Dichotomies and Lyapunov functions

JOURNAL

OF DIFFERENTIAL

EQUATIONS

Dichotomies

52,

and

National Received

University,

(1984)

Lyapunov A.

w.

Australian

58-65

COPPEL

Box 4 GPO,

June 23, 1982;

Functions

revised

Canberra, September

ACT

2601,

Australia

2, 1982

This paper is prompted by an interesting recent article of Kulik 12 1. It appears that some of the arguments given there and in Samoilenko and Kulik [4] can be simplified and strengthened. We consider the homogeneous linear differential equation x’ = A(t)x,

(1)

where x E C” (or, with trivial changes, R”) and the coefficient matrix A(f) is continuous on an open interval .7 = (a, b). Suppose there exists a continuously differentiable Hermitian matrix function H(t) such that H’(t)

+ H(t) A(t) + A*(t)

H(t)

< 0

for

tE.7.

(2)

Then for any nontrivial solution X(I) of (1) the real-valued function p(t) =x*(t) H(r) x(f) is strictly decreasing. Let H(t) be singular for t = t, (1
+ *** +x!+(t),

where xj(t) E YJ, and assume x*(c) H(tJ x(t) > 0 for some t > t,. If we set P.jCr) = ixltt> + *‘* +x,(t)}*

H(t)(x,(t)

+ “. +xj(t)}

(1 US

k),

then v),J~ > 0 and, hence, pk(f) > 0 for t < I. Since wl(t,) = 0 this implies k > 1. Moreover, since qPk(fk) = (Do- ,(f,J it also implies qk- ,(t) > 0 for t < fk. If k > 2 then from ~~-l(t,_,)=40,_,(t,_,) we obtain, similarly, ~~-~(t) > 0 for t < t,- 1. Proceeding in this way we eventually obtain p,(t) > 0 for t < f,, which is a contradiction. We conclude that if t > t, the Hermitian form x*(t) H(t) x(t) is negative definite for x(.) E 7 ; + .. + 7 i. An analogous 58 0022.0396/84 Copyright All rights

$3.00

0 1984 by Academic Press, Inc. of reproduction in any form reserved.

DICHOTOMIES

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FUNCTIONS

argument shows that if t < t, the Hermitian form x*(t) H(t) x(t) is positive definite for x(a) E i”; + ... + T;,. It follows that the subspaces 7’; ,..., F’, intersect trivially and that the sum 7”; + ... + W, is direct. Hence, n, + ... + n, < n. In particular, H(t) is singular for at most n values of t. Thus we may now assume the notation chosen so that H(t) is nonsingular for t # t, (1 < k < m). Choose any t, < t, and let n, be the number of negative eigenvalues of H(t,). Evidently n, is independent of the choice of t,,. Let Y be a subspace of C” of dimension n, on which H(t,) is negative definite and let “t/;, be the subspace of solutions x(t) with x(t,,) E 3. Similarly be the number of positive eigenvalues of choose t,,, + , > t, and let n,,, H(t,+ i). Let 9’ be a subspace of C” of dimension n,, , on which H(t,+ i) is be the subspace of solutions x(t) with positive definite and let T,+, x(t,+ i) E .Y. The same argument as before shows that if t > t, (0 < k < m) then x*(t) H(t) x(t) is negative definite for x(e) E S, + . . . + 7’; and if t < t, (1 < k < m + 1) then x*(t) H(t) x(t) is positive definite for x(.) E Y;+...+“l,+,. It follows that the sum Yi+...+Ti+, is direct. These conclusions still hold, with appropriate interpretations, if P 0, F ,+, , or r; + *.- + Y?i is trivial. On the interval (to. ti) the matrix H(t) has n, negative eigenvalues. On the interval (t,, tJ the matrix H(t) has exactly n, + n, negative eigenvalues. In fact, it cannot have more, by continuity, and it cannot have less, since H(t) is negative definite on 7, + 7;. Proceeding in this way we see that on the interval (fk, t,, i) the matrix H(t) has exactly n, + . . . + nk negative eigenvalues. Hence, n, + ... + n, = n - n,, 1. It follows that the vector space P of all solutions of (1) admits the representation r.=T,@P:;@--.

@7,@3,+,.

