On Decay of Solutions and Spectral Property for a Class of Linear Parabolic Feedback Control Systems

Unlike regular stabilizations, we construct in the paper a specific feedback control system such that   u t decays exponentially with the designated decay rate, and that some non-trivial linear functionals of u decay exactly faster than   u t . The system contains a dynamic compensator with another state v in the feedback loop, and consists of two states u and v . This problem entirely differs from the one with static feedback scheme in which the system consists only of a single state u . To show the essential difference, some specific property of the spectral subspaces associated with our control system is studied.


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][2][3][4][5][6]).In [2][3][4]6], no Riesz basis is assumed, corresponding to the coefficient elliptic operators with complicated boundary operators (see (   k c u linear functionals of u which allow unboundedness but are subordinate to L .The control system also reflects boundary feedback schemes by interpreting k h and the differential equation in weaker topologies.In general stabilization studies, the inputs   k g t are designed as a suitable feedback of the outputs   k c u , so that the state   u t could be stabilized as t   .Then every linear functional of u also decays at least with the same decay rate.This is true in the case where the functional is unbounded and subordinate to L .
We then raise a question: can we find a nontrivial linear functional which decays faster than   u t ?The purpose of the paper is to construct a specific feedback control system such that   u t decays exponentially with the designated decay rate, and that some nontrivial linear functionals of in the product space H H .We note that the above decay property is achieved in a straightforward manner in the static feedback control scheme in which we set M N  and In fact, the static feedback system contains a single state   u t only, and the so-called spectral decomposition of H associated with the elliptic operator enables us to find such an 0 f  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 H H to achieve a faster decay of , , 1 .It is very unlikely and almost denied to find such a vector   , 0 f  f in our control system with state   , u v .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 m  with the boundary  which consists of a finite number of smooth components of   , L be a pair of linear operators defined by x   for some positive  and The closure of L is denoted as L .It is well known (see [7]) that L has a compact resolvent   and that the following estimates hold: where the symbol  also denotes the It is well known (see, e.g., page 285 of [8]) that the set .
It is well known that the ascent i l coincides with the order of the pole i where , L defines an operator * L just as the above L .Then the adjoint of L , denoted by * L , is given as the closure of * L in H .There is a set of generalized eigenpairs , we have the relationship: Let us turn to the characterization of a dynamic compensator.Let H be any separable Hilbert space with inner product ,   H and norm  H . Relabelling an orthonormal basis for H , let  ; 1,  be a sequence of increasing positive numbers: , where 0 1 a   .Let B be a linear closed operator defined as with dense domain The adjoint operator * B of B is described as and thus Our control system has state   , u v H  H , and is described as a differential equation in H H : ,  d  d  ,  d , .
The equation with state v means a dynamic com- pensa-tor equipped with a set of outputs and assume conditions in terms of f L .These conditions have never appeared in the literature.Note that conditions are posed on L for state stabilization.
The functional   , u t f may be regarded as a kind of output of the system.It is worthwhile to refer to our previous results on output stabilization [10][11][12][13]: In [10], the decay of outputs is discussed with lack of observability conditions, but the relationship of the decay between   u t and the outputs is unclear.In [11][12][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: 0   is arbitrary, and, as we see later that the conditions on k  are much less restrictive than those in our preceding works above.In fact, k  only have to be designed, belonging to an infinite dimensional subspace of H .
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 H H 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.

