An oscillation theorem for self-adjoint differential systems and an ...

Math. Proc. Camb. Phil. Soc. (1995), 118, 351 351
Printed in Great Britain
An oscillation theorem for self-adjoint differential systems and an
index result for corresponding Riccati matrix differential equations
BY WERNER KRATZ
Abteilung Mathematik V, Universitat Ulm, .D-89069 Ulm, Germany
e-mail: kratz@dulruu51.bitnet
(Received 9 December 1993; revised 17 August 1994)
Abstract
The main result of this paper is an oscillation theorem on linear self-adjoint
differential systems and a corresponding eigenvalue problem. It establishes a formula
between the number of focal points of a so-called conjoined basis of the differential
system on a given compact interval and the number of eigenvalues which are less
than the given eigenvalue parameter. It extends an earlier result of the author and
generalizes an oscillation theorem of M. Morse. Among others the proof of the
theorem requires a formula on the index of the difference of symmetric solutions of
a corresponding Riccati matrix differential equation. This index formula is the other
new result presented.
1. Introduction
Let nxn matrix-valued functions A(t), B(t), C(t), C0(t) be given together with a
fixed interval [a, b], and In x 2w-matrices S13, S2i, such that A(t), B(t), C(t), C0(t) are
piecewise continuous; B(t), C(t), C0(t) are symmetric; B(t), C0(t) are non-negative
definite on [a,b], and such that rank (S13,S2i) = 2n and S13S2i = S2iS'[3.
We consider the eigenvalue problem, given by the self-adjoint linear system
± = A(t)x+B(t)u, u = (C(t)-AC0(t))x-A T (t)u ( + )
(with eigenvalue parameter A) and the boundary conditions
0
- w e w r i t e

352 WERNER KRATZ
Then Theorem 1 asserts that
where Jf = yK*(A) is a symmetric 3n x 3n-matrix depending on X, U, S13, S2i, and A,
which will be constructed explicitly (see (19)) below; nx = n^X) denotes the number
of focal points of X(t) (i.e. the zeros of detX(t) including multiplicities) in the interval
(a,b): n2 = n2(A) denotes the negative index of Jf (i.e. the number of its negative
eigenvalues), and where n3 = n3(A) denotes the number of eigenvalues of our
eigenvalue problem above (including multiplicities), which are less than A.
The local form of this result generalizes a corresponding result by M. Morse [14,
theorem 24-1] (see also [15]), where it is assumed that B(t) and C0(t) are invertible
on [a, b] (i.e. the case corresponding to quadratic functionals, see [2]). But Theorem
1 extends in particular a recent result of the author, in which (*) is shown
(essentially) under the additional assumption that X(6) = X(b; A) is invertible (cf. [7,
theorem 2]). But this did not allow the derivation of formulae of G. D. Birkhoff[3]
(see also [1]) on the number of zeros of the eigenfunctions in the 'scalar case', i.e. if
n = 1. These results of G. D. Birkhoffare now covered by our new result. Besides the
tools used in [2, 7, 10] we need some new results on the indices of monotone matrixvalued
functions (see [8]) for the proof of our oscillation theorem.
These index theorems on matrix-valued functions also lead to a result on the index
of the difference of solutions of the corresponding Riccati matrix differential
equation
This result is the content of Theorem 2 below.
We briefly describe the contents of this paper. In Section 2 we introduce the
necessary notation, and provide some formulae concerning essentially the factorization
of the crucial matrix Jf above. In Section 3 we state and prove Theorem
1. Finally, in Section 4, we shall derive the index result on Riccati matrix differential
equations (which is also needed for the proof of Theorem 1).
2. Notation and auxiliary formulae
Throughout we denote by ker, Im, rank, def and ind the kernel, image, rank, defect
(that is the dimension of the kernel) and negative index (that is the number of
negative eigenvalues) of a matrix; and / denotes the identity matrix. Moreover, we
write Ql < Q2 if Qt and Q2 are symmetric and if Q2 — Ql is positive definite.
Monotonicity of symmetric matrix-valued functions is then defined accordingly (see,
for example, [8]).
Throughout this paper we shall use the following assumptions:
(A) there are given n x w-matrix-valued functions A{t), B(t), C(t), C0(t), a fixed
interval [a, b] (a < b), and 2n x 2n-matrices S13, SM, such that: A(t), B(t), C{t), C0(t)
are piecewise continuous on [a,b] (we write eC s [a,b]); B(t), C(t), and C0(t) are
symmetric for te [a, b]; B(t) and C0(t) are non-negative definite (i.e. ^ 0) for te [a, b];
and such that rank (S13, S24) = 2n, S13S24 = S2iSf3.
Then we consider the following self-adjoint eigenvalue problem, which consists of
the linear self-adjoint differential system
x = Az+Bu, u = (C-AC0)x-A T u (1)

