Printed in Great Britain PII S0898-1221(98)00210-7 0898-1221(2)

Azk=

Ak(

)xk+,+

Bk(A)uk0< k< N, (A e R), --X0

Auk= Ck(A)x~+, - A~(A)uk J '

(E)

As usual, a number A 6 R is called an eigenvalue of (E) if (Hx) has a nontrivial solution (x, u) satisfying (R), and this solution is then called an eigenfunction corresponding to the eigenvalue A. Moreover, the set of all eigenfunctions is called the eigenspace, and its dimension is referred to as being the multiplicity of the eigenvalue. We shortly s-mmarize some basic definitions and results from[14] on linear Hamiltonian difference systems that will be needed later on. DEFINITION 1. (Conjoined Basis; see[14, Definition 1].) If the n x n-matrices Xk, Uk (instead

of the vectors xk, uk) solve (H) with

r~(X:

U:)=n

and

X:U~=U~X~,

~or~k6J*,

then (X, U) is ca//ed a conjoined basis of (H). Two conjoined bases (X, U) and (X, U) are called normalized whenever X~ Ok - U~ f(k= I (the n x n-identity-matrix), holds. The conjoined bases (X, U) and (X, U) of (H) with for all k e J*

Xo=0o=0

and U o= - R o= I|

are known as the special normalized conjoined bases of (H) at O.

LEMMA 1. (See[19, Coro//ary 3.3.9] and[14, Lemma 3].) For any m E J* and any conjoined basis (X, U) of (H), there exists another conjoined basis (f(, O) o[ (H) such that (X, U) and

(fC, O) are normal~ed and such that f¢m is invextible. Farthexmore, two matrix-valued solutions (X, U) and (X, U) are normalized conjoined bas~

of(H) iffCX*, U* ) withX'=

(0,)x g

and

U

O

0)

'

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

182

M. BOHN~.R

is a conjoined basis of the system

A

<

zt`

=

0 At,

0 0

\ ut`/ '

O<k<N,

Bk/

where the occurring matrix is of size 4n x 4n.

DEFINITION 2. (Disconjugacy; see[14, Defin

ition 2].) The discrete quadratic functionalN t`=O~N+I

T~N+X

is called positive definite (we write jr> O) if~r(z, u)> 0 holds for all admissible pairs (z, u) O.e., t h a t s a t i s f Y A z t`= A t` x t`+ x+ B t` u t` f° r a l l k E J ) w i t h x# O a n d ( - Z°E I m R T ' I f i n t h i s z N+ l ) definition R= 0 and~> O, then (H) is called disconjugate on J*.|

DEFINITION 3. (Controllability; see[12, Definition 3] and[14, Definition 5].) The system (H) is called controllable on J* if there exists k E J* such that for all solutions (x, u) of (H) and for all m E J with m+ k E J*, we have thatXm ---~Xm+l -----''"= X m+ t` -~0

implies x= u= 0 on J*. The m/nimal integer t¢ E J* with this property is then called the controllability index of (H).|

3. S T R I C T C O N T R O L L A B I L I T Y A N D M A I N R E S U L T SWe open this section with the following key definition. DEFINITION 4. (Strict Controllability). The set of s y s t e m s{(HA): )t E R}=: (HR) is called strictly controllable on J* if (i) (HA) is controllable on J* for all,~ E R (see Definition 3), and if (ii) there exists k~ J such that for al/A E R, for al/solutions (z, u) of (HA), and for al/rn E J with m+ k E J

implies z= u= 0 on J*. The minimal integer tcs E J with this property is then called the strict controllability index of (Ha). I

For stating our main results, we wish to label the following assumptions. (V1) (Ha) is strictly controllable on J*. (V2)~1 _<~2 always implies Ht`(~l) _< Ht`(~2) for all k E J. (Vs) There exists~ E R such that (.;~)> 0 and such that )~ _<~ always implies for all k E J Ker Bk(~) C Ker Bk(~) andB~()~){B~(~)-B~(~)}B~(~)>_O.

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

Hamiltonian Eigenvalue Problems

183

Now our main result reads as follows.THEOREM 1. Assume (V~), (V~), and (Vs). Then, ff there exist eigenvalues of (E), they may be arranged by-oo<~AI_~As_~ A3_~""

More precisely, (i) (V1) and (V2) imply that the eigenvalues are/solated, and (ii) (V2) and (Vs) imply that the eigenvalues are bounded below by A (which is not an eigenvalue) provided (HA) is controllable on J* for all A E R.|The remaining sections are devoted to the proof of the above theorem. However, here we wish to make some remarks concerning the concept of strict controllability. REMARK 1. Suppose Ak(A) -: Ae and Bk(A) -: Bk are constant for all k E J. Then condition (ii) of Definition 4 (with strict controllability index s, E J) already implies condition (i), i.e., controllability of (HA) on J* for all A E R, and the controllability indices s(A) of (HA) satisfy maxAeR~(A) _< ss+ 1 E J*. To prove this, assume (ii), let there be given A E R, a solution (z, u) of (HA), and m E J with m+~,+ 1 E J* such thatXn= Xm+l= .Tm+2=

=

Xm+~,+l

--~ 0

holds. Therefore,0 (A)xn+l= 0+l(A)xn+ .....= 0,

andhence(note/-/k(A)=(-C0(A)

0)0 f o r k E J ),

Condition (ii)thus implies x= u= 0 on J* so that controllability of (HA) on J* with controllability

index~(A)<~a+ 1 follows.| R E M A R K 2. Suppose as in the previous remark that Ak(A) and Bk(A) are independent of A E R for all k E J. Furthermore, assume that 0k(A) is nonsingular for all k E J and all A E R. Then controllability of (HA) on J* for all A E R with controllability indices s(A) E J implies strict controllability of (HR) on J" with strict controllability index s,< s:= maxAeR s(A). To show this, let A E R, let (x, u) be a solution of (HA), let m E J with m+ s E J (this yields m+ i …… 此处隐藏：5408字，全部文档内容请下载后查看。喜欢就下载吧 ……