> 0. Also, x[to,t] is bounded. Thus, |x(i)| —> 0. Because (a) is satisfied, the proof of (b) is complete. Proof of (d). By (b), x = 0 is uniformly stable. Find 5 > 0 such that \<j>\ < 5 implies |x(i, (f>)\ < 1. Take r\ = S and let e > 0 be given. We then must find T such that to > 0, \4>(t)\ < 5 on

[0,i0],

imply |x(i,
a n d t > t0 + T

2. LINEAR EQUATIONS

44

(i) Find L > 0 and p > 0 with (e/2kL) + pK + (KeM/L) < re. For that p find S in (d). We show that if |x(t)| < e/L on an interval of length S, then |x(£)| < e always. Suppose \x(t)\ < e/L on an interval [ii,ii + P] with P > S. Then at t = ti + P we have (as |x(t)| < 1) ti />co

/ / ,-tt+P

+ / Jtt

K\C(u,s)\du\x(s)\ds

Jti+p

/-OO

/

K\C(u,s)\du\x{s)\ds

Jti+p

< (e/2kL) + pK + KMe/L < re . As y ' < 0, we have r|x(t)| h + P, so that |x(t)| < e if t > h + P. (ii) Note that there is a Pi > 0 such that the inequality |x(i)| > ejiL on an interval of length Pi must fail because 0 t0 + N(S + Pi), then we will have \x(i)\ < e always. Then taking T = N(S + Pi) completes the proof. Remark. The corollary to Theorem 2.6.1 will show that the conclusion in (b) demonstrates uniform asymptotic stability in the convolution case. Exercise 2.5.1. Consider the scalar equation x' = -x+

* Jo

a(t,s)(t-s

+ l)~nx(s)ds

for n > 1 and a(t, s) a continuous scalar function satisfying \a(t, s)\ < d for some d > 0. Determine conditions on d and n to ensure that each part of

2.5. LIAPUNOV FUNCTIONALS AND SMALL KERNELS

45

Theorem 2.5.1 is satisfied. That is, give different conditions for each part of the theorem. Pay careful attention to (d) and notice how Part (i) of the proof would be accomplished. Exercise 2.5.2. Consider /"* x' = -x +

d(t-s + iynx(s)

ds + sint, Jo with d and n positive constants. Determine d and n such that the variation of parameters formula yields all solutions bounded. There is also a variation of parameters formula for * x' = Ax + C(t,s)x(s)ds +F(t), (2.5.14) Jo namely, x(£) = i?(£, 0)x(0) + / R(t,s)F(s)ds,

(2.5.15)

Jo where R(t, s) is called the resolvent and is an n x n matrix that satisfies dR(t,s)/ds = -R(t,s)A-

R(t,u)C(u,s)du,

(2.5.16)

Js

for 0 < s < t and R(t, t) = I. When C(t, s) is of convolution type so is R(t, s), and in fact, R(t, s) = Z(t — s), where Z(t) is the n x n matrix satisfying * Z'(t) = AZ(t)+ C(t-s)Z(s)ds (2.5.17)

Jo and Z(0) = / . We found conditions for which Jo°° \x(t)\dt < oo for each solution of (2.5.1), so that Jo°° \Z(t)\ dt < oo. Thus, in the convolution case, a bounded F in (2.5.14) produced bounded solutions. But in the general case of (2.5.14), we have too little evidence of the integrability of R(t, s). Thus we are motivated to consider Liapunov's direct method for the forced equation (2.5.14). Extensive treatment of the resolvent may be found in Miller (1971a), in a series of papers by Miller (see also the references mentioned in Chapter 7), and in papers by Grossman and Miller appearing in the Journal of Differential Equations from 1969 to mid-1970s. Additional results and references are found in Grimmer and Seifert (1975). The following is one of their results, but the proof presented here is different.

46

2. LINEAR EQUATIONS

Theorem 2.5.2. Let A by an n x n constant matrix all of whose characteristic roots have negative real parts, let C(t, s) be continuous for 0 < s < t < oo, and let F : [0, oo) —> Rn be bounded and continuous. Suppose B satisfies ATB + BA = —I and a 2 and 01 are the smallest and largest eigenvalues of B, respectively. If L \BC(t, s)\ds < M for 0 < t < oo and 2(3M/a < 1, then all solutions of (2.5.14) are bounded. Proof. If the theorem is false, there is a solution x(t) with limsupj^^ xT(t)Bx(t) = +oo. Thus, there are values of t with |x(t)| as large as we please and [xT(t)Bx(t)~\ > 0, say, at t = S, and xT(t)Bx(t) < xT(S)Bx(S) if t < S. Hence, at t = S we have [x T (t)Bx(t)]'=-x T (t)x(t)+ /

2xT(s)CT(t,s)Bx(t)ds

Jo T

+ 2F (t)Bx(t) >0 or xT(S)x(S)<

,s / 2xT(s)CT(S,s)Bx(S)ds Jo rs

<2|x(5)| / \BC(S,s)\\x(s)\ds Jo

+2FT(S)Bx(S) +2xT(S)BF(S)

<2\x(S)\ f \BC(S,s)\[(xT(s)Bx(s))1/2/a]ds Jo + 2xT{S)BF(S) < (2/a) |x(5)| (xT(5)JBx(5))1/2 [

\BC(S,s)\ds

Jo

+ 2xT{S)BF(S) < (2/a) |x(5)| [3 \x(S)\M + 2xT(S)BF(S) = (2(3M/a) \x(S)\2 + 2xT(S)BF(S). As 2f3M/a < 1 we have a contradiction for (x(S')l sufficiently large. The proof of the last theorem is a variant of what is known as the Liapunov-Razumikhin technique, which uses a Liapunov function (rather than a functional) to show boundedness and stability results for a functional differential equation. An introduction to the method for general functional differential equations is found in Driver (1962).

2.5. LIAPUNOV FUNCTIONALS AND SMALL KERNELS

47

Detailed adaptations of the Razumikhin method to Volterra equations may be found in Grimmer and Seifert (1975) and in Grimmer (1979). Most of those results are discussed in Chapter 8. Halanay and Yorke (1971) argue very strongly for the merits of this method over the method of Liapunov functionals. Notice that the main conditions in the last two theorems are very different. In Theorem 2.5.1 we mainly ask that Jt°° \C(u, t)\ du be small, where the first coordinate is integrated. But in Theorem 2.5.2 we ask that Jo \BC(t, s)\ ds be small, where the second coordinate is integrated. Under certain conditions on C(t, s) it is possible to obtain a differential inequality when considering (2.5.6), (2.5.7), and (2.5.1). That is, we differentiate V(t,x(-)) along a solution of (2.5.1) and attempt to find a scalar function rj(t) > 0 with V'(t,x(-))
(2-5.18)

When that situation occurs, owing to the global Lipschitz condition in x(t) that V satisfies, it turns out that the derivative of V along a solution of the forced equation (2.5.14) results in the inequality y'(i,x(-)) < -77(t)F(t,x(-)) + K\F(t)\.

(2.5.19)

It then follows that for a solution x(£, (f>) on [to, oo)

y(t,x(.))
\F{s)\i

Jtu

e x p - / r](u)du\\ds,

I

L Js

(2.5.20)

JJ

which can ensure boundedness, depending on the properties of r\ and F. Equation (2.5.20) becomes a substitute variation of parameters formula for (2.5.14), acting in place of (2.5.15). In fact, it may be superior to (2.5.15) in many ways even if much is known about R(t,s). To see this, recall that, for a system of ordinary differential equations x' = P(t)x + Q(t) with P{t) not constant, if Z(t) is the n x n matrix satisfying

Z'(t) = P(t)Z(t),

Z(0) = I,

then the variation of parameters formula is ft x(i) = Z(t)x{0) + / Jo

Zit)Z-1is)Q{s)ds.

48

2. LINEAR EQUATIONS

Even if Z(t) is bounded, Z~1(s) may be very badly behaved. One usually needs to ask that / tr P(s)ds > -M > -oo Jo

to utilize that variation of parameters formula; and this condition may imply that Z(t) -» 0 as t —> oo. In that case, the hope of concluding that bounded Q produces bounded x(i) vanishes. To achieve (2.5.19) we examine r V'<\-k+K

I

Jt

i \C(u,t)\du\\x\-(K-K)

J

*

Jo

\C(t,s)\ |x(s)| ds

(2.5.7)

once more and observe that we require a function A:[0,oo)^[0,l] with />OO

\C(t,s)\>\(t)

\C(u,s)\du,

(2.5.21)

Jt for 0 < s < t < oo. For, in that case, if K > K and k
K /

|C(M,t)|dM

with k positive, then from (2.5.7) we have

V <-k\x\ - (K - K)\(t) <

-2kkX(t)[xTBX]1/2 -[{K -K)/K]\(t)K

/ Jo Jt / Jo Jt

\C(u,s)\du\x(s)\ds

\C(u,s)\du\x(s)\ds

<-v(t)V(t,x(-)), where r](t) = X(t) min [2kk; (K - K)/R] . These calculations prove the following result.

(2.5.22)

2.5. LIAPUNOV FUNCTIONALS AND SMALL KERNELS

49

Theorem 2.5.3. Suppose that the conditions of Theorem 2.5.1 (b) hold and that (2.5.21) and (2.5.22) are satisfied. If x(t, 0) is a solution of (2.5.14) on [t0, oo) and ifV is defined by (2.5.6), then

V'(t,x(-)) < -v(t)V(t,x(-))

+ K\F(t)\,

and therefore,

V(t,x(-))
*

r

+ -K~ / |F(s)K exp

r*

ii

- / 7?(u)du ^ds.

Exercise 2.5.3. Verify that (2.5.19) holds. Exercise 2.5.4. Consider the scalar equation

x'(t) = -x{t)+

ft

/ C{t,s)x{s)ds + a cost, Jo

where \C(t, s)\ < ci{ exp [ — h(t — s)] } with c\ and h being positive constants. Find conditions on h and c\ to ensure that the conditions of Theorem 2.5.3 are satisfied and n(t) is constant. Your conditions may take the form rj(t) = h({3 - 1)/13 < 1 for some

(3 > 1

and [3c\/h < a

for some

a < 1.

In the convolution case there is a natural way to search for rj(t). Exercise 2.5.5. Consider the vector system * x'=^x+/

D(t - s)x(s) ds

(2.5.23)

Jo in which the characteristic roots of A have negative real parts and |-D(£)| > 0 on [0, oo). Let B, k, and K be denned as before and suppose that there is a d > K and k\ > 0 with k>h>d

\D(u-t)\du, Jt

Prove the following result.

0
50

2. LINEAR EQUATIONS

Theorem 2.5.4. If there is a continuous and nonincreasing scalar function X : [0, oo) —> (0, oo) with />oo

\D(v)\ >\(v) / Jv

\D(u)\du,

then there is a constant q > 0 such that for x(t) a solution of (2.5.23) and rt

V(t,x(-)) = [xTBx}1/2+d

/>oo

/ Jo Jt

\D(u-s)\du\x(s)\ds

we have V'(t,X(-))<-q\(t)V(t,X(-)). In our discussion of the variation of parameters formula for an ordinary differential equation x' = P(i)x + Q(i) with P not constant, but P and Q continuous on [0,oo), we looked at Z(t)Z-1(s) where Z'(t)=P(t)Z(t),

Z(0)=I.

Jacobi's identity [or the Wronskian theorem [see Hale (1969; pp. 90-91)] states that det Z(t) = exp fQ tr P(s) ds, so that det Z(t) never vanishes. However, if Z(t) is the principal matrix solution of x' = Ax+

r* Jo

B(t-s)x(s)ds

(2.5.24)

with A constant and B continuous, then det Z(t) may vanish for many values of t. Theorem 2.5.5. Suppose that (2.5.24) is a scalar equation with A < 0 and B(t) < 0 on [0, oo). If there exists ti > 0 such that ,-t ,-u I I B(u — s) ds du —> —oo as t —> oo , Jti Jo then there exists ti > 0 such that if x(0) = 1, then x(^) = 0.

2.6. UNIFORM ASYMPTOTIC STABILITY

51

Proof. If the theorem is false, then x(t) has a positive minimum, say xi, on [0,ti]. Then for t > t\ we have i

x'{t)<

Jo ft!

< / Jo

j-t

B(t-s)x(s)ds+

.It i

B(t - s)x(s) ds

B(t-s)xids

implying, upon integration, that ft ft! x(t) < x\ + / / B(u — s)x\ ds du —> —oo Jtt Jo as t —> oo, a contradiction. This completes the proof.

2.6

Uniform Asymptotic Stability

We noticed in Theorem 2.5.1 that every solution x(i) of (2.5.1) may satisfy O

/ Jo

|x(t)|di
(that is, x is L1 [0, oo)) under considerably milder conditions than those required for uniform asymptotic stability. However, in the convolution case * x' = ^ x + / D(t - s)x(s) ds

(2.6.1)

Jo with D(t) continuous on [0, oo) and A being an n x n constant matrix, then O

\ Jo

f'OO

\D(t)\dt < oo and

/ Jo

|x(t)|dt
(2.6.2)

is equivalent to uniform asymptotic stability of (2.6.1). This is a result of Miller (1971b), and we present part of it here. Theorem 2.6.1. If each solution x(i) of (2.6.1) on [0, oo) is L ^ o o ) , if D(t) is L1 [0, oo), and if A is a constant n x n matrix, then the zero solution of (2.6.1) is uniformly asymptotically stable. Proof. If Z(t) is the n x n matrix with Z(0) = I and

Z'(t) = AZ{t) + then Z(t) is Ll[Q,oo).

/"*

Jo

D(t- s)Z{s) ds ,

52

2. LINEAR EQUATIONS Let x(t,to,(p)

x'(t,to,(f))

be a solution of (2.6.1) on [to,oc). Then

= Ax{t,to,(f)) +

D(t-s)cj){s)ds+

D(t-s)x(s,to,(j))ds, Jtlt

Jo so that

x ' ( t + t0, t0, ) = A x ( t + t0, t 0 , < / > ) + / D ( t + t0- s)(s) ds Jo ft + D(t-s)x(to+s,to,4>)ds Jo or x(t + to, to, 4>) is a solution of the nonhomogeneous equation D(t - s)x(s) ds +

D(t + t0 - s)<j>(s) ds , Jo

which we write as y' = Ay+ [ D(t - s)y(s) ds + F(t) Jo with y(0) = x(to,to,(f)) F(t)=

(2.6.3)

= <j)(t0) and

D(t +10 - s)cj}{s) ds.

(2.6.4)

Jo By the variation of parameters formula [see Eq. (2.3.5) in Theorem 2.3.1] we have y(t) = Z(t)(t>(t0) + I Jo

Z(t-s)F(s)ds

or ft

x{t + to,to,(f>) = Z(t)(f)(to)+ Jo so that

f

rk,

Z(t-s){

/ I Jo

^

D(s +

to-u)(j)(u)du}ds, )

x{t + to,to,4>) = z(t)4>(t0) ft

r

fU,

>

/ D(s + u)4>(t0 - u) du \ ds . (2.6.5) + / Z{t-s)\ Jo I Jo ) Next, notice that, because A is constant and Z(t) is Z/1[0,oo), then AZ(t) is L 1 [0,oo). Also, the convolution of two functions in L1^, <x>) is

2.6. UNIFORM ASYMPTOTIC STABILITY

53

Ll[0, oo), as may be seen by Fubini's theorem [see Rudin (1966, p. 156).] Thus Jo D(t - s)Z(s) ds is L^O, oo), and hence, Z'(t) is L^O, oo). Now, because Z'{t) is i 1 [0, oo), it follows that Z(t) has a limit as t —> oo. But, because Z(t) is L1^, oo), the limit is zero. Moreover, the convolution of an L1 [0, oo) function with a function tending to zero as t —> oo yields a function tending to zero as t —> oo. {Hint: Use the dominated convergence theorem.) Thus

Z'(t) = AZ(t) +

r* Jo

D(t- s)Z(s) ds^O

as t —> oo. Examine (2.6.5) again and review the definition of uniform asymptotic stability (Definition 2.4.4). We must show that \4>{t)\ < r] on [O,£o] implies that x(i + to,to,(f>) —> 0 independently of toNow in (2.6.5) we see that Z{t){to) —> 0 independently of to- The second term is bounded by f-to

7] \Z(t-s)\ Jo

/-OO

/'t

\D(s+u)duds
\Z(t-s)\ Jo

\D(v)\dvds, J8

and that is the convolution of an L1 function with a function tending to zero as t —> oo and, hence, is a (bounded) function tending to zero as t —> oo. Thus, x(t + to,to, oo independently of to- The proof is complete Corollary 1. If the conditions of Theorem 2.5.1 (b) hold and ifC(t,s) is of convolution type, then the zero solution of (2.5.1) is uniformly asymptotically stable. Proof. Under the stated conditions, we saw that solutions of (2.5.1) were L^O.oo). Corollary 2. Consider /"* x'=ylx+/ D(t-s)x(s)ds

(2.6.1)

Jo with A being an nxn constant matrix and D continuous on [0, oo). Suppose that each solution of (2.6.1) with initial condition x(0) = xo tends to zero ast —> oo. If there is a function A(s)ei 1 [0, oo) with fQ" \D(s + u)\du < A(s) for 0 < to < oo and 0 < s < oo, then the zero solution of (2.6.1) is uniformly asymptotically stable.

54

2. LINEAR EQUATIONS

Proof. We see that Z(i) —> 0 as t —> oo, and in (2.6.5), then, we have X(t

+ to,toA)\<\Z(t)^(to)\+

max |0(,s)| / \Z(t - s)\X(s) ds . 0<S
Jg

The integral is the convolution of an L1 function with a function tending to zero as t —> oo and, hence, tends to zero. Thus x(i + to,to,4>) —> 0 as t —» oo uniformly in to- This completes the proof. Example 2.6.1. Let D(t) = (t + I)"™ for n > 2. Then /" D(,s + u)rfu= [ (s + u + l)-ndu Jo Jo _ (s + u + l)-n+1 *" < (s + l ) - " + 1 —n + 1 n— 1 0 ~~ which is L 1 . Recall that for a linear system x' = A(t)x

(2.6.6)

with A(t) an n x n matrix and continuous on [0,oo), the following are equivalent: (i) All solutions of (2.6.6) are bounded. (ii) the zero solution is stable. The following are also equivalent under the same conditions: (i) All solutions of (2.6.6) tend to zero, (ii) The zero solution is asymptotically stable. However, when A{t) is T-periodic, then the following are equivalent: (i) All solutions of (2.6.6) are bounded. (ii) The zero solution is uniformly stable. Also, A(t) periodic implies the equivalence of: (i) All solutions of (2.6.6) tend to zero, (ii) The zero solution is uniformly asymptotically stable, (iii) All solutions of x' = A(t)x + F(i)

(2.6.7)

are bounded for each bounded and continuous F : [0, oo) —» Rn.

2.6. UNIFORM ASYMPTOTIC STABILITY

55

Property (iii) is closely related to Theorem 2.6.1. Also, the result is true with \A(t)\ bounded instead of periodic. But with A periodic, the result is simple, because, from Floquet theory, there is a nonsingular Tperiodic matrix P and a constant matrix R with Z(i) = P(t)eRt being an n x n matrix satisfying (2.6.6). By the variation of parameters formula each solution x(t) of (2.6.7) on [0, oo) may be expressed as ft x(i) = Z{t)x{0) + / Jo

Z(t)Z-1(s)F(s)ds.

In particular, when x(0) = 0, then x(i) = /

Jo

P{t)[eR{t-s)}p-1{s)F(s)ds.

Now P(t) and P~1(s) are continuous and bounded. One argues that if x(i) is bounded for each bounded F, then the characteristic roots of R have negative real parts; but, it is more to the point that O

/ \P(t)em\dt < oo. Jo Thus, one argues from (iii) that solutions of (2.6.6) are L1[0,cx)) and then that the zero solution of (2.6.6) is uniformly asymptotically stable. We shall shortly (proof of Theorem 2.6.6) see a parallel argument for (2.6.1). The preceding discussion is a special case of a result by Perron for \A(t) \ bounded. A proof may be found in Hale (1969; p. 152). Problem 2.6.1. Examine (2.6.5) and decide if: (a) boundedness of all solutions of (2.6.1) implies that x = 0 is stable. (b) whenever all solutions of (2.6.1) tend to zero then the zero solution of (2.6.1) is asymptotically stable. We next present a set of equivalent statements for a scalar Volterra equation of convolution type in which A is constant and D{t) positive. An n-dimensional counterpart is given in Theorem 2.6.6. Theorem 2.6.2. Let A he a positive real number, D : [0, oo) —> (0, oo) continuous, Jo°° D(t) dt < oo, -A + Jo°° D(t) dt ^ 0, and x' = -Ax+

ft Jo

D(t - s)x(s) ds.

The following statements are equivalent.

(2.6.8)

56

2. LINEAR EQUATIONS

(a) All solutions tend to zero. (b) -A + /0°° D{t) dt < 0. (c) Each solution is Ll[0, oo). (d) The zero solution is uniformly asymptotically stable. (e) The zero solution is asymptotically stable. Proof. We show that each statement implies the succeeding one and, of course, (e) implies (a). Suppose (a) holds, but —A + JQ" D(t) dt > 0. Choose to so large that /0°° D(t) dt> A and let (i) = 2 on [0, t0]- Then we claim that x(t, ) > 1 on [to,oo). If not, then there is a first t\ with x(t\) = 1, and therefore, x'(ti) < 0. But ft! x'(t1) = -Ax(t1)+ - s)x(s) ds Jo = -A+ > -A+

Jo

D(s)x(ti - s)ds D(s)ds

Jo

> -A+

D(s)ds > 0, Jo a contradiction. Thus (a) implies (b). Let (b) hold and define f-OO

V(t,x(-)) = \x\+ I I D(u-s)du\x(s)\ds, Jo Jt so that if x(t) is a solution of (2.6.8), then r* V'(t,x(-))<-A\x\+ / D(t-s)\x(s)\ds Jo oo

/

r-t

D(u-t)du\x\-

r

D(u-t)du

-A+

\x\

Jt

1

Jo

i D(u)du \x\ J

r =

1

/-oo

= - A+

I D(t-s)\x{s)\ds Jo

= — a\x\ for some

a >0 .

2.6. UNIFORM ASYMPTOTIC STABILITY

57

An integration yields 0 < V(t,x(-))

< V(to,) -a

ft Jto

\x(s)\ds,

as required. Thus, (b) implies (c). Now Theorem 2.6.1 shows that (c) implies (d). Clearly (d) implies (e), and the proof is complete. To this point we have depended on the strength of A to overcome the effects of D{t) in ft

x' = Ax + / D(t - s)x(s) ds Jo

(2.6.1)

to produce boundedness and stability. We now turn from that view and consider a system /"* x'=A(t)x+

C(t,s)-x.(s)ds + F(t),

(2.6.9)

Jo with A being an n x n matrix and continuous on [0, oo), C(t, s) continuous for 0 < s < t < oo and n x n, and F : [0, oo) —> Rn bounded and continuous. Suppose that /"CO

G{t,s) =

C(u,s)du

(2.6.10)

is defined and continuous for 0 < s < t < oo. Define a matrix Q on [0, oo) by Q(t) = A(t) -G{t,t)

(2.6.11)

and require that Q commutes with its integral

(2.6.12)

(as would be the case if A were constant and C of convolution type) and that ft exp / Q(s)ds Ju

< M exp [ - a(t - u)]

for 0 < u < t and some positive constants M and a.

(2.6.13)

2. LINEAR EQUATIONS

58

Here, when L is a square matrix, then eL is defined as the usual power series (of matrices). Also, when Q(t) commutes with its integral, then exp Jt Q(s) ds is a solution matrix of

x' = Q(t)x. Moreover,

Q(t)exp \J Q(s)ds] = | exp \J Q(S)ds] \ Q(t). Notice that (2.6.9) may be written as x' = [A(t) - G(t, t)] x + F(t) + (d/dt) / G(t, s)yi(s) ds .

(2.6.14)

Jo

If we subtract Qx from both sides, left multiply by exp [ — Jo Q(s) ds ], and group terms, then we obtain f * 1 1 ' ^ exp - / Q(s)dsx(t) \

{

Jo

J J

c r * 1 1r * i = { exp - / Q(s) (Is (d/dt) / G(t, s)x(s) ds + F(t) .

I

L Jo

i) I

Jo

J

Let <> / be a given continuous initial function on [0, to]- Integrate the last equation from to to t and obtain <^ exp - / Q(s) ds

\ x(t)

= |exp[-^"Q( S )d S ]|x(t 0 ) + / jexpf- / Q(s)ds] \F(u)du +

\ exp - / Q(s) ds Jtn I

L Jo

^ x (d/du) / G(u, s)x{s) du JJ

JO

2.6. UNIFORM ASYMPTOTIC STABILITY

59

If we integrate the last term by parts, we obtain

jexp[-y Q(s)ds] }x(i) = { e x p [ - j T ^ d s l |x(to) *f \ r ii + / < exp - / Q(s) ds \ F(u) du Jta I

I

JO

JJ

+ | exp I"- / g ( s ) d s | | / G(t,s)x(s)ds

1 1 /"t(>

r /**"

r

— < exp — / Q(s)ds > / I I Jo J J Jo

+ / Q(uW exp - / Q(s)cis Jtt)

{

I

Jo

G(to,s)x(s)ds

\ / G(w, s)x(s) cis du. i J Jo

Left multiply by exp [J o Q(s)ds], take norms, and use (2.6.13) to obtain |x(i)| < M|x(t o )|+ I

\G(to,s)\4>(s)\ds] exp [ - a(t - t0)]

ft

ft

Me-a(-t-u'>\F(u)\du+

+

\G(t,s)\\x(s)\ds

(2.6.15)

Jo

Jta

+ I \Q(u)\Me-a{t-u) I \G(u,s)\\x(s)\dsdu. Jta

Jo

Theorem 2.6.3. If x(i) is a solution of (2.6.9), if\Q(t)\ < D on [0, oo) for some D > 0, a n d if s u p 0 < t < o o J o |G(i, s)\ds < (3, then for (3 sufficiently small, x ( i ) is bounded.

Proof. For the given to and , because F is bounded there is a K\ > 0 with M|x(i o )|+ / Jo +

sup

|G(i o ,s)^(s)|ds / M{ exp [ - a(i - u)] } |F(u) | du < Kx.

t()
60

2. LINEAR EQUATIONS

From this and (2.6.15) we obtain /"* x(t)\


\G(t,s)\\x(s)\ds

Jo

r-u

t

/

D M e x p [ - a ( t - u ) ] / \G(u,

s)\\x(s)\dsdu


0<s<4

= ifi+/3[l + (Z>M/a)] sup |x(s)|. 0<s
Let /? be chosen so that /3[l + (DM/a)] = m < 1, yielding x(£)| < K\ + m sup |x(s)| . 0<s
Let isT2 > maxo to with |x(ii)| = K-2- Then K2 = |x(ti)| < Kx + mif2 < K2 , a contradiction. This completes the proof. Exercise 2.6.1. Let a > 0 and /"* x' = a(t-s + l)~3x(s) ds + F(t). Jo

Work through the entire sequence of steps from (2.6.10) to (2.6.15). Then state Theorem 2.6.3 for this equation, let F(t) = sint and (t) = 1 on [O,£o], and follow the proof of Theorem 2.6.3 to find M, K\, D, a, K2, and

P. Exercise 2.6.2. Interchange the order of integration in the last term of (2.6.15), assume \G(t, s)\ < Le^1^'^ for L and 7 positive, and use Gronwall's inequality to bound |x(£)| under appropriate restrictions on the constants. Theorem 2.6.4. In (2.6.9) let F(t) = 0, \Q(t) < D on [0,oo) and Jo \G(t,s)\ds < (3. If P is sufficiently small, then the zero solution is uniformly stable. Proof. Let e > 0 be given. We wish to find S > 0 such that £o>0,

\(t>(t)\<5on[0,t0],

imply |x(i,
and t>t

0

2.6. UNIFORM ASYMPTOTIC STABILITY

61

Let 5 < s with S yet to be determined. If \
0<s
= (M + p)5 + /3[l + (DM/a)] sup |x(s)|. 0<s
First, pick [3 so that [3[l + ( f l l / a ) ] < | . Then pick 5 so that (M + (3)6 + j£ < s. If \4>(t)\ < 5 on [0, to] and if there is a first t\ > t0 with |x(ti)| = e, we have e = |x(t 1 )|<(M + /8)<J+ j | x ( t i ) | < e , a contradiction. Thus, the zero solution is uniformly stable. The proof is complete. Naturally, one believes that the conditions of Theorem 2.6.3 imply that the unforced equation (2.6.9) is uniformly asymptotically stable. We would expect to give a proof parallel to that of Perron showing that the resolvent satisfies sup 0 < t < o o JQ \R(t,s)\ds < oo, where R(t,s) is defined in (2.5.16). We would then hope to use the variation of parameters formula (2.5.15) and prove a theorem similar to Theorem 2.6.1 showing that R(t, s) in L^O, oo) implies uniform asymptotic stability. Now we proceed with the convolution case. Theorem 2.6.5. Let the conditions of Theorem 2.6.3 hold and let F = 0. Suppose also that A is constant and C(t,s) = D(t — s). If su Po
Z(t-s)F(s)ds.

For x(0) = 0 this is x(t) = [ Z(t- s)F(s) ds , Jo

62

2. LINEAR EQUATIONS

which is bounded for every bounded F. One may repeat the proof of Perron's theorem [Hale (1969, p. 152)] for ordinary differential equations to conclude that O

/ \Z(t)\dt
By Theorem 2.6.1 the zero solution is uniformly asymptotically stable. The proof is complete. Remark. Theorems 2.6.3-2.6.5 are found in Burton (1983a). We return now to the n-dimensional system /"* x' = Ax +

D(t - s)x(s) ds ,

(2.6.1)

Jo

with A constant and D continuous. Our final result of this section is a set of equivalences for systems similar to Theorems 2.6.2 for scalar equations. These two results may be found in Burton and Mahfoud (1983), together with examples showing a certain amount of sharpness. Let Z(t) be the n x n matrix satisfying * Z'(t)=AZ(t)+

D(t-s)Z(s)ds, Z(0) = I.

(2.6.16)

Jo

Theorem 2.6.6. Suppose there is a constant M > 0 such that 0 < to < oo and 0 < t < oo we have ft fta

I

I

\D(u + v)\dudv < M.

(2.6.17)

Jo Jo

Then the following statements are equivalent. (a) (b) (c) (d) (e)

Z(t) -> 0 as t -> oo. All solutions x(t, to, 0) of (2.6.1) tend to zero as t —> oo. The zero solution of (2.6.1) is uniformly asymptotically stable. Z(t) is in L1[0,(x) and Z(t) is hounded. Every solution x(t,O,xo) of x' = Ax+

ft

./o

D(t-s)x(s)ds + F(t)

(2.6.18)

on [0, oo) is bounded for every bounded and continuous F : [0, oo) —> Rn.

2.6. UNIFORM ASYMPTOTIC STABILITY

63

(f) The zero solution of (2.6.1) is asymptotically stable. Furthermore, the following is a second set of equivalents: (g) Z(t) is bounded. (h) All solutions x(t,to,4>) of (2.6.1) are bounded. (i) The zero solution of (2.6.1) is uniformly stable. (j) The zero solution of (2.6.1) is stable. Proof. Let (a) hold. Then a solution x(t, to,
x' = Ax+ I

D(t - s)4>(s) ds+ I D(t - s)x(s) ds

Jo

Jtn

for t > to with the second term on the right treated as a forcing term. If we translate the equation by y(t) = x(t + to), we obtain rt

y'(t)=Ay(t)

+

r-U,

D(t - s)y(s) ds + Jo

Jo

D(t + t0 - s)(s) ds .

We may now apply the variation of parameters formula and write rt

rU,

y(t) = Z{t)(j}{to) + / Z(t-u) Jo

I D(u + t0Jo

s)(j){s) ds du .

The substitution s = to — v yields

y(t) = Z(t)4>(t0) + I Z(t-u) Jo

I

Jo

D(u + v)4>(t0 - v) dv du .

Because \4>(t)\ < K, K > 0, on [0,t0], we have rt

\y(t)\
+K

/"to

\Z(t-u)\ Jo

\D(u + v)\dvdu. Jo

The last term is the convolution of an L1[0, oo) function (Jo" \D(u + v)\ dv) with a function tending to zero (Z(t)), and so it tends to zero. Thus, (a) implies (b).

64

2. LINEAR EQUATIONS

Suppose that (b) holds. Then, in particular, all solutions of the form x(t, 0, xo) tend to zero, which implies that Z(t) —> 0 as t —> oo. Now

[ [ \D{u + v)\dudv<M Jo Jo uniformly in to. Thus the integral / \Z{t-u)\ Jo

f Jo

\D{u + v)\dvdu

tends to zero uniformly in to and hence x(t,to,(f>) tends to zero uniformly in to for bounded (f>. Thus, (b) implies (c). Let (c) hold. Then Miller's result implies that Z(t) is Ll[Q,oo). Also, the uniform asymptotic stability implies that Z(t) is bounded. Hence, (d) holds. Suppose (d) is satisfied. Then solutions x(t,O,xo) of (2.6.18) on [0,oo) are expressed as x(t) = Z(t)x(0) + [ Z(tJo

s)F(s) ds .

Because Z(t) is L1 [0, oo) and bounded and because F is bounded, then x(t) is bounded. Hence, (e) holds. Suppose (e) is satisfied. Then the argument in the proof of Perron's theorem [see Hale (1969, p. 152)] yields Z(t) being L ^ o o ) . This, in turn, implies uniform asymptotic stability. Of course, uniform asymptotic stability implies asymptotic stability, so (e) implies (f). Certainly, (f) implies (a). This completes the proof of the first set of equivalences. Let (g) hold. The variation of parameters formula implies that x(t, ) is bounded. Thus, (h) holds. Suppose (h) is satisfied. Then \Z{t)\ < P and /„* /ot(l \D(u + v)\dudv< M imply that whenever \4>{t)\ < 5 on [0,to] we have x ( t + t o , 0 ) | < P\
Jo + 5PM < e,

v)\dvdu

Jo

provided that 5 < s/(P + PM). Hence, x = 0 is uniformly stable. Thus, (h) implies (i). Certainly, (i) implies (j). Finally, if x = 0 is stable, then Z(t) is bounded, so (j) implies (g). This completes the proof of the theorem.

2.7. REDUCIBLE EQUATIONS REVISITED

2.7

65

Reducible Equations Revisited

In Section 1.5 we dealt with Volterra equations reducible to ordinary differential equations. That discussion led us to a large class of solvable equations. But that is only a small part of a general theory of equations that can be reduced to Volterra equations with .L1[0, oo) kernels. Though all this can be done for vector equations, it is convenient to consider the scalar equations x(t)=f(t)+

/ C{t- s)x(s)ds Jo

(2-7.1)

or /"* x1 = Ax + / C{t - s)x(s) ds + f{t)

(2.7.2)

Jo in which / and C have n continuous derivatives on [0, oo) and A is constant. Definition 2.7.1. Equation (2.7.1) or (2.7.2) is said to be (a) reducible ifC(t) is a solution of a linear nth-order ordinary differential equation with constant coefficients L(y) = a o y ( n ) + a i y ( n - 1 } + + OnV = F(t) (2.7.3) with F continuous on [0, oo) and O

\F(t)dt
/

(2.7A)

Jo (b) y-reducible if it is reducible and if r-t

/>OO

/ / \F(u- s)\duds exists on [0,oo); Jo Jt (c) t-reducible if it is reducible and if

(2.7.5)

/ \tF{t)\dt 0 with

I I \F(u + v)\dudv<M Jo Jo

(2.7.7)

for 0 < t < oo and 0 < t0 < oo; (e) completely reducible if reducible and F{t)=0.

(2.7.8)

66

2. LINEAR EQUATIONS

We have discussed (e) in Section 1.5. The vast majority of stability results for (2.7.2) concern one of the reducible forms. In most cases the results are stated for n = 0, so that one is directly assuming at least one of (2.7.4)-(2.7.7) with F replaced by C, so that (2.7.3) is just L(y) = y = C(t). Of course, when (2.7.1) or (2.7.2) is reducible, then we operate on it using (2.7.3) to obtain a higher-order integro-differential equation with F as the new kernel. Thus, (2.7.4) is the basic assumption for Miller's results on uniform asymptotic stability (U.A.S.); (2.7.5) was used in Theorem 2.5.1 and elsewhere; (2.7.6) is a basic (but unstated) assumption of Brauer (1978) in deriving certain results on U.A.S.; and (2.7.7) was just used in Theorem 2.6.6. We now give examples that will be of interest in later chapters. Example 2.7.1. Let A and a be constants and suppose Jo°° |C"(w)| dv < oo. If we differentiate the scalar equation i-t x'

= Ax+

[a + C(t-s)]x{s)ds,

(2.7.9)

Jo we obtain x" = Ax' +[a + C(0)] x + / C'(t- s)x(s) ds, Jo

(2.7.10)

which we may write as x' = y y'

ft =[a + C(0)] x + Ay + / C'(t - s)x(s) ds , Jo

or in matrix form as

U ) = ( a + C(0) A){y)+JQ(c'(t-s)

Oj{y(s))ds>

which we finally express as ft

X ' = J B X + / D{t - s)X(s) ds , Jo

(2.7.11)

in which D is in L 1 [0,oo) and (2.7.5) is satisfied so that (2.7.9) is Vreducible. Now, it is possible to investigate (2.7.11) by means of Theorem 2.5.1 (and others) because (2.7.9) is not covered by Theorem 2.5.1.

2.7. REDUCIBLE EQUATIONS REVISITED

67

Let us look more closely at (2.7.10). The integral is viewed as a small perturbation of x" - Ax'- [a + C(0)}x = 0

(2.7.12)

because C'(t) is in L^O, oo). Also —A is the coefficient of the "damping," whereas — [a + C(0)] is the coefficient of the "restoring force." From wellestablished theory of ordinary differential equations we expect (2.7.10) to be stable if -A > 0, - [a + C(0)] > 0, and D(t) is small. Example 2.7.2. Consider the scalar equation ft x'

= Ax+

Bln(t- s + a)x(s)ds

(2.7.13)

Jo

with a > 1, A < 0, and b < 0. Differentiate and obtain ft x" = Ax' + b(ln a)x + / b(t - s + a)~1x(s) ds Jo and ft / b(t - s + a)-2x(s) ds.

x'" = Ax" + b(lna)x' + (b/a)x-

Jo l

Now the kernel is L [0, oo) so we express it as a system x' = y

y' = z

ft b(t - s + a)~2x(s) ds ,

z' = (b/a)x + b(\na)y + Az Jo

which may be written as ft X' = BX+ / D(t-s)X(s)ds Jo with D in L1[0, oo). By the Routh-Hurwitz criterion, the characteristic roots of B will have negative real parts if |aj41na| > 1. We expect stability if b is small enough. Exercise 2.7.1. Consider the scalar equation ft = Ax+

b[cos(t-s)](t-s + ay1x(s)ds.

(2.7.14) Jo Can (2.7.14) be reduced to an integro-differential equation with L1 kernel? x'

68

2. LINEAR EQUATIONS

These problems and theorems give us a good start in our understanding of stability. We will see various parts of them again in Chapters 5, 6, 7, and 8. The problems with arc length will become central. The constructions for Liapunov functionals will generalize in a natural way to systems. Chapter 3 will provide details on existence, uniqueness, and continuation which were simply assumed in this introductory chapter.

Chapter 3

Existence Properties 3.1

Definitions, Background, and Review

From our point of view there is a close parallel between the existence theory of ordinary differential equations and that of integral equations. Indeed, ordinary differential equations are frequently converted to integral equations to prove existence results. We hasten to add, however, that some writers consider integral equations under such general conditions that the similarities are lost. Consistent with our aim to make a very gentle transition from differential equations to integral equations, we first state the standard results for ordinary differential equations and briefly sketch the concept of proof as a motivation, a comparison, and, sometimes, a contrast with integral equations. Definition 3.1.1. Let {fn(t)} be a sequence of functions from an interval [a, b] to real numbers. (a) {/«(£)} is uniformly bounded on [a, 6] if there exists M such that n a positive integer and t £ [a, 6] imply \fn(t)\ < M. (b) {/«(£)} is equicontinuous if for any e > 0 there exists S > 0 such that [n a positive integer, t\ £ [a, b], £2 S [a, b], and \t\ — t^\ < S] imply \fn(h) - fn(t2)\ < £ . 69

70

3. EXISTENCE PROPERTIES

Part (b) is sometimes called uniformly equicontinuous. Also, some writers consider a family of functions (possibly uncountable) instead of a sequence. Presumably, one uses the axiom of choice to obtain a sequence from the family. If {/„(£)} is a uniformly bounded and L e m m a 3.1.1. Ascoli-Arzela equicontinuous sequence of real functions on an interval [a, b], then there is a subsequence that converges uniformly on [a, b] to a continuous function. Proof. Because the rational numbers are countable, we may let t\,t2, be a sequence of all rational numbers on a, b] taken in any fixed order. Consider the sequence {/ ra (ti)}. This sequence is bounded, so it contains a convergent subsequence, say, {/^(ii)}, with limit 4>(t\). The sequence {fn(^)} also has a convergent subsequence, say, 1/^(^2)}, with limit 4>{^2)If we continue in this way, we obtain a sequence of sequences (there will be one sequence for each value of m): /"(£),

m=l,2,...;

n= l,2,...,

each of which is a subsequence of all the preceding ones, such that for each m we have f™(trn)

- (tm) a s H - > 00 .