An oscillation theorem 353
and the 2n linearly independent, self-adjoint boundary conditions
where A is the (real) eigenvalue parameter.
We need the following notions summarized in
= 0, we write (x, u)e8%, (2)
Definition 1. (i) Two nxn matrix-valued functionsX(t), U(t) are called a conjoined
basis of (1) (for any fixed AeIR), if they solve (1) and if rank (X T (t), U T (t)) = n,
U T (t)X{t) = X T (t) U(t) for te[a, b]; and then toe [a, b] is a focal point oiX(t) of multiplicity
seN if s = deiX(t0) (so that X(t0) is non-invertible).
(ii) The pair (A,B) is controllable or normal (on [a,b]), ifw = —A T u,Bu = 0 almost
everywhere on some non-degenerate interval [t1,t2] c [a,b] always implies that
u(t) = 0 almost everywhere on f^,^]-
(iii) The triple {A,B, Co) is strongly observable (on [a,b]), if x = Ax+Bu, Cox = 0
on some non-degenerate interval [t1,t2] a [a,b] implies that x(t) = 0 on [tl,t2] for all
Remark 1. The concept of conjoined bases and focal points was studied by M.
Morse[13,14] and W. T. Reid[15,16] (see also [2,10]). Concerning the notion of
controllability (called normality by Morse and Reid) see [5, 14, 15, 17]. Strong
observability was introduced in [6]. It can be characterized by a rank condition in the
time-invariant case (see [9]).
Moreover, it is well known (see [2, 10, 15, 16]) that, if X, £7 is a conjoined basis of
(1), then (whenever it exists) Q(t) = U(t)X~ l (t) is symmetric and solves the Riccati
matrix differential equation
= 0. (3)
Let us mention also the so-called Wronskian identity, which says that
is constant for any two (matrix) solutions of (1); so that the conditions of Definition 1
on a conjoined basis are satisfied if they hold for some toe[a, b].
Now, given any conjoined basis X, U of (1), there always exists another conjoined
basis X2, U2, such that Xlt Ux (with X1 = X, U1 = U) and X2, U2 are so-called
normalized conjoined bases (see [2, 10]), which means that the following identities
hold (for all te[a,b]), where AeIR is fixed):
X? Ut = UfXt for t = 1,2, XXX^ = X2Xf, U, VI = U2 U\
and X T XU2-V T XX2=XXU T 2-X2V1=1
Moreover, we shall use the special normalized conjoined bases Xlt C/j andX2, U2 of (1),
which satisfy the initial conditions (see [10, (8)])
X,(a) = 02(a) = 0, U^a) = -X*(a) = 1. (5)
(4)