(3)

The hypotheses will now be strengthened. We will assume that A(t) and H(t) are bounded on 3 and we will replace (2) by H’(t)

+ H(t)A(t)

+ A*(t)

H(t) < -71

for

t E 3,

(4)

where y > 0 is a constant. Moreover, we will take 3’ = (a, oo), where --oo < u < 0. Then, by [ 1, Proposition 7.21, Eq. (1) has an exponential dichotomy on the half-line R + . Moreover, by the proof of this proposition, a nontrivial solution x(t) is bounded on R + if and only if x*(t) H(t) x(t) > 0 for all I E Z. If it is unbounded then Ix(t)1 + co as t + co. Let P ; denote the subspace of solutions x(t) which are bounded on R + and set p = dim W ; . We will show that p = n,, , . It then follows that in the previous discussion we can take Y’,+,=Yi. The values x(s) corresponding to solutions x(t) E Y’ ‘+ form a pdimensional vector space Z, on which H(s) is positive definite. Hence Moreover, if p < n,, L there is an n,, ,-dimensional vector space P
60

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A. COPPEL

E$ containing Z, on which H(s) is positive definite. The values x(tO) corresponding to solutions x(t) with x(s) E P?s form an n,, ,-dimensional vector space fs containing Et0 on which H(t,) is positive definite. Give s a sequence of values s, tending to +co and choose a unit vector <, E-zs,, orthogonal to Zt,. Without loss of generality, we can assume that r, -+ c as v + co. If x,(t) is the solution such that x,(t,) = <, and x’(t) the solution such that .?(t,,) = f then x,(t) -+ x”(t) for every t E 3 as v -+ co. Since x:(t) H(t) x,(t) > 0 for t < s, it follows that x’*(t) H(L) f(t) > 0 for all t E 3’. Thus L?(I) E Y+ and f= x’(tJ E ZtO. Since cis a unit vector orthogonal to ZlO, this is a contradiction. Suppose now that 3’ = (-co, co). Then, by the same argument, Eq. (1) also has an exponential dichotomy on the half-line R- . A nontrivial solution x(t) is bounded on R _ if and only if x*(t) H(t) x(t) < 0 for all t E 3’. If it is unbounded then Ix(t)1 + co as t -+ --TV). The subspace P L of solutions bounded on R _ has dimension q = n,, and we can take 7; = “t” L. We have seen that o(t) = x*(t) H(t) x(t) is a strictly decreasing function for any nontrivial solution x(t). If o(t) > 0 for all real t then Ix(t)/ --t 0 as t+ +co and lx(t)1 + 00 as t + -co. If o(t) < 0 for all real t then /x(t)1 -+ co as t + +oo and Ix(t)l+ 0 as t + -co. If o(t) vanishes for some t then Ix(t)1 -+ co both as l-+ +co and as t + -co. Hence no nontrivial solution of (1) is bounded on R. Thus, if A(t) is bounded and if (4) holds on 3’ = R for some bounded continuously differentiable Hermitian matrix function H(t), then Eq. (1) has exponential dichotomies on R, and R- and no nontrivial solution is bounded on R. From (4), or even from (2), we obtain at once (a)

if H(s)r

= 0 for some nonzero vector < and some s E I then C*H’(s)C-

< 0.

Moreover, (a) alone implies that H(t) is singular for at most n values of t. If A(t) and H(t) are bounded then (4) also implies (b)

there exists a constant p > 0 such that H’(t)

< PI

for

t E.7.

If, in addition, 3’ = R then (4) further implies (c)

there exists constants

T > 0, 6 > 0 such that

I det H(t)/ > 6

for

I cl > T.

For H(t) can be replaced by H(t) f EI for some sufficiently small E > 0 without altering the values of n,,, =p and n, = q. Thus all eigenvalues of

DICHOTOMIES

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H(t) have absolute value greater than E for large It I. Kulik points out that, conversely, if a bounded continuously differentiable Hermitian matrix function H(t) has the properties (a)-(c) then there exists a bounded continuous matrix function A(t) and a constant y > 0 such that (4) holds. Indeed, one can take A(t) = -AH(t) for any large 1 > 0. We return now to the previous setup. Put r = n, + ee. + n,,,, so that p + q + Y = n. Then r is the number of negative eigenvalues of H(t) for large positive t less the number of negative eigenvalues for large negative t. Moreover, r = 0 if and only if H(t) is nonsingular for all real t. In this case we have 7”=71@7;. If P is the projection whose range is the subspace of initial values x(0) with x(t) E 7”; and whose nullspace is the subspace of initial values x(0) with x(t) E 7 1, then (1) has exponential dichotomies on R + and R- with common projection P and, consequently, an exponential dichotomy on R with projection P. This proves the second part of Theorem 1 below. THEOREM 1. If the equation (1) has an exponential dichotomy on 7 = R, then there exists a nonsingular bounded. continuously differentiable Hermitian matrix function H(t) which satisfies (4). Conversely, suppose A(t) is bounded and there exists a bounded continuously dtrerentiable Hermitian matrix function H(t) which satisfies (4). Then (1) has an exponential dichotomy on R tf and only tf H(t) is nonsingular for all real t.

