1. Introduction

In the framework of the so-called geometric approach, many control problems with state feedback and/or incomplete-state feedback (e.g., controllability and observability problems, decoupling problems, and disturbance-rejection problems, etc.) have been studied for finite-dimensional systems (see, e.g., [1, 2]). Further, the concept of -pairs was first introduced by Schumacher [3], and this concept has been used successfully to design dynamic compensators. After that Curtain extended the geometric concepts to infinite-dimensional systems and various control problems have been studied (see, e.g., [4–11]). On the other hand, from the practical viewpoint, Ghosh [12] and Otsuka [13] studied the concepts of simultaneous -pairs and of generalized -pairs, respectively, for finite-dimensional systems, and the parameter-insensitive disturbance-rejection problems for uncertain linear systems were studied. Then, Otsuka and Inaba [14–16] extended the concepts of simultaneous invariant subspaces and simultaneous -pairs to infinite-dimensional systems. Further, Otsuka and Hinata [17] studied the concept of generalized invariant subspaces for infinite-dimensional systems.

The objective of this paper is to investigate the concept of generalized -pairs for infinite-dimensional systems and to study the parameter-insensitive disturbance-rejection problem with dynamic compensator.

The paper is organized as follows. Section 2 gives the concept of generalized -pairs and its properties. In Section 3, the parameter-insensitive disturbance-rejection problem with dynamic compensator is formulated and its solvability conditions are presented. Section 4 gives an example to illustrate our results. Finally, some concluding remarks are given in Section 5.

2. Generalized -pairs

First, we give some notations used throughout this investigation. Let denote the set of all bounded linear operators from a Hilbert space into another Hilbert space ; for notational simplicity, we write for . For a linear operator the domain, the image, the kernel, and the -semigroup generated by are denoted by , Im, , and , respectively. Further, the dimension and the orthogonal complement of a closed subspace are denoted by dim() and , respectively.

Next, consider the following linear systems defined in a Hilbert space :

where , , are the state, the input, and the measurement output, respectively. Operators , and are unknown in the sense that they are represented as the forms: where :=, :=, :=, is the infinitesimal generator of a -semigroup on , , and . Here, in the system and mean the nominal system model and a specific uncertain perturbation, respectively.

Since are bounded linear operators, we remark that always generates a -semigroup and has the domain for all . Further, from the practical viewpoint it is assumed that the dimensions of input and output are finite.

Now, introduce a compensator defined in a Hilbert space of the form : where is the infinitesimal generator of a -semigroup on a Hilbert space with the domain , , and

If a compensator of the form is applied to the system , the resulting closed-loop system with the extended state space is easily seen to be where means the direct sum of and . For the closed-loop system , define with domain D

For the system , we give the following invariant subspaces.

Definition 2.1. Let be a closed subspace of .
(i) is said to be a generalized -invariant if there exists an such that Also for all
(ii) is said to be a generalized -invariant if there exists an such that Also is a generalized -invariant and is contained in a given closed subspace
(iii) is said to be a generalized -invariant if there exists a such that Also for all
(iv) is said to be a generalized -invariant if there exists a such that Also is a generalized -invariant and contains a given closed subspace . for all and all

Remark 2.2. (i) For the system a generalized -invariant subspace has the property that if an arbitrary initial state , then there exists a state feedback which is independent of and such that the state trajectory for all .
(ii) If is a bounded linear operator on (i.e., ), then the statements (i), (ii) and (iii), (iv) in Definition 2.1 are equivalent, respectively. Further, in this case and

Theorem 2.3 (see [17, 18]). Suppose that are arbitrary fixed real numbers such that and . Then, the following three statements are equivalent. (i) is a generalized -invariant.(ii)There exists an such that and and (iii)There exists an such that for all and (iv)There exists an such that for all and

The following theorem is the dual version of Theorem 2.3.

Theorem 2.4 (see [17, 18]). Suppose that are arbitrary fixed real numbers such that and . Then, the following three statements are equivalent. (i) is a generalized -invariant.(ii)There exists a such that and and (iii)There exists a such that for all and (iv)There exists a such that for all and

For finite-dimensional systems, Schumacher [3] first introduced the concept of -pair. The following definition is a generalized and infinite-dimensional version of -pair.

Definition 2.5. Let and be closed subspaces of . A pair of subspaces is said to be a generalized -pair if the following three conditions hold.(i) is a generalized -invariant.(ii) is a generalized -invariant.(iii)

For closed-loop system , we give the following definition.

Definition 2.6. Let be a closed subspace of .(i) is said to be a generalized -invariant if for all (ii) is said to be a generalized -invariant if for all and all

The following lemma was shown by Zwart.

Lemma 2.7 (see [11]). Let be a closed subspace of and the following three subspaces are introduced: where is the projection operator from onto along . Then, the following statements hold. (i)(ii)(iii)If ( then is a closed subspace of and

Lemma 2.8 (see [4, 6, 11]). Suppose that is the infinitesimal generator of a -semigroup on , and is a closed subspace of and Then, the following statements hold. (i)If for all , then (ii)If D() and , then for all (iii)If for all , then (iv)If there exists a such that for all and then for all (v)If there exists a such that for all and then for all and all

The following two lemmas are extensions of the results of Otsuka [13] to infinite-dimensional systems.

Lemma 2.9. If a pair of subspaces of is a generalized -pair such that then there exist , , , and such that for all

Proof. Suppose that a pair is a generalized -pair satisfying the stated above conditions. Since we remark that
Claim 1. for all and
To prove Claim 1, choose an arbitrary element Then, by Lemma 2.8(iii) there exists an such that Now, noticing that and we have Since dim that is a closed subspace. Further, noticing that are bounded operators, which proves Claim 1.
Next, Claims 2 and 3 hold as follows.
Claim 2. There exists a such that
To prove Claim 2, choose a and such that and with Define a linear map by Then, for some which proves Claim 2.
Claim 3. There exists a and such that for all
To prove Claim 3, let be a basis of Then, it follows from Claim 2 that there exists an and such that Define linear maps and by Then, which proves Claim 3.
Claim 4. There exists an such that Im
In fact, it follows from Claim 2 and hypotheses of this lemma that there exists a such that
Let be an arbitrary element. Then, there exist and such that Define such that Hence, which implies Then, we can easily obtain which proves Claim 4.
Now, choose and define such that
Then, the following claim holds.
Claim 5. One has for all and
In fact, at first, is obvious. Therefore, we prove the first one. Since implies , it follows from Lemma 2.8(v) that it suffices to show in order to prove .
Now, let be an arbitrary element. Then, there exists and such that Then, we have Now, it follows from Lemma 2.8(iii) that there exists a such that .
Hence, Noticing that is bounded linear operator, it follows from (2.28) and Claim 4 that which proves Claim 5.
Finally, define a bounded linear operator such that Then, the following claim holds.
Claim 6 (). In fact, which proves Claim 6. This completes the proof of Lemma 2.9.

Lemma 2.10. If a pair of subspaces of is a generalized -pair such that then there exist a compensator on and a subspace of such that , , and is generalized -invariant, where and are given in Lemma 2.7.

Proof. Suppose that there exists a pair of subspaces satisfying the stated above conditions. Since and are closed subspaces and we have Define and Let