We select the diagonal. That is, consider the sequence of functions Fk{t)

=

fkk{t).

It is a subsequence of the given sequence and is, in fact, a subsequence of each of the sequences {/™(t)} ; for n large. As /™(t m ) —* <j>(tm): it follows that Fk(tm) —> 4>{tm) as k —> oo for each m. We now show that {Fk(t)} converges uniformly on [a, b]. Let E\ > 0 be given, and let e = Si/3. Choose S with the property described in the definition of equicontinuity for the number e. Now, divide the interval [a, b] into p equal parts, where p is any integer larger than (6 — a)/5. Let £j be a rational number in the jih part (j = 1 , . . . ,p); then {Fk(t)} converges at each of these points. Hence, for each j there exists an integer Mj such that \Fr(£j) -Fs(£j)\ < e if r > Mj and s> Mj. Let M be the largest of the numbers Mj.

3.1. DEFINITIONS, BACKGROUND, AND REVIEW

71

If t is in the interval [a, b], it is in one of the p parts, say the jth, so \t - £j\ < 6 and \Fk(t) - Fk(£j)\ < s for every k. Also, if r > M > Mj and s > M, then \Fr(£j) - Fs(£j)\ < e. Hence, if r > M and s > M, then \Fr(t) - Fs(t)\ = \(Fr(t) - Fr(^)) + (Frfo) - F a &)) - (F8(t) - Fs(^))\ < \Fr(t) - F r &)| + |F r &) - Fs(^)\ + \Fs(t) - F8(t:)\ < 3e = £i.

By the Cauchy criterion for uniform convergence, the sequence {Fk(t)} converges uniformly to some function (£). As each Fk(t) is continuous, so is 4>{t). This completes the proof. The lemma is, of course, also true for vector functions. Suppose that {Fn(t)} is a sequence of functions from [a, 6] to i?p, for instance, Fn(t) = (fn(t)i, , fn(t)P)- [The sequence {Fn(t)} is uniformly bounded and equicontinuous if all the {fn(t)j} are.] Pick a uniformly convergent subsequence {fkj(t)i} using the lemma. Consider {fkj(t)2} a n d use the lemma to obtain a uniformly convergent subsequence {fkjr(t)2}- Continue and conclude that {F^jr...s(t)} is uniformly convergent. We are now in a position to state the fundamental existence theorem for the initial-value problem for ordinary differential equations. Theorem 3.1.1. Let (£o,xo) £ Rn+1 and suppose there are positive constants a, b, and M such that D = {(£, x) : \t — to\ < a, |x — xo| < 6}, Then there is G : D -^ Rn is continuous, and |G(i,x)| <Mif{t,x)eD. at least one solution x(t) of x' = G(i,x),

x(i o )=x o ,

and x ( t ) is defined for \t-to\
(3.1.1) with T = min[a, b/M}.

Sketch of Proof. Let j be a fixed positive integer and divide [to, to + T] into j equal subintervals, say, to < t\ < < tn = to + T, with tk+i — tk = Sj = T/j. (We construct the solution only on [to, to + T]; the rest being similar, its construction is left to the reader.) Define a continuous function Xj(t) on [to, to + T] as follows. Let x3(t) =xo + (i-i o )G(io,x o ) for t0 < t < h , and define Xj(ii)

=yi.

72

3. EXISTENCE PROPERTIES

Then Xj(4)=yi + (*-*i)G(*i,yi)

o n

[*i.*2]

and

Xj(i2) =Y2-

We may continue in this way and define a piecewise linear function on [to, to + T]. Because of the properties of M and T, the graph will not leave D. One then shows that {xj(t)} is uniformly bounded and equicontinuous. This follows from the fact that each Xj(t) stays in D and the slopes are bounded by M. Thus, there is a uniformly convergent subsequence. The differential equation is then converted to an integral equation containing the approximating sequence {xj(t)}: ft Xj(i)=x(to)+ / {G(s,xj(s)) + Aj(s)}ds, Jtit

where

By the uniform convergence, the limit may be passed through the integral to show the existence of a solution. This is called the Cauchy-Euler-Peano method, and we shall see that it requires drastic changes for integral equations. Exercise 3.1.1. Given the scalar initial-value problem x' = x1/3,

x(0)=0,

notice that x(t) = 0 satisfies the problem. Separate variables and obtain a second solution. Note that there is a whole "funnel" of solutions to the problem. Nonuniqueness of solutions of the initial-value problem is most easily "cured" by requiring that G satisfy a local Lipschitz condition in the second argument.

3.1. DEFINITIONS, BACKGROUND, AND REVIEW

73

Definition 3.1.2. Let U C Rn+1 and G : U -> Rn. We say that G satisfies a local Lipschitz condition with respect to x if for each compact subset MofU there is a constant K such that (£,x,) in M implies |G(t,xi)-G(t,x2)| < i f | x i - x 2 | . Frequently we need to allow t on only one side of to Exercise 3.1.2. Show that G(t,x) = x1^3 does not satisfy a local Lipschitz condition when U = {{t,x) : -oo < t < oo, | x | < l } . Exercise 3.1.3. Let A be an n x n real constant matrix and G(t,x) = Ax, x e Rn. Show that there is a K > 0 with G(t,xi)-G(t,x2)| < ^ | x i - x 2 | for all x, e Rn. In Chapter 2 (Section 2.1) we stated and proved Gronwall's inequality. It was used to obtain uniqueness of solutions of the linear initial-value problem. We use it here in the same way, and state it (without proof) for convenient reference. Theorem 3.1.2. Let f and g be continuous nonnegative scalar functions on [a, b] and let A > 0. If

f(t)
a
(3.1.2)

Theorem 3.1.3. Let the conditions of Theorem 3.1.1 hold and suppose there is a constant L such that (t,Xj) e D implies |G(t,Xl)-G(t,x2)| < L | x i - x 2 | . Then (3.1.1) has one and only one solution.

74

3. EXISTENCE PROPERTIES

Proof. Theorem 3.1.1 yields one solution. If x(i) and y(i) are two solutions with (i,x(t)) G L> and (t,y(t)) e -D for t0 < * < Ti < t0 + T, then x(i) = xo + / G(s,x(s))cis and y(i) = x o + / G(s,y(s))ds, -'to

so that

ftL\X(s)-y(s)\ds,

|x(i)-y(i)|< Jta

yielding |x(t)-y(4)l <0e L(t -* ()) by Gronwall's inequality. Continual dependence of solutions on initial conditions is easily obtained in exactly the same way. For if x(i) and y(t) are solutions of (3.1.1) on [to,Ti], then |x(i) - y(i)| < |x(i0) - y(*o)| +L f |x(s) - y(s)\ds , so that |x(t)-y(t)|<|x(to)-y(to)|e L ( t - t ( l ) . Under the conditions of Theorem 3.1.3, the existence of solutions may be proved as in Chapter 2 (Theorem 2.1.1) by using Picard's successive approximations. Sketch of Alternative Proof of Theorem 3.1.3 . Write (3.1.1) as ft x(t) = x 0 + / G(s,x(s))ds

,

and define a sequence inductively by xi(i) = x 0 xn+i(i)=x0+/

G(s,xn(s))ds,

n>l.

3.1. DEFINITIONS, BACKGROUND, AND REVIEW

75

Notice that x n + i(t) is the typical partial sum of the series CO

Xl (t) + ]T(x n + 1 (i)-x n (i)) n=l

and that series converges uniformly and absolutely to a function x(t) by comparison with a series for Aek^~tu\ Thus, in the definition of x n +i(i), we take the limit through the integral and obtain x(t)=xo+/ Jtn

G(s,x(s))ds.

This is almost identical to the proof of existence of solutions of linear Volterra equations given in Chapter 2. We shall see that few changes are necessary for nonlinear Volterra equations. The successive approximation technique has been traced to Liouville (1838, p. 19) and, in its general form, to Picard (1890, p. 217). Banach (1932) recognized that Picard's result was actually a fixed point theorem, because the operator

T(x)(t)=xo + y G(s,x(s))ds has a fixed point x(i). A good setting for this is a complete metric space. Definition 3.1.3. A pair (S,p) is a metric space if S is a set and p : S x <S —> [0, oo) such that when y, z, and u are in S then (a) p(y, z) > 0, p(y, y) = 0, and p(y, z) = 0 implies y = z, (b) p(y,z) = p(z,y), and (c) p(y,z) < p(y,u) + p{u,z). The metric space is complete if every Cauchy sequence in (S, p) has a limit in that space. Definition 3.1.4. Let (S,p) be a metric space and A : S —> S. the operator A is a contraction operator if there is an a € (0,1) such that x £ S and y s S imply p[A(x),A(y)}


Theorem 3.1.4. Contractive Mapping Principle Let (S,p) be a complete metric space and A : S —> <S a contraction operator. Then there is a unique (f> <E S with A(i = A(ip) and ipn+i = A(tpn), then tpn —> , the unique fixed point. That is, the equation A() = 4> has one and only one solution.

76

3. EXISTENCE PROPERTIES

Proof. Let XQ G S and define a sequence {xn} in S by xi = A(XQ), X% = A(x{) = A2xo, ,xn = Axn-i = AUXQ. To see that {xn} is a Cauchy sequence, note that if m > n, then p(xn,xm) = p{Anx0,Amx0)
<

anp(x0,xm-n)

< an{p(x0,Xi) +p(xi,X2)-\ < an

o,xi)

= anp(x0,

X1){1

+

\-p{xm-n-i,Xm-n)} +

hara^V(2o,2:i)}

+ a™-™-1}

+a +

oo. Thus, {xn} is a Cauchy sequence, and because (<S, p) is complete, it has a limit x G S. Now A is certainly continuous, so A(x) = A( lim xn) = lim A(xn) = lim xn+i = x, n—^oo

n—>oo

n—^oo

and x is a fixed point. To see that x is unique, consider A(x) = x and A(y) = y. Then p(x, y) = p(A(x), A(y)) < ap(x, y), and because a < 1, we conclude that p(x,y) = 0, so that x = y. This completes the proof. With the contractive mapping principle, the proof of Theorem 3.1.3 becomes very simple, as we shall see in the next section.

3.2

Existence and Uniqueness

Let x, f, and g be n vectors and let ft x(t) = f(t)+

g(t,s,x(s))ds. (3.2.1) Jo Recall from Chapter 1 [see Eq. (1.1.2)] that an integro-differential equation with initial conditions can be put into the form of (3.2.1).

3.2. EXISTENCE AND UNIQUENESS

77

Theorem 3.2.1. Let a, b, and L be positive numbers, and for some fixed a G (0,1) define c = a/L. Suppose (a) f is continuous

on [0,a],

(b) g is continuous on U = {(t,s,x)

: 0 < s < t
(c) g satisfies a Lipschitz condition with respect to x on U |g(M,x)-g(M,y)| < i | x - y | if(t,s,x),

(t,s,y) G U.

IfM = max[/ |g(t, s, x)|, then there is a unique solution of (3.2.1) on [0, T], where T = min[a, b/M, c]. Proof. Let S be the space of continuous functions from [0, T] —> Rn with

||V - f || = ' oma<xr |V(t) - f (t)| < 6 (The norm defines the metric p.) Define an operator A : S —> S by AW(() = f(()+

/ g(t,s,il>(s))ds. Jo

To see that A : <S —> <S notice that ip continuous implies A(if>) continuous and that ||A(^) - f || = Qmaxr |A(^)(t) - f(t)| = max / g(t,s,t/j{s))ds

< MT < b.

To see that A is a contraction mapping, notice that if

an

d ip G S,

s(t,s,if>(s))ds

\s(t,S,(S))\dS ft

< max L / \4>{s) - ip(s)\ ds (s) - ib(s)\ ~ 0
= TLU-^\\
78

3. EXISTENCE PROPERTIES Thus, by Theorem 3.1.4, there is a unique function x e <S with A(x)(t) = x(t) = f(t) + / Jo

g(t,s,x(s))ds.

This completes the proof. The interval [0,T] can be improved by using a different norm. The conclusion can be strengthened to T = min[a,b/M]. On an interval [0,T] one may ask that ||0||=

maxTAe-B'\
for A and B positive constants [see Hale (1969, p. 20)]. Using a continuation result, we shall find that for most theoretical purposes a short interval of existence of a solution is as good as a longer one. There are, however, times when one needs a long interval. Example 3.2.1. Consider a system of ordinary differential equations x' = F(£,x) with F : (—00,00) x Rn —> Rn continuous. Suppose that solutions are unique. Also suppose that there exists T > 0 with F(£ + T, x) = F(i,x) for all (t,x) and that F(—t,x) = —F(t, x) for all (£, x). If for each xo the successive approximations {Xra} defined inductively by X o = x0

and ,-t

s.n-\-\ = x o + / -r \s, j\.n{s)) as Jo converge uniformly on [0, T], then every solution of the initial-value problem J^.

A I V, -A. I ,

J^.\\J I

^*-0

is T-periodic. Proof. Proceed by verifying that Xi is even and T-periodic. Thus, F(i,Xi(t)) is odd and T-periodic. If follows by induction that Xra is even and T-periodic for each n. By the uniform convergence we may pass the limit through the integral and conclude that the solution is T-periodic. This completes the proof.

3.2. EXISTENCE AND UNIQUENESS

79

Technical problems arise if the successive approximations converge on an interval smaller than [0,T]. If we look at the proof of the contraction theorem (Theorem 3.1.4) and our existence result (Theorem 3.2.1), we see that the latter simply uses Picard's successive approximations, which we outline in the sketch of the alternative proof of Theorem 3.1.3 for ordinary differential equations. However, there are interesting differences in the sets over which we are working. For ordinary differential equations we look at the set in Fig. 3.1, whereas for integral equations we look at the set in Fig. 3.2.

Figure 3.1: Set for ordinary differential equations.

Figure 3.2: Set for integral equations.

80

3. EXISTENCE PROPERTIES

For existence without uniqueness, we ask that f and g be continuous and use an integral version of the Cauchy-Euler-Peano method. We emphatically point out that this is not the usual treatment, and it avoids many of the interesting theoretical problems in integral equations. Ordinarily, much less than the continuity of g is required. For such discussions, consult the excellent book by Miller (1971a). There are two reasons for our choice of treatment. First, this is a book on stability and our main methods will require that the integral equation be converted to an integro-differential equation; and for that it is most convenient that g be continuous. The second reason is our wish to present the theory as nearly as possible as if it were ordinary differential equations, so the investigator from that area may parlay existing expertise into a knowledge of Volterra equations in a very direct manner. Theorem 3.2.2. Let f : [0, a] -> Rn and g : U -> Rn both be continuous, where U = {(t, s, x) : 0 < s < t < a and |x - f (t) \ < b} . Then there is a continuous solution of x ( i ) = f ( t ) + I g(t,s,x(s))ds Jo

(3.2.1)

on [0,T], where T = mm[a,b/M] and M = maxy |g(i,s,x)|. Proof. We construct a sequence of continuous functions on [0, T] such that xi(t) = f(t), and if j is a fixed integer, j > 1, then Xj(t) = f(i) when t G [0,T/j] and ft-(T/j)

x,(t) = f(t) + / Jo

g(i - (T/i), S , X j (s)) ds

for T/j 1, then x,-(4) = f(t) on [0,T/j] and rt-(T/j) Xj -(4) = f ( t ) + / Jo

g{t-(T/j),s,f(s))ds

3.2. EXISTENCE AND UNIQUENESS

81

for t G [T/j,2T/j\. At t = 2T/j, the upper limit is T/j, so the integrand was defined on [T/j, 2T/j], thereby defining Xj(t) on [T/j, 2T/j]. Now, for t £ [2T/j, 3T/j] we still have ft-(T/j) Xj (t)

= f(t) + / Jo

g(t-(T/j),S,^(S)) ds,

and the upper limit goes to 2T/j on that interval, so Xj(s) is defined, and hence the integral is well defined. This process is continued to [(j — l)T/j, T], obtaining Xj(t) on [0,T] for each j . A Lipschitz condition will be avoided because Xj(i), j > 1, is independent of all other Xfe(t), Notice that ft-(T/j)

|Xj-(t)-f(*)l< / Jo

\s(t-(T/j),S,^(s))\ds

<M(t-(T/j))<M(b/M) = b.

This sequence {xj(t)} is uniformly bounded because \Xj(t)\<\f(t)\+b<maxT\f(t)\+b.

To see that {xj(t)} is an equicontinuous sequence, let e > 0 be given, let n be an arbitrary integer, let t and v be in [0, T] with t > v, and consider |xn(i)-x»|<|f(i)-f(t;)| pv-(T/n)

+ / [g(t-(T/n),s,x n (s)) Jo -g(v-(T/n),s ) X n (s))]ds )

+

/

S(t-(T/n),s,xn(s))ds

Jv-(T/n)

<

\t(t)-t(v)\+M\t-v\ fV-(T/n)

+

|g(t-(T/n), S)Xn (s)) Jo

-g(v-(T/n),s)Xn(s))|ds. By the uniform continuity of f, there is a <5i > 0 such that \t — v\ < 5\ implies |f(i) -i{v)\ < e/3.

82

3. EXISTENCE PROPERTIES By the uniform continuity of g on U, there is a 82 > 0 such that

\t — v\ < 5-2 implies

T\g(t-(T/n),s,xn(s))

- g ( w - ( T / n ) , s , x n ( s ) ) | < e/3 .

Let 5 = mm[5i,52,£/3M]. This yields equicontinuity. Apply Lemma 3.1.1.

3.3

Continuation of Solutions

Theorems 3.2.1 and 3.2.2 are local existence results. They guarantee a solution on an interval [0, T] with T obtained as follows: (i) f is continuous on [0,a], (ii) g is continuous on U=

{{t,s,x)

:0<s
|f (t) - x | < b} ,

(iii) |g(t, s, x)| < M on U, and (iv) T = min[a,b/M]. Suppose, however, that f is continuous on [0, 00) and g is continuous for 0 < s < t < 00 and all x e Rn. Then the number a drops out of the definition of T and M need not exist. In some way we must manufacture both a and b to apply the existence theorem. We wonder just how large the interval of existence can be made. Presumably, if we take a or b too large, then M becomes so large that b/M is small. To be definite and to start the search for the interval of definition, let a = b = ir and find M. This yields T, and we have at least one well-defined solution, say (f>(t), on [0,T]. Thus, we would like to start a solution at (T, 4>(T)) and apply the existence theorem again. But that theorem starts solutions at t = 0. Thus, we must translate our equation x(t) = f(t)+ / g(t,s,x(s))ds Jo

3.3. CONTINUATION OF SOLUTIONS

83

by letting y(t) = x(t + T) (as in Chapter 1 with initial functions), so that

y(t) = x(i + T) j-t+T

= f(t + T) + / Jo

g(t + T,s,x{s))ds

rT

,-t+T

= {(t + T)+ / g(i + T, s, (f)(s)) ds + Jo JT d =h(t)+ f g(t + T,s + T,y(s))ds Jo

g(t + T, s, x(s)) ds

d

^h(t) + f q(t,s,y(s))ds. Jo The function h : [0, oo) —> Rn is continuous and q is continuous for 0 < s < t < oo and y G Rn. Thus, the existence theorem may be applied to

y(t)=h(t)+

f q(4,s,y(s))ds.

Jo One may take a = b = ir again, but q is a translation of g, so a new M, say, Mi, will be obtained and an interval [0,Ti], with Ti = mm[n,7r/Mi], will result. Any solution y(i) on [0, Ti], translated to the right by y(t — T), so as to be denned on [T, T + T\], will be called a continuation of (p. Naturally, there may be many continuations of on intervals T, Ti, T2,.... With slight modifications the process can be used if f is defined on [0, 00) and g is defined and continuous for 0 < s < t < 00 and x e D, where D is an open and connected subset of Rn. The number b will change each time, becoming the distance from the new f (t) to the boundary of D. Now, there are two possibilities. The intervals T, Xi, T2,..., when translated, may exhaust [0, 00). This would happen if, for example, |g(i, s,x)| is bounded and D = Rn, because M and Mi would be the same. [Is it also true if |g(i, s, x)| < a + /3|x|? If g is continuous for 0 < s < t < 00 and x G D, then x(i) may approach the boundary of D and Tn may approach zero too quickly. If D = Rn, then x(t) may become unbounded as t approaches some number L from the left. One thing is clear, however, from our translation argument: if D = Rn, then bounded solutions can be continued tot = 00, because the bounds Mi on |g| are then bounded on any interval [0,-ftT]. The next result formalizes this.

84

3. EXISTENCE PROPERTIES

Theorem 3.3.1. Let f : [0, oo) -> Rn and g : U -> Rn be continuous where U = {(t, s,x) : 0 < s < t < oo , xe Rn). Ifx(t) is a solution of x(t) = f(t) +

/"*

g(t,s,x(s))ds

(3.3.1)

Jo

on an interval [0, T) and if there is a constant P with \x(t) — f (£)| < P on [0,T), then there is a f > T such that x(t) can be continued to [0,T]. Proof. If we can show that lim t ^j-- x(t) exists, then an application of the existence theorem starting at t = T will complete the proof. Let {tn} be a monotonic increasing sequence with limit T. We shall show that {x(i n )} is a Cauchy sequence. Now |x(tm)-x(tn)| < |f(tm)-f(tn)| I

rt,,

rtm

+ \ / I Jo

g(tm,s,x(s))ds

- / Jo

g(tn,s,x(s))ds

,

and, because f is continuous and tn —> T, the first term on the right tends to zero as n, m —> oo. Also, if tm > tn, then / Jo

g(tm,s,x(s))ds -

/ Jo

g(tn,s,x(s))ds

rt,,.

<

/ Jo

[g(tm,s,x(s))ds

-g(t n ,s,x(s))] ds

I ,*,„ +

/ g(t m ,s,x(s))ds . I Jt,, The function g is uniformly continuous on the set U={(t,s,x):0<s
|x-f(i)|
so for a given e > 0 there exists <5 > 0 such that | ( i m , s , x ) — ( i n , s , x ) | < 5 implies |g(t TO ,s,x) — g(tn,s,x)\ < e. Thus, in the last integral inequality, the jirst term on the right tends to zero as n, m —> oo; because g is bounded on U and tm,tn —> T, the second integral also tends to zero. Hence {x(i n )} is a Cauchy sequence with limit, say x(T), making x(t) continuous on [0, T]. Now, translate by y(t) = x(t + T) and apply the existence theorem again. That will complete the proof. From this it readily follows that / continuous on [0, oo) and g continuous for 0 < s < t < oo and all x e Rn implies that solutions that remain bounded are continuable to all of [0, oo).

3.3. CONTINUATION OF SOLUTIONS

85

There is one important difference in the continuation problem of integral equations and that of differential equations. A solution x(t) of (3.3.1) can be continued to [0, oo) unless there is a T with x(£) denned on [0, T) and limsup t ^ T - |x(i)| = +oo, assuming that f is continuous on [0, oo) and g is continuous for 0 < s < t < oo and x e Rn. But for x' = G(t,x),

(3.3.2)

x(0)=xo

with G continuous on [0, oo) x Rn, a solution x(i) can be continued to all of [0,oo) unless there is a T with x(i) defined on [0, T) and lim t ^ T - |x(i)[ = +oo. The reason for this is well worth explaining. Suppose that x(i) is a solution of (3.3.2) on [0,T) with limsup t ^ T - |x(i)| = +oo, but \x(t)\ -» oo as t —> T~. Then there is an R > 0 and a sequence tn tending monotonically upward to T with |x(i n )| < R. Because limsup t ^ T - |x(t)| = +oo and x(t) is continuous, there is a sequence {5n} of positive numbers with |x(i n + Sn)\ = 2R and |x(t)| < 2R on [tn,tn + 5n}. Now G(t,x.) is continuous on [0,T x | x G Rn : |x| < 21?}, and hence, there is an M with |G(t, x)| < M on that set. Moreover, R< |x(£n + <5 n )-x(t n ) rt,,+S,,

=

/

G(s,x(s))ds

<M5n,

so that Sn > R/M. This implies that T = oo. That is, the speed of \x.{t)\ in moving from R to 2R is bounded by M as the argument of G is confined to a compact set, and hence, at least R/M time units elapse during the journey. But the situation is different for (3.3.1). If we use the same notation with a solution x(t) of (3.3.1), we have

x(t n + Sn) = f{tn + 5n)+ /

g(tn + 5n, S, x(s)) ds

Jo and

x(tn) = i(tn)+

Jo

g(tn,s,x(s))ds,

86

3. EXISTENCE PROPERTIES

so that R< x(i n +<J n )-x(i n )| rt,,+5n

= t(tn + 5n) - f (*„) + / Jo - I g(tn,s,x(s))ds . Jo

g(tn + 5n,s,x(s)) ds

There is no way to exclude the values of x(s) when |x(s)| > 2R from the consideration. Thus, these integrands may become arbitrarily large even though 0 < t < T and |x(s)| < 2R for £„ < s < tn + Sn. There are (very complicated) examples showing that limsup t ^ T - |x(t)| = +oo, but |x(i)| -v> oo as t -^ T~ for various delay equations. See, for example, Herdman (1980) or Myshkis (1951). The ability to look at general classes of integral equations and decide if solutions are continuable to all [0, oo) is indispensable. In the theory of ordinary differential equations one of the primary tools of this investigation is Liapunov functions. It turns out that much less is required of the Liapunov function to prove continuation of solutions than was required in Chapter 2 for boundedness. The following is the main result of this type for ordinary differential equations. If G is fairly nice, then its converse is also true. It can be traced back to Conti, Kato and Strauss, and Yoshizawa. Theorem 3.3.2. Let G : [0, oo) x Rn —> Rn be continuous, and suppose there is a function V : [0, oo) x Rn —> [0, oo) having continuous Erst partial derivatives. If on [0, oo) x Rn we have V'(t,x)=gradV-G

+ dV/dt<0,

(3.3.3)

and if for each T > 0 the relation F(i,x) -> oo as |x| -> oo uniformly for 0 < t < T

(3.3.4)

holds, then every solution of (3.3.2) can be continued to +oo. Proof. If the theorem is false, then there is a solution x(t) of (3.3.2) defined on some interval [to,T) with lim_|x(i)| = +oo.

(3.3.5)

According to (3.3.3), V'(t,x(t)) < 0 so F(t,x(t)) < F(i o ,x(t o )) on [to,T). But by (3.3.4) and (3.3.5) we have V(t,x(t)) -> oo as t -> T~, a contradiction. This completes the proof.

3.3. CONTINUATION OF SOLUTIONS

87

The drawback of this result is, of course, the necessity of finding the function V. We illustrate such a function in the proof of the following classical result known as the Conti-Wintner theorem. Theorem 3.3.3. Let G : [0, oo) x Rn —> Rn be continuous and suppose there are continuous functions X : [0, oo) —> [0, oo) and u : [0, oo) —> [1, oo) with f™[ds/cj(s)] = +oo. If |G(i,x)| < A(i)w(|x|), then any solution of

x' = G(i,x),

x(io)=xo

(3.3.2)'

can be continued to [to,oo). Proof. Let x(i) be a solution of (3.3.2) and define

U

|x|

ft

>

"I

[ds/co(s)} + 1

exp - / A(s) ds . ) Jo (A differentiable norm may be chosen for i ^ O . Because we are concerned with limt_>y- |x(i)| = +oo, we will not be bothered with the nondifferentiability of |x| at 0.) We find that

V(3.3.2)(*»x) < -\(t)V(t,X)+

[|G(t,x)/a;(|x|)]exp [ - jT A(s)ds] r

< -X(t)V(t,

x) + X(t) exp

*

i

- / A(s) ds

L io

< 0.

J

The result now follows from Theorem 3.3.2. When ui is nondecreasing the result may be extended to (3.3.1). Definition 3.3.1. Let h : [0, oo) —> (—00,00) and for U = {(t,s,x)\0

< s
xeR],

let p : U -^ (—00,00). Let x(t) be a continuous solution of the scalar equation x(t)=h(t)+

/ p(t,s,x(s))ds

(3.3.6)

Jo on [0, A] with the property that ify(t) is any other solution, then as long as y(t) exists and t < A we have y(t) < x(t). Then x(t) is called the maximal solution of (3.3.6). Minimal solutions are defined by asking y(t) > x(t).

88

3. EXISTENCE PROPERTIES

Of course, if solutions are unique, then the unique solution is the maximal and minimal solution. Much can be proved concerning maximal solutions, and we shall repeat little of it here. The interested reader may consult Hartman (1964, pp. 2531) for some interesting properties of ordinary differential equations and integral equations. Extensive results of this type are also found in Lakshmikantham and Leela (1969, e.g., pp. 11-31). Theorem 3.3.4. Let the maximal solution x{t) of the scalar equation x{t) = B +

p(s,x(s))ds Jo exist on [0,^4], where B is constant, and letp: [0, A] x R —> R be continuous and nondecreasing in x when 0 < t < A. If y(t) is a continuous scalar function on [0,A] satisfying y(t)
p(s,y(s))ds, Jo theny(t) < x(t) on [0,A]. Proof. Let

yo
*

Y(t) = yo +

p(s,y(s))ds, Jo so that y(t) < Y(t) and Y'(t)=p(t,y(t)) 0. This completes the proof. Unfortunately, the monotonicity in x cannot be dropped for integral inequalities as it can be for differential inequalities. For this reason the Conti-Wintner theorem for integral equations is not as satisfactory as for differential equations. Theorem 3.3.5. Let the conditions of Theorem 3.3.1 hold (f and g are continuous) and suppose that for each T > 0 there is a constant K{T) > 0 and a continuous function LO : [0,oo) —> [l,oo) with to nondecreasing, g(t,s,x)| < K(T)LO(\X\) ifO<s
3.3. CONTINUATION OF SOLUTIONS

89

Proof. Because x(t) is defined on [0, a), take T = a and have |x(i)|<|f(i)|+ / Jo

\g(t,s,x(S))\ds

<M + K(a) I u{\x(s)\) ds, Jo where \f(t)\ < M on [0,a]. Because LJ is monotone, |x(t)| is bounded by the maximal solution of

y = M+ / K(a)uj{y{s))ds, Jo or of the initial-value problem y' = K(a)ij{y),

y{0) = M .

Separating variables yields

rv(t) / [ds/cj(s)]=K(a)t. JM

Because JM [ds/co(s)} = oo, y(a) exists; hence x(a) exists. This completes the proof.

The next result is not as strong as Theorem 3.3.5, but the proof is very simple and u> is specified. Theorem 3.3.6. Let g : [0, oo) x [0, oo) x Rn -> Rn be continuous, and suppose that for each T > 0 there is a continuous scalar function M(s,T) with | g ( M , x ) | < M ( s , T ) ( l + |x|) if 0 < s < t < T. Lett:

be continuous. Ifx(t)

[0,oo) -^ Rn

is a solution of

x(i) = f ( i ) + f g(t,s,x{s))ds Jo

(3.3.7)

on some interval [0, a), then it is bounded, and, hence, is continuable to all of [0,oo).

90

3. EXISTENCE PROPERTIES

Proof. Let x(i) be a solution on [0, a) and let |f(i)| + I Jo

M(s,a)ds
Then for 0 < t < a, we have |x(i)| < |f(*)| + / M(s,a)(l + |x(s)|)ds
r* Jo

M(s,a)\x(x)\ds,

(3.3.8)

so that r * / M(s,a)ds] I Jo by Gronwall's inequality. This completes the proof. |x(i)|
Theorem 3.3.6 showed that if g is small enough, then the sign of g has nothing to do with continuation. The next result shows that if the "signs are right," then the growth of g has nothing to do with continuation. We then show that if g grows too fast and if the "signs are wrong," then solutions cannot always be continued. Examples of the counterparts for differential equations are x' = x

(continuable),

x' = —x3

(continuable),

x1 = x3

(not continuable).

and

Theorem 3.3.7. Let f : [0, oo) -> (-00,00), let C(t,s) be a scalar function defined for 0 < s < t < 00, and let fit), Ct(t,s) and C(t,s) be continuous on their domains. Suppose g : (—00,00) —> (—00,00) with xg(x) > 0 ifx ^ 0, and for each T > 0 we have C(t, t)+J^ \Cu(u, t)\du< 0. Then each solution of x(t) = f(t) + f C(t, s)g(x(s)) ds Jo can be continued for all future times.

(3.3.9)

3.3. CONTINUATION OF SOLUTIONS

91

Proof. We show that if a solution x(t) is defined on [0, a), it is bounded. Let |/'(i)| < M on [0,a] and define

V(t,x(-)) = e-Mt\l

+ \x(t)\+ j I

\Cu(u,S)\du\g(x(s))\ds],

so that V'(t,x(-)) < e-Mt

-M-

M\x\ + \f'(t)\ + C(t,t)\g(x)\

+ f \Ct(t,s)\\g(x(s))\ds + I" \Cu(u,t)\du\g{x)\ Jo Jt

- J*\Ct(t,s)\\g(x(s))\dsj <0. This shows that V, and hence |x(i)|, is bounded on [0,a). This completes the proof. Example 3.3.1. Let C(t, s) = -(t-s Then (3.3.9) becomes x(t)

=el -

ft

(t-s

+

+ I)" 2 , g(x) = x3, and f(t) = e*.

l)-2x3(s)ds.

Jo

We have C(t,t)+ Jt

\Cu(u,t)\du = -l+

2(u-t + l)-3du Jt

=-1 - (u - t + l)-2\^ = - 1 -(T-t+1)'2

+ 1 < 0.

Thus, solutions are continuable by Theorem 3.3.7. It is clear that Theorem 3.3.7 actually concerns an integro-differential equation when we differentiate (3.3.9) and obtain x'(t) = f(t) + C(t, t)g(x) + f Ct(t, s)g(x(s)) ds . ./o Exercise 3.3.1. Generalize (3.3.10) to x'(t) = h(t)+A(t)g(x)+

ft

(3.3.10)

B(t,s)r(x(s))ds. (3.3.11) Jo Examine the statement and proof of Theorem 3.3.7, and place conditions on (3.3.11) so that Theorem 3.3.7 can be stated and proved for (3.3.11).

92

3. EXISTENCE PROPERTIES Now consider the second-order scalar differential equation x" + a(t)f(x) = 0 ,

(3.3.12)

where a : [0, oo) —> (—00,00), / : (—00,00) —> (—00,00), both are continuous, and xf(x) > 0 for x 7^ 0. Write

F{x)= f f{s)ds. Jo

The following is a fundamental continuation result for (3.3.12) that we wish to generalize, in some sense, to encompass integral equations. Theorem 3.3.8. Suppose a(t\) < 0 for some t\ > 0. If either J0+c°[l+F(x)]-^dx<^or

(a)

(b) J-ao[l +

F(x)]-1/2dx>-oo,

then (3.3.12) lias solutions not continuable to +00. Moreover, ifa(t) < 0 on an interval [ti,^), then (3.3.12) has a solution x(t) defined at t\ satisfying limt_>y- \x(t)\ = +00 for some T satisfying t\ < T < £2 if and only if either (a) or (b) holds. Proof. Because a(ti) < 0 and a is continuous, there are positive numbers <5, m, and M such that -M < a(t) < -m < 0 if tx < t < tx + 5. In (3.3.12) let x' = y to obtain the system x' = y,

y'= -a(t)f(x).

(3.3.12)'

Assume that condition (a) holds. Denote by (x(t), y{t)) a solution satisfying x{t\) = 0 with y{t\) large and to be determined. As long as (x(t),y(t)) is defined on [f 1, t\ + 5) we have both x(t) and y(t) monotonically increasing. From (3.3.12)' we have yy' = —a(t)f(x)x', which, upon integration, yields

y2(t)-y\tl)

= -2 f a(S)f(x(S))x'(s)ds. Jt!

Because x(t) is increasing, there is a £ satisfying t\ < t < t such that y\t)-y

rx(t)

2

(tl)

= -2a(i)

f{u)du. Jx(tt)

3.3. CONTINUATION OF SOLUTIONS

93

And, because x{t\) = 0, we obtain y2(t)-y2(t1)

=

-2a(t)F(x(t)),

as long as (x(t),y(t)) is denned and t\
y(i)= so

[y2(h) + 2mF(x(t))]1/2 < y(t) < [y2(h) + 2MF(x(t))]1/2 . Because x' = y, we have

x'(t)>

[y2(t1)+2mF(x(t))]1/2

or [y2(t1) + 2mF(x)y1/2

dx> dt.

Integrating both sides from t\ to t and recalling that x{t\) = 0, we have / Jo

[y2(h)

+ 2mF{s)}

1/2

ds>t-h.

Because (a) holds, we may choose y'2{t\) so large that the integral is smaller than <5. It then follows that x(t) —> co before t reaches t\+8. This completes the proof of the first part of the theorem when (a) holds. If (b) holds, then a similar proof is carried out in Quadrant III of the xy plane. The details showing that the integral can be made smaller than 5 and the proof of the second part of the theorem can be found in Burton and Grimmer (1971). That paper also contains results on the uniqueness of the zero solution that may be extended to integral equations. We return now to our integral equation and show that if a grows too fast and if C(t, s) becomes positive at one point, then there are solutions with finite escape time. It is convenient to introduce an initial function on an initial interval [0,a and show that the solution generated by this initial function has finite escape time. As discussed in Chapter 1, it is possible to translate the equation by y(t) = x(t + a), so that the initial function becomes a forcing function.

94

3. EXISTENCE PROPERTIES

Theorem 3.3.9. Consider the scalar equation x(t) =xo+

[ C(t, s)g(x(s)) ds , Jo

(3.3.13)

where g is continuous and positive for x > 0 and C(t,s) and Ct(t,s) are continuous on 0 < s < t < t\ for t\ > 0. Suppose also (a) there exist e > 0 and CQ > 0 with C(t, s) > CQ if t\ — e < s < t < t\, (b) g(x)/x —> oo as x —* oo, and (c) f™[dx/g(x)] < o o . Then there is a t% G (0, t\) and an initial function : [0, £2] —> [0, 00) sucii thai a solution x(t, <j>) has finite escape time. Proof. Because C(t, s) > CQ if t\ — e < s < t < t\, there is a K > 0 with \Ct(t, s)\ < KC(t, s)foit1-e<s 0 with cog(x) - Kx = f h(x) > 0 for x > R . Note that g(x)/x —> 00 as x —> 00 implies that /i(x) > Mg(x) for some M > 0 and x large. Thus j^[dx/h(x)] < 00, and we may choose Ri > R with f™[dx/h(x)] <e/2. Now, pick xo = i ? i . Define an initial function

be linear from (ti — £,0) t o [t\ — (S/2),XQ). Then as long as t h e solution x(t) = x{t,) exists on t\ — (e/2)
x'(t) = C(t,t)g(x)+ f Ct(t,s)g(x(s))ds Jo >C(t,t)g(x)-K

f C(t,s)g(x(s))ds, Jo

because \Ct(t,s)\ < KC(t,s) for ti - e < s < t < ti. But from (3.3.13), we have *

x(t) — xo = / C(t,s)g(x(s)) ds ,

Jo

so that on t\ — (s/2) < t < t\ we have x'(t)>C(t,t)g(x)-K[x-x0] > cog(x) - Kx = h(x) > 0.

3.4. CONTINUITY OF SOLUTIONS

95

Thus dx/h(x) > dt, so Ads/M*)] > < - ( /

[dz/ft(a:)] = /

J Ri

[ds//i(s)] > / [ds/h(s)}

J xo

J X{)

> * - ( * ! - (e/2)) Thus, if x(t) exists to t = t\, then e/2 > e/2, a contradiction. Roughly speaking, this theorem tells us that if C(t, t) > 0 at some t = ti, if g(x) > 0 for x > 0, and if jx [ds/g(s)\ < oo, then

x(t) = f(t)+ f C(t,s)g(x(s))ds Jo

has solutions with finite escape time. Exercise 3.3.2. Study the statement and proof of Theorem 3.3.9. (a) State and prove a similar result for x{t) =f{t)+

{ Jo

g(t,s,x(s))ds.