If H(t) is nonsingular for all real t then G(t) = -H-‘(t) by (c), and there exists a constant y’> 0 such that G’(t) - G(f) A *(t) -A(t)

G(t) < +Z

for

is also bounded, t E3.

(5)

It remains to prove the first part. Let X(t) be the fundamental matrix of (1) such that X(0) = Z and suppose there exists a projection P and constants K> 1, 01> 0 such that IX,(t, s)l < Ke-a+s’

for

--oo
IX,(t, s)l < Kepa(Spt’

for

-m
where X,Q, s) =X(t)

Px-‘(s),

X,(t, s) = X(t)(Z - P) x- I(s). If we set H(t) = ,(“’ X::(s, t) X,(s, t) ds - 1’

“J

X,*(s, t) X,(s, t) ds,

62

W. A.COPPEL

then (cf. [ 1, p. 601) H(t) IS a bounded continuously differentiable Hermitian matrix function which satisfies (4) with y = f. We will show that H(t) is nonsingular for all real t. SupposeH(t)r = 0 for some real t and some vector l. Then

Premultiplying by P*X*(t) we see that the common value of both sides is zero. It follows that X,(s, t)< = 0 for s > t and X,(s, t) < = 0 for s < t. Taking s = t we obtain r = 0. It is instructive to us to recall that Massera and Schliffer [3, p. 3321 state that Lyapunov’s method does not permit a characterisation of exponential dichotomies on R “in any natural way.” It may be remarked also that the roughness of exponential dichotomies on R (see [ 1, p. 341) follows at once from Theorem 1, at least for systems with bounded coefftcient matrix. For completenesswe include here a more detailed proof of the following result of Kulik [2]. THEOREM 2. Suppose the coefficient matrix A(t) is bounded on ,Y = R. Then the following statements are equivalent:

(i)

the inhomogeneousequation x’ = A(t)x +f(t)

has at least one solution bounded on R for every continuousfunction is bounded on R, (ii)

(6)

f which

the homogeneousadjoint equation x’ = -A *(t)x

(7)

has exponential dichotomies on R + and R- and no nontrivial solution is bounded on R, (iii) there exists a bounded continuously differentiable Hermitian matrix function G(t) and a constant F > 0 such that (5) holds. Proof: By [ 1, Propositions 8.1, 3.31, (i) holds if and only if (1) has exponential dichotomies on R + and R _ corresponding to projections P, and P- such that the range of P, and the nullspace of P- together span C”. Equivalently, (7) has exponential dichotomies on R + and R _ corresponding to projections Q, = I - PT and Q- = I - PT such that the range of Q, and the nullspace of Q- intersect trivially. This proves (i) u (ii). Our previous discussion, applied to (7) instead of (I), shows that (iii) * (ii). We will now show that (ii) * (iii).

DICHOTOMIES

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Thus we suppose that (1) has exponential dichotomies on R + and R with corresponding projections P, and Pp such that the range of P, and the nullspace of P- together span C”. Since P, and P- can be replaced by any projections with the same range and nullspace, respectively, we may assume that the nullspace of P, is contained in the nullspace of P_ and the range of P- is contained in the range of P, . That is, P-P,

=P,P-

=P-.

Set P, = P, -Pp. Then P_, P,, and I-P, are mutually orthogonal projections with sum I. Let P ;, 7 ;, and 7; denote the subspaces of solutions x(t) of (1) with x(0) = P_ x(O), (I - P+) x(O), and Pox(O), respectively. Then there exist constants K > 1, a > 0 such that if x,(.) E 7;

then ix,(t)] < KF”(‘pS’ lx,(s)1

for --co
if x,(.)Ey;

then Ix,(t)1 < Kt?e(sp” Ix2(s)I

for -m
if x3(.)E7;

then lxj(t)l

co,

< K.?(‘+’

Ix~(s)~

for O

Ix&)l

for -co < t < s < 0.