354 WERNER KRATZ
Next, assuming (A) and (4), (5) (which will be done throughout) we introduce further
notation and state formulae which can easily be verified by using mainly the
Wronskian identity (e.g. XfU1-UfX1 =Xf(a) by (5)), the formulae (4), (5), and
their consequence:
X1=X1U1(a)-X2Xl(a),
Finally, let
Xl = -X1X^(a)+X2X^(a) on [a,b].
MS) = ( {xfl}* 2 c^-i) F-l W = x ir«
with
and correspondingly
and
MS) =
I-
\t),
-X2(a) U2(a)\
)
Then, as follows directly from (1), (2) and (4) (see [10, lemma 4]), A is an eigenvalue
of (1), (2), if and only if A is non-invertible, and s = def A is its multiplicity.
Remark 2. When applying this theory the so-called Picone identity was of
particular importance (see [10, theorems 1 and 2] and [2, proposition 6 - l]). In the
formula of Picone the following term is essential
where
T = {F(t,x(t))-x T (t)u(t)}\ b a = f-x T (t)u(t)\ b a,
F(t,x) = x T
- ^ X~ l (t)X2{t)a
for some fixed (parameter) vector aeU n and 'admissible' functions x(t), u(t). This
leads to the most important matrices Ji1 and Jt'. Namely, if X^a) and X^b) are
invertible, then
T=c T Jixc for c T = (a T ,x T (a),-x T (b)),
and if, in addition, {x,u)e!%, i.e.
x(a)
(see [10, proposition A 2]), then
T = d T Jld for d T = (a T ,cf),
where the symmetric 3n x 3w-matrices Mx and dt are given by
(XS(a)X2{a)-X-\b)X2{b) -X?(a) -
-{X?} T (a) -U,{a)Xl\a)
~{X^} T (b) 0
(6)
(8)

and
An oscillation theorem 355
= RlRf+RiJ[1R? with (9)
Remark 3. Observe that these matrices Jtx and Ji are simpler than the
corresponding matrices in [2, (3-6)], but they are congruent (see [4, p. 296]) to those
matrices via an easy transformation, provided that X^a) and X^b) are invertible, as
was assumed. Of course, the formulae above and subsequently are always valid if the
inverse matrices involved exist. Moreover, observe that Mx and consequently also Jt
do not depend on the particular choice of the conjoined basis X2, U2 at the beginning
but only on the given conjoined basis X = Xx, U = U^; a fact which follows also from
formula (14) below.
Furthermore, we need some more formulae and notation arising from (8) and (9),
which may be verified as before (use (4) and (5) in particular).
where
A=X- l (a)X2(a)-X- l (b)X2(b)
rTWJ
7 0 0
with «=|[/,(a) Ux{a) U2(a)\,
0 U^b) U2(b)
( -Xx(a)
-I
-X2(a)\,
X2(b)
so that
A -X-^a)
SC-* = \X-\b)X2{b) 0
-
-7 0 0
0 0
X = \-
-X,(a) -Xx(a)
0 Xx(b)
/
0
f*R T -l A
1 2 ~\0 8lti
C — (Cj ,C2,C3 )
where (15)
0
/
(10)
(11)
(12)
(13)
(14)

356 WERNER KRATZ
The formulae (12) and (13) defining X and A yield
rank#" = n + rank Xx(a) +rank Xx(b), def & = defX^ + defX^b),)
rank A = n + rank A, def A = def A. '
3. The oscillation theorem
This theorem concerns mainly the behaviour of the matrix Ji', when A varies
(which was arbitrary, but fixed, throughout the last section). To study the
dependency of the terms of the previous section on the parameter A, we assume
toe(a,b) is fixed, X,(«0;A) =X4(«0), Ut(t0\ A) = Ut(t0)\
for t=l,2,
and X1(a;A) = U2(a;A)=0, Ux(a; A) = ~X2(a; A) = /
(as in (5)) are fixed with respect to AelR.
Then, of course, the functions of Section 2 depend on A as well, e.g. X((t) = Xt(t; A),
= Ut(t; A), Mx(t) = Mx(t; A), etc. A = A(A), Mx = Jtx(A), Jt = Jt(A), etc.
We state the following results from [2] (or [10]), which will be needed.
LEMMA 1. Assume (A), let the pair (A,B) be controllable, and let the triple (A,B,C0)
be strongly observable. Then the following statements hold:
(i) the focal points ofXx(t; A) {and also ofXx(t;A)) are isolated with respect to te[a,b],
t =(= t0 (when A is fixed), and also with respect to AelR (when t #= tQ is fixed)
(by [2, (2-5) and (2-7)]);
(ii) the singular points o/A(A) (i.e. the eigenvalues) are isolated (by [2, (3 - 5)]);
(iii) nx(X + )— nx(X —) = defXx(a;A)+defXx(b;A) for AelR, where nx(A) denotes the
number of focal points of Xx(t;A) (including multiplicities) in the interval (a,b)
(by [2, (2-8)]);
(iv) Mx(b;A), —Mx(a;A), and Mx(b;A) are strictly decreasing (where they exist)
(by [2, (2-9)]);
(v) Jtx(A), Jf?(A), J/(A) and J?*(A) are decreasing (see (8), (9), (14), (15));
(vi) n3(A + ) — n3(A —) = def A(A) for AelR, where n3(A) denotes the number of the
eigenvalues of (1), (2), which are less than A.
Now, [8, theorem 2] yields the local oscillation theorem, which may be obtained from
[2, theorem 1] as well (see also [14, theorem 24-1]).
PROPOSITION 1. Assume (A), let the pair (A,B) be controllable, and let the triple
(A, B, Co) be strongly observable. Then, if Xx, Ux, A and M are according to (4), (7), (9)
and (17), we have
ind Jt(A +) - ind M(A -) = def A(A) - defX^a; A) - def Xx(b; A) for AeU.
Proof. By (A), (4), (9), (12), (13), (16) and particularly by Lemma 1 (v), the main
result of [8] may be applied to 3C(A),

