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Actuator Dynamics Compensation in Stabilization of Abstract Linear Systems

This is the first part of four series papers, aiming at the problem of actuator dynamics compensation for linear systems. We consider the stabilization of a type of cascade abstract linear systems which model the actuator dynamics compensation for linear systems where both the control plant and its actuator dynamics can be infinite-dimensional. We develop a systematic way to stabilize the cascade systems by a full state feedback. Both the well-posedness and the exponential stability of the resulting closed-loop system are established in the abstract framework. A sufficient condition of the existence of compensator for ordinary differential equation (ODE) with partial differential equation (PDE) actuator dynamics is obtained. The feedback design is based on a novelly constructed upper-block-triangle transform and the Lyapunov function design is not needed in the stability analysis. As applications, an ODE with input delay and an unstable heat equation with ODE actuator dynamics are investigated to validate the theoretical results. The numerical simulations for the unstable heat system are carried out to validate the proposed approach visually.


Introduction
System control through actuator dynamics can usually be modeled as a cascade control system which has been intensively investigated in the last decade. An early infinite-dimensional actuator dynamic compensation is the input time-delay compensation for finite-dimensional systems in stable heat system with ODE actuator dynamics. We point out that the considered stabilization of heat-ODE cascade system is an interesting and challenging problem because the actuator dynamics is finite-dimensional yet the control plant is of the infinite-dimension. In other words, what we need to do is to control an "infinite-dimensional" system via a "finite-dimensional" compensator.
Compared with the ODE system with PDE actuator dynamics, the results about PDE system with ODE actuator dynamics are still scarce.
The rest of the paper is organized as follows: In Section 2, we demonstrate the main idea through an ODE cascade system. Sections 3 and 4 give some preliminary results about the similarity of operators and the Sylvester equation. Section 5 is devoted to the dynamics compensator design.
The well-posedness and the exponential stability are also established. In Section 6, we apply the proposed method to the input delay compensation for an ODE system. Stabilization of an unstable heat equation by finite-dimensional actuator dynamics is considered in Section 7. Section 8 presents some numerical simulations, followed up conclusions in Section 9. For the sake of readability, some results that are less relevant to the dynamics compensator design are arranged in the Appendix.
Throughout the paper, the identity operator on the Hilbert space X i will be denoted by I i , i = 1, 2, respectively. The space of bounded linear operators from X 1 to X 2 is denoted by L(X 1 , X 2 ).

Finite-dimensional dynamics
In order to introduce our main idea clearly, we first consider system (1.1) in the finite-dimensional case. Suppose that X 1 , X 2 , U 1 and U 2 are the Euclidean spaces, C 2 : X 2 → U 1 and B 1 : U 1 → X 1 .
We shall design a full state feedback to stabilize the cascade system (1.1). Although this problem can be achieved completely by the pole assignment theorem provided system (1.1) is controllable, when we come across that at least one of A 1 and A 2 is an operator in an infinite-dimensional Hilbert space, the problem would become very complicated. We thus need an alternative treatment that can be extended to the setting of infinite-dimensional framework.
We first divide the controller into two parts: where K 2 ∈ L(X 2 , U 2 ) is chosen to make A 2 + B 2 K 2 Hurwitz and u c (t) is a new control to be designed. The main role played by the first part K 2 x 2 (t) of the controller is to stabilize the x 2subsystem. If (A 2 , B 2 ) is controllable, such a K 2 always exists. Under (2.1), the control plant (1.1) becomes   ẋ 1 (t) = A 1 x 1 (t) + B 1 C 2 x 2 (t), (2.2) Now, we decouple system (2.2) by the block-upper-triangular transformation: where S ∈ L(X 2 , X 1 ) is to be determined. Evidently, the system matrix of (2.2) is block-diagonalized if the matrix S solves the matrix equation which is a well known Sylvester equation. An immediate consequence of this fact is that the controllability of the following pairs is equivalent: (2.5) Owing to the block-diagonal structure, the stabilization of the second system of (2.5) is much easier than the first one. Indeed, since A 2 + B 2 K 2 is stable already, the controller u c (t) in (2.2) can be designed by stabilizing system (A 1 , SB 2 ) only: x 2 (t) = K 1 Sx 2 (t) + K 1 x 1 (t), (2.6) where K 1 ∈ L(X 1 , U 2 ) is chosen to make A 1 + SB 2 K 1 Hurwitz. In view of (2.1), the controller of the original system (1.1) is therefore designed as which leads to the closed-loop of system (1.1): (2.8) Lemma 2.1. Let X 1 , X 2 , U 1 and U 2 be Euclidean spaces, and let A j ∈ L(X j ), B j ∈ L(U j , X j ), Hurwitz, and S ∈ L(X 2 , X 1 ) is the solution of the Sylvester equation (2.4). Then, the closed-loop Proof. Let and  20]) Let X 1 , X 2 , U 1 and U 2 be Euclidean spaces, and let A j ∈ L(X j ), B j ∈ L(U j , X j ), C 2 ∈ L(X 2 , U 1 ) and K 2 ∈ L(X 2 , U 2 ), j = 1, 2. Suppose that Then, the Sylvester equation (2.4) admits a unique solution S ∈ L(X 2 , X 1 ) which can be represented where Γ is a smooth contour around σ(A 1 ) and separated from σ(A 2 + B 2 K 2 ), with positive orientation.