(b) Do the same for ft

x' = h(t,x) + / g(t,s,x(s))ds . Jo

3.4

Continuity of Solutions

In Chapter 1 we saw that the innocent-appearing f (t) in x ( i ) = f ( i ) + / g(4,s,x(s))ds Jo

(3.4.1)

may, in fact, be filled with complications. It may contain constants x'(0), x"(0),..., x(™)(0), all of which are arbitrary, or (even worse) a continuous

96

3. EXISTENCE PROPERTIES

initial function 0 : [0, to] —> Rn, where both 0 and to are arbitrary. Recall that for a given initial function 0 we write x(t) = f (t) + / " g(t, s, 0(s)) ds + f g(t, s, x(s)) (is , JO

(3.4.2)

./to

and ask for a solution of the latter equation for t > to. To change it into the form of (3.4.1) we let y(t) = x(t + t 0 ) = f (t + t 0 ) + / g(t +1 0 , s, 0(s)) ds Jo t

+ /

g(to+t,s,x(s))ds,

Jtu

and define h(t) = f(t + t o ) + /

g(t + to,s,0(,s))
(3.4.3)

Then g(to+t,s,x(s))ds ,)

= / g(t0 + t, u + t o , x ( « + t o ))du Jo = / g(t0 +t,s + to,y{s))ds, Jo

and if g(t0 +t,s + t0, y(s)) =f G(t, s, y(s)),

(3.4.4)

then (3.4.2) becomes (3.4.5) y ( t ) = h ( t ) + / G(t,s,y(s))ds, Jo an equation of the form of (3.4.1). In particular, 0 is consolidated with the forcing function f. Consider (3.4.3) and note how h(t) will change as to and 0 change. We shall want to see precisely how a solution x(t) of (3.4.2) will change as to and 0 change. For future reference we note the role of t in the integrand in (3.4.3). Physical problems frequently demand that a solution x(t) of (3.4.1) take into account its past history; thus, the term 0 appears in the integral. However, events of long ago tend to fade from memory, and hence, in (3.4.3) as t —> oo, the term Jo" g(t + to, s, 0(s)) ds may reasonably be expected to tend to zero and, frequently, even be L1[0, oo).

3.4. CONTINUITY OF SOLUTIONS

97

Example 3.4.1. Consider the scalar equation ft

q(t,s)e-^-s)x(s)ds,

x(t) = l+ Jo

where \q(t, s)\ < 1. Let : [0,1] -> [-L, L] for some L > 0. Then

r1 h(t)-l=

q(t +

l,s)e-it+1-s)(s)ds

Jo = e" ( * +1) / q(t + l,s)es
I M * ) - 1 |
= Le-(t+1\e

r1

/ Jo

esds

- 1).

In the literature, theorems on the continual dependence of solutions on initial conditions (and parameters) take many different forms. Our treatment here will be quite brief. Much can be found on the subject in Miller (1971a). Basically, we want to say that if x(i) and y(i) are solutions of x(t) = f(t)+ f g(t,s,x(s))ds Jo

(3.4.1)

and y ( t ) = f i ( t ) + / g(t,s,y(s))ds (3.4.6) Jo on an interval [0, T], with |fi(i) — f(i)| small on [0, T], then for g Lipschitz we also have |x(i) — y(i)| small on [0,T]. It is worthwhile to state the result locally with a local Lipschitz condition, but the details tend to obscure the basic simplicity. Theorem 3.4.1. with U = {(t,s,x)

Let f, fi : [0,a] -> Rn and g : U -> Rn be continuous :0<s
x G Rn} .

Suppose there exists L > 0 with |g(i,s,x) — g(i, s,y)\ < L\x — y| on U. Let x(t) and y(t) be solutions of (3.4.1) and (3.4.6), respectively, on an interval [0,T] and let S = max o < (
for 0 < t < T .

98

3. EXISTENCE PROPERTIES

Proof. From (3.4.1) and (3.4.6) we have |x(t)-y(t)l<|f(*)-fi(*)l+

I

|g(t,s,x(S))-g(t,S,y(5))|dS

Jo

<S + L f |x(s)-y(s)| ds, Jo so the result now follows from Gronwall's inequality. Exercise 3.4.1. State and prove Theorem 3.4.1 using a local Lipschitz condition. Begin with f continuous on [0, a] and g continuous on U = {(i,s,x) : (0 < s 0, say /3 < a. Now let U = {(t,s,x) : 0 < s < t < / 3 , | x - f ( i ) | < b} and let g satisfy a Lipschitz condition on U with constant L. Formulate and prove the local form of Theorem 3.4.1. From Theorem 3.4.1 and the exercise we see that the continuity of f and g, together with a local Lipschitz condition in x on g, yield existence, uniqueness, and continual dependence of solutions on initial conditions. (It is, perhaps, surprising to learn that the uniqueness of solutions implies a property of this general kind.) Before proceeding with integral equations, it is worth noting the standard results for ordinary differential equations. For simplicity we consider only an autonomous equation. Throughout the remainder of this section we assume a strong growth condition on the equation simply to avoid protracted arguments concerning continuation of solutions. The reader will recognize that (3.4.8) can be replaced by a Conti-Wintner equation. We consider the system x' = g(x), n

x(0) = Co ,

(3.4.7)

n

in which g : R —» R is continuous and for some K > 0 we have |g(z)|
allxei?™.

gn =£ F means that g n converges uniformly to F.

(3.4.8)

3.4. CONTINUITY OF SOLUTIONS

99

Theorem 3.4.2. Consider a sequence of functions g^ : Rn —> Rn being continuous and satisfying |gfc(x)| < K(l + |x|) on Rn. Suppose (a) for each compact set B c Rn, then gfe(x) =4 g(x) on B; (b) for each k, xp^it) is a solution of ^' = g f c W0, ^ f c (o) = £fc on [0, oo) with £fc —> £0. Tiien for any T > 0 there is a subsequence kj —> oo with j such that ^fc.(i)^^(i)

on [0,T]as j ^ ex,,

and ^(t) is a solution of €' = g ( € ) , V'(O) = Co Proof. Write (b) as Jo so that on [0, T] we have |^ f c (i)|<|^ f c | + A T + / K\t/>k(s)\ds. Jo By Gronwall's inequality we obtain \ipk(t)\ < (|£fe| +KT)eKT on [0,T]. As ^fc -^ | 0 , then |^ fe (i)| < M for some M, for all k, and for 0 < t < T. Let BM = {€ III < M}, and on BM we have gfe(£) =4 g(£). Because g and the gfc are continuous, there is a Q with |gfc(£)| < Q on B M for all fc, and because ^fe(s) € -BM on [0, T], if £ and ti £ [0,T], then |^ fe (t)-V fe (ii)l = | ^ gfe(Vfe(s))ds


Thus, {V'fe} is a uniformly bounded and equicontinuous sequence on [0, T]. Hence, there is a subsequence tpk. =4 ^(t), for some ip. Therefore gfe,(^ fe ,(0)^g(^(*))

on

[0,T].

As ^fc7-(*)=€fc7.+ / gk:ie oo we obtain ^ ( * ) = £ o + / g(^(«))ds, Jo so that XJ) is some solution of (3.4.7), as required.

100

3. EXISTENCE PROPERTIES

Theorem 3.4.3. Let g : Rn -> Rn be continuous with |g(£)| < K(l +|£|) on Rn. If for each £0 s _Rn the problem

has a unique solution i/)(t,£0), if £k —> £0 and £& —> £o; ^ > 0, then we have ^(tfc,£ fc )^V(*o,£o)That is, uniqueness implies continuity of the solution in all variables. Proof. Let T > 0 be such that 0 < tk < T, and identify gfc(^) with g(£) and tpk(t) with ip(t, £fe). By Theorem 3.4.2, there is a subsequence &;., —> ex) such that ^(*,^)^^(*)

on

[0,T] as j ^ oo.

Because ip{i) is a solution of €' = g(O.

^(o) = € o .

then t/>(£) = ^(t,€o) b y uniqueness. Thus t/>(t,£k.) =t tp(t,£0) on [0,T]. For, by way of contradiction, suppose there were a subsequence for which this were not true. Then by Theorem 3.4.2 there would be a subsequence of that one tending to a solution ip* of £' = f ( £ ) ,

V>*(0)=^

with ip*{t) ^ ip(t,£o). This contradicts uniqueness. Thus ift(t,£k) =4 ^ ( t , | 0 ) on [0,T], so V(tfc,€fc) ^ ^(*o,€o) b e c a u s e ^(*fc,Co) ^ ^(*o,€o) and ijj(t,^0) is continuous in i. This completes the proof. When we set out to formulate a counterpart to Theorem 3.4.2 for x(t) = f(£)+ / Jo

g(t,s,x(s))ds,

it is clear that we want a sequence gk{t, s,x) n£ g(t, s,x) on compact subsets of [0,oo) x [0,oo) x Rn. But f(i) contains the initial conditions, and we therefore desire a sequence of functions ffe(t) — f (t). However, the type of convergence needed is not very clear. The fact that £fc —> £0 in Theorem 3.4.2 is of little help for functions ffc(i). A simple solution is to ask for equicontinuity of {fk} and a form of equicontinuity of {gk(t, s,x)} in t. In particular, if there is a P > 0 with |g/t(£, s,x) — gfe(^i, s, x)| < P\t — ti\ on compact sets, this works very well.

3.4. CONTINUITY OF SOLUTIONS

101

Theorem 3.4.4. Let {gk} be a sequence of continuous functions with gk : [0, a] x [0, a] x Rn —> Rn satisfying \gk(t, s, x)| < K(l + |x|) on its domain. Suppose that {ik} is a sequence of uniformly bounded and equicontinuous functions with ik : [0, a] —> i?" and ffc(t) =1 f(t) on [0,a]. Suppose also (a) for each compact subset B C i?", then gfc(i,s,x) =^ g(t,s,x) on [0,a] x [0,a] x B; (b) for eaci k,ipk{t) is a solution of *jjk(t) = fk(t)+

I Jo

Sk(t,s,il>k(s))ds,

0 < t < a; (c) for each £ > 0 and M > 0, there exists S > 0 such that [fc an integer, s e [0,a], \t - t i | < 6, t , t i G [0,a], \x\ < M] imply

|gfc(t,s,x)

-

g fe (ti,s,x)| < e | t - t i | . Then there is a subsequence kj —> oo with j such that ij}k.(t) =£ tZ'(t) on [0, a] as j —> oo, and tp satisfies V(t) = f(t)+ / g(t,s,iP(s))ds Jo on [0,a]. Proof. If |ffe(i)| < J, then from (b) we have |^ f c (4)l<J+ / K(l + \tj,k(s)\)ds Jo <J + aK + K I \ipk(s)\ds, Jo so that \^k(t)\<(J

+ aK)eaKd=

M

on

[0,o].

Let BM = { ( i , s , x ) : 0 < s < a , 0 < t < a , | x | < M } . Now on BM we have gfe(t, s, x) =t g(t, s, x) Also, by assumption, there is a Q > 0 with

102

3. EXISTENCE PROPERTIES for all k. If t, tx e [0,a] with i > *i, as |i/>fc(s)| < M

|gfc(t,s,x)| < QonBM we have

hM*)-^fe(*i)l = ffc(t) - ffe(ti) + I gk(t,s,if>k(s))dsJo < \ik(t) - ik(h)\ + \J^ +

I

[ Jo

gk(h,s,ipk(s))ds

[gk(t,s,t/>k(s)) - gk(h,s,tl>k(s)j\ ds

Sk(t,s,il>k(s))ds

Jtx

< |ffe(i) - ffc(ti)| + ea\t - h\ + Q\t - h\. Because the ik are equicontinuous, so is {ipp.}- Hence, there is a subsequence of the ipk, say ipk again, with il)k(t) =S i/>(t) on [0, a]. We have t/jk(t) = fk(t)+

I Jo

Sk(t,s,i>(s))ds,

and as k —> oo we obtain

*P(t) = t(t)+ f Jo

g(t,s,Ms))ds,

as required. Notice that if gfc(t, s,x) = g(t, s,x) and if solutions are unique, then the result states that as the initial functions ik(t) converges to f(t), then the solutions converge. That is, solutions depend continuously on initial functions. In short, uniqueness implies continual dependence of solutions on initial conditions. Quite obviously, continual dependence of solutions on initial conditions implies uniqueness.

Chapter 4

History, Examples, and Motivation 4.0

Introduction

This chapter is devoted to a selection of problems and historical events that have affected the development of the subject. Many of the formulations are quite different from the traditional derivations seen in mathematical physics, which proceed from first principles. At least in the early development of the subject, problems were formulated from the descriptive point of view; a physical situation was observed and a mathematical model was constructed that described the observations. The aim was to discover properties from the mathematical model that had not been observed in the physical situation, which could assist the observer in better understanding the outside object. Much of mathematical biology has proceeded in this fashion. And though its critics abound, the successes have been marked and important. An authoritative case for proceeding in this way is made by the eminent biologist J. Maynard Smith (1974, p. 19) in a modern monograph on mathematical biology. In this chapter we briefly discuss numerous problems related, in at least some way, to Volterra equations. In some cases we present substantial results; in other cases we formulate the problems so they may be solved using methods of later chapters; and finally, some problems are briefly introduced as examples to which the general theory applies. In all cases we attempt to provide references, so that the interested reader may pursue the topic in some depth. 103

104

4.1

4. HISTORY, EXAMPLES, AND MOTIVATION

Volterrra and Mathematical Biology

In this section we study the work that went into the formulation of a pair of predator-prey equations x' = x[a — by — dx],

r

r*

i

(4.1.1)

y' = y\ - c+ kx + / K(t - s)x(s) ds I 1 form treated in the general and then transform theseJo equations into the theory discussed in Chapters 5 and 6. The study of Volterra's work on competing species is a fundamental example of the progressive improvement of a model to explain a description of a physical process. It shows, in particular, how a description of observable facts can lead to the suggestion of new information. Fairly accurate records had been kept by Italian port authorities of the ratio of food fish to trash fish (rays, sharks, skates, etc.) netted by Italian fisherman from 1914 to 1923. The period spanned World War I and displayed a very curious and unexpected phenomenon. The proportion of food fish markedly decreased during the war years and then increased to the prewar levels. Fishing was much less intense during the war; it was hazardous, and many fisherman were otherwise occupied. Intuitively, one would think fishing would be much improved after the slow period. Indeed, rare is that person who has not dreamed of the glorious fishing to be had in some virgin mountain lake or stream. The Italian biologist Umberto D'Ancona considered several possible explanations, rejected all of them, and in 1925 consulted the distinguished Italian mathematician Vito Volterra in search of a mathematical model explaining this fishing phenomenon. The problem interested Volterra for the remainder of his life and provided a new setting for his functionals. Moreover, his initial work inspired such widespread interest that by 1940 the literature on the problem was positively enormous. A brief description of his concern with it is quite worthwhile. It was, to begin with, quite clearly a problem of predator and prey. The trash fish fed on the food fish. Moreover, the literature on descriptive growth of species was not at all empty. In 1798 Thomas Robert Malthus, an English economist and historian, published a work (of inordinate title length) contending that a population increases geometrically (e.g., 3, 9, 27, 81, . . . ) whereas food production increases arithmetically (e.g., 3, 6, 9, 12, . . . ) . He contended that population will always tend to a limit of subsistence at which it will be held by

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105

famine, disease, and war. (See Encyclopaedia Britannica, 1970, vol. 14 for a synopsis.) This contention, although far from ludicrous, has been under attack since its publication. One proceeds as follows to formulate a mathematical model of Malthusian growth. Let p(t) denote the population size (or density) at a given time t. If there is unlimited space and food, while the environment does not change, then it seems plausible that the population will increase at a rate proportional to the number of individuals present at a given time. If p{t) is quite large, it may be fruitful to conceive of p(t) as being continuous or even differentiable. (Indeed, the science philosopher Charles S. Peirce (1957, pp. 57-60) contends that the "application of continuity to cases where it does not really exist . . . illustrates . . . the great utility which fictions sometimes have in science." He seems to consider it a cornerstone of scientific progress.) In that case we would say dp(t)/dt = kp(t),

p(to)=po,

(4.1.2)

where k is the constant of proportionality. As it is assumed that the population increases, k > 0. This problem has the unique solution p(t)=poexp[k(t-to)}.

(4.1.3)

Notice that when time is divided into equal intervals, say, years, this does yield a geometric increase. Obviously, no environment could sustain such growth, and by about 1842 the Belgian statistician L. A. J. Quetelet had noticed that a population able to reproduce freely with abundant space and resources tends to increase geometrically, while obstacles tend to slow the growth, causing the population to approach the upper limit, resulting in an S-shaped population curve with a limiting population L (see Fig. 4.1). Such curves had been observed by Edward Wright in 1599 and were called logistic curves, a term still in use. The history of the problem of modeling such a curve mathematically is an interesting one. A colleague of Quetelet, P. F. Verhulst, assumed that the population growth was inhibited by a factor proportional to the square of the population. Thus, the equation for Malthusian growth was modified to p'(t)=kp(t)-rp2(t)

(4.1.4)

for k and r positive. This has become known as the logistic equation and rp2{t) the logistic load. It is a simple Riccati equation, which is equivalent to a second-order, linear differential equation. Its solution is p(t) = m / [ l + Mexp(-fet)],

(4.1.5)

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4. HISTORY, EXAMPLES, AND MOTIVATION

Figure 4.1: S-shaped population curve with limit L. which may be obtained by separation of variables and partial fractions. Its limiting population is m = k/r, called the carrying capacity of the environment. It describes the curve of Fig. 4.1 and, moreover, if p(to) > k/r, it describes a curve of negative slope approaching k/r as t —> oo. Thus, for example, if a fishpond is initially over stocked, the population declines to k/r. With the proper choice of k and r, (4.1.5) describes the growth of many simple populations, such as yeast [see Maynard Smith (1974, p. 18)]. Although the logistic equation is a descriptive statement, it has received several pseudo derivations. The law of mass action states, roughly, that if m molecules of a substance x combine with n molecules of a substance y to form a new substance z, then the rate of reaction is proportional to [x]m[y]ra, where [x] and [y] denote the concentrations of substances x and y, respectively. Thus, one might argue that for population x{t) with density p(t), the members compete with one another for space and resources, and the rate of encounter is proportional to p(t)p(t). So, one postulates that population increases at a rate proportional to the density and decreases at a rate proportional to the square of the density p'(t) = kp(t) - rp2(t). Derivations based on the Taylor series may be found in Pielou (1969, p. 20). One may ask: What is the simplest series representation for p'(t) = f(p),

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107

where / is some function of the population? To answer this question, write f(p) = a + bp + cp2 +

.

First, we must agree that /(0) = 0, as a zero population does not change; hence, a = 0. Next, if the population is to grow for small p, then b must be positive. But if the population is to be self-limiting and if we wish to work with no more than a quadratic, then c must be negative. This yields (4.1.4). Detractors have always argued that the growth of certain populations are S-shaped, and hence, any differential equation having S-shaped solutions with parameters that can be fitted to the situation could be advanced as an authoritative description. Enter Volterra: Let x{t) denote the population of the prey (food fish) and y{t) the population of the predator (trash fish). Because the Mediterranean Sea (actually the upper Adriatic) is large, let us imagine unlimited resources, so that in the absence of predators, x' = ax

a>0,

(4.1.6)

which is Malthusian growth. But x(t) should decrease at a rate proportional to the encounter of prey with predator, yielding x' = ax — bxy ,

a > 0 and b > 0 .

(4-1-7)

Now imagine that, in the absence of prey, the predator population would decrease at a rate proportional to its size y' = -cy,

c > 0.

But y should increase at a rate proportional to its encounters with the prey, yielding y' = -cy + kxy ,

c > 0 and k > 0 .

(4.1.8)

We now have the simplest predator-prey system x' = ax — bxy , (4.1.9) y = -cy + kxy , and we readily reason that a, b, c, and k are positive, with b > k, because y does not have 100% efficiency in converting prey.

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4. HISTORY, EXAMPLES, AND MOTIVATION

Incidentally, (4.1.9) had been independently derived and studied by Lotka (1924) and, hence, is usually called the Lotka-Volterra system. The system may be solved for a first integral as follows. We have dy/dx = (—c + kx)y/(a — by)x ,

(4.1.10)

so that separation of variables and an integration yields (ya/eb*)(xc/ekx)=K,

(4.1.11)

K a constant. The solution curves are difficult to plot, but Volterra (1931; p. 29) [See Davis (1962, p. 103)] devised an ingenious graphical scheme for displaying them. The predator-prey system makes sense only for x > 0 and y > 0. Also, there is an equilibrium point (x' = y' = 0) in the open first quadrant at (x = c/k, y = a/b), which means that populations at that level remain there. May we say that the predator and prey would "live happily ever after" at that level? The entire open first quadrant is then filled with (noncrossing) simple closed curves (corresponding to periodic solutions), all of which encircle the equilibrium point (c/k, a/b) (see Fig. 4.2). We will not go into the details of this complex graph now but a simplifying transformation presented later will enable the reader easily to see the form.

Figure 4.2: Periodic solutions of predator-prey systems.

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We are unable to solve for x(t) and y(t) explicitly, but we may learn much from the paths of the solutions, called orbits, displayed in Fig. 4.2. An orbit that is closed and free of equilibrium points represents a periodic solution. Each of those in Fig. 4.2 may have a different period T. Let us interpret the action taking place during one period. We trace out a solution once in the counterclockwise direction starting near the point (0,0). Because there are few predators, the prey population begins to increase rapidly. This is good for the predators, which now find ample food and begin to multiply, but as the predators increase, they devour the prey, which therefore diminish in number. As the prey decrease, the predators find themselves short on food and lose population rapidly. The cycle continues. Although we cannot find x(t), y(t), nor T, surprisingly, we can find the average value of the population densities over any cycle. The average of a periodic function / over a period T is

/ = ( i / r ) f f(t)dt. Jo From (4.1.9) we have

(1/T) / [x'(t)/x(t)}dt = (l/T) f Jo Jo

[a-by(t)}dt

= (l/T)aT-(b/T)

f y(t)dt; Jo

furthermore (1/T) f [x'(t)/x(t)\ dt = (1/T) In [x(T)/x(0)\ = 0 , Jo because x(T) = x(0). This yields

y = (l/T) f y(t)dt =a/b. Jo A symmetric calculation shows x = c/k. Thus, the coordinates of the equilibrium point (c/k, a/b) are the average populations of any cycle. Notice that statistics on catches would be averages, and those averages are the equilibrium populations. To solve the problem presented to Volterra (in this simple model), we must take into account the effects of fishing. The fishing was by net, so the

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4. HISTORY, EXAMPLES, AND MOTIVATION

densities of x and y are decreased by the same proportional factor, namely, —ex and — sy, respectively. The predator-prey fishing equations become x' = ax — bxy — ex ,

y = -cy + kxy - ey .

(4.1.12)

[The reader should consider and understand why b ^ k, but e is the same in both directions.] The new equilibrium point (or average catch) is /c + e a

-e\

{— In other words, a moderate amount of fishing (a > e) actually diminishes the proportion of predator and increases the proportion of prey. If one believes the model (and not even Volterra did, he continued to refine it), there are far-reaching implications. For example, spraying poison on insects tends to kill many kinds, in the way the net catches many kinds of fish. Would spraying fruit trees increase the average prey density and decrease the average predator density? Here, the prey are leaf and fruit eaters and the predators are the friends of the orchard. The controversy rages, and we will, of course, settle nothing here. Let it be said, however, that elderly orchardists in southern Illinois claim that prior to 1940 they raised highly acceptable fruit crops without spraying. Chemical companies showed them that a little spraying would correct even their small problems. Now they are forced to spray every 3 to 10 days during the growing season to obtain marketable fruit. In a more scientific vein, there is hard evidence that the feared outcome of spraying did occur in the apple orchards of the Wenatchee area of Washington state. There, DDT was used to control the McDaniel spider mite, which attacked leaves, but the spraying more effectively controlled its predator [see Burton (1980b, p. 257)]. We return now to Volterra's problem and consider the effect of logistic loads. Thus we examine x' = ax — dx2 — bxy ,

y = -cy + kxy ,

(4.1.13)

with equilibrium at ( c ak - dc\ def ,_ _, requiring ak > dc, so that it is in the first quadrant.

,. , , ,s

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111

It is easy to see that any solution (x(t), y(t)) in the open first quadrant is bounded, because kx' + by' = kax — kdx2 — bey is negative for x2 + y2 large, x and y positive. Thus, an integration yields kx(t) + by(t) bounded. In fact, one may show that all of these solutions approach the equilibrium point of (4.1.14). To that end, define u = In [x/x]

and

v = In [y/y] ,

(4.1.15)

so that x = xeu

and

y = ye".

(4.1.16)

Then using (4.1.13)-(4.1.16) we obtain u1 = dx(l - eu) + by(l -

ev),

and

(4.1.17) v' = kx(eu - 1).

If we multiply the first equation in (4.1.17) by kx{eu — 1) and the second by by(ev — 1), then adding we obtain kx(eu - l)uf + by{ev - l)v' = -dkx2{eu

- I) 2

or (d/dt) [kx{eu -u)

+ by{ev - v)} < 0 .

Thus the function V(u, v) = kx(eu - u) + by(ev - v) is a Liapunov function. It has a minimum at (0, 0) (by the usual derivative tests), and V(u, v) —> oo as u2+v2 —> oo. As V/4 17\(u, v) < 0, all solutions are bounded. Moreover, if we examine the set in which V'(u,v) = 0, we have eu - 1 = 0, or u = 0. Now, if u = 0, then v' = 0 and u' = —by(ev — 1), which is nonzero unless v = 0. Thus, a solution intersecting u = 0 will leave u = 0

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4. HISTORY, EXAMPLES, AND MOTIVATION

unless v = 0 also. This situation is covered in the work of Barbashin and Krasovskii (see our Section 6.1, Theorems 6.1.4 and 6.1.5). We may conclude that all solutions of (4.1.17) tend to (0,0). But, in view of (4.1.16), all solutions of (4.1.13) approach the equilibrium (x,y) of (4.1.14). [Incidentally, transforming (4.1.9) by (4.1.15) will simplify the graphing problem.] It seems appropriate now to summarize much of this work in the following result. Theorem 4.1.1. Consider (4.1.13) and (4.1.14) with a, b, c, and k positive, ak > dc, and d > 0. (a) If d > 0, then all solutions in Quadrant I approach [x, y). (b) If d = 0, all solutions in Quadrant I are periodic. The mean value of any solution (x(t),y(t)) is (c/k,a/b). The predator-prey-fishing equations become x' = ax — dx2 — bxy — ex ,

(4.1.18)

y' = -cy + kxy - ey , so that the new equilibrium point is / c + £ (a - e)k - d(c + e) \

V k '

bk

)'

Thus, the asymptotic population of prey increases with moderate fishing and the predator population decreases. Much solid scientific work has gone into experimental verification of Volterra's model, with mixed results. A critical summary is given in Goel et al. (1971, pp. 121-124). The next observation is that, although the prey population immediately decreases upon contact with predator, denoted by —bxy, it is clear that the predator population does not immediately increase upon contact with the prey. There is surely a time delay, say, T, required for the predator to utilize the prey. This suggests the system x' = ax — bxy , y< = -Cy + kx(t - T)y(t - T),

(4.1.19)

which is a system of delay differential equations. Actually, (4.1.19) does not seem to have been studied by Volterra, but rather by Wangersky and Cunningham (1957). Yet, the system seems logically to belong here in the successive refinement of the problem given Volterra.

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113

The initial condition for (4.1.19) needs to be a pair of continuous initial functions x(t) = {t - T),

(4.1.20)

which is a linear, first-order equation with the initial condition y(0) = tp(O) having a unique solution, say, rj(t), on [0,T]. Then on [0, T] we have x' = ax - bxi](t),

x(0) = (p(0).

(4.1.21)

An integration yields x(t) = j(t) on [0,T]. Now the new initial function is x(t) = j(i) and y{t) = r](t) on [0,T], so system (4.1.19) can be solved on [T, 2T]. One may continue by quadratures alone and obtain a solution on any interval [0, £]. The more general system x' = ax — dx2 — bxy , y'

= -cy + kx{t - T)y{t - T)

(4.1.22)

may be solved in the same way when we note that x' = ax - dx2 - bxr)(t)

(4.1.23)

is simply a Bernoulli equation. Obviously, the integrations quickly become fierce, and one longs for a qualitative approach. Wangersky and Cunningham (1957) used computer simulation on (4.1.22) with somewhat inconclusive results. (Careful consideration of (4.1.22) under the transformation (4.1.15) may well lead to a Liapunov functional.) The computer simulations indicated that for d = 0 there are large amplitude oscillations. When d > 0, then increasing d tends to stabilize the behavior, whereas increasing T tends to create instability. Before returning to Volterra's own formulation, we point out that several alternative forms of the predator-prey system have been seriously studied. Leslie (1948) suggested the system x' = ax — dx2 — bxy ,

y' = -ey -

{ky2/x).

(4.1.24)

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4. HISTORY, EXAMPLES, AND MOTIVATION

Rosenzweig and Mac Arthur (1963) studied the general system x' = f(x)

-<j>(x,y),

(4.1.25)

y' = -ey + kcj>(x,y)

and conjectured various forms for / and . Later, Rosenzweig (1969) analyzed, in some detail, the biological significance of the shape of the curve for x = 0, f(x) =
(4.1.26)

In the fourth chapter of his book, Volterra (1931) formulates the object of that section of this book. We describe it here in his own notation and with his explanation. He begins with the Lotka-Volterra system written in differential form cWi = s1N1 dt - 7i7Vi7V2 dt,

(4.1.27)

dN2 = -£2N2 dt + -f2N1N2 dt,

with £j > 0 and 73 > 0. Let us assume that, in the second species at least, the distributions by age of the individuals remains constant, and let (£)d£ be the ratio of the number of ages lying between £ and £+
N2(t)

m<%=N2(t)f(e),

/>o,/#o.

Je

Among N2(t) individuals existing at time t, there are then N2f which existed already at the previous T. By the law of mass action, one can see then that the quantity of nourishment in individuals of the first species absorbed during the interval (T, T + dr) by the individuals of the second species who existed at both times T and t, is 7/(t-T)Wi(r)iV 2 (i)dT. We can assume that this nourishment creates an increment of ij}{t - T)dt-yf(t - T^WNzWdT

,

V > 0 , V ^ 0,

individuals of the second species during the interval (t,t + dt), so that, in adding these supposedly independent effects, one obtains

N2(t)dt I J—00

7 V'(4-T)/(t-r)iVi(r)dr.

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We then replace the second equation in (4.1.27) by ft

dN2

= s 2 N 2 d t +N2dt /

F ( t - T ) JVi ( r ) d r ,

F > 0 F^O.

J-co

We then have the system N[ = N^t)^

-

7lN2(t)]

I N^=N2(t)

* -e2 +

1 F(t-T)iVi(r)dT

(4-1.28)

J

J — oo

or the more symmetric system

N[ = L - 7lJV2(t) - /" Fi(t - r)7V2(r) dr] N^t), L

J-oo (4.1.29)

iVa = [ - ^2 + 72iVi(t) + /

F2(t- T)N!(T) drj iV2(t)

with £i,£2)7i)72 > 0, Fi > 0, F2 > 0, and especially 71 > 0 and F2 ^ 0. Volterra emphasizes that these integrals may take the form of

/'

"

/ '

depending on the duration of the heredity. Although the complete stability analysis of the problems formulated by Volterra has not been given, it is enlightening to view some properties of equations of that general type. We might call r x' =x\a-bx+

*

1 K{t,s)x(s)ds

,

(4.1.30)

with a and b positive constants and K continuous for to < s < t < ex) a scalar Verhulst-Volterra equation. Here, to may be —00, in which case we would, of course, write to < s < t < 00. Thus we are taking into account the entire past history of x. Frequently, K(t, s) is discontinuous and the integral is taken in the sense of Stieltjes as described, for example, by Cushing (1976). We shall suppose for the next two results that to ^ 0 and that we have an initial function on [to,O]. Thus, we shall be discussing the solutions for t > 0. We could let to be any value and let the initial function be given on any interval [to,ti], but the historical setting of such problems tends to be of the type chosen here.

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4. HISTORY, EXAMPLES, AND MOTIVATION

Theorem 4.1.2. Let r, R, and m be positive constants with

I \K(t,s)\ds <m to, and suppose that for to < s < 0 we have 0 < r < x(s) < R with a - bR + mR < 0 and a - br - Rm > 0. Then r < x(t) < R for 0
x'(ti)/x{ti)
\K(ti,s)\ds

0. Now suppose x(t\) = r. Then x'{t\)/r > a — br — Rm > 0, a contradiction because we must have x'(t\) < 0. This completes the proof. Roughly speaking, if m is quite small, then solutions are bounded and extinction does not occur. It would be very interesting to learn precisely how such solutions behave. For example, can carrying capacity be defined, and do solutions oscillate around that carrying capacity? Next, we consider the system r x' = x\a — bx — cy I

ft

i K\(t, s)y(s) ds , Jtu J

(4.1.31)

y' = y\-a + (3x+ I K2(t, s)x(s) ds . Theorem 4.1.3. If Ki(t, s) > 0 and continuous for to < s < t < oo, if a, b, c, a, and (3 are positive constants, and if there is an e > 0 with a > (a/b) / K2(t, s)ds + s for t > to , then all solutions remaining in the first quadrant and satisfying x(t) < a/b on the initial interval [to,O] are bounded.

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117

Proof. First notice that x' < 0 if x > a/b, hence, x(t) < a/b if t > toNext, /

fjx' + cy' < Px(a -bx-cy)

\

ft

+ cy[ - a + fix + / K2(t, s)x(s) ds \

i'1

r

/

Jtu

i

< [3x{a - bx) + cy\ -a + (a/b) / K2(t, s) ds < [3x(a — bx) — cey , which is negative if cey > afix — b(5x2 .

Hence, if the line j3x + cy = constant lies above the parabola cey = af3x — bf3x2 ,

then [Px(t) + cy(t)]' < 0 , so that y(t) is bounded. It would be very interesting to obtain information about the qualitative behavior of these bounded solutions. It is our view that one of the real deficiencies of the attempts to analyze (4.1.31) is the absence of a true equilibrium of any type. For example, Cushing (1976) considers the system x[ =x1(a1 r

-cix2) r*

x'2 = X2\ — a,2 + /

] k2(t - s)xi(s)

ds

and speaks of the equilibrium point [a^j \b\ + Jo°° k2(s) ds ] , a\/ci), where b\ comes from another equation. Clearly, x2 = ai/ci, but that value in x'2 does not yield x'2 = 0 for any constant x\. Similarly, (4.1.31) does not have a constant equilibrium solution. It seems that one needs to locate an asymptotic equilibrium and then work on perturbation problems. Volterra's derivation suggests that we consider K\ = 0. Thus, let us consider x' = x[a — dx — by], r

y' = y\ -c + kx+

L

ft

-,

K(t-

^o

(4.1.32)

s)x(s) ds ,

J

in which a, b, c, d, and k are positive constants, K is continuous with Kit) > 0, and Jo°° K(s) ds = r < oo.

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4. HISTORY, EXAMPLES, AND MOTIVATION

To locate an asymptotic equilibrium we write ,t

,t

/ K(t-s)ds= / K(s)ds Jo Jo o

= / Jo def

/-oo

K(s)ds

- / Jt

K(s)ds

,,-.

= r - 7(t). Then we write (4.1.32) as x' = x[a — dx — by] f * 1 y' = y< -c + kx + rx - xj(t) + / Kit - s)[x(s) -x]ds { Jo J

(4.1.33)

where x is denned by —c + kx — rx = 0 or x = c/(k + r). Then a — dx — by = 0 yields y = (a — dx)/b. Let u = In [x/x]

and w = In [t//y] ,

so that (4.1.33) becomes u' = dx(l - eu) + by(l - ev) rt v1 = kx{eu - 1) - xj(t) + / xK(t - s)(eu('s) - 1) ds. Jo Now the linear approximation of this is

(4.1.34)

v! = —dxu — byv , ?t v' = kxu + / xK(t — s)u(s) ds — xj(t), Jo

(4.1.35)

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119

which, in matrix form, is

fu\'=f-dx \v)

-by\fu\

\ kx

0)

\v)

Jo \xi^(i - s) Oj \v(s)J

\-x-y(t)J

or X' = ^ X + / C ( i - s ) X ( s ) d s + T ( i ) ,

(4.1.37)

where all characteristic roots of A have negative real parts, Jo°° |C(s)| ds < oo, and T(t) —> 0 as t -^ oo. Moreover, it is consistent with the problem to ask that roo

/

\l{t)\dt < oo.

(4.1.38)

We then find a matrix B = BT satisfying ATB + BA = —I and form a Liapunov functional for (4.1.37) in the form

F(i,X(-)) = j p ^ B X ] 1 / 2 + K j r \C(u -s)\du \X(s)\ds + 1 j r

x e x p \-L

L

/

]

* |T(S)|

Jo

rfs .

J

These forms are precisely the ones considered in Chapter 6. See the perturbation result Theorem 6.4.5 for both (4.1.37) and (4.1.34). Now return to the nonlinear form (4.1.34) and consider u' = dx{\ - eu) + by(l - ev), v' = kx(eu - 1).

(4.1.39)

The work leading to Theorem 4.1.1 yields uniform asymptotic stability in the large. Under these conditions we shall see in Chapter 6 (Theorem 6.1.6) that there is a Liapunov function W for (4.1.39) that is positive definite and satisfies W'{iAm){u,v)<-cW{u,v),

(4.1.40)

where c > 0. We then show in Chapter 6 (see Theorems 6.4.1-6.4.3 and 6.4.5) how W may be used to show global stability for the nonlinear system (4.1.34).

120

4. HISTORY, EXAMPLES, AND MOTIVATION

In addition, Brauer (1978) has an interesting discussion of such equilibrium questions as raised here. He applies certain linearization techniques of Grossman and Miller to systems of the form of (4.1.34) with (4.1.38) holding.

4.2

Renewal Theory

The renewal equation is an example of an integral equation attracting interest in many areas over a long period of time. An excellent account, given by Bellman and Cooke (1963), contains 41 problems (solved and unsolved) of historical interest. Consider the scalar equation

u(t)=g(t)+ f u(t-s)f(s)ds, Jo

(4.2.1)

where / and g : [0, oo) —> [0, oo) are continuous. Note that we are assuming that / and g are nonnegative. Our discussion here is brief and is taken from the excellent classical paper by Feller (1941), which appeared at an interesting time historically. The work of Volterra in Section 4.1 had been well circulated and had received much attention. Moreover, just two years earlier Lotka (1939) had published a paper containing 74 references to the general questions considered in Section 4.1. The work by Feller represents an attempt to synthesize, simplify, and correct much of the then current investigation. He gives very exact results concerning the behavior of solutions of (4.2.1). This is in contrast to the stability objective of this book. His work is important here in revealing the kinds of behavior one might attempt to prove in qualitative terms. Moreover, he provides two excellent formulations of concrete problems. The entire paper is strongly recommended to the interested reader. Although we make no use of it here, Feller points out that (4.2.1) can be put into another form of particular interest when / is not continuous. We have frequently differentiated an integral equation in order to use the techniques of integro-differential equations. By contrast, one can integrate (4.2.1) and obtain a new integral equation.

4.2. RENEWAL THEORY

121

Define U, F, and G by ft U(t) = / u(s)ds, Jo F(t)=

f Jo

f(s)ds,

and

(4.2.2)

G(t)= f g(s)ds, Jo

so that we may write U(t) = G(t) +

ft Jo

U(t- s) dF(s).

