0387986995Basic TheoryC

where 9(17) is a real-valued and continuously differentiable function of g on the real line R. Equation (VIII.1.9) is equivalent to the system

d

If g(a) = 0, then system (VIII.1.10) has a constant solution yt = a, Y2 = 0. Set yi = zi + a and y2 = z2. Then, system (VIII.1.10) becomes

The eigenvalues of A are ±(-g'(a))I/2. Note also that

Then, G(n) takes a local minimum (respectively a local maximum) at n = a if g(a) = 0 and g'(a) > 0 (respectively g'(a) < 0). Also, set

if z(t) is a solution of (VIII.1.11).

Case when g'(a) < 0: If g'(a) < 0, there exists a positive number PO such that G(rl) < G(a) for 0 < 177 - al < Po Let t;(t) be the unique solution of (VIII.1.11) satisfying the initial condition z1(0) = 0, z2(0) = Co > 0. Since t;2(t) = (1(t),

we obtain ('(t) = 2(G(a) - G((1(t) + a)) + co >- Co as long as 0 < (1(t) < po.

Hence, there must be a positive number to depending on (o such that (I (to) = Pa no matter how small to may be. This implies that the trivial solution of (VIII.1.11) is not stable.

Case when g'(a) > 0: For a given positive number p, let V (p) be the connected component containing 0 of the set {[; : G(( +a) < G(a)+p}. Then, V (pl) C V (P2) if p1 < p2. Furthermore, if g(a) = 0, g(a) > 0, and if a positive number p is sufficiently small, there exist two positive numbers e1(p) and e2(p) such that

(1) e2(P) < e1(P),

(2) Plmoe1(P) = 0,

FIGURE 2.

Observe also that G(a) < G(( + a) < G(a) + p for [; E V (p) if p is a sufficiently small positive number.

For a given positive number e, choose another positive number p so that ei(p) < e

and p:5 e. Choose also the initial value z(0) so that Jz1(0)j < e2 (2) and z2(0)2 < p.

Then, G(z1(0)+a) < G(a)+ since z1(0) EP (2). Hence, (VIII.1.14) implies that

G(zi(t) + a) < G(a) + p for all t. Observe that the set {z1(t) : all t} is connected

2

and zi(0) E D (2) C V (p). Therefore, zi(t) E D(p) for all t. Hence, Iz1(t) < ei (p) < e. On the other hand, G(a) < G(z1(t) + a) < G(a) + p since z1(t) E D (p)

for all t. Hence, (VIII.1.14) implies that2(tz2< G(z1(0) +a) - G(a) + < p < e.

This proves that the trivial solution of system (VIII.1.11) is stable as t -+ +oo.

2

The analysis given in Example VIII-1-6 is an example of an application of the

Liapounoff functions to which we will return in Chapters IX and X.

The third example is the following result.

Theorem VIII-1-7. If f (x, y) and g(x, y) are real-valued continuously differen- tiable functions such that

(i)f (0, 0) = 0 and g(0, 0) = 0,

(ii)(f (x, y), g(x, y)) 0 (0, 0) if (x, Y) # (0, 0),

(iii) 8x (x, y) + (x, y) = 0,

49Y

then, the trivial solution (x, y) = (0, 0) of the system

2. A SUFFICIENT CONDITION FOR ASYMPTOTIC STABILITY 241

VIII-2. A sufficient condition for asymptotic stability

In this section, we prove a basic sufficient condition for asymptotic stability. Let us consider a system of differential equations

(i)A is a constant n x n matrix,

(ii)the entries of the 1R"-valued function §(t, y-) are continuous and satisfy the

estimate

where, vo > O, co > O, ro > O, k(t) > O and bounded fort > 0, t+xlim k(t) = 0, and A(ro) = {(t, yj : 0 < t < +oo, Iy1 < ro).

The following theorem is the main result in this section.

Theorem VIII-2-1. If the real part of every eigenvalue of A is negative, the tnvial solution g = 0 of (VIII. 2.1) is asymptotically stable as t -i +oo.

Proof.

We prove this theorem in four steps.

Step 1. Let A, (j = 1, 2,... , n) be the eigenvalues of A. Then, RIA,I (j = 1, 2, ... ,

n) are Liapounoff's type numbers of the system t d= Ay (cf. Example VII-2-8).

