Printed in Great Britain PII S0898-1221(98)00210-7 0898-1221(4)

k u.(A) )

k u.CA) )$

so that )Co(A)= Co(A)= 0 prove the validityof our assertion.

L E M M A 5. Suppose (X(A), U(A)) and ()[(A),0(A)) are norma!i~,ed conjdned

bases of (HA) for

A e R with Xo(A) -- 0o(A)=~o(A)= bo(A)= o. Let~ a'. A~meinvertible on some nontrivial open interva/2[. Put

that X~(A)

QkCA):=

('

Uk(A)

0kCA)

0)(0

Xk(A)

2k(A)

.),

'

A e2[.

Then (V2) implies that Q~(A) decreases on 2[. Moreover, (V1) and (V~) imply that Qk(A)decreases strictly on 2[ provided k>~s holds, where~, E J is the strict contro//ab///ty index of

(HR).PROOF. Let k E J*\{0} and A E 2[. We may apply Lemma 4 with the conjoined basis

(X'(A),U*CA))

x(~)

~(~)

and

U'(~)=

U(~)

0(~)

of the"big" system from Lemma 4 so that for d E R 2n

bohner~mr, edu Abstract--This paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict control-lability of a discrete system, that imply isolated

Hamiitonian Eigenvalue P r o b l e m s

189

"* JCk(~)a= J{ O;(:,)x;-' (:,) - u;(:,)x;-' (~)x~(:,)x~ * - '--

{x:'/~>¢{x~"~>~-.:~>x~>}x~-'~>~~=o× x; - ' (:,)d

=-~'x;,-'(~,)d:~./

/xa+~(:,) T\ u~(~)

>(!00 o)}-em(~)0

(A)} d

A~(~)

o o

o

o

\ u;~(~)]

X,,,+~(A)~=o× X;-'k-1 ra=O Um

U~(~)

um(~) )

\

um(~)

(A)dT~rn

holds provided we assume (V2) and use the solution (x, u) of (HA) defined by

u~

:=

\ um(~) v~(~)

-

x~(~)

1), R~(~,)

d.

Now we assume (V 0 and (V2), let k> ss, and suppose dV(~(A)d= 0. This yields/:/re(A)\(x'~+lum) It follows that

= O,

for all0 _< m _< k - 1.

x'),o

x'),l .....

.. J

0

holds. Strict controllability of (HR) on J* with strict controllability index~s E J now forces x= u= 0 on J* so that d= 0 and hence Qk(A)< 0 follows.| LEMMA 6. (V1) and (V2) imply (I). PROOF. For every A E R, we denote the special normalized conjoined bases of (HA) at 0 by (X(A), U(A)) and (X(A), 0(A)). Let Ao E R. We pick a conjoined basis (.~, 0) of (HAo) such that (X(Ao), U(A0)) and (X, 0) are normalized and such that )(N+I is invertible (observe Lemma 1). Let (X(A), 0(A)) be the conjoined basis of (HA) with Xo(A)= )(o and 0o(A) -= 0o, A e R. Due to continuity, XN+I(A) is invertible on some nontrivial open interval that contains Ao, and on this interval we have strict monotonicity of I (-0N+I(A) 0 0 U~+~(,x)) (-RN+~(,X)XN+I(A),

)-'

by L e m m a 5 so that .~ I ( A ) X N+ I ( A ) is strictlydecreasing on this interval also. Thus, there exists~> 0 such that XN+I(A) is invertibleon[Ao-6, Ao+6]\{Ao}. W e now may apply L e m m a 5 once again to obtain that

(,

o)(0

decreases strictly on[Ao - e, Ao+ e]\{Ao}. This shows that (I) holds and hence the proof of Theorem l(i) is done.|

bohner~mr, edu Abstract--This paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict control-lability of a discrete system, that imply isolated

190

M. BOHNZR

7. L O W E R

BOUNDEDNESS

OF EIGENVALUES

The purpose of this section is to provide a proof of Theorem l(ii). We need the following auxiliary result. LEMMA 7. Let there be given m x m-matrices A, A, B, B, C, C such that

H= are symmetric. Suppose that

and

H=

A

B

H_>~ H,hold. Then, we have

KerBCKerB~

and

B(Bt-Bt)

B>O

xTc2, nc u T B ufor a/l z, u, u e R m with B u - B u= (A - A)z.

>_ z T C z+ uT B u

PROOF. By[19, Lemma 3.1.10], H> H implies B> B and the existence of a matrix D with A- A= (B- B)D

and

D T ( B - B ) D< C - C.

According to[14, Remark 2(iii)], Ker B C Ker B is equivalent toB= B

BtB=

BBtB.

Let x,u,u 6 R m with B u - B u= ( A - A)z= (B - B)Dz. Then,

>_

- B)D

+,:B,,-{B,,+ (B -

= x'rD'r(B _ B B t B ) D z+ uT(B - B B t B ) u+ 2zTDT(B _ BBtB)u= (mx+ u)V(B - B B t B ) ( D z+ u)= (Dz+ u)rB(B t -

Bt)B(Dx+

u)> o.

T H E O R E M 3. (Comparison Theorem.) Suppose that conditions (%'2) and (Vs) hold.~'(.;A)> 0 for all A< A.

Then,

bohner~mr, edu Abstract--This paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict control-lability of a discrete system, that imply isolated

HamiltoniaaEigenvalue ProblemsPROOF. Suppose~'(.;A)> 0 and let A< A. By (V2) and (Vs), we have for all k J Hk(A)>_ Hk(A), Ker Bk(A) C Ker Bk(A),

191

Bk( )> O.

Let (x,u) be such that ( -x0~ i m R T, x~ 0, and Axk= Ak()t)X,k+l+ Sk()t)~l,k, k J.\ XN+I/ Define Uk:= Bt(A)Bk(A)Uk -{ I - Btk(A)Bk(A)} DkXk+l, k J, where Ak(A) - Ak(A)={Bk(A) - Bk(A)} Dk according to the proof of Lemma 7. Then,

Bk()t)~tk- Bk(~)~k~--(Bk(~)- Bk()t)} DkXk+l= (Ak(~)- Ak()t)} . T k+ land thus Axk= Ak(A)xk+l+ Bk(~)Uk for all k J, so that an application of Lemma 7 yields

o< y(z,~; !)k----O N k----0 XN+I~~ XN+I T X XN+I

= Y'(z, u;~).Hence~r(.;A)> 0 also.|

Now we are able to finish the proof of Theorem l(ii)--and hence, of Theorem 1--as follows. Assume (V2), (Vs), and controllabilityof (H),) on J* for all A R. For A R, let (X(A), U(A)) and ()[(A),0(A)) be the special normalized conjoined bases of (HA) at 0 and define

M(A):=R'RT+R

(

UN+I(A) UN+I(A)

0)(0

XN+I(A)

XN+I(A)

whenever the inverseexists. Now we pick A0 _< A. Thus,~'(.;Ao)> 0 according to …… 此处隐藏：6372字，全部文档内容请下载后查看。喜欢就下载吧 ……