00,

Let X(t) be a fundamental matrix for (1) whose columns belong in order to 9 ;, 7”;, y3. By the Gram-Schmidt process we can write X(t) = U(t) Y(t), where U(t) is a continuously differentiable unitary matrix and Y(t) is an upper triangular matrix with positive main diagonal elements. The change of variables x = U(t)y replaces (1) by

Y’ = WY, where

is also upper triangular with real main diagonal elements. Thus B has the form

Since B is bounded, by a further kinematic similarity we can obtain a system of the same form, whose coefficient matrix will still be denoted by B, with B 12= 0 (cf. [ 1, pp. 11, 4 1I). Moreover, if Yk(t) is a fundamental matrix for the system Y; = Bdf) Y, 3

64

W.

A. COPPEL

and if we set Yk(f, s) = Y,(t) Y,‘(s) L > 1 such that

(k = 1, 2, 3), then there exists a constant

) Y,(C, s)l < Le”“-S’

for

--co
1Y2(fr s)l < LP’S -O

for

--co
/ Y,(f, s)l < LP”“-S’

for

O
< Lg4s-t’

for

-a3
co,

(To obtain the inequality for Y,(t, s) one uses also the equivalence of (i) and (ii).) By replacing y, by ey,, where E is a small positive constant, we may assume also that B13 and B,, have arbitrarily small norm. Take

G”(t) = diag [G, (t), G,(f), G,(t) 1, where

G,(t) = -1'

-co

Y,(f, s>Y;"(t, s) ds,

G,(t) = la Y>(t, s) Y;(t, s) ds, f G,(t) = -1

Y&, s) YT(t, s) ds. 0

Then G(t) is a bounded continuously differentiable Hermitian matrix function and G’(t) - G”(t) B*(t) -B(t) c”(t) < -+1 for all real t if B,, and B,, have sufficiently small norm. Returning to the original variables, we obtain a matrix G(f) which satisfies (5). It is easily verified that G’(t) is also bounded. Suppose now that A(t) is not only bounded but also uniformly continuous on .7 = R. Then, by Theorem 2 and [ 1, Proposition 9.21, the following statements are equivalent: (i)

for every d(t)

in the hull of A(t), the equation x’ = @)x

has no nontrivial

(8)

solution bounded on R,

(ii) Eq. (1) has no nontrivial solution bounded on R and has exponential dichotomies on R + and R ~, (iii) there exists a bounded continuously dlrerentiable Hermitian matrix function H(t) and a constant y > 0 such that (4) holds.

DICHOTOMIES

AND LYAPUNOV FUNCTIONS

65

An analysis of the proof of Theorem 2 shows that it is possibleto choose H(t) in (iii) so that both it and its derivative are bounded and uniformly continuous. If A(t) is positively or negatively Poisson stable, i.e., if there exists a sequenceh, + f co such that A (t + h,) converges to A(t) for every t as v + co, then (iii) implies that H(t) is nonsingular for all real t. For in this case (ii) implies that (1) has an exponential dichotomy on R. I take this opportunity of pointing out an error in [ 1, Proposition 7.4). In the statement of the proposition “if and only if’ should be replaced by “if,” as counterexamples show. The argument which purports to prove that G singular implies H singular is valid only if w/i is invariant under H. This is true if all eigenvaluesof A have negative real part and, in this case,there is a valid converse. I am grateful to Professor Hans Schneider of the University of Wisconsin for drawing my attention to this oversight.

REFERENCES 1. W. A. COPPEL, “Dichotomies in Stability Theory,” Lecture Notes in Mathematics No. 629, Springer-Verlag, Berlin, 1978. 2. V. L. KULIK, Quadratic forms and dichotomy of solutions for systems of linear differential equations, Ukrain. Mat. 2. 34 (1) (1982), 43-49. [Russian] 3. J. L. MASSERA AND J. J. SCH.&FFER, “Linear Differential Equations and Function Spaces,” Academic Press, New York, 1966. 4. A. M. SAMOILENKO AND V. L. KULIK, Exponential dichotomy of an invariant torus of dynamical systems, Dzferenciafnye Uraunenija 15 (1979) 1434-1443. [Russian]