Therefore, choosing a positive number p so that 0 < p < -9RIA,I f o r j = 1, 2, ... , n, we obtain

IeXpIAtll < Ke-"t on the interval To = {t : 0 < t < +oc}

for some positive constant K.

Step 2. Fixing a non-negative number T, change system (VIII.2.1) to the integral equation

g(t) = e(t-T)Ag(T) + fTte(t-')Ag(s,g(s))ds.

If Ig(t)I < b for some positive number b on an interval T < t < T1, then

on the same interval T < t < Ti. Setting p(T) = sup k(s), change this inequality

a>T

to the form

for T:5 t < Tl (cf. Lemma 1-1-5).

Step 3. Fix a non-negative number T and two positive numbers 6 and 61 in such a way that

p-K(p(T)+co6'o) > 0, 0<61 <6, K61 <6.

Assume that 19(T)J < 61. Then, inequality (VIII.2.3) holds for T < t < T1 as long as Iy(t)I < bon the interval T < t < T1. This, in turn, implies that

This is true for all T1 not less than T. Hence, inequality (VIII.2.3) holds for t > T.

Step 4. If Jy(t)J < b < 1 for 0 < t < T, there exists a positive number is such that I d t) I < n19(t)j, for 0 < t < T. This implies that Jy-(t)J < Jy-(0)JeK' as long

as Jy(t)I < 6 < I for 0 < t < T. Therefore, Jy(T)J is small if Jy(0)j is small. Thus, it was proven that (VIII.2.3) holds for t > T if Iy(0)J is small. This completes the proof of Theorem VIII-2-1. O

Example VIII-2-2. For the following system of differential equations d = A#+

the trivial solution y = 0 is asymptotically stable as t -' +oc. In f act, the char- acteristic polynomial of the matrix A is PA(a) _ (A + 0.3)2 + 1.99. Therefore, the real part of two eigenvalues are negative.

Remark VIII-2-3. The same conclusion as Theorem VIII-2-1 can be proven, even if (VIII.2.2) is replaced by

(VIII.2.4) 19(t, y-)J < (k(t) + h(t)Jyj")Jy7 for (t, y) E A(ro),

where p is a positive number, and two functions h(t) and k(t) are continuous for

t > 0 such that k(t) > 0 for t > 0, lim k(t) = 0, h(t) > 0 fort > 0, and

Liapounoff's type number of h(t) at t = +oo is not positive (cf. [CL, Theorem 1.3 on pp. 318-319]).

Remark VIII-2-4. The converse of Theorem VIII-2-1 is not true, as clearly shown in Example VIII-1-5. In that example, the eigenvalues of the matrix A are -1 and 0, but the trivial solution is asymptotically stable as t -+ +oo. Also, in that example, solutions starting in a neighborhood of 0 do not tend to 0 exponentially as t -+ +oo.

Remark VIII-2-5. Even if the matrix A is diagonalizable and its eigenvalues are all purely imaginary, the trivial solution is not necessarily stable as t -+ +oo. In such a case, we must frequently go through tedious analysis to decide if the trivial solution is stable as t -+ +oo (cf. the case when g'(a) > 0 in Example VIII-1-6).

We shall return to such cases in IR2 later in §§VIII-6 and VIII-10 .

Remark VIII-2-6. If the real part of an eigenvalue of A is positive, then the trivial solution is not stable as it is claimed in the following theorem.

Theorem VIII-2-7. Assume that

(i)A is a constant n x n matrix,

(ii)the entries of y(t, yam) are continuous and satisfy the estimate

(iii) the real part of an eigenvalue of the matrix A is positive.

Then, the trivial solution of the system dydt = Ay + §(t, y-) is not stable as t -. +oo.

A proof of this theorem is given in (CL, Theorem 1.2, pp. 317-3181. We shall prove this theorem for a particular case in the next section (cf. (vi) of Remark VIII-3-2).

An example of instability covered by this theorem is the case when g'(a) < 0 of

Example VIII-1-6. The converse of Theorem VIII-2-7 is not true. In fact, Figure 3 shows that the trivial solution of the system

is not stable as t - +oo. Note that, in this case, eigenvalues of the matrix A are

-1 and 0.

Y2

Yt

FIGURE 3.

