On Decay of Solutions and Spectral Property for a Class of Linear Parabolic Feedback Control Systems ()
1. Introduction
Stabilization problems for linear parabolic control systems have the history of more than three decades. Although some difficult problems are left unresolved, it seems that the study has reached a degree of maturity in a sense. The so-called dynamic compensators are introduced in the feedback loop to cope with the most difficult case such as the scheme of boundary observation/boundary input (see the literature, e.g., [1-6]). In [2-4,6], no Riesz basis is assumed, corresponding to the coefficient elliptic operators with complicated boundary operators (see (1) below). Let 
be a separable Hilbert space with the inner product 
and the norm
. A standard control system with state
, 
consists of a finite number of inputs
, 
, and outputs
, 
, and is described by the following linear differential equation in
:
Here, 
denotes a linear closed operator with dense domain 
such that the resolvent 
is compact; 
actuators through which the scalarvalued inputs 
are inserted in the equation; and 
linear functionals of 
which allow unboundedness but are subordinate to
. The control system also reflects boundary feedback schemes by interpreting 
and the differential equation in weaker topologies. In general stabilization studies, the inputs 
are designed as a suitable feedback of the outputs
, so that the state 
could be stabilized as
. Then every linear functional of 
also decays at least with the same decay rate. This is true in the case where the functional is unbounded and subordinate to
.
We then raise a question: can we find a nontrivial linear functional which decays faster than
? The purpose of the paper is to construct a specific feedback control system such that 
decays exponentially with the designated decay rate, and that some nontrivial linear functionals of
, say
, decay definitely faster than 
for any initial state. To achieve this property, our control scheme contains a dynamic compensator with state 
in another separable Hilbert space 
in the feedback loop to connect 
and
. Thus the control system has state 
in the product space
. We note that the above decay property is achieved in a straightforward manner in the static feedback control scheme in which we set 
and
,
. In fact, the static feedback system contains a single state 
only, and the so-called spectral decomposition of 
associated with the elliptic operator enables us to find such an 
in some spectral subspace. Such typical examples are the Fourier coefficients corresponding to higher frequencies. In our control system, however, it is indispensable in the spectral decomposition method to ensure a vector of the form 
in the spectral subspace of 
to achieve a faster decay of
, where 
1. It is very unlikely and almost denied to find such a vector 
in our control system with state
. To make the paper clearer and more readable, unlikeliness of the above vector 
is discussed in detail in Section 3, which turns out to be a new spectral feature of the control system, and has never appeared in the literature; this spectral property also justifies the relevance of our problem setting.
Let us begin with the characterization of the controlled plant. Let 
denote a bounded domain in 
with the boundary 
which consists of a finite number of smooth components of 
-dimension. Let 
be a pair of linear operators defined by
(1)
where 
for
,
;
, 
, 
for some positive 
and
being the unit outer normal at
. Henceforth set
. The pair 
defines an operator 
closable in 
as 
for
.
The closure of 
is denoted as
. It is well known (see [7]) that 
has a compact resolvent
; that the spectrum 
lies in the complement 
of some sector
, where 
; and that the following estimates hold:
(2)
where the symbol 
also denotes the 
-norm. Thus 
is an infinitesimal generator of an analytic semigroup
,
. The fractional powers
, 
are defined in a standard manner, where 
and 
is sufficiently large. It is not very clear on how the domain 
is characterized by the fractional Sobolev spaces, since the Dirichlet boundary is continuously connected by the Robin boundary. Such a characterization is, however, neither essential nor necessary in our study. There is a set of generalized eigenpairs 
such that
1)
;
2)
for
; and
3)
.
It is well known (see, e.g., page 285 of [8]) that the set 
spans
, but does not necessarily form a Riesz basis for
. Let 
be the (not necessarily orthogonal) projector corresponding to the eigenvalue
. The restriction of 
onto the invariant subspace 
is, according to the basis
, equivalent to the 
upper triangular matrix
:
(3)
By setting
, the matrix 
is nilpotent, that is,
. The minimum integer 
such that 
is called the ascent of
. It is well known that the ascent 
coincides with the order of the pole 
of 
(see Theorem 5.8-A of [9] for more details). Let 
be the formal adjoint of
:
(4)
where
. The pair 
defines an operator 
just as the above
. Then the adjoint of
, denoted by
, is given as the closure of 
in
. There is a set of generalized eigenpairs 
such that
1)
; and
2)
.