(4.2.3)

The main objective is to study the mean value of u(t), namely, ft u*(t) = (1/t) / u(s)ds. Jo

Equation (4.2.1) has at least two practical applications. The first is Lotka's formulation. In the abstract theory of industrial replacement, each time an individual drops out that individual is replaced by a new one of age zero. (One may think, at times, of light bulbs, for example.) Here f(t) denotes the probability density at the moment of replacement that an individual of age t will drop out. Now let i](s) denote the age distribution of the population at time t = 0. Thus the number of individuals between ages s and s + (Ss) is r)(s)(6s) + O(Ss). Then g(t) denned by 9(t)= I V(s)f(t-s)ds (4.2.4) Jo represents the rate of removal at time t of individuals belonging to the parent population. The function u(t) gives the removal rate at time t of individuals of the total population. Note that each individual dropping out at time t either belonged to the parent population or entered the population by the process of replacement at some time t — s for 0 < s < t. Hence, u(t) satisfies (4.2.1). Because / is a probability density function, we have / f(t)dt=l. Jo

(4.2.5)

122

4. HISTORY, EXAMPLES, AND MOTIVATION

The next formulation is for a single species and is akin to Volterra's own derivation of the predator-prey system of Section 4.1. Let f(t) denote the reproduction rate of females of a certain species at age t. In particular, the average number of females born during a time interval {t,t + {5t)) from a female of age t is f(t)(5t) + O(5t). lir](s) denotes the age distribution of the parent population at t = 0, then Eq. (4.2.4) yields the rate of production of females at time t by members of the parent population. Then u(t) in (4.2.1) measures the rate of female births at time t > 0. This time / is not a probability density function, and we have />OO

/ f(t) dt being any nonnegative number. (4.2.6) Jo This integral is a measure of the population's tendency to increase or decrease. We list without proof some sample results by Feller. Theorem 4.2.1. Suppose that JQ f(t)dt = a and jQ g(t)dt = b, both are finite, / > 0, and g > 0. (a) In order that u*(t) = (1/t) / uis)ds ->c Jo as t —> oo, where c is a, positive constant, it is necessary and sufficient that a = 1 and that Jo°° tfit) dt = m, a finite number. In this case, c = b/m. (b) If a < 1, then /0°° u(t) dt = 6/(1 - a). Notice that according to Theorem 2.6.1, this result concerns uniform asymptotic stability. Theorem 4.2.2. Let Jo°° f(t) dt = 1 and Jo°° g(t) dt = b < oo. Suppose there is an integer n > 2 with O

tkf\t)dt

m k=

for fe = l , 2 , . . . , n

Jo all being Unite and that the

functions

f(t),tf(t),...,tn-2f(t) are of bounded total variation over (0, oo). Suppose also that O

lim tn~2git) = 0 and

lim tn~2 / t

g(s) ds = 0 .

4.3. EXAMPLES

123

Then limbec u(t) = b/m\ and limi™-2[u(t)-(6/mi)] = 0 .

4.3

Examples

In this section, we give a number of examples of physical processes that give rise to integro-differential or integral equations. In most cases the examples are very brief and are accompanied by references, so that the interested reader may pursue them in depth. The main point here is that applications of the general theory are everywhere. If i(t, x) is smooth, then the problem x' = f(£,x),

x(t0) = x 0

has one and only one solution. If that problem is thought to model a given physical situation, then we are postulating that the future behavior depends only on the object's position at time to. Frequently this position is extreme. Physical processes tend to depend very strongly on their past history. The point was made by Picard (1907), in his study of celestial mechanics, that the future of a body in motion depends on its present state (including velocity) and the previous state taken back in time infinitely far. He calls this heredity and points out that students of classical mechanics claim that this is only apparent because too few variables are being considered. A.

Torsion of a Wire

In the same vein, Volterra (1913, pp. 138-139, 150) considered the first approximation relation between the couple of torsion P and the angle of torsion W as W = kP. He claimed that the elastic body had inherited characteristics from the past because of fatigue from previously experienced distortions. His argument was that hereditary effects could be represented by an integral summing the contributions from some to to t, so that the approximation W = kP could be replaced by W{t) = kP{t) + I K(t,s)P{s)ds. He called K(t, s) the coefficient of heredity.

(4.3.1)

124

4. HISTORY, EXAMPLES, AND MOTIVATION

The expression of W is a function of a function, and Volterra had named such expressions "functions of lines." Hadamard suggested the name "functionals," and that name persists. [This problem is also discussed by Davis (1962, p. 112) and by Volterra (1959, p. 147).]

B.

Dynamics

Lagrange's form for the general equations of dynamics is

d or TtW* where qt,-

d(T-n) d^—=Q*> n

(4 3 2)

-'

are the independent coordinates,

i

s

the kinetic energy,

~n= ~ 2 5Z Yl bisqiqs the potential energy, and Qi, , Qn the external forces. See Rutherford (1960) or Volterra (1959, pp. 191-192) for details. When aiS and b{S are constants, then the equations take the linear form s

s

Volterra (1928) shows that in the case of hereditary effects (4.3.3) becomes

J2a^

+

J2bisqs+J2

s

s

s

J

*is(t,r)qa(r)dr = Qi.

(4.3.4)

-°°

If the system has only one degree of freedom, if $ is of convolution type, and if the duration of heredity is T, then the system becomes the single equation

q" + bq+ f

&{s)q(t-s)ds = Q.

(4.3.5)

Jo If we suppose that «&(s) is continuous, nonpositive, increasing, and zero for s > T and if b > 0, then b + fQ 3>(s) ds = m > 0. In this way we may write (4.3.5) as q" + mq-f

&(s)[q(t) - q(t - s)] ds = Q . Jo

(4.3.6)

4.3. EXAMPLES

125

Then \mq2 - \ f z

z

*(s) [q(t) - q(t - s)]2 ds

(4.3.7)

Jo

is called the potential of all forces. Potentials are always important in studying the stability of motion. Frequently a potential function can be used directly as a Liapunov function, an idea going back to Lagrange (well before Liapunov). See Chapter 6 and the discussion surrounding Eq. (6.2.4) for such construction. A suggestion by Volterra (1928) concerning energy enabled Levin (1963) to construct a very superior Liapunov functional. C.

Viscoelasticity

We consider a one-dimensional viscoelastic problem in which the material lies on the interval 0 < x < L and is subjected to a displacement given by u(t,x)

= f(t,x)-x,

(4.3.8)

where / : [0, oo) x [0, L] —> R. If po : [0, L] —> [0, oo) is the initial density function, then, from Newton's law of F = ma, we have ax(t,x)

= [p0(x)}[ftt(t,x)},

(4.3.9)

where a is the stress. For linear viscoelasticity the stress is given by

r*

a(t,x)= / G(t-s,x)uxt(s,x)ds,

(4.3.10)

Jo

where G : [0, oo) x [0,L] —> [0,oo) is the relaxation function and satisfies Gt < 0, Gtt > 0. If we integrate (4.3.10) by parts we obtain a(t,x) = G(0,x)ux(t,x) — G(t,x)ux(0,x) + / Gt(t-s,x)ux(s,x)ds. Jo Because po(x)fu(t,x)

= ax(t,x)

r po(x)ua = \G(0,x)ux(t,x) L

(4.3.11)

it follows that -

* Gt(t- s,x)ux(s,x)ds\ Jo

i . ix

(4.3.12)

126

4. HISTORY, EXAMPLES, AND MOTIVATION

If the material is homogeneous in a certain sense, then we take po(x) = 1 and G to be independent of re, say, G(t,x) = G(t). This yields utt = G{0)uxx(t,

x)+ [ Gt{tJo

s)uxx(s,

x) ds .

(4.3.13)

With well-founded trepidation, one separates variables

u(t,x)

=g(t)h(x)

and obtains (where the overdot indicates d/dt and the prime indicates d/dx for this section only) g(t)h(x) = G(0)g(t)h"(x) + f G(t - s)g(s)h"(x) ds ,

(4.3.14)

Jo

so that h(x)/h"{x)

= \G(0)g(t) + f G(t-

s)g(s) ds 1 /g(t)

(4.3.15)

K a constant (which may need to be negative to satisfy boundary conditions). This yields g(t) = KG{0)g{t) + K

ft . G{t-s)g{s)ds Jo

(4.3.16)

Let g = y, g = z, and obtain

(y\_( \z)

o

A(y\+ft(

\KG{Q) 0) \z)

+

.o

°)(y^)ds

Jo \KG(t - s) o) {z(s)J

as

'

which we write as

X' = AX+ /

Jo

C(t-s)X{s)ds.

If K < 0, then the characteristic roots of A have zero real parts and the stability theory developed in Chapter 2 fails to apply. A detailed discussion of the problem may be found in Bloom (1981, Chapter II, especially pp. 2931, 73-75). Stability analysis was performed by MacCamy (1977b).

4.3. EXAMPLES

D.

127

Electricity

Even the very simplest RLC circuits lead to integro-differential equations. For if a single-loop circuit contains resistance R, capacitance C, and inductance L with impressed voltage E(t), then Kirchhoff's second law yields LI' + RI+(l/C)Q = E(t),

(4.3.17)

with Q = j t I(s) ds. Although this is usually thought to be a trivial integro-differential equation, if E is too rough to be differentiated, then the equation must be treated in its present form, perhaps by Laplace transforms. At the other end of the spectrum, Hopkinson (1877) considers an electromagnetic field in a nonconducting material, where E = (Ei,E2,Es) is the electric field and D = (Di,D-2,Ds) the electric displacement. He uses Maxwell's equations (indeed, the problem was suggested by Maxwell) to write /"* D(i) = eE(t) +

(t- s)E(s) ds,

(4.3.18)

J — oo

where e is constant and

(4.3.19)

with an exponential kernel and F" + G(x)F = 0,

(4.3.20)

a Hill equation with bounded solutions. This problem does not fit into the theory of Chapter 2 because (4.3.20) is never asymptotically stable. However, in certain cases (4.3.19) can be stabilized by the methods of Chapter 5, especially Theorem 5.3.2.

128 E.

4. HISTORY, EXAMPLES, AND MOTIVATION Reactor Dynamics

Levin and Nohel (1960) consider a continuous-medium nuclear reactor with the model du/dt=-

a(x)T(t,x)dx,

(4.3.21)

J — OO

aTt = bTxx + 7]{x)u ,

(4.3.22)

for 0 < t < oo, and satisfying the initial condition «(0) = Mo ,

T(0, x) = f(x),

—oo < x < oo .

Here u(t) and T(t,x) are the unknown functions, a, 77, and / given, realvalued functions, UQ a real constant, and a, b given, positive constants. The various quantities are interpreted as follows: t = time, x = position along the reactor, regarded as a doubly infinite rod, u(t) = logarithm of the total reactor power, T(t, x) = deviation of the temperature from equilibrium, —a(x) = ratio of the temperature coefficient of reactivity to the mean life of neutrons, r](x) = fraction of the power generated at x, a = heat capacity, and b = thermal conductivity. When / , a, and r] are L2[0, 00), then an application of Fourier transform theory [see Miller (1968)] shows that u(t) satisfies ft

u'(t) = -

Jo

mi(t - s)u{s) ds - m2(t),

where O

nij(t) = (1/TT) / Jo

exp[—s2t]hj(s) ds

w(0) = u0 ,

(4.3.23)

4.3. EXAMPLES

129

with /ii(s) = ReT]*(s)a*(-s),

h2(s) =

Rer(s)a*(-s),

and the asterisk being the L2 Fourier transform. Notice that when we can differentiate m\{t) through the integral then m[(t) < 0, m'{(t) > 0, and m'{'(t) < 0. Also, (4.3.23) is linear, so that we can consider the homogeneous form and then use the variation of parameters theorem. F.

Heat Flow

In many of the applications we begin with a partial differential equation and, through simplifying assumptions, arrive at an integral or integrodifferential equation. If one casts the problem in a Hilbert space with unbounded nonlinear operators, then these problems appear to merge into one and the same thing. A particularly pleasing example of the merging of many problems and concepts occurs in the work of MacCamy (1977b) who considers the problem of one-dimensional heat flow in materials with "memory" modeled by ft

ut(t,x) = / a(t - s)ax(ux(s,x))ds °

u(t,0)

+ f(t,x),

0 < x < 1 t>0

= u(t,l)

(4.3.24)

= 0 ,

and

u(0,x) = uo(x). Now (4.3.24) is an example of an integro-differential equation ,-t

u'(t) = -

a(t-s)g(u(s))ds

+ f(t),

Jo

(4.3.25)

M(0) = M0

on a Hilbert space with g a nonlinear unbounded operator. Moreover, (4.3.25) is equivalent to u"(t) + k(0)u'(t)+g(u(t))+

ft Jo

k'{t-s)u'(s)ds

=4>{t)

(4.3.26)

130

4. HISTORY, EXAMPLES, AND MOTIVATION

for some kernel k, and the damped wave equation Utt + OlUt — Cx(ux) u(0,x)

= UQ(X) ,

= 0,

—OO < X < OO , t > 0 ,

(4.3.27) ut(O,x)

= u\(x),

a>0

is a special case of (4.3.26). Finally, the problem of nonlinear viscoelasticity is formally the same as (4.3.26). Thus, we see a merging of the wave equation, the heat equation, viscoelasticity, partial differential equations, and integro-differential equations. The literature is replete with such merging. In Burton (1991) there is a lengthy, detailed, and elementary presentation of the damped wave equation as a Lienard equation. The classical Liapunov functionals for the Lienard equation are parlayed into Liapunov functions for the damped wave equation and corresponding stability results are obtained.

G.

Chemical Oscillations

The Lotka-Volterra equations of Section 4.1 are closely related to certain problems in chemical kinetics. The problem discussed here was also discussed by Prigogine (1967), who gives a linear stability analysis of the resulting equations. Consider an autocatalytic reaction represented by A + X ^ IX , h

X + Y^2Y,

(4.3.28)

h 1

Y^B, h

where the concentrations of A and B do not vary with time. Here, all kinetic constants for the forward reactions are taken as unity and the reverse as h. The reaction rates, Vl=AXV2

hX2 ,

= XY - hY2 ,

v3 = Y -hB

(4.3.29)

4.3. EXAMPLES

131

are based on the law of mass action [see the material in Section 4.1 following Eq. (4.1.5)]. Thus, the differential equations are X' = AX-XY

- hX2 + KY2 , (4.3.30)

Y' = XY - Y - hY2 + hRA.

Note that as h —> 0 we obtain the Lotka-Volterra equations (4.1.9) with a = b = c= k = 1. The total affinity of the reaction is A = - log h3R with

R = B/A .

Although it is difficult to find even the equilibrium point in the open first quadrant for (4.3.30), much can be said about the system. Solutions starting in the open Quadrant I remain there, according to our uniqueness theory. Also, X' + Y' = AX - Y - hX2 + hRA < 0

(4.3.31)

if X2 + Y2 is large. Hence, all solutions starting in the open Quadrant I are bounded. Moreover, if we write (4.3.31) as X' = P and Y' = Q, then (dP/dX) + (dQ/dY) = A-Y

- 2hX - 1 - 2hY < 0

(4.3.32)

in Quadrant I provided that h >-

a n d A<1.

(4.3.33)

By Bendixson's criterion [see Lefschetz (1957, p. 238)], there are no closed orbits in Quadrant I. Because solutions are bounded, one may argue from the Poincare-Bendixson theorem [see Lefschetz (1957, p. 230)] that all solutions in the first quadrant approach the equilibrium point. Prigogine (1967, pp. 122-124) gives a linear discussion of some interest and discusses the case h —> 0, yielding the Lotka-Volterra system X' = AX - XY, (4.3.34) Y' =

XY-Y,

which we recall, has all periodic solutions in Quadrant I. One wonders if for very small h > 0, Eq. (4.3.30) may also possess at least one nontrivial periodic solution in Quadrant I. Also recall that such a solution follows a simple closed curve, along which X(t) and Y(t) alternate between large and small values. Indeed,

132

4. HISTORY, EXAMPLES, AND MOTIVATION

even if a solution (X(t),Y(t)) spirals toward the equilibrium point, then X(t) and Y{t) alternate between larger and smaller values. That is, the concentrations of the products are changing. This may be called chemical oscillation. Some chemical oscillators have attracted enormous interest. The best known such oscillator was discovered by Belousov (1959). It is now referred to as the Belousov-Zhabotinski chemical reaction and consists of a ceriumion-catalyzed oxidation by bromate ion of brominated organic material. The medium is kept well stirred, so that oscillations occur in the ratio of the oxidized and reduced forms of the metal ion catalyst. The oscillations are visible as sharp color changes caused by a redox indicator. A nice discussion, together with differential equations involved, is given by Troy (1980).

Chapter 5

Instability, Stability, and Perturbations 5.1

The Matrix ATB + BA

The following result was obtained in Section 2.4. It is listed here for handy reference. Theorem 5.1.1.

Let A be a real n x n constant matrix. The equation

ATB + BA = -I

(5.1.1)

can be solved for a unique, positive definite symmetric matrix B if and only if all characteristic roots of A have negative real parts.. Thus, given the system of ordinary differential equations x.' = Ax,

(5.1.2)

if the characteristic roots of A all have negative real parts, then one forms the Liapunov function V(x)=x T 5x,

(5.1.3)

finds that V(5.i.2)(x) = - x T x ,

(5.1.4)

and easily concludes uniform asymptotic stability as seen in Chapter 2. 133

134

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Now consider the possibly nonlinear system x' = G(£,x),

(5.1.5)

in which G : [0, oo) x D —> Rn, where D is an open set in Rn and 0 6 B . We suppose that G is continuous and G(t, 0) = 0, so that x(t) = 0 is a solution. As in Chapter 2, we define stability. Definition 5.1.1. The zero solution of (5.1.5) is stable if for each e > 0 and to > 0, there is a S > 0 such that |x o | < 0 and t >t0 imply |x(i,t o ,xo)| < e. We have seen substantial work with stability and Liapunov's direct method in Chapter 2, but the basic stability result for (5.1.5) may be stated as follows. Theorem 5.1.2. Suppose that V : [0, oo) x D —> [0, oo) lias continuous Erst partial derivatives, W : D —> [0, oo) is continuous with W(0) = 0, W(x) > 0 if x ^ 0, V(t,x) > W(x) on D, and V(t,O) = 0. If VL 1 5s(i,x) < 0, then the zero solution of (5.1.5) is stable. Proof. Let e > 0 and to > 0 be given. Assume e so small that |x| < e implies x G D. Because W is continuous on the compact set L = {x G Rn : |x| = e}, W has a positive minimum, say, a, on L. Because V is continuous and V(t,O) = 0, there is a 5 > 0 such that |xo| < 5 implies V(to,xo) to implies W(x(t,to,xo)) < V(t,x(t,to,xo)) < V(to,xo) < a. This implies that |x(t,to: x o)| < £, completing the proof. As previously noted, when G(t, x) = Ax. with all characteristic roots of A having negative real parts, then one chooses B = BT with ATB + BA = —I, and V(t,x) =xTBx = W{x). This condition on A actually implies uniform asymptotic stability. We negate Definition 5.1.1 and obtain the definition of instability.

5.1. THE MATRIX ATB + BA

135

Definition 5.1.2. Tie zero solution of (5.1.5) is unstable if there exists e > 0 and there exists to > 0 such that for any S > 0 there is an xo with |xo| < S and a t\ > to such that \x(ti,to,xo)\ >eNotice that stability requires all solutions starting near zero to stay near zero, but instability calls for the existence of some solutions starting near zero to move well away from zero. Ordinary differential equations frequently have a property that is not seen in most types of functional differential equations. Definition 5.1.3. The zero solution of (5.1.5) is completely unstable if there exists a io > 0 and an e > 0 such that for any 5 > 0 if 0 < |xo| < 8, then there is a t\ > to with x(ii, to,xo)\ > e. Here, t\ depends on e, to, and xo; however, every solution starting near zero (but not at zero) at to moves well away from zero. Theorem 5.1.3. Consider (5.1.5) and suppose there is a function V : D —> (—oo, oo) having continuous Erst partial derivatives and a sequence {x n } converging to zero with V(xn) < 0 and V(0) = 0. Suppose also that there is a continuous function U : D —> [0, oo) with U{x) > 0 if x ^ 0 and V(5.i.5)(*.x)<-tf(x)

on

[0,oo)x£>.

Then the zero solution of (5.1.5) is unstable. Proof. Assume, by way of contradiction, that x = 0 is stable. Choose £ > 0 such that |x| < e implies x e D. Choose to = 0 andfix8 using the definition of stability. Now, select xo with |xo| < S and T^(xo) < 0. Let x(t) = x(t, 0,xo). We shall show that |x(£)| cannot be bounded by e. Because V(x(t)) < 0, we have V(x(t)) a for some a > 0. Also, there is a [3 > 0 with U(x) > (3 if ct <\x\ < e. Because V'(x(£))<-t/(x(i))<-/3, we have V(x(i)) < V(xo) - /3t -+ -oo as i -^ oo, so that |x(i)| < e is impossible. This completes the proof.

136

5. INSTABILITY, STABILITY, AND PERTURBATIONS

One may generalize the result to allow a V(t, x) with V(t, x) > —H(x), where H is continuous and positive definite. If the characteristic roots of A in (5.1.2) all have positive real parts, then the V of Theorem 5.1.3 is easily selected. Theorem 5.1.4. Suppose that the characteristic roots of A all have positive real parts. Then there is a unique, symmetric, positive definite matrix B with ATB + BA = I. Proof. All characteristic roots of —A have negative real parts, so that {-A)TB + B(-A) = -I has a unique, positive definite, symmetric solution matrix B. Thus, ATB + BA = /, as required. The function V for (5.1.2) that satisfies Theorem 5.1.3 may be chosen as V(x) = - x T B x , so that V('5.1.2)(a;) = - x T x = - y ( x ) . Clearly, it is excessive to require that all characteristic roots of A have positive real parts to prove instability. With Theorem 5.1.3 in mind one might attempt to find a matrix B = BT with ATB + BA = —I whenever A has at least one characteristic root with a positive real part. However, such a matrix B will never exist if A has a characteristic root with a zero real part. Theorem 5.1.5. If A has a characteristic root with a zero real part, then ATB + BA = -I has no solution. Proof. Suppose there is a solution B. If A is a characteristic root of A with a zero real part, and if X is a characteristic vector belonging to A, then x(i) = XeAt is a solution of (5.1.2). Because A is real, the real part of x(i) and the imaginary part of x{t) are both solutions of (5.1.2); moreover, at least one of the solutions is nonzero. This means (5.1.2) has at least one solution, say, z(t), that is bounded and bounded away from zero, say, z(i)| > a, for a > 0.

5.1. THE MATRIX ATB + BA

137

If we write V(x) = x T 5x, then V(z(t)) = zT(t)Bz(t), and because z(t) is a solution of (5.1.2), we have V'(z(t)) = -zT(t)z(t)

< -a2

for 0 < t < oo. Thus, V(z(t)) < F(z(0)) - a2t -+ -oo as t —> oo is a contradiction to F(z(i)) being bounded. This completes the proof. Remark 5.1.1. Barbashin(1968) notes that for any given matrix C = CT, the equation ATB + BA = C may be uniquely solved for B = BT provided that the characteristic roots of A are such that Aj + Xk does not vanish for any i and k. To give a unified exposition, we have consistently taken C = —I. However, any negative definite C would suit our purpose, and when A, + \k = 0 , we sometimes must take C ^ —/. That is, if Aj + A^ = 0, then we may still be able to solve ATB + BA = C with C being negative definite; but the solution may not be unique. The Barbashin result is an interesting one. We give three examples and a general construction idea. Example 5.1.1. Let

A=(1

°)

B=(h

b

A

and solve ATB + BA = -I for B. We have

,R, f^h o \ / - i o\ A B + B A = ^ Q _ 2 h ) = { 0 _ : J, / B

so that b\ = — j , 63 = 5, and 62 is not determined. Any choice for 62 will produce a matrix B such that V(x) = x T Bx will satisfy Theorem 5.1.3 for x' = Ax. Thus, B exists for C = —I, but B is not unique.

138

5. INSTABILITY, STABILITY, AND PERTURBATIONS

When B can be determined, but not uniquely, it usually best serves our purpose to make \B\ a minimum. See, for example, Theorems 5.3.1 and 5.3.3. The next example has B unique, and A an unstable matrix. Example 5.1.2. Let AA=

1 h b f*\ BB= f { - 1 2)> [ b 2 6 3AJ '

and solve for B in ATB + BA = -I. We have (-2{b1+b2) [b1+b2-b3

h + b2-b3\ 262 + 463 )

(-1 \Q

0\ -l) '

so that 61 + b2 = \ , hi + 62 = £*3 , and 262 + 463 = - 1 . The determinant of coefficients is 1 1 0

1 0 1 - 1 = 4 - [-2 + 4] ^ 0 , 2 4

so the solution &3 = \ ,

b2 = - § ,

and

61 = 2

is unique. Because Aj + Xj ^ 0, this was predicted by the Barbashin result. If we write x = {x\, x2)T, we find V(x) = xTBx = 2xj ~ 3.T1.X2 + I x\. Along x\ = x2 we have V(x) = — \x\, so that the conditions of Theorem 5.1.3 are satisfied. Our next example shows that there are cases when we must select C other then —/.

5.1. THE MATRIX ATB + BA

139

Example 5.1.3. Consider A

-{0

-l)

'

B

-[b2

bs) '

and try to solve ATB + BA = -I. We obtain ATB + BA-(2bl

h

)

which cannot be —/; however, if b\ = — 1, b2 = 0, and 63 = 1, then

— i.j r x2 '

and

V = -2(x21+x1x2 +xl) S -{Xi

+ x2)

The conditions of Theorem 5.1.3 are satisfied when x2 = (l/n,0). The Barbashin result, stated in Remark 5.1.1, takes care of all matrices A with Xi + \k ^ 0. We know that there is no possible B when A has a characteristic root with a zero real part. However, we do wish to find some V satisfying Theorem 5.1.3 when no root of A has a zero real part. The following procedure shows that it is possible to do so. Some of the details were provided by Prof. L. Hatvani. Let x' = Ax with A real and having no characteristic root with zero real part. Transform A into its Jordan form as follows. Let x = Qy so that x1 = Qy1 = AQy or y' = Q~lAQy. Now

where Pi and P2 are blocks in which all characteristic roots of Pi have positive real parts and all roots of P2 have negative real parts. Let M* denote the conjugate-transpose of a matrix M and define ,0

Bi = - I

(exp P*t)(exp Pit) dt

J — GO

and O

B2 = +

Jo

(exp P*t) (exp P2t) dt.

140

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Notice that -/=(expP*t)(expP 1 t)|° o o = - f

(d/dt) [(exp P1*t)(exp P^)] dt

J — OO

= /

-P*(expP*t)(expPii)dt

./ — OO

- /

(expP1*t)(expPit)Pidt

J-co

= PfB, + BiPi . In the same way, - 7 = +(expP2*t)(expP2t)|~ O

= / Jo

(d/dt) [(exp P2*t) (exp P 2 t)] dt OO

P 2 *(expP 2 *i)(expP 2 i)dt

/

O

+ / Jo

(expP 2 *t)(expP 2 t)P 2 dt

= P2*B2 + B2-P2 Next, form the matrices „

p =

(Pi

0\

(Bx

and B =

( o ftj

0\

(o B J '

and notice that

(PIB1 ^ 0 =

0 \ P1B2)

fP1*B1 + B1P1 ^

0

+

(BXPX ^ 0 0

0 \ BsPs^ A

P2*B2 + B2P2J

5.1. THE MATRIX ATB + BA

141

Thus y' = Q-lAQy = Py, and so V(y) = yTBy yields V'{y) = -yTy. We then have {Q-lAQ)*B + B{Q~lAQ) = -I or Q*AT(Q-1)*B + BQ-1AQ = -I. Now, left multiply by (Q*)^1 and right multiply by Q^1 obtaining AT(Q~1)*BQ-1

+ {Q*)-lBQ~lA

=

-{Q*)-lQ~l

and, because (Q^1)* = (<3*)~\ we have AT(Q-1)*BQ-1

+ ((Q-1)* BQ-1)*A = -(Q^TQ-1

d

= -C.

Now Q~l is nonsingular and x T (Q- 1 )*Q- 1 x=(Q- 1 x)*(Q- 1 x). Because (Q" 1 )*^" 1 is Hermite-symmetric, this quadratic form is real and we have V = xT(Q-1)*BQ-1x with V' = - x T C x = -x T (Q- 1 )*Q- 1 x being negative definite. Thus when A has no characteristic root with a zero real part, then a positive definite matrix C can be found so that ATB + BA = -C can be solved. Remark 5.1.2. We will continue to write ATB + BA = —I for the sake of definiteness, even though we understand that when Xl + X k = 0

then —/ may have to be replaced with a different negative definite matrix.

142

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Naturally, if we wished only to determine stability or instability properties of x' = Ax, we would not bother with a Liapunov function. We have in mind a perturbed system x' = ^x + G(i,x)

that we expect to inherit the stability properties of x' = Ax. Thus, we construct a Liapunov function for x' = Ax, say V(x) = xTBx, and apply it to x' = Ax + G(t,x) for |x| small. This is developed in Chapter 6, Eq. (6.2.27) (Liapunov's Theorem). In the same way, we present this material for x' = Ax, but we shall apply it to r* x' = Ax+ / C(t, s)x(s) ds Jo in later sections. Theorem 5.1.6. Let A be a real matrix, C = CT a negative definite matrix, and B = BT any solution of ATB + BA = C. Then the zero solution of x' = Ax is stable if and only if xTBx > 0 for each i ^ O . Proof. If V(x) = xTBx > 0 for each x ^ 0, then V is positive definite and V'(x) = xTCx < 0, so that stability readily follows. Indeed, x = 0 is uniformly asymptotically stable. Suppose there is some xo ^ 0 with x^Bxo < 0. If X^BXQ = 0, let x(t) = x(i,0,x 0 ) and consider V(x(t)) = xT(t)Bx(t) with V'(x(t)) = xT(t)Cx{t) < 0 at t = 0. Thus, V decreases, so if tx > 0, V(x.{ti)) = xT(ti)Bx(ti) < 0. Thus, we may suppose X^BXQ < 0. If xo = nyn defines yn, then X^BXQ = n2y^Byn < 0, so {yn} converges to zero and V(yn) < 0. All parts of Theorem 5.1.3 are satisfied and x = 0 is unstable. This completes the proof.

5.2

The Scalar Equation

The concept in Theorem 5.1.6 is a key one, and it will be extended to systems of Volterra equations after we lay some groundwork with scalar equations.

5.2. THE SCALAR EQUATION

143

Consider the scalar equation x' = A(t)x +

ft

Jo

C(t,s)x(s)ds

(5.2.1)

with A{t) continuous on [0, oo) and C(t, s) continuous for 0 < s < t < oo. Select a continuous function G(t,s) with dG/dt = C(t,s), so that (5.2.1) may also be written as /"* x'= Q(t)x + (d/dt)

G(t,s)x(s)ds

(5.2.2)

Jo with Q(t)+G(t,t) = A(t). Note that (5.2.1) and (5.2.2) are, in fact, the same equation. For reference we repeat the definition of stability of the zero solution of (5.2.1) and then negate it. Definition 5.2.1. The zero solution (5.2.1) is stable if for each e > 0 and each to > 0 there exists a 5 > 0 such that 4> : [O,io] — R, 4> continuous, \ to imply x(t, to, <j>) < £. Definition 5.2.2. The zero solution of (5.2.1) is unstable if there exists an e > 0 and there exists a to > 0 such that, for each 5 > 0, there is a continuous function cj> : [O,to] —> R with \(j>(t)\ < 5 on [O,io] and with \x(ti,to,(p)\ > s for some ti > toTheorem 5.2.1. Suppose there are constants Mi and Mi, such that for 0 < t < oo we have rt

/>oo

/ \C(t,s)\ds + Jo Jt

\C(u,t)\du

<M1<M2<2\A(t)\.

Then the zero solution of (5.2.1) is stable if and only if A(t) < 0.

(5.2.4)

144

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Proof. First suppose that A(t) < 0. We shall show that x = 0 is stable. Define

V1(t,x(-))=x2 + f f

\C{u,s)\dux2{s)ds,

Jo Jt

so that the derivative of V\ along a solution of (5.2.1) satisfies V{{b.2A)(t1x(-))<2Ax2

+2 f

\C(t,s)\\x(s)x(t)\ds

Jo oo

/

i-t

\C(t,s)\x2(s)ds

\C(u,t)\dux2 Jo ft

2

<2Ax +

Jo

\C(t,s)\

[x (s)+x2(t)]ds 2

oo

/ =

,-t

\C{u,t)\dsx2

2A+ / \C(t,s)\ds Jo

- / \C(t,s)\x2{s)ds Jo + \C(u,t)\du \x2 Jt J

< [2A + Mx\x2 < \-M2 + M{\x2 d

=-ax\ a>0. As V\ is positive definite and V[ < 0, it readily follows that x = 0 is stable. Now suppose that A(t) > 0. (Note that Mx < M2 < 2\A(t)\ implies that A(t) ^ 0). Define \C{u,s)\dux'2{s)ds,

V2(t,x(-)) =x2 - / / Jo Jt so that V2'(521)(t,x(-))>2A(t)x2-2

I

\C(t,s)\\x(s)x(t)\ds

Jo rt

OO

/

\C{u,t)\dux2 + / Jo

\C(t,s)\x2(s)ds

ft

> 2A(t)x2 - / \C(t,s)\ [x2{t) +x2(s)] ds Jo oo

/

i-t

\C(u,t)\dux2+

/ Jo

\C(t,s)\x2(s)ds

5.2. THE SCALAR EQUATION

[

r-t

2A(t)-

145

rCO

/ \C(t,s)\ds Jo

- / Jt

\C(u,t)\du

x2

> [2A(t) - M J ] I 2 > \M2 - Mx\x2 d

^ax2,

a>0.

Now, if x(t) = x(t,to,V2(t,x(-))>V2(t0A(-))+a

/

x2(s)ds.

Jtu

Given any to > 0 and any 5 > 0 we can pick a continuous initial function <j) : [0,t0] -^ -R with V^io, ) > 0. Thus, x2(t) > V2(t0, so that x2(t)>V2(t,x(-)) >V2(t0, = ^(io,

+a /

F2(t0,

) ds

) + aV2(to, (-))(t -

t0),

and so \x(t)\ —> oo as i —> oo. This completes the proof.

Corollary 1. Let (5.2.4) hold, let A(t) be bounded, and let A{t) < 0. Then x = 0 is asymptotically stable. Proof. We showed that Vr1'/521j(t,2;(-)) < —ax2, so we have x2(t) in L1[0, oo) and x2(t) bounded. Note also that x'(t) is bounded. Thus, x(t) —> 0 as t —> oo.

Exercise 5.2.1. Try to eliminate the requirement that J4(£) be bounded from the corollary. In Chapter 6 we develop three ways of doing this. Corollary 2. Let (5.2.4) hold and let A{t) > 0. Then the unbounded solution x(t) produced in the proof of Theorem 5.2.1 satisfies \x(t)\ > c\ + ci(t — to) for c\ and c2 positive.

146

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Theorem 5.2.2. R<2, and

Suppose there are constants J, Qi, Qi, and R with

0 < Qi < \Q{t)\ < Q2 , ft

2\A\+ Jo

(5.2.5) />OO

\C(t,s)\ds + Jt

\C(u,t)\du < J ,

(5.2.6)

and [ Jt

\G(u,t)\du + f \G(t,s)\ds
(5.2.7)

for 0 < t < oo. Suppose also that there is a continuous function h : [0, oo) -> [0, oo) with

\G(t, s)\ < h(t - s) and h(u) - ^ O a s m o o .

(5.2.8)

Then the solution of (5.2.2) is stable if and only if Q(t) < 0. Proof. First suppose that Q(i) < 0 and define V3(t,x(-))=

x- f G{t,s)x(s)ds L Jo

+Q2 [ [ Jo Jt

\G(u,s)\dux2{s)ds,

so that along a solution x(t) of (5.2.2) we have *3(5.2.2) (*> ^(O) =2(X-

G

J

^

S

)

X

^ dS ) Q®X

|G(u,t)|dua; 2 -Q2 / |G(t,s)|2;2(s) ds Jo

+ Q2 / Jt

r-t

< 2Q{t)x2 + Q2 / \G(t, s)\ [x2(s) + x2(t)] ds Jo pt

oo

/

\G{u,t)\dux2 -Q2

= \2Q(t) + Q2(l < [-2Qi + RQi]x2 d

=-f3x2,

/3>0.

\G(t,s)\ds+J

/ \G(t,s)\x2(s)ds Jo

\G(u,t)\du)]x2

5.2. THE SCALAR EQUATION

147

Recall that (5.2.1) and (5.2.2) are the same and consider V\{t, x{-)) once more. It is certainly true that VL52 1 j(t,x(-)) = V-[,52 2Jt,x(-)). Hence, Vj'/g 2 2) m a Y be obtained by taking V;^2A)(t,x(-))<2\A(t)\x2 ft + / \C(t,s)\ [x2{s)+x2{t)]ds Jo O

+

Jt

pt

\C{u,t)\dux2 -

\C(t,s)\x2(s)ds

Jo

= |2|A(t)|+ I \C(t,s)\ds+

\C{u,t)\du]x2

I

< Jx2 . Hence, if we let ^ 1 (i,x(-)) = (/3/2J)T/1(t,x(-)) + T/3(t,x(-)), then we have ^l(5.2.2)(*^(0) < (P/2)X2-PX2 =

-pX2/2.

Because W\ is positive definite, it follows that x = 0 is stable. Now, let Q > 0 and define \G(u,s)\dux2(s)ds,

V4(t,x(-))= (x- I G(t,s)x(s)ds) -Q2 I I V Jo J Jo Jt so that *4(5.2.2) (*- <-)) = 2 U - j

G^'SMS)

ds

) Q(t>

f-t

r-OO

+ Q 2 / \G{t,s)\x'2{s)ds - Q 2 / Jo Jt > 2Q(t)x2 -Q2

\G(u,t)\dux2

I \G(t, s)\ [x2(s) + x2(t)] ds Jo

f-t

f'OO

+ Q 2 / \G{t,s)\x2{s)ds Jo

-Q2

/ Jt

= \2Q(t) - Q2( I \G(t,s)\ds+f > [2Q(t) - RQ^x2 > [2Qi - RQi]x2 = 72; ,

7>0.

\G{u,t)\dux2 \G(u,t)\du\\x2

148

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Now, by way of contradiction, we suppose x = 0 is stable. Thus, given £ > 0 and 0, there is a 5 > 0 such that for any continuous 4> : [0, to] —> R with \)\ < £ for t > t0. For this t0 and this S, we may choose such a cf) with V 4 (io,0(-))>O and let

x{t)=x{t,to,4>). As J o |G(£, s)| ds is bounded and x(t) is bounded, so is JQ G(t, s)x(s) ds. If x2{t) is not in £ 1 [0, oo), then x(t) is unbounded, and we have

[ * ( * ) - / G(t, 3)1(3) ds

>F4(t,x(-)) >F 4 (t o ,0(O)+7 / a;2(s)ds. o

1

2

Hence, we suppose x' {t) is in X [0, 00). Next, note that

< / |G(t,s)|ds / Jo Jo

\G(t,s)\x2(s)ds

by the Schwartz inequality. Moreover, \G(t, s)\ < h(t — s) and h(u) -^ 0 as M -^ 00. Thus,

/ \G{t,s)\x2{s)ds<

Jo

Jo

h{t-s)x2{s)ds,

which is the convolution of an L1 function with a function tending to zero, and hence, the integral tends to zero. We then have x-

I G(t,S)x(s)ds Jo

> [V4(t0,(P(-))]1/2.

As the integral tends to zero, it follows that \x(t)\ > a for large t and some a > 0. This contradicts x2(t) in L1, thereby completing the proof.