Decay Estimates of Solutions
To ensure well-posedness of our control system (8), let us begin with the operator Equation (10).Adjusting   and i  , (see (2) for 0  ).In (9), let us express the actuators k  , , , where the overline denotes the complex conjugate.Before stating our first result, let us define the matrices i  , and i W , , and respectively.Then our first result is stated as follows: Theorem 1. 1) By assuming (14), the operator equation (11) 2) Assume further that   rank ; 0, , Then we have The first condition in (18), called the complete observability condition, is fulfilled with 1 N  in the case where   1 i  , the condition is fulfilled.In the case where 0  , 1 i  , which requires that N be equal to or greater than : this is the case, for example, where L is selfadjoint.
2) The condition: is the so called finite multiplicity condition.In the case of 1 m  , we know that 1 i m  , 1 i  .Thus, by choosing 1 N  , the complete observability condition is automatically fulfilled.As another example, let L be a self-adjoint operator defined by [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][3][4][5], so that we give here only an outline of the proof.1) Expression (17) and uniqueness of X are examined in a straightforward manner. 2) and each k , we introduce a series of meromorphic functions  by the recursion formula:  ; and 2) the algebraic growth rate  of these zeros, i    is smaller than 2 by (14).These properties combined with Carleman's theorem [15,16] imply that Following [5], we calculate the residue at each i  .Then, we see that , and thus, The complete observability in (18) implies that i  u 0 for 1 i  .In view of (5), we see that , 0 ij u   for every i and j .Since the set   ij  spans the whole space H , we conclude that 0 u  .
Q.E.D. Decay of solutions to Equation ( 9): In view of (11), it is (see Proposition 2).Equation ( 9) contains various parameters:   be given arbitrarily.Then, 0 X  by Theorem 1.We find a non-trivial vector f    There is a variety of choice of such an f .In fact, this is simply possible, e.g., by finding The integer R may be chosen arbitrarily large.The vectors k  will be determined in terms of the operator f L .
The state   , u t  in (9) satisfies the equation: An assumption on these k  will be discussed later (see (28)).Then, it is immediately seen that By the decay (22), we obtain the estimate: The operator f L defined in (13) has a compact resolvent, and    may admit generalized eigenfunctions.Let f i P be the projector corresponding to the eigenvalue i  , which is calculated as being the small counterclockwise circle with center i  .
Then, Proposition 2. 1) The number are orthogonal to each other.
Proof. 1) Suppose that there is a 0  , and The integer k varies over a finite set of positive integers depending on  .
Q.E.D. We now choose the actuators k  in (9) such that by the above proposition.All parameters except for k  in (9) are determined.
Rewrite the equation for   u t in (9) as Let us introduce an operator ff L as The following proposition is just a simple version of the result in [17].
Proposition 3. Let P be the projector defined by Thus, e e ff tL t C     , 0 t  .We are ready to state the non-standard deay of solutions to Equation (9). , where ˆ0 C       .The decay estimate for the functional is already obtained in (25).The control system (30) is derived in the following , and apply the projector n P to the equation for v in (9).By noting 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 X  are close to k c : When both ff L and ff L  are selfadjoint, it is well known-via the min-max principle (see [18])-that each eigenvalue of ff L is continuous relative to the coefficient parameters.In our problem, the following result holds: Proposition 5.The minimum of In view of (32), the left half-plane: Re . Let  be the counterclockwise circle: 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 ff L .Choose 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 u be an eigenfunction of ff L , corresponding to the eigenvalue  .Then, The right-hand side is, however, estimated as follows: which is a contradiction.Therefore, the spectrum   As easily seen from Equation (29), the function is the solution to Equation ( 9) with the initial value   0 0 , u v .In view of the reduction process to Equation (30), the function   where , and be an operator defined, for u H  , by Clearly T defines an isomorphism in H . Let us consider the case where  is of the third kind, i.e.,   , which defines a self-adjoint operator L  with dense domain , a is unchanged; b and c 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

 forms a Riesz basis. Any u H
 is uniquely expressed as . In our problem, we know that 1 i m  , 1 i  .Thus we choose 1 N  , so that the output of the system is a single observation at the end point 0   1 0 w  .Let us examine some assumptions in Theorem 4 in this example.Most important is the complete observability (18).The matrices i W in ( 16) are now Thus, we see that the complete observability is satisfied.Since f L in ( 13) is a one dimensional operator, the multiplicities of the eigenvalues i  are equal to 1.This enables us to choose 1 M  in (9).In Proposition 3, the controllability condition on the actuator , the controllability condition is simply that 0 In the case where  is of the first kind, the output is a single observation at the end point 0 . Proposition 6 also holds in this case.Since the complete observability is similarly satisfied.

Spectral Property of the Coefficient Operator
We go back to the problem raised in Section 1: Unlikeliness of a vector of the form   T 1 0 f .The basic control system is Equation (30) in the product space n H P H .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 H as well as B in ( 6), we can choose where A is defined as , (see (37)).The operator A is sectorial, and every solution to (30) or ( 38) is expressed as , , , , , 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 0 : such conditions are very easily broken.Thus we may henceforth assume that where i   and i f  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 * L might be equal to 0. Thus we may also assume that , 0, , 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.
Then any linear combination of these eigenvectors  