Similarly, the set 
spans
. Setting 
, we have the relationship:
(5)
Let us turn to the characterization of a dynamic compensator. Let 
be any separable Hilbert space with inner product 
and norm
. Relabelling an orthonormal basis for
, let 
be a new orthonomal basis for
. Every vector 
is then expressed in terms of the basis 
as a Fourier series:
,
. Let 
be a sequence of increasing positive numbers:
, and set
, where
. Let 
be a linear closed operator defined as
(6)
with dense domain
. Then, 1)
; and 2)
, 
,
. Thus 
is the infinitesimal generator of an analytic semigroup
, which is expressed by
. The semigroup 
satisfies the decay estimate
(7)
The adjoint operator 
of 
is described as
(8)
and thus
. Let
, 
be the projector in 
such that
.
Our control system has state
, and is described as a differential equation in
:
(9)
The equation with state 
means a dynamic compensator equipped with a set of outputs 
and
, 
, where α > 0. The parameters 
and 
denote given actuators of the controlled plant, and 
actuators of the compensator to be designed. The outputs 
of the controlled plant are considered on the boundary, and defined as
(10)
where 
denotes observation weights. The operator
, specified later, denotes a unique solution to the operator equation on
,
(11)
Given a suitable vector
, the control law is to construct such that the number 
determines the decay rate of a functional
, where
. As we see later, the roles of the outputs 
and 
are, respectively, to determine the decay of 
and the decay of
. In state stabilization problems only, the output 
does not appear. More precisely, let 
be a number such that
. We seek a new feedback scheme such that the decay estimates
(12)
hold for every initial value and
, such that the decay of 
is no longer improved. To achieve the non-standard decay (12), introduce a new operator
(13)
and assume conditions in terms of
. These conditions have never appeared in the literature. Note that conditions are posed on 
for state stabilization.
The functional 
may be regarded as a kind of output of the system. It is worthwhile to refer to our previous results on output stabilization [10-13]: In [10], the decay of outputs is discussed with lack of observability conditions, but the relationship of the decay between 
and the outputs is unclear. In [11-13], the problem is discussed, based on a different principle, i.e., a finitedimensional pole assignment theory with constraint. The controlled plants are, however, limited to those equipped with Riesz basis, and the actuators of the controlled plant, corresponding to our
, are restrictive, and must be subject to a strong constraint: the actuators have to be designed so that their spectral elements in each spectral subspace are orthogonal to the weights of the outputs at infinity. As we have seen, the controlled plants in the present paper do not necessarily allow a Riesz basis, although a somewhat stronger condition, i.e., complete observability, is assumed. An example of systems equipped with complete observability is illustrated in the end of Section 2.
The feedback law in (9) contains parameters:
, 
, 
, 
, 
, 
, and 
to achieve the nonstandard decays (12). They are designed in the following manner: 1) The actuator 
is given in advance arbitrarily. 2) Given the parameters 
and
, the operator solution 
is ensured. 3) Then, a vector 
is chosen among a very wide range of sets in
. 4) Finally, 
, as well as
, are designed to satisfy a finite number of controllability conditions associated with the new operator
. It is generally desirable to pose less restrictive assumptions on the actuators of the controlled plant. In fact, 
is arbitrary, and, as we see later that the conditions on 
are much less restrictive than those in our preceding works above. In fact, 
only have to be designed, belonging to an infinite dimensional subspace of
.
Our main results consist of Theorem 4 in Section 2 and a series of assertions in Section 3 (Theorem 7, Propositions 8, and Theorem 9): The former is on the control law ensuring the non-standard decays (12), and the latter on the relevance of the problem setting with Equation (8), which discusses a spectral property of the invariant subspaces in 
associated with the coefficient operator in (9), that is, unlikeliness of a vector of the form, 
in these subspaces. In Section 3, the spectrum of the coefficient operator is characterized (Theorem 9). To the best of the author’s knowledge, the latter has never been discussed so far, and clarifies a new property of internal structures of control systems.
2. Decay Estimates of Solutions
To ensure well-posedness of our control system (8), let us begin with the operator Equation (10). Adjusting 
and
, 
, we may assume that
(14)
(see (2) for
). In (9), let us express the actuators
, 
as Fourier series in terms of
:
where the overline denotes the complex conjugate. Before stating our first result, let us define the matrices
, and
, 
by
(15)
and
(16)
respectively. Then our first result is stated as follows:
Theorem 1. 1) By assuming (14), the operator equation (11) on 
admits a unique operator solution
, which is expressed as
(17)
2) Assume further that
(18)
Then we have
. Here, 
denotes the closure of 
in
.