5.2. THE SCALAR EQUATION

149

Corollary 1. Suppose that A is constant, C(t,s) = C(t — s), G(t,s) = G(t - s), Jo°° \C(t)\dt < oo, and G(t) = - / t °° C(s) ds. Also suppose that O

\G(u)\du
(5.2.9)

Proof. We verify the conditions of Theorem 5.2.2. Because A is constant and G is of convolution type, it follows that Q is constant. As Q ^ 0, (5.2.5) holds with Q1=Q2 = \Q\. Now ,t

,t

/ \C(t-s)\ds= Jo

/ \C(s)\ds < oo Jo

and />CO

OO

/

\C(u-t)\du=

/ Jo

|C(v)|
so that (5.2.6) holds. Finally, /"CO

OO

/

|G(u-i)|du= / Jo

\G(s)\ds < 1

and

f \G(t-s)\ds=

Jo

f \G(S)\ds<

Jo

Jo

r\G(s)\ds
so that (5.2.7) holds. This completes the proof. Exercise 5.2.2. Formulate a condition from Theorem 5.2.2 ensuring asymptotic stability. Exercise 5.2.3. Theorem 5.2.1 says that if C(t,s) is "small", then the stability of (5.2.1) depends on the sign of A. Theorem 5.2.2 says that if G(t, s) is "small", then the stability of (5.2.2) depends on the sign of Q{t). Can other fundamental quantities be found besides A and Ql Theorems 5.2.1 and 5.2.2 are very closely related. The covering assumption in Theorem 5.2.1 relates to the conclusion in Theorem 5.2.2. A necessary and sufficient condition is strong or weak depending on the mildness or severity of the covering assumptions. One would call Theorem 5.2.1 a strong result, for example, if we could say the following: Suppose A(t) < 0; then the zero solution of (5.2.1) is stable if and only if the

150

5. INSTABILITY, STABILITY, AND PERTURBATIONS

covering assumption (5.2.4) holds. In the present generality, that seems too complicated to check. However, when we check for it under simplifying assumptions, then we see the relation between the two results. To that end, we suppose that A is constant, C(t, s) = C(t — s), C(t) > 0, G(t) = - Jt°° C(v) dv, and O

/

\G(v)\dv
(5.2.10)

Jo This last condition requires the "memory" of the initial conditions to fade, a concept discussed in some detail in Chapters 6 and 8. Then Eq. (5.2.1) becomes

r* x' = Ax+

C(t-s)x(s)ds;

(5.2.11)

Jo (5.2.2) becomes ft x' = Qx+(d/dt)

G{t-s)x{s)ds;

(5.2.12)

Jo (5.2.3) becomes O

Q = A+

C(v)dv;

(5.2.13)

Jo (5.2.4) becomes O

/ Jo

C(v)dv<\A\;

(5.2.14)

and (5.2.7) becomes (5.2.10). We negate (5.2.14) in two steps. Theorem 5.2.3. Let (5.2.10) hold, let C{t) > 0, and let A < 0. If /

Jo

C(v)dv = \A\,

then the zero solution of (5.2.11) is stable. Proof. Write (5.2.11) as (5.2.12). Because A < 0, f™C(v)dv = -A or 0 = A + f™ C{v) dv = Q. Let e > 0 and t0 > 0 be given. Let cj>: [0, t0] -> R

5.2. THE SCALAR EQUATION

151

be a continuous initial function with \4>(t)\ < S on [0, to], where 8 < e is yet to be determined. Integrate (5.2.12) from to to t > to and obtain /.to

i-t

x(t) = x{t0) +

Jo

G{t- s)x{s) ds -

Jo

G(t 0 - s)(f)(s) ds .

If we let M = \x(t0) - /0*° G(t0 - s)
[ Jo

\G{t-s)\\x{s)\ds.

Now, if \x(t)\ < 5 on [0,oo), there is nothing to prove, because we may take 5 = e. So suppose that there is a t\ > to with |.x(ti)| the maximum of \x(t)\ on [0,ti]. Then fix

\x{h)\ <2<5+|a,-(ti)| / Jo <25+\x(h)\

I Jo = 2<5+|x(ti)|P,

\G{tx-s)\ds \G(v)\dv 0
Hence, |x(ti)| [1 - P] < 25 or |x(ti)| < 25/[l - P] < e if 8 < e[l - P]/2. This proves stability. We continue and negate (5.2.14). Theorem 5.2.4. Let (5.2.10) hold, let C{t) > 0, and let A < 0. If O

/ Jo

C(v)dv>\A\,

then Q > 0, so the zero solution of (5.2.11) is unstable. Proof. Because A < 0, /0°° C{v) dv > -A or Q = A + /0°° C(v) dv > 0. Clearly, both (5.2.5) and (5.2.6) are satisfied. Because (5.2.7) reduces to (5.2.10), the conditions of Theorem 5.2.2 are satisfied with Q > 0, so the zero solution is unstable. This completes the proof. Theorem 5.2.5.

G

Let A be constant, A<0,

C s ds and let

C{t) > 0, /0°° C(s) ds < oo,

(0 = - i r '( ) ' rOO

/

\G(v)\dv
(5.2.10)

The zero solution is asymptotically stable if and only if Eq. (5.2.14) holds.

152

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Proof. If (5.2.14) holds, then we do have asymptotic stability by Corollary 1 to Theorem 5.2.1. If Jo°° C(v) dv = \A\ = -A then A + Jo°° C(v) dv = Q = 0. Thus, by (5.2.12), we have upon integration ft x(t) = x0 + / G(t — s)x(s) ds , Jo

x0 > 0.

Because C > 0, G < 0, then as long as x(t) > 0, we have x(t) decreasing and ft x(t) > xo + xo / G(t — s) ds Jo >xo(l+

f

G{s) ds V = f x 0 ( l -a),

a>0.

Hence, x(t) is bounded strictly away from zero. If /0°° C(v) dv > \A\ = -A, then Q = A + Jo°° C(v) dv > 0, so x = 0 is unstable by Theorem 5.2.2. This completes the proof. Exercise 5.2.4. Obtain the counterparts of Theorems 5.2.3 and 5.2.4 under the assumption that A > 0. Exercise 5.2.5. Give an example of a function C{t) > 0 with /0°° C(t) dt > 10 but /0°° \G{v)\ dv < 1. We pass from (5.2.1) to (5.2.2) in an effort to make A and C more tractable; thus, we integrate C to obtain G. We have seen before that the same end may sometimes be accomplished by differentiation. For example, if C'{t) is continuous, then differentiation of (5.2.11) yields ft x" = Ax' + C(0)x + / Jo

C'(t-s)x(s)ds

or the system

X

'=(4)) $X + f*(c'(?-s) o ) X ^ '

which can be treated using the results of the next section.

5.2. THE SCALAR EQUATION

153

The theorems here concerning A and Q are boundary results and an infinite number of results lie in between. Write C(t, s) = Ci(t, s) + C2(t, s) and consider x' = A(t)x+

C1(t,s)x(s)ds Jo

+

C2(t, s)x(s) ds,

(5.2.15)

Jo

so that if G(t, s) now satisfies dG(t, s)/dt = C2(t, s)

(5.2.16)

Q(t) = A(t) - G(t,t),

(5.2.17)

and if

then ,t

x' = Q{t)x +

,t

C1{t,s)x(s)ds

+(d/dt)

G{t,s)x(s)ds Jo Jo is the same as (5.2.15). Thus, we consider the scalar equation C1(t,s)x(s)ds

+(d/dt)

,-t

x' = L(t)x+

(5.2.18)

,-t

H(t,s)x(s)ds, (5.2.19) Jo Jo in which L(t) is continuous for 0 < t < oo, and Ci(t,s) and H(t,s) are continuous for 0 < s < t < oo. Here, L, C\, H, and x are all scalars. We assume that Jt \C\{u,t)\ du is defined for t > 0, and let P = sup / |Ci(t,s)|ds, t>o Jo

(5.2.20)

J = s u p f \H{t,s)\ds,

(5.2.21)

t>0 Jo

and agree that 0 x P = 0. Theorem 5.2.6. and

Suppose that J < 1, J | i ( i ) | < ^Q, for some Q > 0,

/ [10^,3)1+Q\H(t,s)\]ds Jo + / [(l + J)\C1(u,t)\ + (Q + P)\H(u,t)\]du Jt

-2\L(t)\<-a,

for some a > 0 .

(5.2.22)

In addition, suppose there is a continuous function h : [0, oo) —> [0, oo) such that \H(t, s)\ < h(t — s) and h(u) —> 0 as u —> oo. Tien tie zero solution of (5.2.19) is stable if and only if L(i) < 0.

154

5. INSTABILITY, STABILITY, AND PERTURBATIONS

The proof is left as an exercise. It is very similar to earlier ones, except that when using the Schwartz inequality one needs to shift certain functions from one integral to the other. Details may be found in Burton and Mahfoud (1983, 1985). Numerous examples and more exact qualitative behavior are also found in those papers.

5.3

The Vector Equation

We now extend the results of Section 5.2 to systems of Volterra equations and present certain perturbation results. Owing to the greater complexity of systems over scalars, it seems preferable to reduce the generality of A and G. Consider the system x' = ^ x + / C(t,s)x(s)ds,

(5.3.1)

Jo in which A is a constant n x n matrix and C an n x n matrix of functions continuous for 0 < s < t < oo. We suppose there is a symmetric matrix B with ATB = BA= -I.

(5.3.2)

Moreover, we refer the reader to Remark 5.1.2 and to Theorem 5.1.6, which show that (5.3.2) can be replaced by the more general condition that ATB + BA = -C has a solution B for some positive definite matrix C. T h e o r e m 5.3.1. Let (5.3.2) hold and suppose there is a constant M > 0 with \B\[

\ \C(t,s)\ds

\ Jo

+

\C(u,t)\du)

Jt

<M < 1 .

(5.3.3)

J

Then the zero solution of (5.3.1) is stable if and only ifxTBx

> 0 for each

5.3. THE VECTOR EQUATION

155

Proof. We define V1(t,x(-))=xTBx+\B\

/

/

\C(u,s)\dux2(s)ds,

Jo Jt

where x 2 = x T x. Then VH5.3.i)(tM-)) = ^TAT

j\T(s)CT(t,s)ds^BX

+

+ xTB\Ax + / C(t,s)x(s)ds I Jo \C(u,t)\dux2 - I \B\\C{t,s)\x2{s)ds Jo rt

= - x 2 + 2x T B / C(t, s)x{s) ds Jo ft

OO

/

\C(u,t)\dux2 -\B\ / \C(t,s)\x2ds Jo < - x 2 + \B\ I \C(t, s)\ [x2(s) + x2(i)] ds Jo

nt

OO

/

\C{u,t)\duy? -\B\ / \C(t,s)\yi2{s)ds

= -1 + \B\ ( f \C(t,s)\ds + < [-l+M]x2=f-ax2, T

Jo r\C(u,t)\du\]x2

a>0.

Now, if x Bx > 0 for all x / 0 , then V\ is positive definite and V{ is negative definite, so x = 0 is stable. Suppose there is an xo ^ 0 with x^Sxo < 0. Argue as in the proof of Theorem 5.1.6 that there is also an xo with xj£?xo < 0. By way of contradiction, we suppose that x = 0 is stable. Thus, for e = 1 and to = 0, there is a S > 0 such that |xo| < 5 and t > 0 implies |x(i,0,xo)| < 1. We may choose xo with |xo| < S and x^Bxo < 0. Let x(t) = x(i,0, xo) and have Vi(0,x0) < 0 and V{(t,x(-)) < -ax2, so that xT(t)Bx(t)
< Fi(0,x0) -a / x2(s)ds Jo

r-t

= x^Bxo-a

Jo

x2(s)ds.

156

5. INSTABILITY, STABILITY, AND PERTURBATIONS

If there is a sequence {tn} tending to infinity monotonically such that x(tn) —> 0, then xT (tn)Bx{tn) —> 0, and this would contradict xT(t)Bx(t) < x^Bx0 < 0. Thus, there is a 7 > 0 with x2(t) > 7, so that xT(t)Bx(t)

< X^BXQ - ccyt,

which implies that |x(i)| —> 00 as t —> 00. This contradicts |x(i)| < 1 and completes the proof. In Eq. (5.3.1) we suppose C is of convolution type, C(t, s) = C(t — s), and select a matrix G with G'(t) = C(t). Then write (5.3.1) as ft

x' = Qx+{d/dt)

Jo

G{t-s)x(s)ds,

(5.3.4)

where Q + G(0) = A. Note that (5.3.1) and (5.3.4) are the same under the convolution assumption. We suppose there is a constant matrix D with DT = D and QTD + DQ = -I.

(5.3.5)

Refer also to Remark 5.1.2. Theorem 5.3.2. Let (5.3.5) hold and suppose G(t) —> 0 as t —> 00. Suppose also that there are constants N and P with 2\DQ\

\G(t)\dt
<1

(5.3.6)

|C(i)|dil
(5.3.7)

Jo and

2UA\+ f

Then the zero solution of (5.3.4) is stable if and only if xTDx x^O.

> 0 for all

Proof. Define

1 rt i T r r* V2(t,x(-))= x G(t-s)x(s)ds\ D x - / I Jo J L Jo + \DQ\ / / Jo Jt

G(t-s)x(s)ds

\G(u-s)\dux2(s)ds,

5.3. THE VECTOR EQUATION

157

so that V2'(5.3.4)(*.X(-))

r

*

i

= xTQTD x - / G(t - s)yi(s) ds

L

Jo

J

+ [ x - / G(t - s)x(s) ds\ DQx I Jo J /"t

O

+ \DQ\

\G(u - t)\ dux2 -\DQ\ \G{t - s)|x 2 (s)ds Jo

Jt

< - x 2 + |DQ| / |G(t - s)| [x2(t) + x2(s)] ds Jo

/-t

OO

/

\G{u-t)\duy? -\DQ\ / |G(t-s)|x2(s)ds

\G{u-t)\d,u\\*2

= -i + \DQ\( I \G(t-s)\ds + r \G(t)\dt]x2

< -1 + 2\DQ\ I
+ N}x2d=-fix2,

fi>0.

Now consider

\C{u-s)\dux2{s)ds

y3(i,x(-)) = x 2 + / / Jo Jt

and find V^b3 4j by computing

^3(5.3.1) («,x(-))< [ x T ^ T + / r

XT(S)CT(t-S)dS]x *

i

+ x T U x + / C(t - s)x(s) (is

\C(t-s)\x.2(s)ds

\C{u-t)\duy? Jo ft

<2|A|x 2 + / \C{t-s)\ [x2(s)+x2(t)]ds Jo + [ \C{u-t)\duy? - I \C(t-s)\x.2(s)ds Jo Jt <2 \A\+ f

Jo

\C(t)\dt]x2 < P x 2 . J

158

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Define W{t, x(-)) = (li/2P)V3(t, x(-)) + F2(t, x(-)) and obtain W / (5.3.4)(i,x(-))<-W2)x 2 . We now suppose that xTDx > 0 for all x ^ 0, so that Gu/2P)x2 < W(t,x(-)) <W(to,0(-))-(/V2) / x 2 (s)ds Jtu

and conclude that x = 0 is stable. Next, we suppose there is an xo / 0 with x^Dxo < 0, arguing as before that we may suppose X^DXQ < 0. Hence, (kxo)TD(kxo) < 0 for each k > 0, so we may suppose |xo| to be as small as we please. By way of contradiction, we suppose x = 0 is stable. Thus, for e = 1 and to = 0, there is a 5 > 0 such that |xo| < S and t > 0 imply |x(t,O,xo)| < 1. Choose |x o | < 5 and x^Dx0 < 0. Then let x(t) = x(i, 0, XQ), so that V^',5 3 4^(i, x(-)) < —jix2(t). We then conclude that ^ 2 (£,x(-))<^2(0,x 0 )-/i / x2(s)ds Jo rt = x^Dxo — fJ, / x 2 (s) ds pt

Jo

x2(s) ds ,

= —T] — n

T] > 0 .

Thus,

I

x(i) - / G(t-s)x(s)rfs

Jo


J

D x ( i ) - / G(t - s)x(s) ds\

/"*

L

Jo

I x?(s)ds.

J

(5.3.8)

Use the Schwartz inequality to conclude

[J

\G(t-s)\\x(s)\ds < I \G(t-s)\ds

Jo

/ |G(i-s)|x 2 (s)ds.

Jo

(5.3.9)

5.3. THE VECTOR EQUATION

159

As J o \G(t — s)\ ds is bounded and |x(i)| < 1, we have JQ G(t — s)x(s)ds bounded. Thus, in (5.3.8) if x 2 (t) is not in L^OjOo), then x(t) is unbounded. Hence, we suppose x 2 (t) in L1. Now G[t) —> 0 as t —> oo and x 2 in L 1 implies JQ G(t — s)x(s) ds —> 0 as t —> oo. In (5.3.8) we see that for large t, then xT(t)Dx(t) < —rj/2. Moreover, as x —> 0 it follows that xTDx —> 0; hence, we conclude that x 2 (t) > 7 for some 7 > 0 and all large t. This contradicts x2(£) in i 1 . The proof is now complete. Our conditions on C(t, s) have been fairly exacting and one would want the results to hold for equations similar to (5.3.1). Thus, we consider a perturbed form of (5.3.1). Let U be an open set in Rn with 0 G U, let Hi, Hi b e n x n matrices of continuous functions of x, and consider the system x' = Ax + A\ (t)x + Hi (x)x ft

+ / Jo

[C(t,s) + Ci{t,s) + C2{t,s)H2(x(s))]x(s)ds,

(5.3.10)

in which A is a constant n x n matrix, Ai an n x n matrix of functions continuous on [0, 00), and G\ and C2 n x n matrices continuous for 0 < s < t < 00. We suppose there are constants mi, rri2, and J with \Ci{t,s)\<mi\C{t,s)\

and

\Ai{t)\ < m2

(5.3.11)

and |C2(M)|<J|C(M)|

(5.3.12)

for 0 < s < t < 00 and Hi(0)=0,

i = l,2.

(5.3.13)

Finally, we suppose there is a symmetric matrix B with ATB + BA = -I.

(5.3.2)

Relative to (5.3.2), the reader should review Remark 5.1.2. Theorem 5.3.3. Let (5.3.2) and (5.3.10)-(5.3.13) hold. Suppose there is a constant M > 0 with r

B|

,t

/

LJo

1

,00

|C(t,s)|ds+/

Jt

|C(u,i)|du

< M < 1 .

(5.3.3)

J

Then for the rrii sufficiently small, the zero solution of (5.3.10) is stable if and only if xTBx > 0 for each x ^ O .

160

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Note. The details on the size of the m, are as follows. Because the Hi(x) are continuous and Hi(0) = 0, for each r\ > 0 there is a 7 > 0 such that |x| < 7 implies |ilj(x)| < 77. Let r = (mi + Jrf)\B\ and pick r, nii, and 77 small enough that, for some M < 1, we have \B\\2m2 + 2r]+(l+m1+ Jrj) f \C(t,s)\ds I Jo + [l + (r/\B\)] f

\C(u,t)\du\

<M.

Proof of Theorem 5.3.3. Solutions of (5.3.10) with continuous initial functions in U will exist as long as they do not approach the boundary of U. Let D = \B\ +r, r > 0 and to be determined, and define V(t,x(-))=xTBx + D

/ Jo Jt

\C(u,s)\dux2(s)ds

+ *TAJ+XTH? +

= -LTAT

ftXT(S)[CT(t,S)+Cf(t,s)

+ i?J(x(s))Cj(i, s)] ds XBX + xTBlAx

+ A^t)* + i?i(x)x

[C(t, s) + d(t, s) + C2(t, s)if2(x(S))]x2(s) ds |

+J

\C(t,s)x2(s)ds

\C(u,t)\dux2 -D

Jo < - x + 2|Ai(4)| |S|x + 2\H1(x)\ \B\x2 2

2

+ \B\ f [\C(t,S)\ + \C^t,s)\ Jo + \C2(t,s)\ |ff2(x(s))| [x2(s) +x 2 (i)]] ds ft

OO

/

\C(u,t)\dux2 -D

\C(t,s)\yL2(s)ds

Jo < - x 2 + 2\B\m2x2 + 2\B\ |iJi(x)|x 2 + \B\ f v(|C(i,s)|+mi|C(i,s)| Jo + J\C(t, s)\ |i?2(x(s))|) [x2(s) + x2(t)] ds OO

/

/-t

\C{u,t)\duy? - D I Jo

\C(t,s)\x2(s)ds

5.3. THE VECTOR EQUATION

161

= I" - 1 + |B| ^2m2 + 2|Jffi(x)| + f \C(t,s)\ (1 + mi + J\H2(x(s))\) ds^j] x2 \C(t,s)\[l+m1+J\H2(x(s))\}x2(s)ds

+ \B\ f Jo

r-t

OO

/

\C{t,s)\x2{s)ds. Jo Now, Hi(x) are continuous and Hi(O) = 0, so for each rj > 0 there is a 7 > 0 such that |x| < 7 implies |i?j(x)| < 77. Let 77 > 0 be given and find 7 such that as long as |x(i)| < 7 we have \C(u,t)\dux2 -D

V'(t, x(-)) < j - 1 + |B| \2rn2 + 2V+ f \C(t, s)\ (1 + mi + Jrj) ds OO

/

N

\C(u,t)\du | x 2

rt

+ \B\ \C(t,s)\(l+m1 + Jr])x2(s)ds Jo -D

i-t

Jo

\C(t,s)\x2(s)ds.

If r = |B|(mi + Jrj), then

f

r

r*

V"(i, x(-)) < <^ - 1 + |B| 2m2 + 2r? + / |C(i, s)| (1 + nn + Jrj) ds I I Jo \C(u,t)\du jx 2 < { - l + M}x 2 as long as |x(s)| < 7 for 0 < s < t. Suppose that x T Bx > 0 for all x ^ 0, and let e > 0 and to > 0 be given. Assume £ < 7. Because B is positive definite, we may pick d > 0 with dx2 < x T Bx < F(t,x(-)). Then, for £ > 0 and t0 > 0, we can find 5 > 0 such that |<£(i)| < <5 on [0,i0] implies V{to,{-)) < e2 / d. Thus, as long as |x(i)| < 7, then V'{t,x(-)) < 0. And, as long as V'(t,x{-)) < 0, then dx2{t) < xT(t)Bx(t) < V(t,x(-))
<de2,

162

5. INSTABILITY, STABILITY, AND PERTURBATIONS

so that \x(t)\ < e. That is, for |x(t)| < 7, we have V < 0 and for V < 0, we have |x(t)| < e. Because e < 7, it follows that |x(i)| will always remain smaller than 7. Thus, x = 0 is stable. Now, suppose there is an xo ^ 0 with X^BXQ < 0. Argue as before that there is an xo with X^BXQ < 0. By way of contradiction, we suppose x = 0 is stable and pick s = 7. Then take to = 0 and select 5 > 0 for stability. Let xo be chosen with |xo| < 5 and x^Bxo < 0. Argue as in the proof of Theorem 5.3.1 that |x(i,0,xo)| = 7 for some t\ > 0. That will complete the proof. There is another interesting perturbation that we wish to study. Consider the system /"* x' = Ax. + f(t, x) + / C(t,s)x.(s)ds,

(5.3.14)

Jo with A and C as in (5.3.1), ATB + BA = -I

(5.3.2)

(taking into account Remark 5.1.2), f : [0, 00) x Rn —> Rn continuous, and |f(t,x)|
(5.3.15)

where A : [0, oo) —> [0, oo) is continuous, /"GO

A(s) ds < oo

/ Jo

and

X(t) -^ 0 as t -> oo .

(5.3.16)

Theorem 5.3.4. Suppose that (5.3.2), (5.3.3), (5.3.15), and (5.3.16) hold. All solutions of (5.3.14) are bounded if and only ifxTBx > 0 for each x ^ 0 . Proof. In the proof of Theorem 5.3.1 we found V-[,5 a > 0. Select L > 0 so that

3 1-)(i,x(-))

< —ax2,

- a x 2 + 2|S| |x| (|x| + l)A(i) - L\(t) < -ax2 , a > 0, for all x when t is large enough, say, t > S. Next, define V(t,x(-)) r

=

xTBx + l + |5| /

L

r-t /.oo

/

Jo Jt

|C(M,s)|dMx2(s)ds

I f /

J

1

*

e x p \-L

L

1

/ A(s)ds

io

J

,

5.4. COMPLETE INSTABILITY

163

so that V(' 5 .3.14)(*>X(-))

< -L\(t)V

+ exp [ - J* LX(s) ds ] [V;(531) (t, x(-)) +21511x1 |f (i, x)|]

<-LX(t)V

+ exp\-L

L r <exp

I

[ - a x 2 + 2\B\ |x|A(i)(|x| + 1)]

\{s)ds

Jo

J

* i - L / A(s)ds [ - a x 2 + 2|B| |x|A(t)(|x| + 1) - LX(t)]

Jo

J

< -/3x 2 , /? > 0 if t > S. Suppose that x^Bxo > 0 for all xo =/= 0. If x(i) is any solution of (5.3.14), then, by the growth condition on f, it can be continued for all future time. Hence, for t > S we have V(t,x.(t)) < V(S,x.(-)), so that x(i) is bounded. Suppose that x^Bxo < 0 for some xo =/= 0. Because B is independent of f, one may argue that x^Bxo < 0 for some xo. Pick to = S and select cj> on [0,t0] with V{to,(f>) < 0. Then F'(t,x(-)) < 0 implies V(t,x(-)) < V(to, 0(-))- One may argue that |x(t)| is bounded strictly away from zero, say, |x(i)| > /i, for some fi > 0. As X(t) -* 0, if t > S, then Vr/(t,x(-)) < -/3x 2 , so y'(i,x(-)) < - / V for large t. Thus, V(t,x(-)) -> -oo and x(i) is unbounded. This completes the proof. Exercise 5.3.1. Review the proof and notice that when B is positive definite one may take L so large that the condition A(i) —> 0 is not needed. Exercise 5.3.2. Formulate and prove Theorems 5.3.3 and 5.3.4 for Eq. (5.3.4)

5.4

Complete Instability

In this section we focus on three facts relative to instability of Volterra equations. First, when all characteristic roots of A have positive real parts, then an instability result analogous to Theorem 5.3.1 may be obtained by integrating only one coordinate of C(t,s), as opposed to integrating both coordinates in (5.3.3). Next, we point out the existence of complete instability in this case. Indeed, this also could have been done in Section 5.2 or 5.3. Finally, we note that Volterra equations have a solution space that is far simpler than one might expect. Generally, complete instability is impossible for functional differential equations.

164

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Consider the system /"* x' = Dx+ / C{t,s)x{s)ds, (5.4.1) Jo where D is an n x n constant matrix whose characteristic roots all have positive real parts. Find the unique matrix L = LT that is positive definite and satisfies DTL + LD = I,

(5.4.2)

and find positive constants m and M with x| > 2m(x T Lx) 1 / 2

(5.4.3)

and |ix| < M(xT£x)1/2

.

(5.4.4)

Theorem 5.4.1. Let (5.4.2)-(5.4.4) hold and let f™ \C(u,s)\du be continuous. Suppose there is an H > M and 7 > 0 with m — H/f°° \C(u,t)\du > 7. Then each solution x(i) of (5.4.1) on [0,oo) with x(0) 7^ 0 satisfies |x(t)| > ci + C2t for 0 < t < 00, where ci and C2 are positive constants depending on x(0). AJso, if to > 0, then, for each S > 0, there is a continuous initial function cf) : [0,to] —> Rn with \4>{t)\ < 8 and x(t, to, 0 ) | > c\ + C2(t — to) for to < t < 00, where c\ and c-i are positive constants depending on (p and to Proof. Let H > 0, define V(t,x{-)) = (x T Lx) 1 / 2 -i7 / / Jo Jt

\C(u,s)\du\x(s)\ds,

and for x / 0 , obtain V('5.4.1)(*.X(-))

= I \*TDT + f xT(s)CT(t, s) dsj i x + x T i ii)x + T C(t, s)x(s) dsj I /{2(xTix)1/2} ,00

-iJ/ it

,t

|C(u,t)|du|x| +iJ / |C(t,s)||x(s)|rfs io

5.4. COMPLETE INSTABILITY

165

> {x T x/2(x T Lx) 1/2 } {Xr(t)LC(t,s)X(s)/[xr(t)LX{t)}1/2}ds

+ [ Jo

/ Jo

\C(u,t)\du\x\+H

\C(t,s)\\x(s)\ds

roo

> m | x | -HI

\C(u,t)\du\x. Jt

+ H f \C(t,s)\\x(s)\ds Jo r

= \m-H I Jt I

- M f Jo

\C(t,s)\\X(s)\ds

i

*

\C{u,t)\du\\x\ + (H-M) J Jo

\C(t,s)\\x(s)\ds

> 7 | x | + ( f f - M ) / |C(M)||x(s)|ds. Jo Hence, there is a /i > 0 with V('5.4.i)(*.x(-))>M[|x(t)l + |x'(t)l](5-4.5) From the form of V and an integration of (5.4.5), for some a > 0, we have a|X(£)|>[xT(i)Lx(£)]1/2>^x(-)) + / M|x(s)|ds, Jto

where x(i) is any solution of (5.4.1) on [to,t) with to > 0. If i 0 = 0, then

|x(t)| > j ^ ^ L x W l ^ + jT M|x(s)|ds| / a > [x^OjixfO)] 1 / 2 /^ so that |x(i)| > {[x T (O)ix(O)] 1/2 +t/i[x T (O)ix(O)] 1/2 /a}/a def

= Ci + C2t . If io > 0, select on [0, io] with /"to

[<j>T{to)Lct>{to)\1/2>

/"OO

/ JO

|C(u,s)|du|^(s)|ds

Jto

and draw a conclusion, as before, to complete the proof.

166

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Roughly speaking, a functional differential equation is one in which x'(t) depends explicitly on part or all of the past history of x(t). Such dependence is clear in (5.4.1). Explicit dependence is absent in x'(t)=f(t,x(t)), although it may become implicit through continual dependence of solutions on initial conditions. Conceptually, one of the most elementary functional differential equations is a scalar delay equation of the form x'{t) = ax(t) + bx(t - 1),

(5.4.6)

where a and b are constants with 6 ^ 0 . Recall that we encountered a system of such equations in Section 4.1. To specify a solution we need a continuous initial function R. We may then integrate x'[t)

= ax{t) + b{t - 1 ) ,

x{t0) = (*o)

on the interval [to, io + 1] to obtain a solution, say, tp(t). Now on the interval [to, to + 1] the function ip becomes the initial function. We then solve x'[t) = ax{t) + 6V(* " 1),

x(t0 + 1) = V(*o + 1),

on the interval [to + 1, to+2] We may, in theory, continue this process to any point T > to- This is called the method of steps and it immediately yields existence and continuation of solutions. We can say, with much justice, that (5.4.6) is a completely elementary problem whose solution is within the reach of a college sophomore. Indeed, letting a and b be functions of t does not put the problem beyond our grasp. By contrast, (5.4.1) is exceedingly complicated. Unless C(t,s) is of such a special type that (5.4.1) can be reduced to an ordinary differential equation, there is virtually no hope of displaying a solution in terms of integrals, even on the interval [0,1 . Yet, it turns out that the solution space of (5.4.6) is enormously complicated. With a and b constant, try for a solution of the form x = ert with r constant. Thus, x' = rert, so rert =aert + ber(t-l) or r = a + be-r,

(5.4.7)

which is called the characteristic quasi-polynomial. It is known that there is an infinite sequence {rn} of solutions of (5.4.7) [see El'sgol'ts(1966)].

5.5. NON-EXPONENTIAL DECAY

167

Moreover, R e r n —> —oo as n —> oo. Each function x(i) = cer"* is a solution for each constant c. Because we may let c be arbitrarily small, the zero solution cannot be completely unstable. As simple as (5.4.6) may be, its solution space on [0, oo) is infinitedimensional, whereas that of (5.4.1) on [0, oo) is finite-dimensional. This contributes to the contrast in degree of instability. The infinitedimensionality would appear to have a stabilizing effect. Roughly speaking, any n-dimensional linear and homogeneous, functional differential equation whose delay at some to reduces to a single point and that enjoys unique solutions will have a finite-dimensional solution space starting at to- For example, the delay equation x'(t) = ax(t) + bx[t - r(t)}

(5.4.8)

with r(t) continuous, r(t) > 0, and r(io) = 0 for some to, should have exactly one linearly independent solution starting at to-

5.5

Non-exponential Decay

In this section we discuss work of J. Appleby and D. Reynolds on a linear scalar equation z'(t) = -az(t) +

ft

Jo

k(t-s)z(s)ds,

t >0,

z(0) = 1,

(5.5.1)

whose solutions decay slower than exponential. We make the assumption that k is a continuously differentiable, integrable function with k(t) > 0 for all t > 0. Then (5.5.1) has a unique continuous solution on [0, oo). It is known that z e L 1 (0, oo) if and only if a > /0°° k(s) ds, and that in this case z(t) —> 0 as t —> oo. On the other hand, if z(t) —> 0 as t —> oo, then a > /0°° k(s) ds > 0. We ask that the kernel further satisfy

lim ffl = 0 ,

(5.5.2)

V t^oo k(t) ' which forces k(t) —> 0 as t —> oo more slowly than any decaying exponential. To see this, put p(t) = k'(t)/k(t), and let e > 0. Then there is a T > 0 such that pit) > - e / 2 for all t > T. Since

k(t) =

k(T)ef'rp{s)ds,

it follows by multiplying both sides by e£t, that e£tk(t) > k(T)e£(-t-J">/2 -> oo as t —> oo. Hence (5.5.2) implies that lim e£tk(t) = oo , t—>oo

for every

e > 0.

(5.5.3)

168

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Suppose that the solution of (5.5.1) obeys z(t) —> 0 as t —> oo. Then it satisfies the ordinary differential equation z'(t)

= -az(t)

+ f(t),

t>0,

with the forcing term given by f(t)=

/ k(t-s)z(s)ds Jo

-> 0

as

t -> oo .

Since a > 0 and the solution can be represented using the variation of parameters formula, t-t

z(t) = e~at+

e-a(t-s)f(s)ds,

t>0,

(5.5.4)

Jo the asymptotic behaviour of f(t) as t —> oo influences the rate at which z(t) —> 0 as £ —> oo. This is brought out in the proof of the following result. Theorem 5.5.1. Suppose that k is an integrable and continuously differentiable function on [0, oo), with k(t) > 0 as t —> oo. Moreover assume that k'(t)/k(t) —> 0 as t —> oo. If the solution of (5.5.1) obeys z(£) —> 0 as i —> oo, then

liminf^>4t^oo

fe(t)

(5-5.5)

az

Consequently limj^oo e6tz(t) = oo for every e > 0. Proof. Firstly note that z(t) > e~a* for all t > 0. Since z(0) = 1, t0 = inf {t e [0, oo) : z(t) = 0} . Since k(t) > 0 and z(t) > 0 for all 0 < t < t0, f{t) > 0 for all 0 < t < t0. If to is finite, it follows from (5.5.4) that o

0 = z(t0) = e-at" + / e-a^'-s)f{s) Jo

ds > e~at" > 0 ,

giving a contradiction. Therefore z(i) > 0 for all t > 0. Employing the positivity of k, f(t) > 0 for all t > 0, and hence (5.5.4) implies that z(t) > e~at for all t > 0. Consequently

/ ( * ) = f k{t - s)z(s) ds > [ k{t - s)e~as ds . Jo Jo

5.5. NON-EXPONENTIAL DECAY

169

Thus /(<) > g(t) for all t > 0, where g(t) = e~a* /„* easfc(s) ds is independent of z. Hence using (5.5.4) again, rt

z{t)>e-at

rt

easf(s)ds>e-at

Jo

easg(s)ds,

t>0,

Jo

and consequently, using the positivity of k(t), \teasg(s)ds

z(t)

-7T > ——T^A—>

i > 0

-

at

k(t) ~ e k(t) By L'Hopital's rule, (5.5.2) and (5.5.3),

(5-5-6) v

;

g(t) S* ea'k(s) ds 1 1 hm —H- = hm — —— = hm ,,,,, = - . t-oo k(t) t-oo e°*fc(t) t-oo ( ^ + f f l ) a Using L'Hopital's rule again,

This and (5.5.6) establish that (5.5.5) holds. Due to (5.5.3) and (5.5.5),

z{tyt

=

W)k{t)eSt ^ °°

as t —> oo if s > 0, completing the proof. We conclude with some remarks. Remark 5.5.1. (5.5.3) implies that, for each T > 0, v

;

-^ 1 as t -> oo, uniformly for 0 < s < T.

(5.5.7)

fc(i) If this condition is assumed instead of (5.5.3) and a > Jo°° k(s) ds, the lower bound in (5.5.5) can be improved to lim inf / N > . ,„ . , t^oo k(t) - a(a - fo k(s) ds) where the right hand side is interpreted as infinity if a = Jo°° k(s) ds [see Appleby and Reynolds (2004)]. It turns out that (5.5.7) also implies (5.5.3).

170

5. INSTABILITY, STABILITY, AND PERTURBATIONS

Remark 5.5.2. Theorem 5.5.1 asserts that the solution z does not decay to zero faster than the kernel k. Positive, integrable, continuous functions satisfying (5.5.7) and f* Jo

kit V - s)k(s) w ds

, .; k(t)

>2

f°°

Jo

k(s)ds as i ^ o o ,

(5.5.8)

are called subexponential in Appleby and Reynolds (2002). It is shown in Appleby and Reynolds (2002, 2003) that if the kernel k is subexponential and a > Jo°° k(s) ds, then z and k decay at exactly the same rate: indeed lim ^ 4 =

*-~ *(*)

o

(a-J^k(s)ds)2

Remark 5.5.3. At first glance the conditions (5.5.7) and (5.5.8) seem very restrictive. However if k is a positive, continuous and integrable function which obeys k(Xt)k(t)^1 —> AQ as t —> oo for all A > 0 for some a < —1, then k is subexponential. An example is k(t) = (1 +12)~1. Another example outside this class is k(t) = exp( — (t + I)13) with 0 < (3 < 1.

Chapter 6

Stability and Boundedness 6.1

Stability Theory for Ordinary Differential Equations

Consider a system of ordinary differential equations x'(i) = G(i,x(i)),

G(i,O) = O,

(6.1.1)

n

in which G : [0, oo) x D —> R is continuous and D is an open set in Rn with 0 in D. We review the basic definitions for stability. Definition 6.1.1. The solution x(t) = 0 of (6.1.1) is (a) stable if, for each e > 0 and to > 0, there is a S > 0 such that |xo| < <5 and t > to imply |x(i,i o ,x o )| < e, (b) uniformly stable if it is stable and 5 is independent of to > 0, (c) asymptotically stable if it is stable and if, for each to > 0, there is an T] > 0 such that |xo| < 77 implies |x(i,io,xo)| —> 0 as £ —> 00 (If,

in addition, all solutions tend to zero, then x = 0 is asymptotically stable in the large or is globally asymptotically stable.), (d) uniformly asymptotically stable if it is uniformly stable and if there is an 7] > 0 such that, for each 7 > 0, there is a T > 0 such that |xo|<77,

*0>0,

and t>to

+T

imply |x(£,£o,xo)| < 7. (If r] may be arbitrarily large, then x = 0 is uniformly asymptotically stable in the large.) 171

172

6. STABILITY AND BOUNDEDNESS

Under suitable smoothness conditions on G, all of the stability properties except (c) have been characterized by Liapunov functions. Definition 6.1.2. A continuous function W : [0, oo) —> [0,oo) with W(0) = 0, W(s) > 0 if s > 0, and W strictly increasing is called a wedge. (In this book wedges are always denoted by WorWi, where i is an integer.) Definition 6.1.3. A function U : [0, oo) x D -> [0, oo) is called (a) positive definite if U(t,O) = 0 and if there is a wedge W\ with

t/(i,x)> 1^(1x1), (b) decrescent if there is a wedge W2 with U(t,x) < W2(|x|), (c) negative definite if — U(t,x) is positive definite. (d) radially unbounded if D = Rn and there is a wedge Ws(\x.\) < U(t, x) and Wz{r) - ^ o o a s r ^ 00, and (e) mildly unbounded if D = Rn and if, for each T > 0, U(t,x) —> 00 as |x| —> 00 uniformly for 0 < t < T. Definition 6.1.4. A continuous function V : [0, 00) x D —> [0, 00) that is locally Lipschitz in x and satisfies V/61 !)(t,x) =limsup [V(t + h,x + hG(t,x))-V{t,x)]/h<

0 (6.1.2)

on [0, 00) x D is called a Liapunov function for (6.1.1). If V has continuous first partial derivatives, then (6.1.2) becomes V'(t, x) = grad V(t, x) G(t, x) + dV/dt < 0 . Theorem 6.1.1. Suppose there is a Liapunov function V for (6.1.1). (a) If V is positive definite, then x = 0 is stable. (b) IfV is positive definite and decrescent, then x = 0 is uniformly stable. (c) If V is positive definite and decrescent and VL 1 ^(£,x) is negative definite, then x = 0 is uniformly asymptotically stable. Moreover, if D = Rn and if V is radially unbounded, then x = 0 is uniformly asymptotically stable in the large. (d) IfD = Rn and ifV is radially unbounded, then all solutions of (6.1.1) are bounded. (e) If D = Rn and ifV is mildly unbounded, then each solution can be continued for all future time.