An oscillation theorem 357
Remark 4. It should be noted that besides some formulae the quoted index result
from [8] mainly uses the monotonicitj 7 of Jl(A) (Lemma l(v)), which is a simple
consequence of the differential system (1). The proof of the corresponding result in
[2, pp. 345-351] is much more elaborate and less transparent.
As is shown in [2, 7] the local oscillation theorem leads to a global result, if the
behaviour of ind^#(A) is known as A-> —oo. This follows, via the identities (14) and
(15), from the asymptotic behaviour of solutions of the Riccati matrix differential
equation (3) as A-> — oo. These results were obtained in [6], and they require the
following additional assumption on the system (A, B, Oo):
(B) kerB(t), kerC0(t), and A T (t) as linear mapping restricted to kerB(t) are
constant on [a, b].
Of course, (B) holds if the system is time-invariant.
Now, our global result will follow from Proposition 1 and
LEMMA 2. Assume (A) and (B), let the pair (A,B) be controllable, and let the triple (A,
B, Co) be strongly observable. Then, the following statements hold:
(i) Xt(t;A) is invertible for all te[a,b], t #= t0, if A is sufficiently small, and
Qi(a; A) -> — oo, Qt(b; A) -> + oo as A -»• — oo (this means that all eigenvalues of the
matrix Qt(a; A), Qt(b;A), respectively, tend to — oo, + oo, respectively, as A -> —oo)
where Qt = Ut X^ 1 for i = 1,2;
(ii) Xt(t;A) is invertible for all te(a, b], if A is sufficiently small, and Q((b;A) =
Ut(b; A) X^ 1 (b; A) ^ oo, M^b; A) ^oo as A^-oo for i= 1,2;
(iii) ind{Qj(6; A) — Qx(b; A)} = n — defX^t^), if A is sufficiently small (see (17)).
Proof. Except on the matrix Mx(b; A) the statements (i) and (ii) are the contents of
[6, theorem 2] under the same assumptions that we have made here. The assertion
that M^b; A)->oo as A-^— oo follows from this asymptotic behaviour via the
minimum—maximum principle. This fact is shown in [10, corollary 13]. Finally,
statement (iii) follows from (i), (ii) and from the index result, Theorem 2 below,
because Q^t; A) — Q±(t; A)-»- — oo as t^a+ for all fixed A, which are sufficiently small,
by (5) and assertion (i) (observe that Xt, Ux is the so-called principal solution of (1)
at a, see [15]).
Remark 5. (i) By using the identity (11), there is an alternative way to obtain
assertion (iii) of Lemma 2, namely: let AeIR be fixed and consider the.function
H(t) = X- 1 (t)X2(t)-X^ 1 (a)X2(a). Then, H(a) = 0, H(t) is strictly increasing on [a,b]
by (1) and (4) (see [2, (2-6)]), and H(t) = {X^(a)Xf(t)} {-A^H*) by (4) again. Now,
statements (i), (ii), formula (11), and [8, corollary 2] yield statement (iii).
(ii) Of course, M^b; A) ->oo implies that Jf^i; A) ->• 0 as A ->• — oo, and then formula
(6) yields (for example) that U2' 1 (b;A)^>-(i as A->— oo. This is remarkable, since
X\~ l (b\A) ¥+0 (although X2~ 1 (b;A)X1(b;A)^*0, use (6)) as A->—oo in general as is
shown in [11] by examples of certain Sturm-Liouville equations.
Now, we can derive the global oscillation theorem.
PROPOSITION 2. Suppose that the assumptions of Proposition 1 hold, use the same
notation as in Proposition 1, and assume (B).
Then, for all AeUfor which -Xj(a;A), Xt(6;A) and A(A) are invertible, we have
, (18)

