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
:
![](https://www.scirp.org/html/5-5300530\0c81fbf5-faab-46ce-b3fb-dc7506ea3472.jpg)
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
![](https://www.scirp.org/html/5-5300530\cc999ea5-a2b3-4d0e-9a9e-8afe6e68fbed.jpg)
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)
![](https://www.scirp.org/html/5-5300530\859bf3e3-0615-4118-a4c2-5477c8b6c0dc.jpg)
;
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:![](https://www.scirp.org/html/5-5300530\096f690e-f733-4b4e-b2ff-95181c2329d7.jpg)
,
. 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
:
![](https://www.scirp.org/html/5-5300530\a38273b5-7d3d-4773-91ea-c9a90007226b.jpg)
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
![](https://www.scirp.org/html/5-5300530\08bfe27b-a7d9-4785-a56b-44fe3a0c5322.jpg)
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
![](https://www.scirp.org/html/5-5300530\f9a32796-9b44-4b86-bd25-6fa2d8e2b1f1.jpg)
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:
![](https://www.scirp.org/html/5-5300530\870f1d6a-51e1-4cbe-8954-f8df1e8b63f6.jpg)
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,
![](https://www.scirp.org/html/5-5300530\e253971d-18b3-429c-90c5-df9e28142074.jpg)
Now set
, and calculate as
![](https://www.scirp.org/html/5-5300530\6366b386-4784-4d3b-a5da-7bb901cbb30d.jpg)
Thus we see that
![](https://www.scirp.org/html/5-5300530\98624848-feec-4cdd-8272-997a6185582f.jpg)
2) Let
,
, and
,
, possible generalized eigenspaces of
. For a
, we calculate by (23) as
![](https://www.scirp.org/html/5-5300530\4b03c64e-aae3-45cc-8ad6-42f4e61981a3.jpg)
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
![](https://www.scirp.org/html/5-5300530\72fc8895-6892-4641-a00a-9eaa49ba46bb.jpg)
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
![](https://www.scirp.org/html/5-5300530\b3a38d73-574c-483e-9e91-ea9662ddc0af.jpg)
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![](https://www.scirp.org/html/5-5300530\95dd9d7b-52e6-4f55-8120-73e98cdf1af3.jpg)
. 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:
![](https://www.scirp.org/html/5-5300530\9b11266b-63ac-4386-98b1-5fb136982ba6.jpg)
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
![](https://www.scirp.org/html/5-5300530\56f22ce1-db14-42f1-99de-796183cf0c03.jpg)
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,
![](https://www.scirp.org/html/5-5300530\e03ae400-d571-405f-ae00-77b42973ca08.jpg)
The right-hand side is, however, estimated as follows:
![](https://www.scirp.org/html/5-5300530\a4b9d7a6-8312-4c6d-b3c7-fd22abf40f76.jpg)
which is a contradiction. Therefore, the spectrum
also lies in the left-half plane:![](https://www.scirp.org/html/5-5300530\25cf5617-ce69-4fa2-929b-2ac5dbc65d65.jpg)
. As a conclusion, the minimum of
satisfies the estimate:
![](https://www.scirp.org/html/5-5300530\e06f13e9-1073-4af6-a9bc-62d21bd61a90.jpg)
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:
![](https://www.scirp.org/html/5-5300530\3bee6b3d-fc16-46ec-bdb4-d6c5f951c61f.jpg)
Set
. As easily seen from Equation (29), the function
given by
![](https://www.scirp.org/html/5-5300530\db5e3e76-d9ea-4134-904c-83f2633995dd.jpg)
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
![](https://www.scirp.org/html/5-5300530\3b455871-e95c-4f8a-bbe0-3b46948b66c8.jpg)
Clearly
defines an isomorphism in
. Let us consider the case where
is of the third kind, i.e.,![](https://www.scirp.org/html/5-5300530\5019be0f-68c8-46c8-baa9-b21a972c990f.jpg)
. 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:
,![](https://www.scirp.org/html/5-5300530\428eed75-3fc8-4f00-9896-ead64375cfdb.jpg)
.