VIII-3. Stable manifolds

A stable manifold of the trivial solution is a set of points such that solutions starting from them approach the trivial solution as t --. +oo. In order to illustrate such a manifold, consider a system of the form

where Y E iR", 11 E R n, entries of R"-valued function gl and RI-valued function g2 are continuous in (1, y-) for max(IiI, Iyj) 5 po and satisfy the Lipschitz condition

for max(jil, Iy1) < po and max(11 i, Ii1) < po, where po is a positive number and u oL(p) = 0. Furthermore, assume that gf(0, 0) = 0 (j = 1, 2). Two matrices AI

and A2 are respectively constant n x n and m x m matrices satisfying the following

where K, and of (j = 1,2) are positive constants. Condition (VIII.3.2) implies that the real parts of eigenvalues of AI are not greater than -01, whereas the real parts of eigenvalues of A2 are not less than -02. Assume that

Let us change (VIII.3.1) to the following system of integral equations:

I

x(t, c) = e1AI C + J e(1-a)A' gi (Y(s, c-), y(s, c))ds,

0

(VIII.3.4)

At, c ) = Jt e(t 8)A2 2(x(s, c), y(s, c))ds. t o0

The main result in this section is the following theorem.

Theorem VIII-3-1. Fix a positive number c so that a1 > 02 + E. Then, there exists another positive number p(() such that if an arbitrary constant vector c' in

for 0 < t < +oo. Furthermore, this solution (i(t, cl, #(t, c)) is uniquely determined

then

K2L(P)P e

al -o2-E

2

Kl I i < , then, using successive approximations, a solution (i(t, cl , y(t, c)) of

2

(VIII.3.4) can be constructed so that

Details are left to the reader as an exercise.

Remark VIII-3-2.

(i)The positive number a is given to start with and the choice of p depends on e. However, since this solution approaches the trivial solution, the constant e may be eventually replaced by any smaller number, since the right-hand side of (VIII.3.1) is independent of t. This implies that the curve (i(t, c), y'(t, c-)) is independent of a as t -. +oo. More precisely, if a solution (x"(t), y"(t)) of

(VIII.3.1) satisfies a condition

with some positive constants K and eo such that of -o2-eo > 0, then for every positive a smaller than co, there exists to > 0 such that (i(to + t), y(to + t)) _

(i(t, ), y(t, cam)), where c = ?(to).

(ii) The initial value of y(t, c-) is given by

0

00, cl = f e-sA292(i(s, c_), y(s, c_))ds.

+oo

(iii) If ay and 09 exist and are continuous in a neighborhood of (6,6) and if

8 (6, U = 0' and (0, 0) = 0', then i(t, c) and y"(t, cl are continuously

89 differentiable with respect to c.

(iv)If the real parts of all eigenvalues of the matrix Al of (VIII.3.1) are negative and the real parts of all eigenvalues of the matrix A2 are positive, then the stable manifold of the trivial solution of system (VIII.3.1) is given by S =

((6, 9(0, c-)) : 161 < p}, where p is a sufficiently small positive number.

(v)Consider a system

in the following situation:

(a)A is a constant n x n matrix,

(b)the entries of §(y1 are continuous for Iy1 < po and satisfy the Lipschitz

where PO is a positive number and lim L(p) = 0,

p-0

(c) 9(0,6) = 6.

Suppose further that A has an eigenvalue with positive real part. Then, applying

Theorem VIII-3-1 to the system = -Ay' - g(y-), we can construct the stable

manifold U of the trivial solution. This means that if a solution fi(t) of (VIII.3.6) starts from a point on U, then urn Q(t) = 6. This shows that the trivial solution of

t - +oo, and Theorem VIII-2-7 is proved for (VIII.3.6).

(i)Po is an invertible constant n x n matrix and PP, E C",

(ii)the formal transformation y' = P(u") reduces system (VIII 4. 1) to

with a constant n x n matrix Bo and the formal power series g(ui) _ E upgp

Ipi>2

with coeficients gp in Cn such that

(iia) the matrix B0 is lower triangular with the diagonal entries Al.... , An, and the entry bo(j, k) on the j-th row and k-th column of B0 is zero

whenever A, j4 Ak,

(iib) for p with IpI > 2, the j-th entry gp, of the vector gp is zero whenever

A1 n

Proof

Observe that if y" = P(d), then

where 03,k is the entry of B0 on the 7-th row and k-th column.

Let us introduce a linear order pi -< p2 for p. = (p,',... , pin) (j = 1, 2) by the

=j (I5,' p -<p) + yp(Pp',ff: 10 <ItI)