358 WERNER KRATZ
where nx(A) denotes the number of focal points [including multiplicities) ofXx(t\A) in the
interval (a, b), n2(A) denotes the negative index of Jl(A) and where n3(A) denotes the
number of eigenvalues of (I), (2) (including multiplicities) which are less than A.
Proof. First, it follows from (17) and Lemma 2(i) that limA^._oon1(A) = defX1(

An oscillation theorem 359
Now, of c = Cj + c2 with c1eImT T , c2ekerT = ImZ, we have
c T Jfc = cfJ^*cl- cl ZZ T c2.
Hence, ind^T = ind^1 / '* + rankZ, which yields (18). I
Remark 7. As already mentioned in the introduction Theorem 1 for the scalar case
n = 1 may be used to derive the results of G. D. Birkhoff (see [3] or [1]) on the
number of zeros of the eigenf unctions. Moreover, concerning the invariance of indJ^
under any possible choice of the matrices S, T, Z in (19), let us mention the following
assertions, which can be derived quite easily (see [10, proposition A 1] and [12,
Ch. I, 5-7-2]): there exist symmetric matrices S1 = »51(A) and S2 such that
so that
% T (A)

360 WERNER KRATZ
Proof. Since the pair (A,B) is controllable, the focal points of Xx(t) and X2(t) are
isolated by Lemma 1 (i) above (or by [2, (2-5)]), so that the left-hand sides of (20) and
(22) are well-defined. Next, we have (whenever Xt(r) is invertible) that
Qi(r)-Q2{r) = -{X?} T (T) WX?(T),
where the Wronskian W = X T (T) U2(T) — U T (T)X2(T) is constant by Remark 1. Hence,
rank {Q1 — Q2} (T) = rank W is constant as mentioned at the beginning of this section.
Now, we prove (22) by applying [8, corollary 1] with the following setting.
Choose a conjoined basisXJ, U* of (1), such thatXj, Ul andXJ, U* are normalized
conjoined bases of (1) (use [2, (2-3)]), so that (4) holds correspondingly, fix te(a, b),
and put (with the notation of [8])
Rx = U*X* -X T U*, R2 = U2 r X1 -X? U1 (= W T ),
U(T) = -Xf(r), X{T) = X?{T), U = U{t), X = X(t),
A = R^X+R2 U, M(T) = R1R2 T +R2 U{T)X-\T)R^.
Then, with m = n, and Q, S, T as in the corollary, we have the same setting as in [8,
corollary 1]. Now our assertion, i.e. (22), follows directly from that result [8, (22)],
by verifying the assumptions and corresponding quantities as follows:
rank (R^RJ = rank{(C/-f, -Z2 T ) (^ f|)| = rank (Z2 T , U T 2) s n
(observe that R1 and R2 are (constant) Wronskians), and R±R2 = R2Rf (use (4)), so
that [8, (3)] holds;
U{T)X~ r (T) = —X^I^X^T) is symmetric and decreasing (by [2, (2-6)]), so that [8, (5),
(14), (15) and (16)] hold;
M(T)=X^(T){Q1(T)-Q2(T)}X2(T) (use (4)), so that indif(r) is equal to the
ind{Ql — Q2}(T) (whenever X2(T) is invertible); and
A(T) = RtX(T)+R2 U(T) = -X^(T) (use (4) again), so that [8, (12), (17)] hold with the
setting of (21) and so that def A(t + ) = 0.
Now, assertion (22) above coincides with [8, (22)], which completes the proof.
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