Remark. 1) The first condition in (18), called the complete observability condition, is fulfilled with 
in the case where
, 
,
. Actually, by choosing the 
with
, 
, the condition is fulfilled. In the case where
, 
, however, the condition means that
, 
, which requires that 
be equal to or greater than
: this is the case, for example, where 
is selfadjoint.
2) The condition: 
is the so called finite multiplicity condition. In the case of
, we know that
,
. Thus, by choosing
, the complete observability condition is automatically fulfilled. As another example, let 
be a self-adjoint operator defined by 
in 
equipped with the Dirichlet boundary. The eigenvalues of 
consist of
, 
, 
, where 
are the zeros of the Bessel functions 
of m-th order. It is expected that
, if the well known Bourget’s hypothesis (see pp. 484-485 of [14]) is proven. As long as the author knows, this conjecture has not been proven so far.
Proof. The result is a version of the results in [2-5], so that we give here only an outline of the proof. 1) Expression (17) and uniqueness of 
are examined in a straightforward manner.
2) Relation 
is equivalent to
. Assuming that
, we see by (17) that
Since
, we see that 
for 
and
.
For a 
such that 
and each
, we introduce a series of meromorphic functions
, 
by the recursion formula:
(19)
Each function 
has the properties: 1) It has at least the zeros
,
; and 2) the algebraic growth rate 
of these zeros, 
is smaller than 2 by (14). These properties combined with Carleman’s theorem [15,16] imply that 
for
, 
,
. Following [5], we calculate the residue at each
. Then, we see that
(20)
Let
. Note that the restriction of 
on 
is equivalent to the matrix
, and thus,
. The relation (20) is rewritten as
(21)
The complete observability in (18) implies that 
for
. In view of (5), we see that 
for every 
and
. Since the set 
spans the whole space
, we conclude that
. Q.E.D.
Decay of solutions to Equation (9): In view of (11), it is easily seen that
, 
, or
. Thus,
(22)
for
. In (9). let
, and 
(see Proposition 2). Equation (9) contains various parameters:
, 
, 
, 
, 
, 
, and
, among which
, 
and 
are already determined in Theorem 1. Let a non-trivial 
be given arbitrarily. Then, 
by Theorem 1. We find a non-trivial vector 
such that
(23)
There is a variety of choice of such an
. In fact, this is simply possible, e.g., by finding 
such that
for
. Then 
does not belong to the space spanned by
,
. The integer 
may be chosen arbitrarily large. The vectors 
will be determined in terms of the operator
.
The state 
in (9) satisfies the equation:
Actuators 
are chosen so that
,
. An assumption on these 
will be discussed later (see (28)). Then, it is immediately seen that
(24)
By the decay (22), we obtain the estimate:
(25)
for
. Here, 
is non-trivial.
The operator 
defined in (13) has a compact resolvent, and 
consists only of eigenvalues. Let
(26)
where 
and 
for i ≠ j. Each 
may admit generalized eigenfunctions. Let 
be the projector corresponding to the eigenvalue
which is calculated as
, Ci
being the small counterclockwise circle with center
. Then
Proposition 2. 1) The number 
belongs to
. 2) Any generalized eigenfunction of 
in 
and 
are orthogonal to each other.
Proof. 1) Suppose that there is a 
such that
. Then,
. Thus we have, as a necessary condition,
Now set
, and calculate as
Thus we see that
2) Let
, 
, and 
, 
, possible generalized eigenspaces of
. For a
, we calculate by (23) as
This implies that
, or
. Suppose then that
,
. For a
, the function 
is in
, and 
. The same calculation as above immediately shows that
. Thus,
(27)
The integer 
varies over a finite set of positive integers depending on
. Q.E.D.
We now choose the actuators 
in (9) such that
(28)
Then, 
by the above proposition. All parameters except for 
in (9) are determined.
Rewrite the equation for 
in (9) as
(29)
Let us introduce an operator 
as
Let the integer 
be such that
. Then, 
,
. The following proposition is just a simple version of the result in [17].
