劫难与你2. P R E L I M I N A R I E S ON D I S C R E T E E I G E N V A L U E P R O B L E M SFirst of all, let us agree upon some terminology. While KerM, I m M, defM, indM, M T, and M? denote the kernel, the image, the dimension of the kernel, the index (i.e., the number of negative eigenvalues), the transpose, and the
Moore-Penrose Inverse (see, e.g.,[23, Theorem 1.5]) of the matrix M, respectively, M> 0 and M _> 0 mean that the (symmetric) matrix M is positive definite and positive semidefinite, respectively. Let n E N, N E 1~IU (0), J:=[0, N]¢3Z, J*:=[0, N+ 1] N Z. We abbreviate a sequence (zk)keJ. by z and use the forward difference operator A defined by Az~:= zk+l - zk, k E J. Let there be given n x n-matrices Ak, Bk, Ck for all k E J so that
I - Ak is invertible and Hk= ( - C k AkThe system
A~ ) i s s y m m e t r i c f o r a l l k E J. Bk
(where xk, uk E R n for all k E J*) is then called a linear Hamiltonian difference system. If the n x n-matrix-valued functions Ak(A), Bk(A), Ck(A) depend for all k E J continuously and differentiable on a parameter )~ E R (so that the above assumptions are satisfied for I - Ak (A) and for
Hk(~)=\ Ak(~)
Bk(~)
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 for each A 6 R), then we consider the systems A= Hs,(A), 0< k< N.
181
xk
\ u~
(Hx)
Moreover, let there be given 2n× 2n-matrices R and R* with rank ( R R* )= 2n and
RR *T= R*R T.
We are interested in so-called self-conjoined boundary conditions (see[19, Definition 2.1.1 and Proposition 2.1.1])
Now, this paper deals with discrete linear Hamiltonian eigenvalue problems of the form (Hx), i.e., A R and (R), (E)
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