6.1. ODE STABILITY THEORY

173

Proof, (a) We have V('61 1} (t, x) < 0, V continuous, V(t, 0) = 0, and Wi(|x|) < V(t,x). Let s > 0 and t0 > 0 be given. We must find 6 > 0 such that xo| < 5 and t > to imply \x(t,to,xo)\ < e. (Throughout these proofs we assume e so small that |x| < £ implies x e D.) As V(to,x) is continuous and V(to,O) = 0, there is a 5 > 0 such that |x| < S implies V(t0, x) < Wi(e). Thus, if t > t0, then V' < 0 implies that, for |xo| < S and x(t) = x(i,to, x o)i w e have Wi(|x(t)|) 0, we select S > 0 such that W2(5) < VKi(e), where Wi(|x|) < F(t,x) < W2(|x|). Now, if i0 > 0 and ]xo| < 5, then, for x(t) = x(t,to,xo) and t > to, we have Wi(|x(i)|) 0 be given. We must find T > 0 such that |x o |<7?,

io>O,

and

t>to+T

imply |x(£,£o,xo)| < 7- Set x(i) = x(t,to,xo). Pick \i > 0 with W2(JU) < VFi(7), so that if there is a t\ > to with |x(ii)| < fi, then, for t > t\, we have Wi(|x(t)|) /i, then y'(t,x(t)) < -W3(A*); thus y ( t , x ( i ) ) < F ( t o , x o ) - / W3(|x(s)|)ds o

<W2(|xo|)-W3(/i)(t-to)

174

6. STABILITY AND BOUNDEDNESS

which vanishes at t = to + W2(r])/W3(fi) d=t0 + T. Hence, if t > to + T, then |x(i)| > /j, fails, and we have |x(i)| < 7 for all t > t0 + T. This proves U.A.S. The proof for U.A.S. in the large is accomplished in the same way. (d) Because V is radially unbounded, we have V(t, x) > Wi(|x|) —> 00 as |x| -^ 00. Thus, given to > 0 and xo, there is an r > 0 with W\(r) > V(to,xo). Hence, if t > to and x(i) = x(i,io,xo), then Wi(|x(i)|) < V(t,X(t)) < V(t,Xo) < W^r), or |x(£)| < r. (e) To prove continuation of solutions it will suffice to show that if x(i) is a solution on any interval [to,T), then there is an M with |x(i)| < M on [to,T). Now V(t,x) —> 00 as |x| —> 00 uniformly for 0 < t < T. Thus, there is an M > 0 with V(t,x) > V(to,xo) if 0 < £ < T and |x| > M. Hence, for 0 < t < T we have V(t,x(t)) < V(to,xo), so that |x(£)| < M. The proof of Theorem 6.1.1 is complete. Theorem 6.1.1 is an attempt to bring the book into focus as we look back at earlier chapters and forward to Chapters 6 and 8. The continuation question treated in Theorem 6.1.1(e) was considered in detail in Section 3.3 and will be seen throughout the remainder of the book. Next, in Chapters 2 and 5 we have seen examples of (a) and (b) extended to Volterra equations. For ordinary differential equations the concept of a Liapunov function being decrescent is simple, natural, and we can readily find examples. However, it is still not known what type of decrescent condition might be necessary and sufficient for asymptotic stability in this scheme. Suppose we have a Liapunov function which is positive definite, decrescent, and whose derivative is not positive. The decrescent condition does two things. First, it allows us to give an argument yielding uniform stability, as we have seen several times in Chapters 2 and 5. Involved in this is the fact that it allows us to show that a solution which gets close to zero will then stay close to zero; that property does not hold for functional differential equations. For Volterra equations and general functional differential equations the decrescent concept becomes elusive and, to this day, a uniformly satisfactory form is unknown. The concept in (2.5.11) of V' < — /4|x| + |x'|] was a major step in avoiding the decrescent question, as was an upcoming Marachkov condition which regulates the speed of a solution. Sections 8.3 and 8.7 are mainly devoted to additional steps in that direction.

6.1. ODE STABILITY THEORY

175

We have chosen our wedges to simplify proofs. But that choice makes examples more difficult. One can define a wedge as a continuous function W : D -> [0, oo) with W(0) = 0 and W{x) > 0 if x ^ 0. That choice makes examples easier, but proofs more difficult. The following device is helpful in constructing a wedge V^i(|x|) from a function W(x). Suppose W : D -> [0, oo), D = {x e Rn : |x| < l } , W(0) = 0, and W(x) > 0 if x 7^ 0. We suppose there is a function V(t,x) > W(x) and we wish to construct a wedge VKidxl) < V(t,x). First, define a(r) = min r <| x |< 1 W(x), so that a : [0,1] —> [0, oo) and a is nondecreasing. Next, define Wi(r) = /J" a(s) ds and note that Wi(0) = 0, W[(r) = a(r) > 0 if r > 0, and W1(r) < ra(r) < a(r). Thus, if |xi| < 1, then V(t,^)>W(Xl)>

min

W(x)=a(\x1\)>W1(\Xl\).

l*i|<|x|
The next result is the fundamental boundedness result for Liapunov's direct method. It is our view that the extension of this result to Liapunov functionals for Volterra equations is one of the most important unsolved problems of the theory at this time. Theorem 6.1.2. Let D = Rn and let H = {x e Rn : |x| > M,M > 0}. Suppose that V : [0, oo) x H —> [0, oo) is continuous, locally Lipschitz in x, radially unbounded, and VL 1 ^(t, x) < 0 if |x| > M. If there is a constant P > 0 with V{t,x) < P for |x| = M, then all solutions of (6.1.1) are bounded. Proof. As in the proof of Theorem 6.1.1(d), if a solution x(i) satisfies \x(t)\ > M for all t, then it is bounded. Suppose x(t) is a solution with |x(ii)| = M and |x(t)| > M on an interval [ii,T]. Then Wi(|x(t)|) < V(t,x(t)) < V(h,x(h))
176

6. STABILITY AND BOUNDEDNESS

Example 6.1.1. Let g : [0, oo) —> (0,1] be a differentiable function with g(n) = 1 and JQ g(s) ds < oo. We wish to construct a function V(t,x) = a(t)x2 with a(i) > 0 and with the derivative of V along any solution of x'=[g'(t)/g(t)}x

(6.1.3)

satisfying V'(t,x) = -x2. We shall, thereby, see that V > 0 and V negative definite do not imply that solutions tend to zero, because x(t) = g{t) is a solution of (6.1.3). To this end, we compute V('6.i.3)(*.z) = o!(t)x2 + 2a(t)[g'(t)/g(t)]x2 and set V = -x2. That yields a'(t) + 2a(t)[g'(t)/g(t)] = - 1 or

a'(t) =

-2a(t)[g\t)/g(t)]-l

with the solution

a(i)= \a(0)g\0)- J g\s) ds] jg\t). Because 0 < g(t) < 1 and g is in X1[0, oo), we may pick a(0) so large that a(t) > 1 on [0, oo). Notice that V(t,x) > 0 and positive definite; however, V is not decrescent. The first real progress on the problem of asymptotic stability was made by Marachkov [see Antosiewicz (1958, Theorem 7, p. 149)]. Theorem 6.1.3. Marachkov If G(t,x) is hounded for |x| bounded and if there is a positive definite Liapunov function for (6.1.1) with negative definite derivative, then the zero solutions of (6.1.1) is asymptotically stable. Proof. There is a function V : [0, oo) x D -* [0, oo) with Wi(\x\) < V(t,x) and V - (6ii)(^ x ) < —W2(|x|) for wedges W\ and W?.. Also, there is a

6.1. ODE STABILITY THEORY

177

constant P with G(i,x)| < P if |x| < m, where m is chosen so that |x| < m implies x is in D. Because V is positive definite and V < 0, then x = 0 is stable. To show asymptotic stability, let to > 0 be given and let Wi(m) = a > 0. Because V(to,x) is continuous and V(to,O) = 0, there is an r] > 0 such that |xo| < f] implies F(to,xo) < a. Now for x(t) = x(t,to,xo), we have V'(t,x(t)) < 0, so Wi(|x(i)|) < V(t,x(t)) < V(t0,xo) < W^m), implying |x(i)| < m if t > t0- Notice that V'(t,x(t)) < -W 2 (|x(i)|), so that 0
I W2(|x(s)|)ds, Jta

from which we conclude that there is a sequence {tn} -^ oo with x(tn) —> 0. Now if x(t) -+> 0, there is an e > 0 and a sequence {sn} with |x(sra)| > e and sn -^ oo. But because x(tn) -^ 0 and x(t) is continuous, there is a pair of sequences {Un} and {Jn} with Un < Jn < Un+\, |x(C/n)| = e/2, |x(J n )| = e, and e/2 < |x(i)| < e if C/n < t < Jn. Integrating (6.1.1) from Un to Jn we have x(Jn)=x(Un)+

/

G(s,x(s))ds,

so that e/2 < \x(Jn) - x(Un)\ < P(Jn - Un) or Jn-Un>e/2P. Also, if t > Jn, then

0

I W2(\x(s)\)ds Jta


178

6. STABILITY AND BOUNDEDNESS

Definition 6.1.5. The argument given in the final paragraph of the proof of Theorem 6.1.3 is called the annulus argument. That argument is central to 40 years of research on conditions needed to conclude asymptotic stability in ordinary and functional differential equations. Marachkov's work was done in 1940 and was extended in 1952 by Barbashin and Krasovskii [see Barbashin (1968, p. 1099)] in a very significant manner that allows V to be zero on certain sets. Somewhat similar extensions were given independently by La Salle, Levin and Nohel, and Yoshizawa. The following is essentially Yoshizawa's formulation of one of those results, all of which, incidentally, will play a central role in construction of Liapunov functionals for Volterra equations. For our purposes a scalar function / : Rn —> [0, oo) is positive definite with respect to a set A if /(x) = 0 for x € A and for each e > 0 and each compact set Q in Rn there exists 5 = S(Q,e) such that /(x) > S for x e Q n U(A, s)c, where U(A7 e) is the e-neighborhood of A. Theorem 6.1.4. Yoshizawa (1963) Let D = Rn and let G(i,x) be bounded for x bounded. Also suppose that all solutions of (6.1.1) are bounded. If there is a continuous function V : [0, oo) x Rn —> [0, oo) that is locally Lipschitz in x, if there is a continuous function W : Rn —> [0, oo) that is positive definite with respect to a closed set ft, and ifVL l ^ (t, x) < —W(x), then every solution of (6.1.1) approaches Q, as t —» oo. Proof. Consider a solution x(£) on [to, oo) that, being bounded, remains in some compact set Q for t > to- If x(t) -^ Q, then there is an e > 0 and a sequence {<„} —> oo with x(tn) £ C/(O,e)cnQ. Because G(t,x) is bounded for x in Q, there is a K with |G(t, x(i))| < K. Thus, there is a T > 0 with x(i) e U{Q,e/2)c n Q for tn < t < tn+T. By taking a subsequence, if necessary, we may suppose these intervals disjoint. Now, for this e/2 there is a <5 > 0 with

V'(t,x)<-5

on [tn,tn + T\.

Thus for t > tn + T we have

0 < V(t,x(t)) < V(to,x(to)) - [ W(x(s))ds n


pti+T

W(x(s))ds t

'


6.1. ODE STABILITY THEORY

179

Usually, boundedness is proved by showing V to be radially unbounded. Also, we understand from the proof that the requirement that all solutions be bounded can be dropped and the conclusion changed to read that all bounded solutions approach Q. Moreover, some authors let V : [0, oo) x Rn —> (—00,00) and ask that V be bounded from below for x bounded, concluding again that bounded solutions approach Q [see Haddock (1974)]. Example 6.1.2. Consider the scalar system (i.e., x and y are scalars) x' = y, v == —(x ~~\- l)w — x with Liapunov function V(x, y) = x2 + y2, so that \r'

(

\

O/

2

1 -1 \

2 def

TT//

\

and il is the x axis. Because V is radially unbounded and (6.1.4) is bounded for (x, y) bounded, all solutions are bounded and approach the x axis. When (6.1.1) is independent of t, say, x' = G(x)

(6.1.5)

and solutions are unique, then much more can be said. A point y is an LJ-limit point of a solutions x(i) of (6.1.5) if there is a sequence {tn} —> 00 with x(tn) —> y. The set of w-limit points of a solution of (6.1.5) is called the u-limit set. By uniqueness, if y is in the w-limit set of x(i), then the orbit through y, say, {z G Rn : z = x(i, 0, y ) , t > 0} is also in the w-limit set. (Actually, this follows from continual dependence on initial conditions, which, in turn, follows from uniqueness.) A set A is positively invariant if y G A implies x(t, 0, y) G A for t > 0. Theorem 6.1.5. Let the conditions of Theorem 6.1.4 hold for (6.1.5) and let V = V(x). Also, let M be the largest invariant set in Q. Then every solution of (6.1.5) approaches M as t —> 00. Proof. If x(t) is a solution of (6.1.5), then it approaches Q,. Suppose there is a point y in the u;-limit set of x(i) not in M. Certainly, y G £l, and as y ^ M, there is a t\ > 0 with x ( t i , 0 , y ) ^ il. Also, there is a sequence {tn} —> 00 with x(tn) —> x(ii, 0, y), a contradiction to x(i) —> Q as t —> 00. This completes the proof.

180

6. STABILITY AND BOUNDEDNESS

The result can be refined further by noticing that V(x(t)) —> c so the set M is restricted still more by satisfying V(x) = c for some c > 0. The ideas in the last two theorems were extended by Hale to autonomous functional differential equations using Liapunov functionals and by Haddock and Terjeki using a Razumikhin technique. These will be discussed in Chapter 8. They were also extended to certain classes of partial differential equations by Dafermos. Example 6.1.3. Consider Example 6.1.2 once more with O being the x axis. Notice that if a solution starts in O with x\ ^ 0, then y' = —x\ ^ 0, so the solution leaves Q. Hence, M = {(0,0)}. Theorems 6.1.4 and 6.1.5 frequently enable us to conclude asymptotic stability (locally or in the large) using a "poor" Liapunov function. But when (6.1.1) is perturbed, we need a superior Liapunov function so we can analyze the behavior of solutions. For example, suppose D = Rn and there is a continuous function V : [0, oo) x Rn —> [0, oo) with (a) \V(t,xi) - V(t,x2)\

< K\xi - x 2 | on [0, oo) x Rn with K constant,

(b) ^ ( e . i . i ) ^ ) ^ ~cV(t,x),

c > 0, and

(c) V{t,x) > Wi(\x\) -> oo as |x| -> oo. Then for a perturbed form of (6.1.1), say, x ' = G(t,x)+F(t,x)

(6.1.6)

with G, F : [0, oo) x Rn —> Rn being continuous, we have ^('6.i.6)(*.x) < ^ . L D ( * , x ) <

+K\F(t,x)\

-cV(t,x)+K\F(t,x)\,

so that y(i,x(t))
+ / e" c(t " s) J ft:|F(s,x(,s))|(is, Jtn

(6.1.7)

which is a "poor man's" variation of parameter formula. Although it may be extremely hard to find a V satisfying (a), (b), and (c), once we establish that (6.1.1) is uniformly asymptotically stable in the large (possibly by using Theorems 6.1.4 and 6.1.5), then the next result assures us that this superior V does exist. And, although it does not specify

6.1. ODE STABILITY THEORY

181

c exactly, it does claim that c > 0 so that (6.1.7) will yield boundedness when F is bounded; additional results are obtained when F(£,x) < A(£), where A is a continuous function tending to zero or in Lp[0, oo). The following result may be found in Yoshizawa (1966, p. 100). We offer it here without proof. T h e o r e m 6.1.6.

Consider

x' = G ( t , x ) ,

G(t,0) = 0 n

(6.1.1)

n

with G : [0, oo) x R —> R continuous and locally Lipschitz in x. If x = 0 is uniformly asymptotically stable in the large, then there is a V : [0, oo) x Rn -> [0, oo) with where the Wi are wedges and W\(r) —> oo as r —> oo, such that V(6.i.i)(t,x)<-cV(t,x),

c>0.

(b) n

If, in addition, for any compact set K C R there is a constant L(K) > Osuch that G(£,xi)-G(t,x 2 )| < L(K)\xi-x2\ whenever (£,Xj) e [0,oo)x K, then V{t,xl)-V{t,x2)\


(c)

where h(K) is a constant depending on K. There are two important alternatives to asking that |G(£,x)| be bounded for x bounded when we seek to establish asymptotic stability for a V with a negative-definite derivative. These two alternatives are very effective for Volterra equations. Suppose there is a Liapunov function V : Rn —> [0,oo) for (6.1.1). Notice that y('6.u)(x)=grady(x).G(t,x) = |gradV(x)| |G(i,x)| cos 9,

(6.1.8)

where 9 is the angle between the vectors gradF(x) and G(t,x) and | | denote Euclidean length. If V is shrewdly chosen, it may be possible to deduce that V ( 6.i.i)(x)<-<J|G(t,x)|

6>0.

(6.1.9)

In that case, we have 0
*

\x'(s)\ds, (6.1.10) Jto which implies that any solution has finite arc length. The annulus argument (see Definition 6.1.5) would become trivial in that case. Note, however, that

182

6. STABILITY AND BOUNDEDNESS

(6.1.10) alone does not imply that solutions tend to zero. For example, x' = 0 and V{x) = x2 satisfy (6.1.9) and (6.1.10), but all solutions are constant. We shall find that in Theorems 6.1.3 and 6.1.4 we may replace |G(i, x)| bounded for |x| bounded by V(6AA)(t,x)<-5\G(t,x)\,

6>0.

Example 6.1.4. Let x' = ,4x,

(6.1.11)

where A is an n x n constant, real matrix whose characteristic roots all have negative, real parts. Let [xTBx}1/2,

V(x) =

where BT = B and ATB + BA= -I.

(6.1.12)

Note that the zero solution is unique and that V has continuous first partial derivatives for x ^ 0. Thus V('6.i.ii)W = (x T A T 5x + x T Bix)/{2[x T 5x] 1 / 2 } = -x T x/{2[x T Bx] 1 / 2 }, and there is a k > 0 with

|x|/{2[xTSx]1/2}
8>0.

(6.1.13)

These ideas have been developed extensively by Erhart (1973), Haddock (1977a,b), Hatvani (1978) and Burton (1977). They were employed in the proof of Theorem 2.5.1. In many results, such as Theorem 6.1.3, we can weaken the condition V'(t,x) < —W(\x\). Here is a typical way. Definition 6.1.6. A scalar function p : [0, oo) —> [0,oo) is said to be integrally positive if for each 5 > 0 and each sequence {£„} —> oo monotonically, rt,,+S

lim inf / ™^°° Jt,,

p(t) dt > 0 .

6.2. CONSTRUCTION OF LIAPUNOV FUNCTIONS

183

Thus, we are allowed to have p(t) = sin 2 t, but p(t) = sin£| + sint would not qualify as being integrally positive. Exercise 6.1.1. In Theorem 6.1.3, drop the condition that the derivative of V is negative definite. Instead, ask that V'(t,x) < — p(t)W(\x\) where p is integrally positive. Show that the zero solution is still asymptotically stable.

6.2

Construction of Liapunov Functions

Beyond any doubt, construction of Liapunov functions is an art. But like any other art, there are guidelines and there are masters to emulate. Whereas the previous section concentrated on formal theorems concerning consequences of Liapunov functions, this section contains a detailed account of the construction of somewhat special Liapunov functions. Such constructions are fundamental in the construction of Liapunov functionals for Volterra equations. A.

x' = Ax

As we have seen, given x' = Ax

(6.2.1)

we try V(x) = x r B x with B = BT and ATB + BA = -I

(6.2.2)

when no characteristic root of A has a zero real part. [In Chapter 5 we explored the possibility of solving (6.2.2).] Also, if all characteristic roots of A have a zero real part and if the elementary divisors are simple, then the equation ATB + BA = 0

(6.2.3)

may be solved for B = BT. The reader may wish to try this for

Equation (6.2.3) has important consequences for perturbed forms of (6.2.1).

184 B.

6. STABILITY AND BOUNDEDNESS x' = y, y' = f(x,y)y

- g(x)

Long before Liapunov, the mathematician Lagrange noted that equilibrium was stable when the total energy of the system was at a minimum. That idea, applied to the scalar equation x" + f{x,x')x' +g(x) = 0

(6.2.4)

with f(x, x1) > 0 and xg(x) > 0 for x ^ 0, produced the Liapunov function V(x,y) = \y2+ 1

f g(s)ds Jo

(6.2.5)

for the system of the form x' = y, (6.2.6) y' =

-f(x,y)y-g(x).

The result is ^(6.2.6) fa y) = -f(x,y)y'2<0.

(6.2.7)

Equation (6.2.4) may be thought of as a fairly general statement of Newton's second law of motion for an object with one degree of freedom. Equations (6.2.5)-(6.2.7) generated scores, if not hundreds, of research articles between 1940 and the present. Bibliographies may be found in the work of Graef (1972) and Burton-Townsend (1971).

C.

x' = y,y' = -c(t)f(x)

It was recognized very early in the development of Liapunov's direct method that a first integral might serve as a Liapunov function. For example, consider x" + x3 = 0

(6.2.8)

and the equivalent system x' = y, (6-2.9)

y = -x3

We have

dy/dx =

-x3/y,

6.2. CONSTRUCTION OF LIAPUNOV FUNCTIONS

185

so that y dy + x3 dx = 0 , yielding 2y2 + x4 = constant. If we take

V(x,y) = 2y2 + x4, then V

(6.2.9)(x,y)

=°-

Because V is positive definite and V < 0, the zero solution is stable. It is but a small jump, then, to try the Liapunov function

V(t,x,y) = 2y2 + c{t)xi for the system x' = y, y' = -c(t)x3

(6.2.10)

when c(t) > c0 > 0 and c'(i) < 0. From there it is natural to deduce that a Liapunov function for x' = y,

y' =

-c(t)f(x),

(6.2.11)

with xf(x) > 0, if x ^ 0 and differentiable c(t) > 0, may be obtained as follows. Write c(t) = a(t)b(t) with a(t) nondecreasing and b(t) nonincreasing. Then V(t,x,y)

= [y2/2b(t)} +a(t) [ f(s)ds Jo

(6.2.12)

is a Liapunov function, from which investigators have derived reams of information. For a bibliography consult Burton-Grimmer (1972).

186

D.

6. STABILITY AND BOUNDEDNESS

V'
Corduneanu was the first to notice that "poor" Liapunov functions might be made into "good" ones by use of differential inequalities. Consider (6.2.10) again where c{t) > 0 but c'(t) < 0 fails. We let V(t,x,y)=2y2 + c(t)x4 and find ^(6.2.10)(*,x,y) =c'(t)x4 <

[J+(t)/c(t)]V(t,x,y),

where c'+(t) = max[c'(t),0]. Thus, an integration yields ft

V(t,x,y) = V(to,xo,yo)exp

/ [c'+(s)/c(s)j ds , Jto

implying V bounded if the integral is bounded. For extensive use of differential inequalities see Lakshmikantham and Leela (1969). E.

x = G(t,x) +F(t,x)

If V(t,x) is globally Lipschitz in x for a constant L with the derivative of V along x' = G(i,x)

(6.2.13)

satisfying VL2 13N(£,X) < 0, then one may perturb (6.2.13) and write x' = G ( i , x ) + F ( i , x ) , where |F(t,x)| < X(t) and f™\(t)dt

(6.2.14) < oo. Then use ft

W{t, x) = [V{t, x) + 1] exp [ - L / A(s) ds ] . Jo x

We have WL2 14) (^ ) < 0, so that if V is radially unbounded, so is W and solutions of (6.2.14) are bounded. A continuation result is obtained in the same way, as is seen in Section 3.3, Theorem 3.3.3. For if V is a mildly unbounded function satisfying V/62 13Jt,x) < 0 and if |F(t,x)| < A(t) with A continuous, then WL 2 14\ (t, x) < 0 and W is mildly unbounded. Hence, solutions of (6.2.14) are continuable. This is an important principle. If G is smooth enough and if all solutions of (6.2.13) are continuable, the converse theorems show the existence of a mildly unbounded V with VL2 13%(t,x) < 0. Hence, we deduce that, for V Lipschitz, continuability of (6.2.13) plus |F(t,x)| < A(t) imply continuability of (6.2.14). For converse theorems on continuability see Kato and Strauss (1967).

6.2. CONSTRUCTION OF LIAPUNOV FUNCTIONS F.

187

First Integral Solutions

We see that investigators began with a much simplified equation, obtained a first integral, and used that first integral as a spring board to attack their actual problem. That progress can be clearly seen by reviewing scores of papers that proceed from x" + g{x) = 0 , xg{x) > 0 , through the series x" + f{x)x' + g(x) = 0 ,

f(x) > 0 ,

x" + h{x, x')x' + g{x) = 0 ,

h(x, x') > 0 ,

x" + h(x, x')x' + g(x) = e(t),

e(t + T) = e(t),

and x" + k(t, x, x')x' + a(t)g(x) = e(t, x, x'), e bounded, k > 0, and a(t) > 0. For bibliographies see Graef (1972), Sansone-Conti (1964), and Reissig et al. (1963). x' = P(x,y),y' = Q(x,y)

G.

Nor is one restricted to a first integral of a given system. From the point of view of subsequent perturbations the very best Liapunov functions are obtained as follows. Consider a pair of first-order scalar equations x' =

P(x,y), (6.2.15)

y' = Q(x,y), so that dy/dx =

Q(x,y)/P{x,y).

Then the orthogonal trajectories are obtained from dy/dx = —P(x,y)/Q(x,y) or P(x,y)dx + Q(x,y)dy = 0. If we can find an integration factor /i(:r, y) so that /i(x, y)P(x, y) dx + /i(x, y)Q{x, y) dy = 0

188

6. STABILITY AND BOUNDEDNESS

is exact, then there is a function V(x, y) with dV/dx = /J,P and dV/dy = fiQ, so that

V^2A5)(x,y)=n(x,y)[P2(x,y)+Q2(x,y)}

(6.2.16)

.

If V and /i are each of one sign and V\i < 0, then is a Liapunov function for (6.2.15). Moreover, if we review Eqs. (6.1.8)-(6.1.10), we have ^6.2.i5) (x,y) = \gradV(x,y)\ \{P(x,y),Q(x,y))\

cos9 ,

(6.2.17)

and because V is obtained from the orthogonal trajectories, we have cos(7 =

.

(6.2.18)

For this reason, (6.2.15) can be perturbed with comparatively large functions without disturbing stability properties of the zero solution.

H.

x' = Ax + b/(cr), <x' = c T x - r/(<x)

In view of (6.2.2) and (6.2.12), one can quickly see how to proceed with the (n + l)-dimensional control problem x' = Ax + b / ( a ) ,

o' = cTx - rf(a),

(6.2.19)

in which A is an n x n matrix of constants whose characteristic roots all have negative real parts, b and c are constant vectors, r is a positive constant, a and / are scalars, and <jf(<j) > 0 if a ^ 0. This is called the problem of Lurie and it concerns automatic control devices. The book by Lefschetz (1965) is devoted entirely to it and considers several interesting Liapunov functions. Lurie used the Liapunov function F(x, a) = x T 5 x + / f(s)ds, (6.2.20) Jo in which B = BT and ATB+BA = —D, where D = DT is positive definite. Then we have

6.2. CONSTRUCTION OF LIAPUNOV FUNCTIONS I.

189

x ' = A(t)x

It is natural to attempt to investigate x' = A(t)x

(6.2.22)

in the same way that (6.2.1) was treated. Suppose that A is an n x n matrix of functions continuous for 0 < t < oo. A common procedure may be described as follows. If all characteristic roots of A(t) have negative real parts for every value of t > 0, then for each t the equation AT(t)B(t) + B{t)A{t) = -I

(6.2.23)

may be uniquely solved for a positive definite matrix B(t) = BT{t). For brevity, let us suppose B(t) is differentiable on [0,oo). We then seek a differentiable function b : [0, oo) —> [0, oo) such that V(t,x) = b{t)xTB{t)x

(6.2.24)

will be a Liapunov function for (6.2.22). Thus, V('6 2 , 22) (t,x) = b\t)xTB{t)x

+ b(t)xT[ATB + BA + B']x

= xT [b{t){ATB + BA + B') + b'(t)B] x d

=b(t)xTH(t)x,

where H(t) = -I + B'(t) + [b'(t)/b(t)\B(t). If we take a(t) to be the largest root of the equation det [ - / + B'(t) + a{t)B{t)} = 0 , and

P(t) = 6(0) exp \ - j a(s)ds] , then the condition [b'(t)/b(t)\ < [P'(t)/p(t)} = ~a(t) is necessary and sufficient for H{i) to have only negative characteristic roots. In that case, stability and asymptotic stability may be determined from V and V. For more details see Lebedev (1957), Hahn (1963, pp. 2932), and Krasovskii (1963, pp. 56-62).

190 J.

6. STABILITY AND BOUNDEDNESS x' = F(x)

The most common method of attack on a nonlinear system x' = F(x),

F(0) = 0,

(6.2.25)

is by way of the linear approximation. If F is differentiable at x = 0, then it may be approximated by a linear function there. One may write (6.2.25) as x' = Ax + G(x),

(6.2.26)

in which A is the Jacobian matrix of F at x = 0 and limx^o |C(x)|/|x| = 0. For example, if f(x) = /(xi, ,xn) is a differentiable scalar function at x = 0, then

/(x) = /(o) + (df/dXl)Xl +

+ (df/dx2)x2

+ (df/dxn)xn

+ higher-order terms,

where the partials are evaluated at x = 0. One expands each component of F in this way and selects the matrix A from the coefficients of the Xj. It is more efficient to consider x ' = Ax + H(i, x),

(6.2.27)

where A is a constant n x n matrix, H : [0, oo) x D —> Rn is continuous, D is an open set in Rn with 0 in D, and lim |H(t,x)|/|x| = 0

uniformly for 0 < t < oo.

(6.2.28)

Theorem. Liapunov If (6.2.27) and (6.2.28) hold and if all characteristic roots of A have negative real parts, then the zero solution of (6.2.27) is uniformly asymptotically stable. Proof. By our assumption on A we can solve ATB + BA = —I for a unique positive definite matrix B = BT. We form

and obtain ^6.2.27) (x) = ( x T i T + HT)Bx + xTB(Ax + H)

= -xTx + 2HTBx < - x | 2 + 2|ff| |B| |x|,

6.3. A FIRST INTEGRAL LIAPUNOV FUNCTIONAL

191

so that for x ^ 0 we have F'(x)/|x| 2 < - 1 + 2\B\ |H(t,x)|/|x| < - 1 / 2 , if |x| is small enough, in consequence of (6.2.28). The conditions of Theorem 6.1.1(c) are satisfied and x = 0 is U.A.S.

K.

A(x) = Jo J(sx) ds

Much may be lost by evaluating the Jacobian of F in (6.2.25) only at x = 0. If we write the Jacobian of F as J(x) = (dFi/dxk), evaluated at x, then for f1 A(x) = / J(sx) ds Jo we have F(x) = ^(x)x. Investigators have discovered many simple Liapunov functions from A(x) yielding global stability. A summary may be found in Hartman (1964, pp. 537-555). Excellent collections of Liapunov functions for specific equations are found in the work of Reissig et al. (1963) and Barbashin (1968).

6.3

A First Integral Liapunov Functional

We consider a system of Volterra equations ft x' = A(t)x +

(6.3.1)

C(t,s)x(s)ds, Jo

with A and C being nxn matrices continuous on [0, oo) and 0 < s < t < oo, respectively. To arrive at a Liapunov functional for (6.3.1), integrate it from 0 to t and interchange the order of integration to obtain rt

rt

x(i) = x(0) + / A(s)x(s)ds+ Jo

rt

/ C(u,s) dux(s) ds . Jo Js

We then have h(i,x(-)) = x ( i ) + /

\-A(s)-

C{u,s)du\x{s)ds,

(6.3.2)

Jo L Js J which is identically equal to x(0). Hence, the derivative of h along a solution of (6.3.1) is zero. It is reasonable to think of h as a first integral

192

6. STABILITY AND BOUNDEDNESS

functional for (6.3.1). Compare this with Eqs. (6.2.8)-(6.2.12) for constructing Liapunov functions. Now h may serve as a suitable Liapunov functional for (6.3.1) as it stands. Moreover, the changes necessary to convert h to an outstanding Liapunov functional are quite minimal. Suppose that (6.3.1) is scalar, A(t) < 0 , C(t, s) > 0 ,

and

- A(s) - / C(u, s)du>0 Js for 0 < s < t < oo. Consider solutions of (6.3.1) on the entire interval [0, oo) (as opposed to solutions on some [to,oo) with to > 0). Because —x(t) is a solution whenever x(t) is a solution, we need only consider solutions x(t) with x(0) > 0. Notice that when x(0) > 0 and C(t, s) > 0, the solutions all remain positive. Hence, along these solutions the scalar equation

*r

r*

i

h(t, x(-)) = x(t) + / - A(s) - / C(u, s) du x{s) ds Jo L Js 1 is a positive definite functional. In fact, we may write it as

H(t,x(-)) = \x(t)\ + £ ^\A(s)\ - J* \C(u, s)\duj \x(s)\ ds , (6.3.3) and the derivative of H along these solutions of (6.3.1) is zero. Under the conditions of this paragraph, we see that solutions of (6.3.1) are bounded. However, much more can be said. Notice that if l^(s)l " / Js

\C(u,s)\du>a>0,

then boundedness of H implies that x(t) must be i 1 [0, oo). Definition 6.3.1. A scalar functional H(t,x(-)) expands relative to zero if there is a t\ > 0 and a > 0 such that if |x(i)| > a on fa, oo) with t?. > t\, then H(t,x(-)) —> oo as t —> oo. We formally state and prove these observations. Theorem 6.3.1. Let (6.3.1) be a scalar equation with A(s) < 0 and \A(s)\ - Jl \C{u,s)\du > 0 for 0 < s < t < oo. Then the zero solution of (6.3.1) is stable. If, in addition, there is a t 0 and an a > 0 with \A(S)\ ~ Is \C(U' s)\du > a for t2 < s
6.3. A FIRST INTEGRAL LIAPUNOV FUNCTIONAL

193

Proof. Let x(t) = x(t,to, 4>) be any solution of (6.3.1). We compute H'(631)(t,x(-))
f

\C(t,s)\\x(s)\ds

Jo

+ \A(t)\\x\- f

\C(t,s)\\x(s)\ds

Jo

= 0.

(Notice that once we formed H, then it was no longer necessary to ask C(t, s) > 0, nor that only solutions on [0, oo) be considered.) The stability is now clear; for if we are given e > 0 and to > 0, we let : [0, to] —> R be continuous and satisfy \(j>(s)\ < 5 on [O,to], where 5 is to be determined. Then for x(t) = x(t, to, to we have \x(t)\

f" \\A(s)\ - j " \C(u, s)\ du]ds\

< s

from which S = 5(e,to) is readily obtained. If t2 a n d a exist, then ff(*o,tf(-))

)

r* > |x(t)| + / a\x(s)\ds Jt2 for t > t2, so that x is in L1[0, oo). As Jo \C(t,s)\ds, A(t), and x(t) are bounded, it follows that x'(t) is bounded. Hence, x(t) —> 0. This completes the proof. We recall from Section 6.1 that there are two alternatives to asking x'(t) bounded. Whereas the requirement that Jo \C(t,s)\ds be bounded is consistent with the other assumptions, the requirement that A(t) be bounded is not only severe but it conflicts with the intuition that the more negative A(t) is, the more stable (6.3.1) should be. Let us return to the vector equation (6.3.2). If we wish to pass from (6.3.2) to a scalar functional analogous to (6.3.3), we have several options for the norms and each option will yield different results.

194

6. STABILITY AND BOUNDEDNESS

Let us suppose there is a constant positive definite matrix D = DT and a continuous scalar function fj, : [0, oo) —> [0, oo) with xT[ATD + DA}x < - / i ( t ) x T x .

(6.3.4)

The norm we will take on the solution x(t) will be [xT Dx]1/2 and bounds will be needed. There are positive constants s, k, and K with |x| > 2fc[x T Dx] 1/2 ,

(6.3.5)

|Dx|
(6.3.6)

s|x| < [xTDx}1'2.

(6.3.7)

and

With this norm, if we replace (6.3.3) by

P(t,x(-))

= [xTDx}1/'2

+I \ \ A ( s ) \ - J

\C(u,s)\du]\x(S)\ds

and differentiate along solutions of (6.3.1), then we readily see that we need to refine P and write P(*,x(-)) = [xTDx]1''2+

j

\kii(s)-K

I

\C(u,s)\du]\x(s)\ds,

(6.3.8)

with fc/i(s) -

/ K\C(u,s)\du>0

for

0 < s < t < oo .

(6.3.9)

Js

Theorem 6.3.2. Let (6.3.4)-(6.3.9) hold. (a) The zero solution of (6.3.1) is stable. (b) If P(i,x(-)) expands relative to zero and if x' is bounded for x bounded, then x = 0 is asymptotically stable. (c) If there is an M > 0 with / \k/j,(s)-K Jo L

Js

\C(u,s)\du\ds<M J

for 0 < t < oo, then x = 0 is uniformly stable.

(6.3.10)

6.3. A FIRST INTEGRAL LIAPUNOV FUNCTIONAL

195

Proof. For x / O w e compute T T X (S)C (t,S)dS^Dx

^(6.3.1) M O ) = | [ x T A T + ^ + XTD\AX

+ / C(t,s)x(s)ds]

I.

Jo

X /{2[x T Dx] 1 / 2 } JJ /

ft

+ kfi(t)\x\-

K\C(t,s)\\x(s)\ds Jo < -{ M (t)x T x/2[x T J Dx] 1 /2} + f \C(t,S)\\x(S)\ds Jo

{\Dx\/[xTDx}1/2}

r* + kn(t)\x\-

K\C(t,s)\\x(s)\ds<0. Jo

Stability is clear and asymptotic stability follows [See the proof of Theorem 6.3.1]. For uniform stability, let e > 0 be given and S > 0 still to be determined. If t0 > 0, if ), then for i > to we have s|x(i)| < P ( t , x ( - ) ) < P ( * o , 0 ( O ) <[
(6.2.22)

and the selection of B(t) by AT{t)B{t) + B(t)A(t) = -I,

(6.2.23)

followed by defining V(t, x) = b(t)xTB(t)x,

(6.2.24)

it seems clear that one may replace [xTDx]1/2 in (6.3.8) by \b(t)xTB{t)x\1/2 and generalize Theorem 6.3.2. We leave the details as an exercise.

196

6. STABILITY AND BOUNDEDNESS

Next, to secure (6.3.10), we could replace fi(s) by O

(K/k)

\C(u,s)\du, Js

so that the integrand in (6.3.10) becomes

[

O

/ Js

j-t

\C(u,s)\duJs

"I

\C(u,s)\du\=K J

f'OO

\C(u,s)\du Jt

and (6.3.8) becomes t

/-OO

/ / . Jt

\C(u,s)\du\x(s)\ds.

(6.3.11)

Hypotheses (b) and (c) of Theorem 6.3.2 are mutually exclusive. However, by replacing /j(s) as we did, the derivative of W may become negative. Thus, we may obtain asymptotic stability and uniform stability at the same time. Finally, if we write W as V(t,x(-)) = [xTDx]1/2+K

rt

/-OO

/

\C(u,s)\du\x(s)\ds

(6.3.12)

Jo Jt

with K > K, then we may be able to drop the requirement that x' be bounded for x bounded and perform the simplified annulus argument as noted following Eq. (6.1.10). However, with A(t) variable there may still be problems with the annulus argument. Those problems will evaporate when we consider the one-sided Lipschitz conditions introduced in (6.1.16) and (6.1.17) and to be developed in Definition 6.4.1. The foregoing explanation shows in detail how we arrive at the Liapunov functional used to prove Theorem 2.5.1. The reader is urged to review Theorem 2.5.1 and its proof carefully at this time. Moreover, the functional

[ I Jo Jt

\C(u,s)\du\x(s)\ds

of (6.3.11) turns out to be a fundamental part of each Liapunov functional, with, at most, minimal changes needed. The method outlined for constructing a Liapunov functional for the linear system can be extended without difficulty to nonlinear equations.

6.3. A FIRST INTEGRAL LIAPUNOV FUNCTIONAL

197

Consider the system ft x' = g(t,x)+ / p(t,s,x(s))ds, (6.3.13) Jo in which g and p are continuous when g : [0, oo) x U —> Rn, p : [0, oo) x [0,oo) xU -> Rr\ and U = {x e Rn : \x\ < e, e > 0}. We integrate (6.3.13) from 0 to £ and interchange the order of integration to obtain x(t) = x(0) + / g ( s , x ( s ) ) d s + / / Jo Jo Jo ,t

*

p(u,s,-x.(s))dsdu

*

= x(0) + / g(s,x(s))ds + / / p(w, S,X(S)) duds Jo Js Jo ft T ft 1 = x(0)+/ g ( s , x ( s ) ) + / p(u,s,x(s))du rfs, 7o L Js

so that ,t r

,t

r(£,x(-)) = x ( t ) + / - g ( s , x ( s ) ) - / p(u,s,x(s))du ds ^o L ^s = x(0) and hence rL3 13>(£, x(-)) = 0. The same sequence following Eq. (6.3.2) may be repeated. Briefly, in the scalar case we write

R(t,x(-))

=\ x \ + J \\g(s,x(s))\-j

\p(u,s,x(s))\du\ds.