Proposition 3. Let 
be the projector defined by
. Choose a 
such that
. Suppose that the pair 
is a controllable one. Then we find 
such that
Thus, 
,
. We are ready to state the non-standard deay of solutions to Equation (9).
Theorem 4. Let 
such that
. Suppose that
1)
and 
satisfy the rank conditions (18);
2) 
is arbitrarily given;
3) 
is chosen to satisfy (23); and 4) 
satisfiy the controllability condition in Proposition 3.
Then we find a large integer
; vectors
; and a postive 
close to
, and subsequently the control system in the product space
;
(30)
where
. Every solution 
satisfies the decay estimate:
(31)
for
. The estimates for 
and 
can be no longer improved.
Proof. Choose the functions
, 
stated in Proposition 3. In view of Theorem 1, we find 
such that 
arbitrarily approximate 
in the topology of
. Since 
is separable, we may assume with no loss of generality that these 
are constructed in 
for an enough large
.
Let us consider the operator 
and the perturbed
. The right-hand side of (29) is dominated by the decay estimate (22). Since 
are chosen close to
, the semigroup 
is stable, and satisfies the standard estimate:
(32)
where
. Thus every solution 
to Equation (9) satisfies the estimate:
(33)
where
. The decay estimate for the functional 
is already obtained in (25).
The control system (30) is derived in the following manner: Set
, and apply the projector 
to the equation for 
in (9). By noting that
, then, (30) is immediately obtained. Equation (30) is clearly well posed in
: Thus every solution to (30) is derived from the solution 
to (9) with initial value 
such that
, by setting
. Thus the first estimate of (31) is derived from (33). The second estimate of (31) is clear by (25).
Finally we show that the first estimate of (31) for 
is no more improved. The spectrum 
of the perturbed operator 
consists only of eigenvalues. It is expected that the value of 
would be close to 
as long as 
are close to
: When both 
and 
are selfadjoint, it is well known—via the min-max principle (see [18])—that each eigenvalue of 
is continuous relative to the coefficient parameters. In our problem, the following result holds:
Proposition 5. The minimum of 
is continuous relative to
,
.
Proof. Set
. In view of (32), the left half-plane: 
is contained in
. Thus we see that 
. Choose an 
enough small so that
. Let 
be the counterclockwise circle:
, and suppose that
for
. Choose 
such that
. Then, 
belongs to
. In fact, we have the relation:
(34)
Recall that, for
, the (second) resolvent equation:
holds. Then we see that
(35)
The first term of the above left-hand side of (35) is the projector, corresponding to the eigenvalue 
of
. Choose 
closer to
, if necessary, so that
Supposing that 
is contained in the half-plane:
, we then derive a contradiction. If so, the resolvent 
is analytic inside and on
. Thus the second term of the left-hand side of (35) must be equal to 0. Let 
be an eigenfunction of
, corresponding to the eigenvalue
. Then,
The right-hand side is, however, estimated as follows:
which is a contradiction. Therefore, the spectrum 
also lies in the left-half plane:
. As a conclusion, the minimum of 
satisfies the estimate:
as long as
, 
are small. Q.E.D.
Let us turn to the proof of Theorem 4. Choose an 
in Proposition 5 such that
. Let 
be the eigenvalue of 
such that
, and
a corresponding eigenfunction:
Set
. As easily seen from Equation (29), the function 
given by
is the solution to Equation (9) with the initial value
. In view of the reduction process to Equation (30), the function 
is thus a non-trivial solution to (30). This shows that the decay (31) for 
is no longer improved. This finishes the proof of Theorem 4. Q.E.D.
Example. In (1), let us consider the case where 
is a bounded interval
. The pair of differential operators 
is then rewritten as
(36)
where
, and
. Let 
be an operator defined, for
, by
Clearly 
defines an isomorphism in
. Let us consider the case where 
is of the third kind, i.e.,
. Then, 
transforms 
into another pair
, which defines a self-adjoint operator 
with dense domain
. In
, 
is unchanged; 
and 
are changed, respectively, to 0 and
; and 
of the third kind. The idea is a slightly modified version of the well known result (see page 292 of [18]). Based on this, we have
Proposition 6. 1) The spectrum 
consists of real and simple eigenvalues:
,
.
2) The eigenfunctions 
of 
forms a Riesz basis. Any 
is uniquely expressed as
.