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:
![](https://www.scirp.org/html/5-5300530\9eb4dc49-a916-4214-8655-5706e8a54395.jpg)
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, ![](https://www.scirp.org/html/5-5300530\3fe9ee87-10ca-4021-a837-1584222215a1.jpg)
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 ![](https://www.scirp.org/html/5-5300530\9c14dc0e-26b5-41fb-967a-43cad6579377.jpg)
. The adjoint operator
is clearly given by ![](https://www.scirp.org/html/5-5300530\e283f816-1f9d-4e14-b1da-433f8aa75b4e.jpg)
with ![](https://www.scirp.org/html/5-5300530\9c59e58a-16d5-42c4-bbbc-466aff21c836.jpg)
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
![](https://www.scirp.org/html/5-5300530\aff21667-c94d-4d4b-a47a-f74596d21e46.jpg)
Then the adjoint
is expressed as
![](https://www.scirp.org/html/5-5300530\af3d1d8b-8f13-4155-8cac-04891ae9a709.jpg)
where
,
. Thus,
(48)
Let us find the equation for
. For
and
, we calculate through Green’s formula, (47), and the boundary condition (48) as
![](https://www.scirp.org/html/5-5300530\600e7c16-706d-4a18-97d7-51eb3b35d469.jpg)
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: ![](https://www.scirp.org/html/5-5300530\6d157165-4ced-421b-9543-abd597501e87.jpg)
,
is calculated as
![](https://www.scirp.org/html/5-5300530\cdb0e65c-69ef-45d1-9f63-f2db936bb6d1.jpg)
where
is given by
(50)
and ![](https://www.scirp.org/html/5-5300530\20baf024-4952-4ea1-a337-39cce7651024.jpg)
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, ![](https://www.scirp.org/html/5-5300530\f0ea93b1-8bb4-4683-a997-35c0cedb1212.jpg)
exists for a sufficiently large
.
Proof. Given a
, we solve the equation:
, where
. Set
, and define an operator
as
![](https://www.scirp.org/html/5-5300530\5ef0deb2-54a0-454a-a757-68e33c7ff659.jpg)
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 ![](https://www.scirp.org/html/5-5300530\bc734dbd-37ee-4e53-ab5e-92883beb06d2.jpg)
uniquely solves the above equation. Thus the bounded inverse
exists.
Denseness of
: it is enough to show that
![](https://www.scirp.org/html/5-5300530\99ddee19-2c5c-47c5-a01c-fca85903f614.jpg)
implies
. The above left-hand side is calculated for every
as
![](https://www.scirp.org/html/5-5300530\87c49811-4156-4674-97e6-438ab6e48fc6.jpg)
Since
is dense, this means that
![](https://www.scirp.org/html/5-5300530\5fede067-d02c-4609-97a2-a1ca75322798.jpg)
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
![](https://www.scirp.org/html/5-5300530\4161eb97-12f9-4fb2-bf7f-62b743680d53.jpg)
(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
, ![](https://www.scirp.org/html/5-5300530\e71539ed-39d1-4e58-b3ac-2dcd00304846.jpg)
, we see that
![](https://www.scirp.org/html/5-5300530\72501244-c15e-4a1b-92d7-5331df492e9d.jpg)
The number of these
is
. Consider the algebraic equation in
:
![](https://www.scirp.org/html/5-5300530\61b15ce1-4829-4eb9-8976-62c8f99857b7.jpg)
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
![](https://www.scirp.org/html/5-5300530\e20279ed-27a2-45d4-a3ef-1d41ca9413f6.jpg)
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
![](https://www.scirp.org/html/5-5300530\1393ea08-bd34-44a6-a4f3-8f14250eb645.jpg)
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
![](https://www.scirp.org/html/5-5300530\f4c8c497-8e26-4325-bd93-62e47810d789.jpg)
Set
. Then,
. Calculating the Fourier coefficients, we have
![](https://www.scirp.org/html/5-5300530\e659149e-a088-42f6-908d-e4fe55921402.jpg)
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
![](https://www.scirp.org/html/5-5300530\ef9249cf-e8e5-4f37-b22b-798d9e2ee1d1.jpg)
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![](https://www.scirp.org/html/5-5300530\00191aa5-71f1-42b0-b77c-23c1593f2090.jpg)
. Applying
to the both sides of the above second equation, we see that, for
,
![](https://www.scirp.org/html/5-5300530\b00b4f5d-a4ca-49f0-bd7d-7bb31fa18ae8.jpg)
In other words, we have the relation:
![](https://www.scirp.org/html/5-5300530\583d0171-b17f-422a-a602-0b050aa5bb31.jpg)
But
for
. Thus, we immediately find that
,
, i.e., ![](https://www.scirp.org/html/5-5300530\35c8b755-9221-4392-b162-8268bf264304.jpg)
, and that
.
Recall that
,
. Thus,
belongs to
, and
by (42). Each
is found an eigenvalue of
, and
must be an eigenfunction. In addition,
![](https://www.scirp.org/html/5-5300530\410fe9b8-0048-47fd-a987-085fe6b490f8.jpg)
But, this contradicts our assumption (41). Q.E.D.
Proof of Theorem 9. We already know that ![](https://www.scirp.org/html/5-5300530\99d286dc-de94-461f-aa96-636e69279256.jpg)
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
![](https://www.scirp.org/html/5-5300530\55ad2a55-fee4-4e95-84fd-749c3189bb94.jpg)
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:![](https://www.scirp.org/html/5-5300530\1414b46e-f5f3-4d4f-8ad1-934a78f129a7.jpg)
.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