(6.3.14)

If xg(t, x) < 0, then we obtain R{6 3 13)(t^(-))<-\9(t,x)\+

I

\p(t,S,x(S))\ds

Jo

+ \ g ( t , x ) \ - f \p(t,8,x(8))\d8 Jo

= 0.

If ft

\g(s,x(s))\ > /

\p(u,s,x(s))\ds

Js

for 0 < s < t < oo and x an arbitrary continuous function, x : [0, oo) —> £/, then the zero solution of (6.3.13) is stable. If, in addition, the functional ex-

198

6. STABILITY AND BOUNDEDNESS

pands relative to zero, then asymptotic stability may be obtained. Finally, under proper convergence assumptions we write *

V(t,x(-)) = \x\ + / / Jo Jt

\p(u, s,x(s))\ duds .

The situation becomes much more interesting in the vector case. We then interpret the norm of x in (6.3.14) as a norm of solutions of y' = g(i,y).

(6.3.15)

That is, we seek a Liapunov function W(t,y) for (6.3.15) with Wl6.3A5)(t,y)<-Z(t,y)<0. If W^dyl) < W(t,y) < W2(\y\) for wedges Wi, then W acts as a norm for y. We then write (6.3.14) as

V(t,x(-))

=W ( t , x ) + f

\\K(s,x(s))\-J

\p(u,s,x(s))\du\ds.

(6.3.16)

Specific results for this V are developed in the next section.

6.4

Nonlinearities and an Annulus Argument

Consider the system * x' = g ( t , x ) + / p(t,s,x(s))ds Jo

+ F(t),

(6.4.1)

in which g : [0, oo) x U -> Rn, p : [0, oo) x [0, oo) x U -> Rn, F : [0, oo) -> Rn, and where U = {x e Rn : |x| < e, e > 0}. We suppose that F, g, and p are continuous and g(t, 0) = 0. Moreover, we suppose there is a continuous function W : [0, oo) x U —> [0, oo) and a constant L > 0 with \W(t,x1)-W(t,x2)\ < i | x i - x 2 | on [0, oo) x U, along with a wedge W\ and a continuous function Z : [0,oo) x [/ -> [0, oo), such that W{t,0) = 0, VKi(|x|) < W(t,x), and the derivative of W along solutions of y' = g(t,y) satisfies W'(t,y)<-Z(t,y).

(6-4.2)

6.4. NONLINEARITIES AND AN ANNULUS ARGUMENT

199

Theorem 6.4.1. Let the conditions in the last paragraph hold and let Jo |F(i)| dt < oo. Suppose there are constants c\ and c2 with 0 < c\ < 1 and L < c-2 such that ft-s

/

Jo

|p(w + s, s, x(s))| du < c\Z{s, x(s))/c2

if 0 < s < t < oo and x : [0, oo) —> £/ is continuous. Then for each to > 0 and each e > 0 there exists rj > 0 suci that if Jo°° |F(t)| dt < 77 and |0(t)| < rj on [0, to], then any solution x(t) = x(t,to,4>) °f (6.4.1) on [to, 00) satisfies |x(t)| < e for t > to- -ff, Jrj addition, Z(t,x) > 6|g(t,x)| for some b > 0 and if for each xo G U — {0} there exists S > 0 and a continuous function h : [0,oo) —> [0,oo) with Z(t,x) > h(t) for |x — xo| < (5 and J o /i(t) dt = 00, then x(t) —> 0 as t —> 00. (Note that h depends on x 0 .) Proof. Define a functional

V(t, x(.)) = exp I" - (1 + L) I \F(s)\ds 1 | l + W(t, x(t))

fts

/"* r

1 l

+ / \ciZ(s,x(s)) - c2 / p(u + s,s,x(s))| du ds > , Jo I Jo \ ) so that by the Lipschitz condition on W we have

V('6.4.i)(t,x(.)) < I - Z(t,x) +L\F(t)\ + Cl Z(t,x) + ( L - c 2 ) /" |p(t,s,x(s))|ds ^0

- (1 + L)\F(t)\ J exp [ - (1 + L) y |F(s)| dsj < |-(l-ci)Z(t,x)-(c2-i)

/" |p(t,s,x(s))|ds

- I F ^ l j e x p ^ l + ^y'lF^ldt] or

V ( ' 6 . 4 .i)(*,x(-))<-Jz(t,x) + y |p(t, S ,x( S ))|ds+|F(t)|] for some u. > 0.

(6.4.3)

200

6. STABILITY AND BOUNDEDNESS

Because W is positive definite and WLi2At,y) follows that Z(t, 0) = 0. Thus because

< —Z(t,y) < 0, it

,t-s

I Jo

\p(u + s,s,x(s))\du

< ciZ(s,x(s))/c 2 ,

we have p(i, s, 0) = 0. Hence, if to > 0 and if |(s)| is sufficiently small on [0,i0], then

y(to,0(O)<(l+Oexp[(-l+i)^"|F(S)|dS], where r is an arbitrarily small preassigned number. Thus, if t > to, then {1 + W{t,x(t))}

exp \-(l

+ L)J

\F(s)\ds]

) <{l + r}exp\-(l+L)

f

\F(s)\ds],

so that 1 + W(t,x(t))

< { l + r } e x p \(1 + L) I

\F(s)\ds]

< { l + r}exp[(l+i)r/]. Then Wi(|x(t)|)<W(t,x(t)) < {l + r}{exp[(l+L)r ? ]} - 1 ^ 0 as r —> 0 and r/ -^ 0. Thus, for r and r\ small, |x(t)| < £ if t > toWe now show that the additional assumptions imply that x(t) —> 0. For in that case we have V ( ' 6 . 4 . 1) (t,x(.))<-/2[|x / (t)|+Z(t ) x)] for some p, > 0. Let |x(t)| < e on [to, 00) and suppose x(i) -^ 0. Then there exists a 5 > 0 and a sequence {£„} with |x(t n )| > 5 and i n —> 00. First, if x(t) has a limit y, then for large t we have Z(t,x) > h(t), so that l / '(t,x(-)) < —p,h(t). An integration sends V to —00. If x(i) does not have a limit, then there is a sequence {Tn} with |x(t n ) — x(Tra)| > q

6.4. NONLINEARITIES AND AN ANNULUS ARGUMENT 201 for some q > 0 and Tn —> oo. We may suppose tn < Tn < tn+\. Thus, if t > Tn, then

V(tM-))
f ' *'(*)<** Jt

<

+ V(t0,4>(-))

- pnq —> - o o

as n —> oo, completing the proof. Exercise 6.4.1. Define

V(*,x(-)) = jexp - ( 1 + L ) J \F(s)\ds r

*

x <^ 1 + W(t,x) +c / / I Jo Jt

| |p(«, s,x(s))\duds

i

*> , )

differentiate V along a solution of (6.4.1), and then reformulate Theorem 6.4.1 to suit this Liapunov functional. Prove your new result. Exercise 6.4.2. In Eq. (6.4.1) let F(t) = 0, let Wi(|x|) < W(t,x) < W2(\X\), let W('6A2)(4,y) < -W3(\y\), and define ft

/-OO

V(t,x(-)) = W(t,x)+c / / Jo Jt

\p(u,s,x(s))\duds.

Using this V, formulate a theorem for (6.4.1) yielding (a) stability, (b) uniform stability, (c) asymptotic stability, and (d) uniform asymptotic stability. [Refer to Theorem 2.5.1.] Exercise 6.4.3. Consider the scalar equation rt

x' = -2x + x3 + / [l + (tJo

sf]~1x2(s)

ds .

Construct the V of Exercise 6.4.1 and decide if the zero solution is stable.

202

6. STABILITY AND BOUNDEDNESS

Theorem 6.4.1 is one of the weakest forms for an equation like (6.4.1). In asking w(6.4.2)(*.y)

, y)

with W positive definite and Z > 0, we ask only that the zero solution of (6.4.2) be stable. As we demand more and more of (6.4.2), converse theorems will bestow more and more properties on W. In Theorem 6.4.1 we ultimately demanded that Z(t,x) be related to |g(i,x)| to obtain V

< -M[|X'|] ,

and thus, to obtain an annulus argument. We noted in Section 6.3 that such a relation is sometimes hard to obtain and we now present yet a third technique for accomplishing the annulus argument. Consider the system x' = g(i,x) + f p(t, s,x(s)) ds (6.4.4) Jo with g and p continuous, g : [0, oo) x U —> Rn, p : [0, oo) x [0, oo) x U —> Rn, and U = {x e Rn : |x| < e, e > 0}. Let P(i,x(-)) be a continuous functional when 0 < t < oo and x : [0, oo) —> U is continuous. Definition 6.4.1. The scalar functional P(i,x(-)) satisfies a one-sided Lipschitz condition with constant L > 0 if, whenever x : [0, oo) —> U is continuous and 0 < t\
(6.4.5)

P(i 2 ,x(-))-P(*i,x(.)) > L ( t i - t 2 ) .

(6.4.6)

or Example 6.4.1. Let f/ be the interval (—1,1), C(t, s) be a scalar function continuous for 0 < s < t < oo, and define P(t,x{-))=

/ Jo it

|C(u,s)|cHz(s)|ds.

Then / |C(i,s)||x(s)|ds, Jo so that if Jt°° |C(u,t)|du is bounded, then (6.4.5) is satisfied, whereas if Jo \C(t,s)\ds is bounded, then (6.4.6) is satisfied. \C(u,t)\du\x\-

6.4. NONLINEARITIES AND AN ANNULUS ARGUMENT 203 Definition 6.4.2.

The functional P(t,x(-)) is positive semidefinite if

P(t, x(-)) > 0 for 0 < t < oo and x : [0, oo) -> U and if for each tQ>0 and each 7] > 0 there is a S > 0 such that if

U and \<j>(t)\ < 5, then P(t0, 0(-))
Definition 6.4.3. The functional P(t, x(0) is decrescent on [0, oo) x U if there is a wedge W4 such that for each e > 0 and each to > 0, if(f> : [0, to] —> U is continuous and satisfied \(s)\ < e on [0, to], then P(t0,
(6.4.7)

with P satisfying Definitions 6.4.1 and 6.4.2, W7 : [0, 00) x U —> [0, 00) continuous and locally Lipschitz in x, Wi(|x|)<W(i,x)<W 2 (|x|), and V{6AA)(t,X(-))

<-W3(\X\)

(6.4.8)

for wedges W\, W2, and W3. Then the zero solution of (6.4.4) is asymptotically stable. If we replace (6.4.8) by V(' 6 .4.4)(*.X(-))
and if P is decrescent, then x = 0 is uniformly stable. Proof. To show that x = 0 is stable, let e > 0 and to > 0 be given. We must find 5 > 0 such that : [O,to] —> U with \4>{s)\ < 5 on [O,to] and t > t0 imply |x(i,i o ,<^)| < £ Use Definition 6.4.2 to find 5 > 0 such that \4>(t)\ < 5 on [O,to] implies W2(6)+P(t0,(-)) < Wx(e). Then, if \<j>{t)\ < 6, if x(4) =x(t,to,), and if t > to, we have V < 0, so that Wi(|x(i)|) < F(t,x(-)) < V(io,0(O) - /"V 3 (|x( S )|)d S ./to

<W2(|0(io)|) + P(*o,^(-))<Wri(e) and we conclude that |x(i)| < e. Thus, x = 0 is stable. Let (6.4.8) hold and let x(i) = x(t,to,<j>) where |0(i)| < 5 on [0,i0]We suppose x(i) -^ 0 as t —> 00. Then there is a /i > 0 and a sequence {in} — 00 with |x(i n )| > fi. To be definite, we suppose (6.4.5) holds.

204

6. STABILITY AND BOUNDEDNESS

Now determine a > 0 so that Wi(fj.) > 2W2(a). Because V'(t,x(-)) < —W3(|x|) there is a sequence {Tn} —> oo with |x(Tn)| < a. In fact, we may suppose |x(Tn)| = a, |x(i n )| = /i, and a < |x(i)| < \i if tn < t < Tn, by renaming tn and Tn if necessary. Now P(Tn, x(.)) - P(t n , x(-)) < £(T n - t n ) .

(6.4.9)

Also, V'(t,x(-)) < 0 implies that V(t,x(-)) —> c, a positive constant, as x(i) ^> 0. Thus, |y(t n ,x(-)) — V(Tn,x(-))\ may be made arbitrarily small by taking n large. But V(tn,x(-))-V(Tn,x(-)) = W(tn,x(tn))-W(Tn,x(Tn)) + P(tn,x(-))-P(Tn,x(-))

> Wtfa) - W2{a) - [P(Tn,x(-)) - P(*n,x(-))] >2if2(Q)-w2(a)-L(rn-g or F(*n,x(-)) - V(r n ,x(.)) > T^2(a) - L(Tn - tn). As the left side tends to zero, for each r] > 0, there exists TV such that n > N implies V>V(tn,x(-))-V(Tn,x(-)) >W2(a)-L{Tn-tn) or r) + L(Tn-tn)

>W2(a)7

so that L{Tn - tn) > W2(a) -

if 77 < W2(a)/2.

V

>

W2(a)/2

Hence, for n > TV, we have

Tn-tn>W2(a)/2Ld=T.

6.4. NONLINEARITIES AND AN ANNULUS ARGUMENT 205 Because F'(t,x(-)) < -W 3 (|x(i)|), if Tn < t, then 0
f

W 3 (|x(s)|)ds


Jti


Jt

'

= V(tN,x(-)) - (n - N)TW3{a) -* -oo as n -^ oo. This proves asymptotic stability. We now prove the uniform stability. Let e > 0 be given. We must find S > 0 such that to>O,

(p: [0,to]^U

with

\
5 on [0,t0],

a n d t > t0

i m p l y \x(t,to, 0 we may find S > 0 such that W2{5) + W4(5) < Wx{e). Thus, if t 0 > 0 and

(i)| < S on [0,i 0 ], then F ' < 0, so

w^t)])

(-)) = w(to,4>(to)) + P(to,4>(-)) <W2{5) + W4(6) < Wi(e)

implying |x(i)| < e for i > io- This completes the proof. Exercise 6.4.4. In Theorem 6.4.2 replace (6.4.7) by W1(\X\)
(6-4.7)'

and prove the result. See Burton (1979a) for details. The conclusion of this exercise is that V need not be too well behaved, as long as it is bounded above and below by well-behaved functions. Note that it is the same W\ above and below V.

206

6. STABILITY AND BOUNDEDNESS

Consider again y' = g(i,y)

(6.4.2)

and suppose there is a Liapunov function W with W[6Am2)(t,y) <-cW(t,y).

(6.4.10)

Theorem 6.4.3. Consider Eq. (6.4.4) and suppose there is a function W : [0, oo) x U -> [0,oo) with Wi(|x|) < W(t,x) < W2(\x\), \W{t,xx) W(t,*2)\ < L\*i — X 2| on [0,oo) x U for some L > 0, and suppose W satisfies (6.4.10) for some c > 0. Suppose also that Jt \p(u, s, x(s))| du is defined for 0 < t < oo whenever x : [0, oo) —> (7 is continuous. (a) If tiiere is a wedge W3 with O

-cW(t,x) + Lj

|p( U) t,x(t))|dw<-W 3 (|x(t)|)

and if />t

/-OO

/ / |p(u, s,x(s))| duds Jo it satisfies Definition 6.4.2, then x = 0 is asymptotically stable. (b) If ,00

-cW(t,x) + L

Jt

\p(u,t,x(t))\du<0

and if t>t

/-OO

/ / |p(w, s,x(s))| duds Jo Jt satisfies Definition 6.4.3, then x = 0 is uniformly stable. Proof. Define ft

V(t,x(-)) = W(t,x)+L

/.OO

/ / io it

|p(u,s,x(s))|duds,

so that V(64i)(t,x(-))<-cW(t,x)+L

I

\p(t,S,x(S))\ds

Jo

+L f Jt

\p(u,t,x(t))\du-L

[

Jo

\p(tjS,x(s))\ds

o

<-cW(t,x)+L

Jt

\p(u,t,x(t))\du.

6.4. NONLINEARITIES AND AN ANNULUS ARGUMENT

207

If (a) holds, then V < -W3(\x(t)\), but if (b) holds, then V < 0. The conditions of Theorem 6.4.2 are readily satisfied. In Chapter 5 we employed functionals for x'

=Ax+

ft

/ C(t, s)x{s) ds Jo

of the form V(t,x(-)) = x T B x + / / Jo Jt

\C(u,s)\du\x(s)\2ds.

The quadratic term xTBx resulted in the requirement that rt

oo

/

\C(u,t)\du+

/ \C(t,s)\ds Jo be small. That is, both coordinates of C are integrated. When our functional uses [xTBx]1^2, then only Jt°° \C(u,t)\du is required to be small, at the expense of requiring that B be positive definite. In Chapter 2 we used a Razumikhin-type argument to place the smallness burden on Jo \C(t,s)\ds with B positive definite. We now present a modified Razumikhin argument that places the smallness requirement on the second coordinate of C. Consider again x' = g(i, x) + / p(t, s, x(s)) ds (6.4.4) Jo with g and p continuous, g : [0, oo) x U —> Rn, p : [0, oo) x [0, oo) x U —> Rn, and U = {x e Rn : |x| < e, e > 0}. Let W : [0, oo) x U -> [0, oo) with \W(t,yn) - W(t,x2)\ < £|xi - x 2 | on [0,oo) x U for L > 0, Wi(|x)| < W(t,x), W(t,0) = 0, W[6A_2)(t,y) < -Z(t,y), where Z : [0,oo) x U -> [0, oo) is continuous. Theorem 6.4.4. Let the conditions of the preceding paragraph hold. Suppose there is a continuous function q : [0, oo) x U —> [0, oo) with \p(t, s,x(s))|/W / (s,x(s)) < q(t,s) if x(-) is any continuous function in U and if 0 < s < t < oo. Also suppose that Z(t,x) > cW(t,x) for some c > 0 and that there are constants c\ and C2 with 0 < c\ < c and C2 > L, so that jQ q(t, s) ds < C1/C2 if 0 < t < oo. Then for each to > 0 and each e > 0 there exists 5 > 0 such that if \(t)\ < 5 on [0, to] and x(i, to, ) is a solution of (6.4.4), then \x(t,to,4>)\ < e f°r t > to-

208

6. STABILITY AND BOUNDEDNESS

Proof. Define V(t,x(-)) = W(t,x)+ f \ClW(u,x(u))-c2

I

\p(u,s,x(s))\ds]du,

so that along a solution x(t) of (6.4.4) we have V'(t,x(-))<-cW(t,x)+L f \p(t,s,x(s))\ds Jo ft

+ aW(t, x) - c2 / \p(t, s, x(s))| ds < 0 Jo if x 7^ 0. Because V is not necessarily positive, boundedness of x(i) may not yet be concluded. Suppose there is a solution x{t) in U on [ioiT] with the property that W(s,x(s)) < W(T,x(T)) if 0 < s < T. Then W'(t,x(t)) > 0 at t = T so that V'(T, x(-)) < 0 implies that, for t = T, we have (d/dt) I \c1W(u,x(u))-c2 Jo I = ClW(T,x(T))-c2

f Jo

\p(u,s,x(s))\ds\du J

f |p(T,s,x(s))|ds<0. Jo

This will be a contradiction because we have f \p(T,s,x(s))\ds= f Jo Jo

[W(s,x(s))\p(T,s,x(s))\/W(s,x(s))]ds

<W(T,x(T)) f q(T,s)ds Jo <W(T,x(T))Cl/c2. Because W is continuous, W(t,0) = 0, and W(t,x) > Wi(\x.\), the result now follows. Exercise 6.4.5. Construct another alternative type of Razumikhin result. Consider the system

r*

x' = Ax + / C(t, s)x(s) ds Jo with A stable and find B = BT satisfying ATB + BA = -I and |a|x| < [K^BX]1/2 < [3\x\ for a and /3 positive. Take c2 > f3K/a and k/c2 >

6.4. NONLINEARITIES AND AN ANNULUS ARGUMENT

209

JQ \C(U, S)\ ds if 0 < s < u < oo with k and K defined in Theorem 2.5.1. Define V(t,x(-)) = [xTBx\V2+

f

\k-c2

f

\C(u,s)\ds]\x(u)\du,

so that V'{t,x(-))
[ \C(t,s)\\x(s)\ds Jo

- c

2f Jo

\C(t,s)\ds\x\

ft

<{K/a)

/ Jo

\C(t,s)\[xT(s)Bx(s)]1/2ds

-{c2/(3)[xT(t)Bx{t)\ll2

I Jo

\C(t,s)\ds.

Argue that if x T (s)Bx(s) < x T (T)Bx(T) for 0 < s < T, then [x r (t)5x(t)]' > 0 at T and V'(t,x.(-)) < 0 at T, so that (d/dt)

f

\k-c2

f

\C(u,s)\ds]\x(u)\du<0

at T, a contradiction. Argue now that this yields stability. What must be added for asymptotic stability? In his monograph Yoshizawa (1966, pp. 118-153) showed in great detail how the existence of a Lipschitz-Liapunov function with a negative definite derivative implied stability under many types of perturbations. The same types of results hold for Volterra equations, and owing to the greater complexity of functions, there is even more variety. We showed in Theorem 2.5.1 that under certain conditions the derivative of V(t,x(-)) = \y.TBy.}1'2 + K f

f

Jo Jt

along solutions of x' = Ax + / C(t, s)x(s) ds Jo satisfies y ' ( i , x ( - ) < - / i [ | x | + |x'|],

/i>o.

\C(u,s)\du\x(s)\ds

210

6. STABILITY AND BOUNDEDNESS

Because V is Lipschitz in x(t), V will also be negative along solutions of nt

ft

x' = , 4 x + / C(t,s)x(s)ds + D(t,s)r(x(s))ds Jo Jo + q(4, x) + H(i, x(-)) + p(t, x)

(6.4.11)

under the following assumptions: (i) |H(i,x(-))| < /3|x(t)| \f*E(t,s)m(x(s))ds\ m defined below.

where (5 > 0, with E and

(ii) D and E are continuous, n x n matrices on [0, oo) x [0, oo) with

\D(t,s)\ < a\C(t,s)\ and \E(t,s)\ < a\C(t,s)\ for some a > 0 and 0 < s < t < oo. (iii) q : [0, oo) x U -> Rn is continuous, U = {x G Rn : |x| < e, £ > 0}, and |q(i,x)| / |x| —> 0 as |x| —> 0 uniformly for 0 < t < oo. (iv) r and m : U —> i?" are continuous, |r(x)|/|x| —> 0 as |x| -^ 0, and m(x)| < u>\x\ for some u> > 0. (v) p : [0, oo) x U —> i?" is continuous and |p(t,x)| < A(t)|x| where A : [0, oo) —> [0, oo) is continuous and Jo°° \(t) dt < oo. (vi) A is an n x n constant matrix whose characteristic roots all have negative real parts and there is a matrix B = BT satisfying ATB + BA = -I, |x| > 2fc[x r Bx] 1 / 2 , |Bx| < i f j x ^ x ] 1 / 2 , and 7 | x | < [x^Bx] 1 / 2 for 7, k, and K positive. (vii) C(t,s) is continuous for 0 < s < t < oo and Jt°° \C(u,s)\du is continuous for 0 < s < t < oo. (viii) There exists K > K and k > 0 with k < k - K f™ \C(u, t)\ du. Theorem 6.4.5. Consider (6.4.11) and suppose that (i)-(viii) hold. Then for each e > 0 and each to >0 there exists 5 > 0 such that if \(f>(t)\ < S on [0, to]j then |x(t, to, 0 as i —> oo. Proof. Define F(i,x(-)) = j ^ S x ]

1

/^^

r x exp \ — L

L

/* /* *

|C(u,s)|du|x(s)|ds|

i \(s) ds ,

^o

J

6.5. A FUNCTIONAL IN THE UNSTABLE CASE where L satisfies [xTBx]x/2L V('6.4.11) f

211

> K\x\. Then

) r

< \ - LX(t)[xTBx}1/2

-iK-K) +

_

roo

- \k-K

i

\C(u,t)\du

f \C(t,s)\\x(s)\ds + K f

Jo tf|q(i,x)|+*:|H(i,x(.))|

+ K\p(t,x)\\exp\-L < \ -k\x\-(K-K)

I

Jo

Jo

|x

\D(t,s)v(x(s))ds

f X(s)ds] \C(t,s)\\x(s)\ds

+ Ka f \C(t,s)\\r(x(s))\ds + K\
+ Kj3aco\x\ / |C(i,s)||x(s)|ds U x p - i / A(s)ds .

Jo

J

L

io

J

By the conditions on q and r, this is nonpositive for |x(-)| small. Thus

[ x T ( 4 ) B x ( t ) ] 1 / 2 e x P | " - L j r \(s)ds]



and because V(to, 0 as |^>| —> 0, we see that the zero solution is stable. The proof that solutions tend to zero proceeds as in Theorem 2.5.1. Exercise 6.4.6. Show that Theorem 6.4.5 is still true if we add the term b(i,x) to (6.4.11), where b : [0, oo) x U -> Rn is continuous, |b(t,x)| < a(t) with a : [0, oo) —> [0, oo) continuous, and Jo°° a{t) dt < ao where ao is small. We then speak of the stability of the zero function, because x = 0 is not a solution. Most of the results of this section are found in Burton (1979a,c, and 1980a).

6.5

A Functional in the Unstable Case

The construction in Sections 6.3 and 6.4 are based on the assumption that y1 = g{t,y)

212

6. STABILITY AND BOUNDEDNESS

is stable and that x' = g(t, x) + I p(t, s, x(s)) ds (6.5.1) Jo will inherit that stability for small p. If we pursue the ideas developed in Chapter 5, we obtain stability from a combination of the properties of g and p. Our discussion here centers on the scalar case. Thus, we consider (6.5.1) with g and p continuous, g : [0, oo) x U —> i?, p : [0, oo) x [0, oo) x U —> R, and U = {x G R : \x\ < e , e > 0 } . We suppose there is a function P(t, s, x) with dP(t, s, x)/dt = p(t, s, x), so that (6.5.1) may be written as x' = Q(t, x) + (d/dt) f P(t, s, x(s)) ds , Jo

(6.5.2)

where Q(t,x)=g(t,x)-P(t,t,x).

(6.5.3)

Let L : [0, oo) x U —> [0, oo) be continuous and define

V(t,x(-))= \x- I

P(t,s,x(s))ds]

+ / / \P(u,s,x(s))\duL(s,x(s))ds (6.5.4) Jo Jt under the assumption that Jt \P(u,s,x(s))\du is continuous for 0 < s < t < oo and all continuous x(t) in U. Then ^(6.5.2)(*>*(-)) = 2 [ a : - ^

P(t,s,x(s))ds^Q(t,x)

O

+ / Jt

\P(u,t,x(t))\duL(t,x)

- f \P(t,s,x(s))\L(s,x(s))ds, Jo so that VL52)(t,x(-))<2xQ(t,x) + 2\Q(t,x)\ f \P(t,s,x(s))\ds Jo O

+ / Jt

\P(u,t,x(t))\duL(t,x)

- / \P(t,s,x(s))\L(s,x(s))ds Jo

(6.5.5)

6.5. A FUNCTIONAL IN THE UNSTABLE CASE

213

from which stability results are readily drawn, as in Chapter 5. The basic assumption must be that xQ(t,x)<0.

(6.5.6)

We then distinguish three cases: lim |Q(t,x)|/|a;| = 0

uniformly for 0 < t < oo,

(6.5.7)

|a|—»0

\Q(t,x)\/\x\>A>0

for 0 < i < o o

a n d all x G U,

(6.5.8)

and lim \x\/\Q(t,x)\ = 0 uniformly for 0 < t < oo .

(6.5.9)

\x\—>0

Theorem 6.5.1. Consider the scalar equation (6.5.2) with (6.5.3) and (6.5.6) holding. Suppose there is a constant M > 0 such that

f \P(t,s,x(S))/Q(s,x(s))\ds<M Jo if 0 < t < oo and x is any continuous function in U. Also suppose that there is an a < 2 and a wedge W\ such that Wi(\x\)/\x\ —> 0 as \x\ —> 0, Wi(|z|) > \Q(t,x)\, and /'OO

/

\P(u,t,x(t))\du>a\x(t)\

for any continuous x(t) in U and 0 < t < oo. then the zero solution is stable. Proof. Define L(t,x) = \Q(t,x)\ so that (6.5.5) yields

^(6.5.2)(t>x(-)) <2xQ(t,x)+ f \P(t,s,x(s))/Q(s,x(s))\ {Q2(s,x(s)) Jo 2

+ Q (t,x(t)))ds+ I Jt

\P(u,t,x(t))\du\Q(t,x)\

- f \P(t,s,x(s))\\Q(s,x(s))\ds Jo < 2xQ(t,x) + Q2(t,x(t))M + \Q(t,x)\ [ \P(u,t,x(t))\du Jt < 2xQ(t,x) + Q2{t,x)M + a\x\ \Q(t,x)\ <0

214

6. STABILITY AND BOUNDEDNESS

if \x\ is small enough. Now / P(t,s,x(s))ds Jo

< f Jo

\P(t,s,x(s))/Q(s,x(s))\\Q(s,x(s))\ds

Jo Suppose that a given solution x(t) = x(t,to,(f>) satisfies \4>(t)\ < 5 on [0,io] f° r some S > 0, but there is a t\ > to with \x(ti)\ > \x(s,to,(j>)\ if 0 < s < ti. Then x(h)-

f P{ti,s,x{s))ds Jo

>\x(h)\-

[ Jo

\P(tus,x(s))\ds

> \x(h)\

f*1 Pfas^js))

-Jo

QMs))

WMs)l)ds

> k(tl)l [l - (Wl(k(il)l)/k(tl)l) X

^ pfrM*)) ds] JO QM*)) I

> x(tl)\[l > x(h)\/2

(W^xihW/lxih^M]

if |x(ti)| < p, for some p > 0, because VKid^D/jxl — 0 as \x\ —> 0. Also, if 0 < e < p and to > 0, there is a 5 > 0, S < e, such that 4> : [0,t0] -^ R and \cj)(t)\ < 5 on [0,t0] imply V(to,(/>(-)) =

(p(to) o

I

P(to,s,(s))ds

/-oo

+ / / Jo Jtlt

\P(u,s,ct>(s))\du\Q(s,(s))\ds<e2/A.

Suppose that t\ > to has the property that \x(t,to,)\ < \x(ti,to,)\ = £ if 0 < t < t1. Then for x(t) = x(t, t0, 4>) we have z 2 (ti)/4
6.5. A FUNCTIONAL IN THE UNSTABLE CASE

215

Theorem 6.5.2. Let (6.5.2) be a scalar equation with (6.5.3), (6.5.6), and (6.5.8) holding. Suppose that for x in U and 0 < t < oo there are positive constants R, N, and M < 1 with / Jo

\P(t,s,x(s))/x(s)\ds<M, \Q(t,s)\
Jo

\P(u,t,x(t))\du
and -2A + MN2 +

R<0.

Then the zero solution of (6.5.2) is stable. Proof. Let L(t,x) = \x\ and have V{652)(t,x(-))<2xQ(t,x)+2\Q(t,x)\

f

\P(t,s,x(s))\ds

Jo rt

oo

/

\P(u,t,x(t))\du\x\-

/ Jo

\P(t,s,x(s))\\x(s)\ds

ft

<2xQ(t,x)+ / \P{t,s,x(s))/x(s)\(Q2(t,x) + x2(s))ds Jo ft

oo

/

\P(u,t,x(t))\du

-

/ Jo

\P(t,s,x(s))\\x(s)\ds

O

< 2xQ(t,x)+MQ2(t,x) + \x\ /

\P(u,t,x(t))\du

Jt

< -2Ax2 + MN2\x\2 + Rx2 < 0,. Next, we note that

I f P(t,s,x(s))ds I ./o

< f

\P(t,s,x(s))/x(s)\\x(s)\ds

^o < M max x(s)\. Because M < 1, the stability follows as in the proof of Theorem 6.5.1. Exercise 6.5.1. Formulate L(t,x) and the appropriate stability result when (6.5.9) holds.

216

6. STABILITY AND BOUNDEDNESS

We will return to many of these stability questions for functional differential equations in Chapter 8. Avoiding the Marachkov condition will occupy almost all of Section 8.3 and we will see several interesting ways of doing so. In Section 8.7 we will avoid the Marachkov condition by using one Liapunov function and one Liapunov functional.

Chapter 7

The Resolvent 7.1

General Theory

We briefly mentioned the resolvent in Section 2.3. It is used to obtain a variation of parameters formula. We noted, in the convolution case, that it was quite effective for dealing with perturbations because it employs the solutions of the unforced equations about which we frequently know a great deal. The nonconvolution case presents many new difficulties. But recently we have been able to make some good progress through use of fixed point theory and Becker's form of the resolvent. In preparation for later work we will give a brief sketch of the resolvent for both integral and integrodifferential equations. Some of the results are only stated and the proofs are left to the references. Given the integral equation x(t) = f(*)+ / C(t,s)x(s)ds (7.1.1) Jo with f : [0, a] —> Rn being continuous and C continuous for 0 < s < t < a, we define the formal resolvent equation as R(t,s) = -C(t,s)+

R(t,u)C(u,s)du.

(7-1-2)

Js

Assuming that a solution R(t, s) exists as a continuous function for 0 < s < t < a, we note that x(£) may be found with the aid of R(t, s) to be x(t) = f(t)-

/ R(t,u)f(u)du,

(7.1.3)

Jo a variation of parameters formula. 217

218

7. THE RESOLVENT

To verify (7.1.3), left multiply (7.1.1) by R(t,s) to t: / R(t,u)x(u)du-

Jo

,t

/ R(t,u)i(u)du

Jo ru

= / R(t,u) /

Jo

,t

and integrate from 0

C(u,s)x(s)dsdu

Jo

,t

=

R(t,u)C(u,s)dux(s)ds Jo Js ft

= / [R(t,s)+C(t,s)]x(s)ds Jo by (7.1.2). Thus ,t

,t

- / R(t,u){(u)du= / C(t,s)yi(s)ds,

Jo

Jo

which, together with (7.1.1) yields x(t) = f ( i ) - / R(t,u){(u)du, Jo as required. Miller (1971a, p. 200) shows that / R(t,u)C(u,s)du=

/

Js

C(t,u)R(u,s)du,

Js

so that (7.1.2) may be written as R(t,s) = -C(t,s)+

C(t,u)R(u,s)du,

(7-1.4)

s

and if we replace t by t + s in (7.1.4), we have j-t+s

R(t + s,s) = -C(t + s,s)+ /

C(t + s, u)R(u, s) du

Js

or r* R(t + s, s) = -C(t + s, s) + / C(t + s,u + s)R(u + s, s)du.

(7.1.5)

Jo

In this form s is simply a parameter and we may write (7.1.5) as *

L(t) = -D(t)+

C(t + s,u + s)L(u)du, (7.1.6) Jo and the proof of existence and uniqueness may be applied directly to it.

7.1. GENERAL THEORY

219

Equation (7.1.2) is conceptually much more complicated than (7.1.1). Thus, one is often inclined to believe that more progress can be made by attacking (7.1.1) directly without going through a variation of parameters argument. We have already indicated that, in the case of an integrodifferential equation, one may use differential inequalities and Liapunov functionals to bypass the resolvent. Nevertheless, much has been discovered about (7.1.2), both theoretically and technically. The interested reader is referred to Miller [(1968), (1971a, Chapter IV)], Nohel (1973), Becker (1979), and Corduneanu (1971). In particular, when (7.1.1) is perturbed with a nonlinear term, then (7.1.4) can be used to rewrite the equation into a much more manageable form. Recall that the ordinary differential equation x =Ax + f (t, x) may be expressed as rt

x{t) = eAtx0 + / eA{t~s)i(s, x(s)) ds . Jo

Similarly, the solution of

- Jof where h is an appropriate functional, may be expressed with the aid of (7.1.3) as x(4) = h ( t , x ( - ) ) - I Jo

R(t,u)h(u,x(-))du.

For special functionals this may be simplified, as may be seen in Miller (1971a, Chapter IV). Whereas we have seen that integro-differential equations may be expressed as integral equations, there are certain advantages to considering resolvents of integro-differential equations directly. We consider

r* x'(t) = f(t)+A(t)x(t)+

B(t,s)x(s)ds,

x(0)=x 0 ,

(7.1.7)

Jo

in which f : [0, a] —> Rn is continuous, A an n x n matrix continuous on [0,a], and B an n x n matrix continuous for 0 < s < t < a.

220

7. THE RESOLVENT

Then we seek a solution R(t, s) of the formal resolvent (or adjoint equation) /"* Rs(t,s) = -R(t,s)A(s)-

R(t,u)B(u,s)du,

R(t,t) = I,

(7.1.8)

Js

on the interval 0 < s < t. (Here Rs = dR/ds.) A proof of the existence of R may be found in Grossman and Miller (1970). Given R(t,s), the solution of the initial-value problem (7.1.7) is given by x(t) = R(t,O)xo + / R(t,s)t(s)ds, Jo

(7.1.9)

a variation of parameters formula. Assuming the existence of R(t,s), (7.1.9) may be verified as follows. Let x(t) be the solution of (7.1.7) and integrate by parts. We have / R(t,s)yL'(s)ds = R(t,s)x(s)\ssZl-

Jo

Rs(t,s)x(s)ds

Jo

or rt

/ [R{t, s)x'(s) + Rs(t, s)x(s)] ds = R(t, i)x(i) - R(t, 0)x0 Jo = x(t)-i?(t,O)xo as R(t,t) = I. Now, because x(i) satisfies (7.1.7) we write this as x(t) = R(t,0)xo+ / \R{t,s)\f{s)+A(s)x.{s)+

Jo I

L

Jo

B(s,u)x(u) du\

J

+ i? s (t,s)x(s)ids. Changing the order of integration we have /"* fs

/"* /"*

/ / R(t,s)B(s,u)x(u)duds= Jo Jo

/ Jo Ju ,-t

R(t,s)B(s,u)x(u)dsdu

,t

=

R(t,u)B(u,s)x(s)duds. Jo Js

Then x(t) - i?(t, 0)x0 - /

Jo

R(t,s)f(s)ds

ft l ft 1 = / i?(t, s)A(s) + i?s(t, s) + / R(t,u)B(u,s)du x(s)ds. Jo I Js J The integral on the right is zero according to (7.1.8), so (7.1.9) is verified.

7.1. GENERAL THEORY

221

If (7.1.7) is perturbed by a nonlinear functional, then (7.1.9) may simplify the equation. Proceeding formally again, if R(t,s) satisfies (7.1.8), then the solution of x'(t) = f(t)+A(t)x(t)+

/"* / B(t, s)x(s) ds+h(t, x(-)), Jo

x(0) = x 0 , (7.1.10)

for an appropriate functional h, may be expressed by (7.1.9) as x(t) = R(t,0)x0 + / R(t,s)[h(s,x(-)) + f(s)]ds. Jo

(7.1.11)

Such results are considered in detail by Grossman and Miller (1970). But Becker (1979) took a different view. We noted that (7.1.2) is conceptually much more complicated than (7.1.1) and the same case can be made that (7.1.8) is more complicated than (7.1.7). Becker's idea was that R(t, s) could be obtained from (7.1.7) with f = 0 and the lower limit changed. This would allow many of the same techniques used to derive information from (7.1.7) to obtain information about R. Indeed, that is exactly the case for convolution constant coefficient equations and it proves to be correct here. We follow his presentation, but more detail and applications can be found in his work. Return to (7.1.7) with a view to showing that its solution can be expressed in terms of solutions of y'(i) = A(t)y(t) + f B(t, u)y(u) du

(7.1.12)

Js

where 0 < s < t < oo. We will show that for each s > 0 there is a unique n x n matrix Z(t,s) such that Z(s,s) = I which satisfies (7.1.12). A particular solution of (7.1.7) will then be expressed in terms of this matrix and / . In fact, it will be true that Z(t, s) = R(t, s). We will use a contraction mapping argument here. A proof of the contractive mapping principle was given in Section 3.1. Proposition 7.1.1. The solution x(t) of ft

x'(t) = A(t)x(t) +

B(t,u)x(u)du, s

is unique and exists on [s, oo).

x(s)=x 0 ,

(7.1.13)

222

7. THE RESOLVENT

Proof. Write (7.1.13) as *r x ( i ) = x

o

+ / is

r

L4(w)x(f) + / L Js

\dv J

= xo + / ^4(w)x(w) du + / Js

i B(v,u)yi(u)du

/ B(v,u)dvx(u) du

Js Ju

r* r * i = x 0 + / L4(u) + / B(v,u)dv \x(u)du Js L i« J where we have changed the order of integration. For a given T > s and an n x n matrix C(i, w) defined and continuous for s < u < t < T, we define a matrix norm by \C\ to be sup s < u < t < T ,| x | < 1 \C(t,u)x.\. Find a number r with A ( u ) + / B ( w , u ) dv < r - l , Ju

s < u < t < T .