In our problem, we know that
,
. Thus we choose
, so that the output of the system is a single observation at the end point 
as
(37)
that is, 
,
. Let us examine some assumptions in Theorem 4 in this example. Most important is the complete observability (18). The matrices 
in (16) are now
,
. Thus, we see that the complete observability is satisfied. Since 
in (13) is a one dimensional operator, the multiplicities of the eigenvalues 
are equal to 1. This enables us to choose 
in (9). In Proposition 3, the controllability condition on the actuator 
is stated as follows: let 
be eigenfunctions of
. By setting
, the controllability condition is simply that
, 
,
.
In the case where 
is of the first kind, the output is a single observation at the end point 
as
. Proposition 6 also holds in this case. Since
, 
, the complete observability is similarly satisfied.
3. Spectral Property of the Coefficient Operator
We go back to the problem raised in Section 1: Unlikeliness of a vector of the form
. The basic control system is Equation (30) in the product space
. To avoid any unnecessary technical complexity, we limit ourselves to the simple case of one dimensioanl equations raised in (36), 
being of the third kind. In the setting of the space 
as well as 
in (6), we can choose
,
. Thus,
. Equation (30) is simply rewritten as
(38)
where 
is defined as
(39)
and
. Here, 
(see
(37)). The operator 
is sectorial, and every solution to (30) or (38) is expressed as
,
. Let 
be the projector corresponding to a 
with
. In view of the relation:
the right-hand side of which decays as 
with decay rate 
for every initial state. Now we ask: Does the range of the 
contain a vector of the form
? This problem immediately leads to the structure of the eigenspaces of the operator
. In the operator
, the vectors 
and 
of the compensator are the parameters to be designed
. In designing these parameters, they are generally influenced by small perturbations. It is thus implausible to assume that some Fourier coefficients of these parameters would be designed to be
: such conditions are very easily broken. Thus we may henceforth assume that
(40)
where 
and 
denote, respectively, 
and
. The actuators 
and 
of the controlled plant are the given parameters in advance. It is also implausible to assume that some Fourier coefficients of 
and 
relative to 
might be equal to 0. Thus we may also assume that
(41)
The main results in this section are Theorem 7, Proposition 8, and Theorem 9 stated just below. The proof of these results will be given later.
Theorem 7. Let
. Suppose that 
and that
(42)
where 
for
. Then any linear combination of these eigenvectors 
of 
cannot generate a vector of the form, 
,
.
Remark. The adjoint operator 
will be characterized later in (49). Theorem 7 also asserts that there is no eigenvector of the form, 
,
. The restriction on 
is derived from our setting of the operator 
in (39): The setting is made for constructing a finitedimensional compensator. In the original equation (9), however, the parameters are constructed in a more general setting. The operator 
is then replaced by
(39')
Then, the above restriction on the 
is removed: In fact, the integer 
may be chosen arbitrarily large.
We hope to know more on
. The following proposition partly gives concrete informations on what 
consists of. It shows that 
is contained in
, regardless of the assumptions (40) and (41).
Proposition 8. The numbers
, 
belong to
. Actually we have the relations:
(43)
for
. Since the set 
forms an orthonormal system for
, any linear combinations of these eigenvectors cannot generate a vector of the form, 
,
.
To seek eigenvalues of 
other than
, let us recall the operator 
which appeared in (32), where
with 
. The adjoint operator 
is clearly given by 
with 
In the following result, we characterize 
by introducing an operator
, a slightly perturbed operator of
:
Theorem 9. Let 
be an operator defined as
(44)
where
. Then, we have the relation
(45)
Let 
be an arbitrary eigenpair of 
such that 
is not contained in
. Then, the corresponding eigenvector 
of 
is given by
(46)
where
.
In the above assertions, we need to characterize the adjoint operator
, which will be described later by (51). To seek the structure of
, let us begin with the operator equation:
(47)
It is clear that (47) admits a unique solution
, and that the solution is expressed as 
(see (17)). Let 
be a unique solution to the boundary value problem: 
in
, 
on
. We note that 
remains bounded when 
(this fact will be used in Lemma 10 below). For any
, note that
Then the adjoint 
is expressed as
where
,
. Thus,
(48)
Let us find the equation for
. For 
and
, we calculate through Green’s formula, (47), and the boundary condition (48) as
and thus
. Since 
is dense in
, we see that
(49)
Let us calculate the adjoint
. By assuming that 
is in 
and satisfies the boundary condition: 
, 
is calculated as
where 
is given by
(50)
and 
We see that
, and thus
. In order to show that
, we need the following elementary result:
Lemma 10. The operator 
is densely defined, and the bounded inverse, 
exists for a sufficiently large
.