Remark. The adjoint operator *
A will be characterized later in (49).Theorem 7 also asserts that there is no eigenvector of the form,   The restriction on K is derived from our setting of the operator A 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 A is then replaced by Then, the above restriction on the K is removed: In fact, the integer K may be chosen arbitrarily large.
We hope to know more on   system for H , any linear combinations of these eigenvectors cannot generate a vector of the form, , let us recall the operator ff L  which appeared in (32), where In the following result, we characterize where 1 n X X  P .Then, we have the relation   be an arbitrary eigenpair of * g L such that where   In the above assertions, we need to characterize the adjoint operator *  A , which will be described later by (51).To seek the structure of * A , let us begin with the operator equation: It is clear that (47) admits a unique solution  

1
; n X H L PH , and that the solution is expressed as on  .We note that g remains bounded when c   (this fact will be used in Lemma 10 below).For any Then the adjoint Let us find the equation for and We see that , and define an operator B as The function g depends on c .However, since g remains bounded as c   , there is a bounded inverse:   . The above left-hand side is calculated for every   is dense, this means that . By (49), we see that * 1 i

B q q
    , and ), we derive a contradiction.In (42), adding the equations of k q over 1 through K , we see that , . The Fourier coefficients of these vectors relative to the orthonormal system   We show that 1 0 c  .Supposing the contrary, we must have , .
solutions.The number of the solutions  which agree with one of the i    is at most   1 K  .In other words, the determinant is not equal to 0 for the other i    , the number of which is . By (53), this implies that 0 i    , which contradicts our assumption (40).We have shown that 1 0 c  .Thus we have, for Comparing the Fourier coefficients in the equations to k q in (42), we see that The number of the eigenvalues k  which agree with one of the i    is at most   1 K  .In other words, the number of the i    which does not agree with any of the k This implies that the relation (56) holds for any , and thus We go back to the equations to k q in (42) again.Since 0 .
The numbers of these i    and k  are 2R and 1 K  , respectively.Thus, the number of i    which does not agree with any of k In other words, we have the relation:  for i j  .Thus, we immediately find that   , and that But, this contradicts our assumption (41).Q.E.D.
Proof of Theorem 9. We already know that   p X q p X q X p X q X f p X q ˆ0 p X q     , we show a contradiction.In fact, if so, the second equation of (58) becomes A is given by the form     T T * 1 ˆ p q X q q    , where 0 q  .We have also shown that where α > 0. The parameters  and k  denote given actuators of the controlled plant, and k  actuators of the compensator to be designed.The outputs   k c u of the controlled plant are considered on the boundary, and defined as

satisfiy the controllability condition in Proposition 3 .
Then we find a large integer n ; vectors k n  P H ; and a postive     close to  , and subsequently the control system in the product space n H P H ; operator ff L  consists only of eigenvalues.It is expected that the value of small.Q.E.D. Let us turn to the proof of Theorem 4. Choose an r in Proposition 5 such that r hand side of which decays as t   with decay rate   for every initial state.Now we ask: Does the range of the * P contain a vector of the form immediately leads to the structure of the eigenspaces of the operator * A .In the operator A , the vectors n  P H and R f P H of the compensator are the parameters to be designed

8 ..
the assumptions (40) and (41).Proposition The numbers i    , 1 i n   belong to   * A Actually we have the relations:

v
P H , we calculate through Green's formula, (47), and the boundary condition (

Lemma 10 .
The operator † A is densely defined, and the bounded inverse,

1 ˆnB
  L P H for a sufficiently large 0 c  .A straightforward calculation shows that  

B
40), this means that the relation in  : .The situation is the same as in (56).Thus the relation (57) holds for any to the both sides of the above second equation, we see that, for 1


must be an eigenpair of * L .But, this contradicts the assumptions (40) and (41).We have shown that   T p qgiven by (46) is an eigenvector of * A .Q.E.D.Remark.In (45), it is not certain if     some i , 1 i n   , and, in addition, ˆ, , 1) below).Let H be a separable Hilbert space with the inner product ,   and the norm  .A standard control system with state   k g t are inserted in the equation; and but does not necessarily form a Riesz basis for H . Let i P  be the (not necessarily orthogonal) projector corresponding to the eigenvalue i .The restriction of L onto the invariant subspace i in the topology of H . Since H is separable, we may assume with no loss of generality that these k k  H such that * k X  arbitrarily approximate k c , (30) is immediately obtained.Equation (30) is clearly well posed in nH P H : Thus every solution to (30) is derived from the solution 