Let (M, | | r ) be the complete metric space of continuous functions : [s, T] —* Rn with 4>(s) = xo and with the metric induced by the norm \cP\r:=

sup

\cP(t)\e-rt-

s
Define ~P : M ^ M by 4> e M implies that

(P)(i) = x 0 + j

f

\A(U)+

B(v,u)dv\
It is clear that P(fi is continuous and we will show that P is a contraction. To see that P is a contraction, let , rj e M. Then

(P0)(t) - (Pr/)(t)| e - rt < f\r - l)e-rt+ru\4>(u) - V(u)\e-ru du Js ft

< \4>~v\r / {r - l)e-r(t-u)

du

Js


f* B(t,u)Z(u,s)du, JS

Z(s,s) = 1. (7.1.13)

7.2. A FLOQUET THEORY

223

Theorem 7.1.1. Tie solution of (7.1.7) such that x(0) = xo is given by the variation of parameters formula x(t) = Z(t, O)xo+

Z(t,s)f(s)ds.

(7.1.14)

Jo Proof. Define y : [0,T] -> Rn by y(t) = /0* Z{t, s)f(s) ds. Differentiating and using (7.1.13), we have y'(t)=Z(t,t)f(t) + j

^Z(t,s)f(s)ds

= /f(t)+ / L4(t)Z(i,s)+ / B(t,u)Z(u,s)du )f(s)ds Jo \

Js

)

= {{t)+A{t) f Z{t,s)f{s)ds+ f Jo

= f(£) + A(t)y(t) + f f Jo Jo

B(t,u)Z(u,s)f(s)duds

B{t,u)Z{u,s)i{s)dsdu

= f(t) + A{t)y(t) + [ B{t,u) f Jo Jo = A(t)y(t)+

f

Jo Js

I B(t,u)y(u)du Jo

Z{u,s)f{s)dsdu

+ f(t).

Thus, y(t) is a solution of (7.1.7) for 0 < t < T. Since T is arbitrary it follows that it is a solution for all t > s. Moreover, since Z(t, O)xo satisfies the homogeneous equation, Z(t, O)xo + y(i) is the desired solution of the nonhomogeneous equation. This completes the proof. If the lower limit of integration in (7.1.7) is replaced by r, then the solution with x(r) = xo is given by x(t) = Z(t,r)xo+

/

Z(t,s)i(s)ds.

JT

7.2

A Floquet Theory

In Section 2.6 we had a brief look at Floquet theory for ordinary differential equations. Suppose that Z(t) is the principal matrix solution of x'=^(t)x

(7.2.1)

224

7. THE RESOLVENT

so that Z'(t) = A(t)Z(t) and Z(0) = I. Then the variation of parameters formula for x' = A(t)x + f (t)

(7.2.2)

is given by Z(t)Z-1(s)i(s)ds.

x(t,0,xo) = Z(t)xo+

(7.2.3)

Jo Suppose that f (£) is bounded and we want to show that solutions of (7.2.2) are bounded. Even if we know that Z(t) —> 0 and that Z £ i 1 [0, oo), Z~1(s) has terms of Z divided by the determinant of Z so that Z~1(s) can be very large. It requires Draconian conditions on A(t) to ensure boundedness of solutions. But if A(t + T) = A(t) for all t and some T > 0, it is possible to find a constant matrix R and a periodic matrix P with Z(t) = P(t)em ,

P(0)=7.

(7.2.4)

Now, the variation of parameters formula becomes rt x(t)

= P(t)emx0+

P(t)eR(-t-s)p-1(s)i{s)ds.

(7.2.5)

Jo The critical term eRt is preserved. The matrix P is periodic and nonsingular so those terms are bounded. Thus, for bounded / we are asking that roo

/

\em\dt
(7.2.6)

Jo In the last section we studied the resolvent and saw that in the variation of parameters formula we would need to integrate Z(t, s) with respect to s, rather than with respect to t. That did not happen in the convolution case with A constant y' = Ay+ [ B(t - s)y(s) ds + f (t). Jo

(7.2.7)

For in that case we had the resolvent equation as ft Z' = AZ+ / B(t-s)Z{s)ds Jo

(7.2.8)

7.2. A FLOQUET THEORY

225

and the variation of parameters formula as y(t, 0, y 0 ) = Z(t)y0 + f Z(t - s)f (s) ds , Jo

(7.2.9)

so that for bounded / , in order to get a bounded solution we needed O

/

\Z(u)\du
(7.2.10)

Jo a condition equivalent to O

/ Jo

\ z ( t , 0 , e j)\ dt < o o ,

j = l,...,n.

The goal of this section is to show that the variation of parameters formula for periodic Volterra equations allows integration of the resolvent with respect to t instead of with respect to s except on the interval 0 < s < T. This work may be found in Becker-Burton-Krisztin (1988). Consider the system of Volterra equations ^ at

= A(t)y(t) + f B(t, s)y(s) ds + f (t) j0

(7.2.11)

in which A and B are n x n matrices, y and f are vectors, A(t + T) = A(t), and B(t + T, s + T) = B(t, s) for some T > 0. It is also assumed that A and f are continuous on (—00,00) while B is continuous for — 00 < s < t < 00. The main problem on which we focus is that of showing that (7.2.11) has bounded solutions for f bounded. We have used Z in so many contexts above, that here we will use R when denoting the resolvent. In the last section we showed that a solution of (7.2.11) can be written as y(t, 0, y 0 ) = R{t, 0)y0 + / R{t, s)f (s) ds , Jo

(7.2.12)

where R(t, s) is an n x n matrix which is the unique solution of Becker's resolvent

dR(t,s)

=A(t}R^s}+

I B{t,u)R(u,s)du,

R(s,s) = I.

(7.2.13)

226

7. THE RESOLVENT

In (7.2.12) it is a point of major concern that we are integrating with respect to s. Our goal is to follow the ideas in Floquet theory and change that into integration with respect to t as we saw in (7.2.6) and (7.2.10). In particular, we want to characterize the condition

sup/ \R(t,s)\ds < oo. *>o Jo

(7.2.14)

As a corollary we show that (7.2.14) holds if there is an E > 0 such that O

/

\R(t, s)\dt<E

for all s e [0, T ] .

(7.2.15)

Js

In preparation for the main result we first prove a special form of Sobolev's inequality. If g : [a, b] —> 2?" has a continuous derivative, then

/

(\g(u)\ + (b- a)\g'(u)\) du > (b - a) max \g(u)\.

Ja

(7.2.16)

a
A simple proof proceeds as follows. Let uo,«i € [a, 6] and m,M £ i? be defined by m=

min \g(u)\ = \g(uo)\,

M= m a x \g(u)\ = |ff(wi)|.

a
a
Then /

(\g(u)\ + (b-a)\g'(u)\)du

Ja

I fUl >(ba)m + (b-a)\ I g'(u) du I Juu > (b - a)m + (b - a)\g(ui) - g{uo)\ >(b-

a)m + (b- a)(\g(Ul)\ - \g(uo)\) = (b - a)M .

One may verify that (7.2.16) holds for n x n matrices using the induced matrix norm.

7.2. A FLOQUET THEORY

227

Theorem 7.2.1. Let A and B he continuous, R(t,s) satisfy (7.2.13), B(t + T,s + T) = B(t,s) (i)

If there is J > 0 such / \B(u,s)\du<J

and A(t + T) = A(t).

(7.2.17)

that for 0 < s
oo

(7.2.18)

Js

and /

/

JO

Js

\R(t,s)\dtds
(7.2.19)

then sup / \R{t, s)\ ds = M t>o Jo (ii)

for some

M > 0.

(7.2.20)

If there is a K > 0 such that / \B{t,s)\ds
for t > 0 ,

(7.2.21)

then (7.2.20) impiies (7.2.19).

Proof. If we integrate (7.2.13) from s to t, s < t, we obtain /" \dR{u,s)/du\du Js

<\\A\\ I \R{u,s)\du+ I I Js

\B(u,v)\\R(v,s)\dvdu

Js Js

<(||A||+J) f \R(u,s)\du, Js

where ||^4|| = maxo
228

7. THE RESOLVENT

Let t > T be fixed. There exists an integer k > 1 and r\ G [0,T) with t = kT + T). Then, ,-t

/

rkT

\R(t,s)\ds=

J0

/

rkT+r)

\R{kT + r/,s)\ds+

/

J0

|#(/cT + 77, s)| ds

JkT

,-kT p = / \R(kT + r], s)\ds+ / \R(r],u)\du Jo Jo (using R(t + T,s + T) = R(t, s) and a variable change) = / V | i ? ( i T + 77,s)|ds+ / Jo i=1 Jo

\R(r},u)\du

(by induction)
} ^ max |i?(u, s ) | d s , io ^ f f < u < ( l + l ) T ' V

where a = s u p 0 < M < T f™ \R(u, s)\ ds. Applying (7.2.16), we have

I \R(t,s)\ds


\R(u,s)\duds.

Since t is arbitrary, (7.2.19) implies (7.2.20). Now assume that (7.2.20) and (7.2.21) hold. In order to prove (7.2.19), by Fubini's theorem and the continuity of R(t, s), it suffices to show that /-co

i-T

\

I

JT

JO

\R(t,s)\dsdt < 00.

Let r(t) = / Jo

\R(t,s)\ds

for

t > T .

7.2.

A FLOQUET THEORY

229

Then for ( 2 > t , > r w e have |r(*2)-r(ti)|=

/ Jo

< f Jo = f Jo

< I Jo

(\R(t2,S)\-\R(h,s)\)ds

\R(t2,s)-R(t1,s)\ds f\dR(t,s)/dt)dt

ds

Jti

(2 \dR(t,s)/dt\dtds.

Jti

Changing the order of integration yields

|r(* 2 )-r(ti)|< (2 I \dR(t,s)/dt\dsdt / i Jo

< I 2 \\A\\ Jtt

Jtt JO A(t)R(t,s)+

<

dsdt

Js

f \R(t,s)\dsdt+ Jo

< f ' \\A\\ f \R(t,s)\dsdt+ Jti

B(t,v)R(v,s)dv

Jo

I f f

\B(t,v)\\R(v,s)\dvdsdt

Jti Jo Js

f

2

f [

\R(v,s)\ds\B(t,v)\dvdt

Jti Jo Jo

j \\\A\\M + KM) dt < (\\A\\M + KM)(t2 -

h).

Jti

This shows that r(t) is Lipschitz continuous with Lipschitz constant L =

(\\A\\+K)M. Now (7.2.20) and R(t + T,s + T) = R(t, s) imply that oo

^2r(iT + rf) < M i=l

for all r/€ [0,T). Let k > 0 be an integer. It follows from the Lipschitz condition on r that r(iT

+ rj + u)< r{iT + r?) + L(T/k)

230

7. THE RESOLVENT

for i = l , 2 , . . . , r / G [0,T), u G [O,T/kj. Thus, fcT

/ J

fc-lfc-1

,.iT+(j + l)(r/fc)

r(t)dt = J2J2

T

i=1 j=0

r dt

^

JiT+j(T/k)

k-lk-1

^ E E( T / f e )l r ( i T + 3[T/k\) + L(T/k)} < {(T/fc) ^ J2r(tT ^

+ j[T/k])\

j=0 i = l

+ (k-

l)k(T/k)L(T/k)

'

< TM + LT2 < oo . Since k is arbitrary, r e i 1 ^ , oo) and the proof is complete.

Corollary. Let A and B be continuous, R(t,s) satisfy (7.2.13), and Jet (7.2.17) and (7.2.18) hold. If there is E > 0 such that O

/

\R{t,s)\dt<E

for all s e [ 0 , T ] ,

(7.2.22)

Js

then ft

sup / \R(t, s)\ds = M for some t>o Jo is satisfied.

M > 0.

(7.2.20)

Proof. If we integrate (7.2.22) from 0 to T, the value is bounded by ET.

T h e o r e m 7.2.2. Suppose that (7.2.17) and (7.2.18) hold with O

/ Jo

\R{t,0)\dt
Then R(t, 0) -> 0 as i -> oo.

7.2. A FLOQUET THEORY

231

Proof. We showed in the proof of Theorem 7.2.1 that o

r-oo

I \dR(u,0)/du\du< (\\A\\ + J) I \R(u,0)\du. Jo Jo A similar result holds for each jth column of R(t,0), say z(t,O,ej). If the theorem is false, there is a j , an e > 0, and a sequence {tn} —> oo with |z(i ra ,0,ej)| > e . Also, ft z(t,0,ej) = ej + / z'(u,0,ej)du

Jo

so that tn < t < tn + 1 implies that z(i,0,ej)-^(in,0,ej)| < f \z'(u,0,ej)\du<e/2 Jtn, for large n. Hence |^:(t, 0, e^-) | > e/2 for tn < t < tn + 1, contradicting z(t,0,ej) G L1. This completes the proof. Example 7.2.1. Consider the scalar equation

r* z' = -A(t)z+

B(t,u)z(u)du Js

in which (7.2.17) and (7.2.18) hold with A and B continuous. If there is an a > 0 and s* G [0, T] with \B(t-s

+ s*,s*)\>\B(t,s)\

for

0 < s < t < oo

and with -A(t)+

\B(u + s*,s*)\du<-a Jo then the conditions of the corollary hold.

for all

t,

Proof. Define a Liapunov functional by t

/'OO

/

/

\B(v +

s*,s*)\dv\z{u)\du

—u

so that

V'(t,z(-))<-A(t)\z(t)\+

f

\B{t,u)\\z{u)\du

Js O

+ / Jo

\B(v +

s*,s*)\dv\z(t)\

ft -

/

\B(t - u + s*,

Js

<-a\z(t)\.

s*)\\z(u)\du

232

7. THE RESOLVENT

Hence, the single column of R satisfies 0 < V(t, z(-)) < V{s, ei) -a

\z(u, s, ei)| du J s

or v [( s 1}e-i) r°° / \z(U,s,ei)\du< =\el\/a '' Js a

=

l/a.

This completes the proof. Perhaps a more transparent way of doing such an example is to ask for a function C(t - s) with C(t - s) > \B(t,s)\

and with

O

-A(t)+

/ Jo

C(s)ds < -a.

Then V would be defined by

V(t,z(-)) = \z(t)\+ f

^

Js

\C{v)\dv\z{u)\du.

Jt — u

Here is one of the main applications of Theorem 7.2.1. It is known that there are examples of (7.2.11) which do have periodic solutions; indeed, when B = 0 they are very common. But for a general B they are rare. The reason for that is that the right-hand-side of (7.2.11) is generally not periodic even when y is periodic. But (7.2.11) can have solutions which are asymptotically periodic in the sense described below. Under the conditions of Theorem 7.2.1, the following is shown in Burton (1985; p. 102). Suppose that lim

/

\B{t,s)\ds=

f

\B{t,s)\ds

(7.2.23)

is bounded and continuous in t, that /

\R{t,s)\ds<M

for t > 0 ,

(7.2.24)

Jo that R(t,0) -> 0 as t -> oo, and that i(t + T) = f(i). Then there exists a sequence of positive integers {fij} such that the function y defined in (7.2.12) satisfies y(t + n / r , 0 , y o ) - » I J — oo

R{t, s)f(s) ds := x(t),

j -> oo ,

(7.2.25)

7.3. UAS AND INTEGRABILITY OF THE RESOLVENT

233

where x(t) is a T—periodic solution of d

^-=A(t)x+f B(t,s)x(s)ds + f(t) dt J_00

(7.2.25)

on (—oo, oo).

7.3

UAS and Integrability of the Resolvent

The basic perturbation result in the previous chapter, Theorem 6.4.5, concerns the equation ,t

,-t

x ' = , 4 x + / C(t,s)x(s)ds Jo

+ / Jo

D(t,s)r(x(s))ds

+ q(t,x) + H(i,x(-)) + p(t,x).

(6.4.11)

It depends on the Liapunov functional ("i /-co

V(t,x(-)) = [xTBx]1/2+K

/ / \C(u,s)\du\x(s)\ds Jo Jt

for the system r* x' = ^ix + / C(t, .s)x(.s) ds , Jo which is required to have nice properties. Other perturbation results are found in Chapters 2 and 5. In this section we present and discuss the work of Hino and Murakami (1996) and Zhang (1997) who consider a system /"* x'(t) = A(t)x(t) +

B(t,s)x(s)ds (7.3.1) Jo where A, B are nx n matrix functions, A(t) is continuous on [0, +oo), B(t, s) is continuous for 0 < s < t < oo. The classical resolvent equation of (7.3.1) is 8R(t,s) os

=_fl(t

)^( s )_ f R(t,u)B(u,s)du, Js

R(t,t)=I,

(7.3.2)

for t > s > 0, while Becker's resolvent is

^p^-=A(t)R(t,s)+ ot

f B(t,u)R{u,s)du

Js

R(s,s)=I,

(7.3.3)

234

7. THE RESOLVENT

for t > s > 0, where / is the n x n identity matrix. When A is a constant matrix and B{t,s) = B(t — s) is of convolution type, equation (7.3.1) becomes /"* x'(t) = Ax{t) +

B{t- s)x(s) ds

(7.3.4)

Jo and the resolvent equation (7.3.3) is * Z'(t)=AZ(t)+ B(t-s)Z(s)ds, Z(0) = I. Jo If we apply the standard variation of parameters formula to ft x'(t) = A(t)x(t) + / B(t, s)x(s) ds + p(t) then a solution x(i) = x(t,to,4>) °f the perturbed equation may be expressed as

x(Mo,
+ f

Jo

R(t,s)p(s)ds.

Jtu Moreover, p may depend on x; thus, we already have useful tools for dealing with perturbations of (7.3.1) when the resolvent is integrable. In the convolution case with A constant, Miller (1971b) showed that for B G L1[0,oo), the zero solution of (7.3.4) is uniformly asymptotically stable if and only if the resolvent Z £ L1 [0, oo). We now follow the work of Hino and Murakami (1996) and Zhang (1997) to prove an extension of Miller's result to (7.1.1). We require that (Hi) sup \\A(t)\+ t>o [

J

o

f \B{t,s)\ds\
(H2) for any a > 0, there exists a n S = S(a) > 0 such that ft-s

/

Jo

\B(t,u)\du
for all

t>S,

(H3) A(t) and B(t,t + s) are bounded and uniformly continuous in (t, s) e {(t, s) e [0,00) x K

- i < s < 0}

for any compact set K C ( — 00, 0].

7.3. UAS AND INTEGRABILITY OF THE RESOLVENT

235

Let R = (—00,00), R+ = [0,oo), and R~ = (—00, 0] respectively. For x G Rn, I I denotes the Euclidean norm of x. For any n x n matrix A, define the norm \A\ of A by \A\ = sup{|Ax| : \x\ < l } . For any interval J C R, we denote by C(J) the set of continuous functions <j> J ~^ Rn, and set \(f>\j = sup {\4>(s)\ : s G J } . For each t0 G R+ and i?" which satisfies (7.3.1) on [to, +00) with x(s) = 4>(s) for 0 < s < to (see Section 2.1). Such a function x(i) is called a solution of (7.3.1) through (to,4>), and is denoted by x.(t,to,). Definition 7.3.1. The zero solution of (7.3.1) is uniformly stable (US) if for any e > 0, there exists a 5 = 5(e) > 0 such that [to > 0, (p G

C([0,t0]), |0|[o,t,,] < <J] " ^ 7 |x(t,t o ,0)| <£fort>

to.

Definition 7.3.2. T i e zero solution of (7.3.1) is uniformly asymptotically stable (UAS) if it is US and there exists a 80 > 0 with the property that for each e > 0 there exists T = T(s) such that [t0 > 0, (p <E C([0,t 0 ]), |0|[o,t,,] <So, t> to + T] imply |x(t,to,<WI < £ Definition 7.3.3. The zero solution of (7.3.1) is totally stable (TS) if for anye > 0, there exists a 5 = S(e) > 0 such that [t0 > 0,
B(t,s)x(s)ds +p(t)

(7.3.5)

Jo suci that x(,s) = 0(,s) for s £ [0,to]-

Theorem 7.3.1. Zhang (1997) Under (Hi), (H 2 ), and (H 3 ), the zero solution of (7.3.1) is UAS if and only if sup / |i?(£,s)|ds < 00. t>o Jo

(7.3.6)

The proof of this theorem is based on a series of results of Hino and Murakami (1996) and Zhang (1997) on uniform asymptotic stability and total stability of (7.3.1). Lemma 7.3.1. Zhang (1997) If (Hi) and (7.3.6) hold, then there exists a constant K such that \R(t, s)\ < K for all 0 < s < t < 00.

236

7. THE RESOLVENT

Proof. Since R(t, s) is a solution of (7.3.2), we obtain R(t,s)=I

+

R(t,u)A(u)du + I

I

Js

Jv

Js

R(t,u)B(u,v)dudv.

Interchange the order of integration in the last term to obtain R(t,u)A(u)du +

R(t,u) Js

B(u,v)dvdu Js

and \R(t,s)\ < 1+ / \R(t,u)\\A(u)\du Js

,-t

,-u

+ / \R(t,u)\ / Js Js

\B(u,v)\dvdu.

(7.3.7)

By (Hi) and (7.3.6), there are positive constants M and L such that sup(|^(t)|+ / \B(t,s)\ds\ <M t>o [ J o ) and sup //"* \R(t,s)\ ds < L . t>o Jo

It then follows from (7.3.7) that \R(t, s)| < 1 + ML =: K. This completes the proof. Lemma 7.3.2. Zhang (1997) The matrix functions A(t) and B(t, t + s) can be continuously extended to (t, s) e R x R~ with A(t) = A(t) and B(t,t + s) = B(t,t + s) on Q = {(t,s) G R+ x R~ \ - t < s < 0}. Moreover, if (Hi)-(Hs) hold for A(t) and B(t,t + s), then the extensions A{t) and B(t, t + s) satisfy the following conditions: (Hi) sup \\A(t)\+ f \B(t,s)\ds\=:M ten I J-oo J


(H2) for any a > 0, there exists an S = S(cr) > 0 such that /

\B(t,u)\du
for all

teR,

J—00

(S3) A(t) and B(t,t + s) are bounded and uniformly continuous in (t, s) € R x K for any compact set K c R~.

7.3. UAS AND INTEGRABILITY OF THE RESOLVENT

237

We omit the proof of Lemma 7.3.2 here and refer the reader to Zhang (1997) for detailed construction of these extensions. Theorem 7.3.2. Hino and Murakami (1996) Suppose that (Hi) and (H2) hold. If the zero solution of (7.3.1) is TS, then it is UAS. Proof. First, notice by definition that when the zero solution of (7.3.1) is TS then it is US. Let t0 G R+ and cj> e C([0,i0]) with \\<j>\\ < 5(1), S(-) is the one given for the TS of the zero solution of (7.1.1). Then \x(t,to, )\ < 1 for all t > to- Now for any s > 0 (0 < s < 1), a > 0, we set

*

u(t) = u(t,a,e) = < l+sat [l

ift>0 if t < 0,

define y(t) by y(t)

teR+,

= u(t - to)x(t),

and p(t) by ft

p(i) = u'(t - io)x(t) + / B(t,s)x.(s)[u(t-to)-u(s-to)]ds Jo

(7.3.8)

for t > to. One may verify that y(t) satisfies (7.3.5) for t > to with p(i) denned in (7.3.8). Notice also that for t > 0 w

eV

2 + 2£at)

and ,. .

a(2-e)

^=(TT^FThis yields 1 < u(t) < 2/e, \u(t) - u(s)\ < 2a\t - s\ for t,s e R. It follows

from (H2) that for any 77 > 0, there exists an S = S(r]) > 0 such that / Jo

\B(t,u)\du
for all t > S(r]). By (Hi), there exists a constant M* > 0 such that sup / \B(t,s)\ds<M* . t>o Jo

238

7. THE RESOLVENT

Let t > to- Without loss of generality, we may assume that to > S(rj). By (7.3.8) we have /"* p(i)| <2a+ \B(t, s)\ \u(t - t0) - u(s - to)\ ds Jt-S(v) ft-S(rj)

+ / \B(t,s)\\u(t-to)-u(s-to)\ds Jo <2a + 2aM*S{rj)+4f]/e for all r] > 0. Thus, we may choose 77 > 0 and a = a(e) so small that |p(t)| < <5(1). Since the zero solution of (7.3.1) is TS, we obtain |y(t)| < 1 for all t>t0. Hence if t > t0 + (1 - e)/(ea), then |x(t,t o ,0)| = |y(t)/u(t-t o )| < [l+ea(t-to)]/[l

+ 2a(t-to)] <e

which proves the theorem. Next we shall discuss the converse of the above theorem. To do this, we assume that (Hi)-(Hs) holds and study the limiting equation of (7.3.1) with A(t) and B(t, s) replaced by A(t) and B(t, s). By the Ascoli-Arzela's theorem, (S3) implies that for any sequence {t'k} with t'k —> 00 as k —> 00, there exists a subsequence {tk} of {t'k} and functions D(t) and E(t, s) such that A(t + tk) -* D(t) and B(t + tk, t + tk + s) —> E{t, t + s) as k —> 00 uniformly on J x K for any compact sets J G R and if C R~. We denote by F(A,i3) the set all pairs (D,E) which satisfy the above situation for some sequence {tk} with tk —> 00 as k —> CXD. We can easily see that each (D,E) £ T(A,B) also satisfies (Hi)-(Hs) with the same number M and S(a). In particular, (A,B) G T(A,B) whenever A(t) and B(t,t + s) are almost periodic in t £ i? uniformly for s £ i?~ [see Hino and Murakami (1991)]. If (D,E) e T(A,B), then the equation /"* x'(t)=D(t)x(t)+

E(t,sMs)ds

(Loo)

J-co

is called a limiting equation of (7.3.1). [See (7.2.23). In a similar way, one can also define the stability of the zero solution of (Loo) by taking ||(-OO.T] m the place of |<^|[ojT]. It is also known (see Hino and Murakami (1991b)) that if the zero solution of (7.3.1) is UAS, then the zero solution of each limiting equation (Loo) is also UAS with a common triple (5o,5(-),T(-)). T h e o r e m 7.3.3. Hino and Murakami (1996) Suppose that (Hi)-(H 3 ) hold. Then the zero solution of (7.3.1) is UAS if and only if it is TS.

7.3. UAS AND INTEGRABILITY OF THE RESOLVENT

239

Proof. The "if" part has been shown in Theorem 7.3.2. We shall prove the "only if" part by assuming that the zero solution of (7.3.1) is UAS. By Lemma 7.3.2 and the remark above, we may also assume that (Hi)-(Hs) hold and the zero solution of each limiting equation (£<x>) is UAS with the same common triple (5o,S(-),T(-)). We now claim that for any e > 0, there exists an a(e) > 0 and 8(e) > 0 such that T > a(s), ,p)\ < e for all t > r . Then the zero solution of (7.3.1) is TS since it is unique. We prove the claim by the method of contradiction. Suppose there exists an e, 0 < e < 5Q/2, and sequences {rfe} G R+, Tfc —> oo as k —> oo, {rk},rk > 0, and functions G C([0, rfe]), pfe G C([rfe, oo)) and the solution xfe(i) = x(t,Tk, (pk,pk) of (7.3.5) with p = pfc through (Tk,4> ) such that |p fe |[r fc ,oo)<^,

l^l[o,rf:]<^,

|xfe(rfc+rfc)|=e (7.o.9)

and

|x fc (t)| < e for te [0, rk + rk).

Let T = T(e) be given in Definition 7.3.2 for UAS. We first consider the case in which the sequence {rk} is unbounded. Without loss of generality, we assume that A(t + Tk+rk-T)^

D(t)

and B(t + Tfc + rk - T, t + s + Tk + rk - T) - E(t, t + s) as k -^ oo uniformly on any compact set in R x R~ for some (D,E) £ T(A,B). Define yfe(t) = xk{t + Tk+rk-T) for t >T-rk-rk. Thenyfc(t) satisfies {t)=A{t

Tk+rk_T)yk{t)

+

ft B(t + Tk+rk-T,u

+

+ Tk+rk- T)yk{u) du

[T-rk

+

B(t + Tk+rk-T,u

+

Tk+rk-T)

J-rk+T-rk 4>k (u + rk + rk -T)du + pk(t + Tk+rk-T) for t > T—rk. In this case we may assume that {yk} converges to a function y uniformly on any compact set in (—00, T]. Moreover, y is a solution of

240

7. THE RESOLVENT

(Loo) on [0,T]. Letting k -> oo in (7.3.9), we have |y(t)| < a on (-oo,T] and |y(T)| = e. This is a contradiction since |y|(_oo.o] < £ < <5o implies |y(T)| < e. Therefore, the sequence {rk} must be bounded. So, we may assume that rk —> r as k —> oo for some r G i? + and set xk(t) = xk(t + rk) for £ > — Tfe. Then xfc(t) satisfies

d — dt

k

(t) = A(t + Tk)xk(t)+ + /

fl / B(t + Tk,u + Tk)x.k(u)du Jo

B(t + Tk,U + Tk)(pk(u + Tk)du +Pk(t + Tk)

for i > 0. Again, we may assume that the sequence {xfe} converges to a function x uniformly on any compact subset (—oo,r]. By the same reasoning as for y, we see that x is a solution of some limiting equation of (7.3.1). On the other hand, it follows from (7.3.9) that x(i) = 0 on R~ and x(r) = e. This is again a contradiction since we must have x(t) = 0 on R by the uniqueness of solutions of (.Loo) with respect to initial functions. This shows that the zero solution of (7.3.1) is TS if it is UAS. We are now ready to prove Theorem 7.3.1 by applying Perron's theorem [Perron (1930)] and using the properties of the resolvent R(t, s) defined in (7.3.2). It is also verified in Hino and Murakami (1996) that resolvent equations (7.3.2) and Becker's resolvent (7.3.3) are equivalent. Proof of Theorem 7.3.1. First we suppose that the zero solution of (7.3.1) is UAS. By Theorem 7.3.3, it is TS. Let p e C(R+) be bounded and x p G C(R+) satisfy ft x'(t) = A(t)x(t) + / B(t,s)x(s)ds Jo

+p(t)

for t > 0 with xp(0) = 0. By the variation of parameters formula, we obtain xp(i) = / R(t, s)p(s) ds for t > 0 . Jo Since the zero solution of (7.3.1) is TS, we see that x p is bounded on R+. This implies that J o R(t, s)p(s) ds is bounded on R+ whenever p G C(R+) is bounded. Applying Perron's theorem, we obtain that sup tefl + J o \R(t, s)\ds < oo, and hence (7.3.6) holds. Conversely, suppose that (7.3.6) holds with sup teij .+ J o \R(t, s)\ds = L for some L > 0. By Lemma 7.3.1, there exists a constant K > 0 such that

7.3. UAS AND INTEGRABILITY OF THE RESOLVENT

241

\R(t,s)\ < K for alH > s > 0. Let x(i) = x(t,i o ,^,p) be a solution of (7.3.5). By the variation of parameters formula again, we obtain »

*

x(t,t o ,0) = R(t,to)

+ /

B(s,u)4>(u)duds Jo

R(t,s)p{s)ds.

Jtu

This implies that |x(i)|< \R(t,to)\\$(to)\ + f <(K

+

+ / \R(t,s)\ I' Jo Jo

\B(s,u)\duds\(l>\[0M

\R{t,s)\ds\p\[tlhOO) LM*)\4>\{oM+L\V\[U)t0o).

Here we have used condition (Hi) with sup t > 0 J o \B(t,s)\ds < M*. For any e > 0, choose 6 > 0 such that (K + LM* + L)S < e. If |<^|[o,t(l] < £ and |p|[to.oo) < S, then |x(t)| < e for all t > to- Therefore, the zero solution of (7.3.1) is TS. By Theorem 7.3.2, it is UAS. This completes the proof. Remark 7.3.1. The integral condition (7.3.6) on the resolvent seems to be very difficult to verify directly. When (7.3.1) is "periodic", Section 7.2 provides an alternative condition which is rather easy to check. Indeed, under the assumptions that A(t + ui) = A(t), B(t + LJ, S + to) = B(t, s) for some to > 0 and sup s>0 J°° \B(t, s)\dt < oo, condition (7.3.6) follows from sup /

\R(t,s)\dt<

oo.

(7.3.10)

s>0 Js

which is satisfied if there exists a Liapunov functional V(t, ) such that V^^xt)

<-c\x(t)\,

c>0

along a solution x(t) = x(t,to,
242

7. THE RESOLVENT

least negative semi-definite. But when we study stability by means of hxed point theory, then the central problem is to dehne a mapping into a space of functions which would be acceptable stable solutions; one must then show that the mapping has a hxed point and that the hxed point satisfies the differential equation, ft turns out that the union of the two methods is far better than either alone.

Chapter 8

Functional Differential Equations 8.0

Introduction

This chapter contains a survey of results concerning problems in general functional differential equations that we encountered in previous chapters for integral and integro-differential equations. Those problems were extensively discussed earlier, so we suppose the reader to be familiar with the background and, hence, primarily just state and prove theorems. Sections 8.1 and 8.2 deal with existence, uniqueness, continuation, stability, and asymptotic stability for a very general functional differential equation. The reader should consult Chapters 2, 3, and 6 for general facts, problems, and insights concerning these questions, as well as their relations to corresponding problems in ordinary differential equations. The view throughout is that the subject may be developed using Liapunov functionals and Razumikhin techniques. Sections 8.3, 8.5, and 8.6 are concerned primarily with a functional differential equation x' = F(i,x t ), where xt is defined on [t — h, t] for some h > 0. In particular, 8.3 deals with boundedness and stability; 8.5 concerns limit sets of autonomous systems; and 8.6 studies the existence of periodic solutions. Section 8.7 considers limit sets of nonautonomous systems, usually with an unbounded delay. We concentrate on the following problems. 243

244

8. FUNCTIONAL DIFFERENTIAL EQUATIONS

(a) What conditions on a Liapunov functional are needed for uniform asymptotic stability? (b) What are alternatives to the condition x'(t) bounded for x(-) bounded in proving stability and boundedness? (c) If a Liapunov functional satisfies V < 0 for |x(t)| > M > 0, then what more is needed to conclude boundedness, uniform boundedness, or uniform ultimate boundedness?

8.1

Existence and Uniqueness

We consider a system of Volterra functional differential equations x'^t) = fi(t,xi(s),... ,xn(s); a < s < t) for t > to, a > — oo, a < to, and i = 1,..., n. These equations are written as x'(t) =F(t,x(-)),

t>t0,

(8.1.1)

where x(-) represents the function x on the interval a, t] with the value of t always determined by the first coordinate of F in (8.1.1). Thus, (8.1.1) is a delay differential equation. This section and part of the next will closely follow the excellent paper by Driver (1962), which remains the leading authority on the subject of fundamental theory for (8.1.1). As Driver notes, much of his material is found elsewhere in varying forms; in particular, the early work is from Krasovskii, EPsgol'ts, Myshkis, Corduneanu, Lakshmikantham, and Razumikhin. But important formulations, corrections, and general synthesis are by Driver. Notation. (a) If x G Rn, then |x| = maxj = i,... jn \xi\. (b) Hip : [a,b] -> Rn, then \\iP\\^=

sup | | ^ ( s ) | | . a<s<6

(c) For any interval [a, b] and any D C Rn, then C([a,b] —> D) denotes the class of continuous functions ip : [a, b] —> D. Because a can be —oo, one accepts the following. Convention. If a = —oo, then intervals [a,t] and [a, 7) mean (—00, t] and (—00,7), respectively, and ip G C([a,t] —> D) means that there is a compact set L^, C D such that ip G C(( —oo,t] —> L,/,). This implies that ip G C[[a,t] —> D) with t0 < t < 7 means ip G C([a, t] —> L^) for some

compact set L^, C D, regardless of whether a is finite or not.

8.1. EXISTENCE AND UNIQUENESS

245

Definition 8.1.1. The functional F(t,x(-)) will be called (a) continuous in t if F(i, x(-)) is a continuous function oft for to < t < 7 whenever x G C([a, 7) —» Z)), (b) locally Lipschitz with respect to x if, for every 7 G [£0,7) and every compact set L C D, there exists a constant K^_L such that |F(i,x(.))-F(£,y(.))| i ) . Definition 8.1.2. Given an initial function G C([a,to] —> Z)), a solution is a function x G C([a,/?) —> £)), wiere to < /3 < 7, such that x(i) = <£(t) 022 [a, to] and x satisfies (8.1.1) for to < t < (3. We write x(t, to, ej>). A solution is unique if every other solution y(t,to,4>) agrees with x(i, to, 4>) as long as both are defined. Theorem 8.1.1, Theorem 8.1.5, and the first paragraph of Theorem 8.1.6 are the basic results on nondifferentiable Liapunov functions and functionals. Theorem 8.1.1. Driver (1962) Let ui(t,r) be a continuous, nonnegative function of t and r for to < t < (i, r > 0. Let v(t) be any continuous, nonnegative function for a < t < (3 such that limsup

v(t V + At);

-v(t) —


at those t G [to, P) at which v(s) < v(t)

for all s G [a,t].

Let ro > max a < s < 4() v(s) be given, and suppose that the maximal continuous solution r(t), of r'(t) = u>(t,r(t)) for t > to with r(to) = ro exists for to
for

to
Proof. Choose any J3 G (to,/3). Then r(t) can be represented as r(t) = lim r(t,e)

for

to
where, for each fixed sufficiently small e > 0, r(t,e) is any solution of r'(t,e)=u(t,r(t,e))+e

for t o < t < / 3 ,

with r(t o ,e) = r 0 . [See Kamke (1930, p. 83).]

246

8. FUNCTIONAL DIFFERENTIAL EQUATIONS

By way of contradiction, suppose that, for some such e > 0, there exists a t e (t0, (3) such that v(t) > r(t,e). Let t\ = sup {t G [to,/3) : v(s) < r(s,e) for all s e [to,t]}. It follows that to < ti < /?, and by continuity of v(t) and r(t,e), v(ti) = r(ti,e). Because r(t,e) has a positive derivative, v(s) < r(ti,e) = v(t\) for all s G [a,ti], and therefore hmsup^

-4

—

<w(ti,w(ti)).

But, f(ti + At) > r(t\ + At,e) for certain arbitrarily small At > 0. Hence, u(ti + A i ) - u ( i i ) , hm sup -^ -4 -^ > r'(ti, s) = iv(tur(t1,e)) + e = u(t1,v(t1))+e is a contradiction. We then have v(t) < r(t,e) for all sufficiently small s > 0 and all t £ [to,P), so v(t) < r(t) for all t £ \to,(5). This completes the proof. Theorem 8.1.2. Driver (1962) Let the functional F(i,x(-)) be continuous in t and locally Lipschitz in x. Let x(t) = x(t,to,(f>) and x(t) = x(t, to, and 4>e C([a,t0] -^ D). Then, for any (3 G (to,/3), x(t) and x(t) both map [a, J3\ into some compact set H c D, and |x(t) - x(i)| < U - WaM

exp[^; L (t - to)]

for to G). Because x(t) and x(t) are continuous on the compact set [to,/3], it follows that x , x G C(Jto,j3] —* FiJ, where F\ is a compact subset of D.

Let H = GUF1._ For to < t < P, we have |x(t + At) - x(t + At)I! - |x(t) - x(t)| lim sup -1 At^0+

At

|x(t + At) - x(t) - x(i + At) + x(t)| < hm sup — At^o+

= |F(t,x(-))-F(t,£(-))| <^>ff||x(.)-x(.)||'a'*l.

A i

8.1. EXISTENCE AND UNIQUENESS

247

The result now follows from Theorem 8.1.1 by taking v(t) = x(t) — x(t)[, u(t,r)

= KptHr,

a n d r 0 = \\ -

.

Remark 8.1.1. This result is the primary one on continual dependence of solutions on initial conditions. A more detailed set of information may be found in Driver (1963). Theorem 8.1.3. Driver (1962) Let F(t,x(-)) be continuous in t and locally Lipschitz in x and let 0 such that a unique solution x(t) = x(t, to, 4>) exists for a < t < to +h. Proof. Uniqueness is immediate from Theorem 8.1.2. Existence is obtained by Picard's approximations. Let G be a compact subset of D such that