Proof. Given a
, we solve the equation:
, where
. Set
, and define an operator 
as
The function 
depends on
. However, since 
remains bounded as
, there is a bounded inverse: 
for a sufficiently large
. A straightforward calculation shows that 
defined by 
and 
uniquely solves the above equation. Thus the bounded inverse 
exists.
Denseness of
: it is enough to show that
implies
. The above left-hand side is calculated for every 
as
Since 
is dense, this means that
and
, from which we conclude that 
and
. Q.E.D.
In view of the fact that both 
and 
exist as a bounded inverse, it is immediate that 
is contained in
. We have proven that
(51)
Proof of Proposition 8. By setting 
and
, 
, 
belongs to
. By (49), we see that
, and 
, which shows (43) Q.E.D.
Proof of Theorem 7. Assuming that 
in (42), we derive a contradiction. In (42), adding the equations of 
over 1 through
, we see that
(52)
where
, 
, and
. The Fourier coefficients of these vectors relative to the orthonormal system 
satisfy
(53)
where
. Note that 
for
. We show that
. Supposing the contrary, we must have
(54)
Set
, 
for simplicity. Then,
. In the equation for
, 
, we see that
The number of these 
is
. Consider the algebraic equation in
:
The equation admits 
solutions. The number of the solutions 
which agree with one of the 
is at most
. In other words, the determinant is not equal to 0 for the other
, the number of which is atleast
. Thus for these
, we must have
. By (53), this implies that
, which contradicts our assumption (40). We have shown that
. Thus we have, for
,
(55)
Comparing the Fourier coefficients in the equations to 
in (42), we see that
The number of the eigenvalues 
which agree with one of the 
is at most
. In other words, the number of the 
which does not agree with any of the 
is at least
. For these
, we see from (55) that
Since
, this means that the relation in
:
(56)
holds for the above
, the distinct number of which is at least
. This implies that the relation (56) holds for any
. calculating the residue at each
, we find that
, and thus
, too.
We go back to the equations to 
in (42) again. Since
, 
, we have
Set
. Then,
. Calculating the Fourier coefficients, we have
The numbers of these 
and 
are 
and
, respectively. Thus, the number of 
which does not agree with any of 
is at least
. For these
, we see by the relation (54) that
But, since 
by (40), this means that the relation in
:
(57)
holds for the above
, the distinct number of which is at least
. The situation is the same as in (56). Thus the relation (57) holds for any
. Calculating the residue at each
, we similarly find that
, 
, and thus
, too. Since
, 
, we have finally obtained from (42) and (55) that 
, 
, and
. Applying 
to the both sides of the above second equation, we see that, for
,
In other words, we have the relation:
But 
for
. Thus, we immediately find that
, 
, i.e., 
, and that
.
Recall that
,
. Thus, 
belongs to
, and 
by (42). Each 
is found an eigenvalue of
, and 
must be an eigenfunction. In addition,
But, this contradicts our assumption (41). Q.E.D.
Proof of Theorem 9. We already know that 
by Proposition 8. Let 
be in
.
In view of (51), the relation:
, 
means that
(58)
where
. The calculation of: (the first equation) + 
× (the second equation) yields that
By noting that 
(see (48)), the function 
belongs to 
. Thus we see that
. Supposing that
, we show a contradiction. In fact, if so, the second equation of (58) becomes
. Since
, however, we see that
, and
, or
. Thus, 
belongs to
, and the corresponding eigenvector of 
is given by the form
, where
. We have also shown that 
.
Conversely, let 
be an arbitrary eigenpair of 
such that
. Then we solve the equation: 
the unique solution of which is given by
.
By setting
, the vector 
means (46), and clearly satisfies the relation:
.To show that
, we suppose the contrary:
, or
. Then, 
, and
. Thus, 
must be an eigenpair of
. But, this contradicts the assumptions (40) and (41). We have shown that 
given by (46) is an eigenvector of
. Q.E.D.
Remark. In (45), it is not certain if
. This problem seems a pathological one. If 
, 
for some
, 
, and, in addition, 
, then the equation 
admits a (non-unique) solution 
(see the second equation of (58)). By setting
, the vector 
belongs to the eigenspace of 
for
.
NOTES