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:

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

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:

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:

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:

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

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

The adjoint operator of is described as

and thus. Let, be the projector in such that.

Our control system has state, and is described as a differential equation in:

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

where denotes observation weights. The operator, specified later, denotes a unique solution to the operator equation on,

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

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

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.

To ensure well-posedness of our control system (8), let us begin with the operator Equation (10). Adjusting and, , we may assume that

(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

and

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

2) Assume further that

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:

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

Let. Note that the restriction of on is equivalent to the matrix, and thus,. The relation (20) is rewritten as

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,

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

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

By the decay (22), we obtain the estimate:

for. Here, is non-trivial.

The operator defined in (13) has a compact resolvent, and consists only of eigenvalues. Let

where and for i ≠ j. Each may admit generalized eigenfunctions. Let be the projector corresponding to the eigenvaluewhich is calculated as, C_i

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,

The integer varies over a finite set of positive integers depending on.                          Q.E.D.

We now choose the actuators in (9) such that

Then, by the above proposition. All parameters except for in (9) are determined.

Rewrite the equation for in (9) as

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;

where. Every solution satisfies the decay estimate:

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:

where. Thus every solution to Equation (9) satisfies the estimate:

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:

Recall that, for, the (second) resolvent equation:

holds. Then we see that

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

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

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.

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

where is defined as

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

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

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

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

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:

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

where. Then, we have the relation

Let be an arbitrary eigenpair of such that is not contained in. Then, the corresponding eigenvector of is given by

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:

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,

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

Let us calculate the adjoint. By assuming that is in and satisfies the boundary condition:

, is calculated as

where is given by

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

where, , and. The Fourier coefficients of these vectors relative to the orthonormal system satisfy

where. Note that for. We show that. Supposing the contrary, we must have

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,

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:

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:

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

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.