Preliminaries on abstract systems
In order to extend the results in Section 2 to infinite-dimensional systems, we present some preliminary background on abstract infinite-dimensional systems. We first introduce the dual space with respect to a pivot space, which has been discussed extensively in [27] particularly for those systems with unbounded control and observation operators.
Suppose that X is a Hilbert space and A : D(A) ⊂ X → X is a densely defined operator with It is evident thatÃx = Ax for any x ∈ D(A). Hence,Ã ∈ L(X, [D(A * )] ′ ) is an extension of A.
Since A is densely defined, the extension is unique. By [27,Proposition 2.10.3], for any β ∈ ρ(A), A generates a C 0 -semigroup e At on X, then, so is for its Suppose that Y is an output Hilbert space and C ∈ L(D(A), Y ). The Λ-extension of C with respect to A is defined by Define the norm a Banach space and C Λ ∈ L(D(C Λ ), Y ). Moreover, there exist continuous embeddings: For other concepts of the admissibility for both control and observation operators, and the regular linear systems, we refer to [29,30,31].

Definition 1.
Suppose that X is a Hilbert space and A j : D(A j ) ⊂ X → X is a densely defined operator with ρ(A j ) = ∅, j = 1, 2. We say that the operators A 1 and A 2 are similar with the transformation P , denoted by A 1 ∼ P A 2 , if the operator P ∈ L(X) is invertible and satisfies P A 1 P −1 = A 2 and D(A 2 ) = P D(A 1 ). (3.6) Suppose that A 1 ∼ P A 2 . Then, A * 1 ∼ P −1 * A * 2 and in particular, P * D(A * 2 ) = D(A * 1 ). Obviously, A 1 ∼ P A 2 implies that A 1 generates a C 0 -semigroup e A 1 t on X if and only if A 2 generates a C 0 -semigroup e A 2 t on X. More specifically, P e A 1 t P −1 = e A 2 t . Lemma 3.1. Let X and U be Hilbert spaces. Suppose that the operator A j : then, the following assertions hold true: Proof. We first prove (i). For any f ∈ L 2 loc ([0, ∞); U ) and φ ∈ D(A * 2 ), it follows from (3.7) that for any t > 0 and 0 ≤ s ≤ t. Define the operator Φ j (t) ∈ L(L 2 Then, it follows from (3.8) that (3.10) When B 1 is admissible for e A 1 t , we have Φ 1 (t)f ∈ X and thus Combining (3.10) and (3.11), we arrive at and hence, due to the arbitrariness of φ. (3.11) and (3.13) imply that Φ 2 (t)f ∈ X for any t > 0 which means that B 2 is admissible for e A 2 t .
When B 2 is admissible for e A 2 t , Φ 2 (t)f ∈ X and thus Combining (3.10) and (3.14), we have (3.15) and hence, due to the arbitrariness of φ. (3.14) and (3.16) imply that Φ 1 (t)f ∈ X for any t > 0 and hence B 1 is admissible for e A 1 t .
We next prove (ii). By the proof of (i), the equality (3.13) always holds provided B 1 is admissible for e A 1 t or B 2 is admissible for e A 2 t . Since P ∈ L(X) is invertible, we conclude that Ran(Φ 1 (t)) = X if and only if Ran(Φ 2 (t)) = X ( or Ran(Φ 1 (t)) = X if and only if Ran(Φ 2 (t)) = X ). This completes the proof of the lemma.
Remark 3.1. When B 1 and B 2 are bounded, (3.7) implies that B 2 = P B 1 . In this case, systems (A 1 , B 1 ) and (P A 1 P −1 , P B 1 ) have the same controllability, which is the same as the finitedimensional counterpart.
By the separation principle of the linear systems, a fair amount of closed-loop systems resulting from the observer based output feedback can be converted into a cascade system. The same thing also takes place in the actuator dynamics compensation. At the end of this section, we consider the well-posedness and stability of general cascade systems, which is useful for the well-posedness and stability analysis of the closed-loop system. Moreover, it can simplify and unify the proofs in [3,4,35].
Then, the operator A generates a C 0 -semigroup e At on X 1 × X 2 . In addition, if we suppose further that e A j t is exponentially stable in X j , j = 1, 2, then, e At is exponentially stable in X 1 × X 2 .
Proof. The operator A is associated with the following system:  and (3.19), it follows that the solution of the x 1 -subsystem satisfies x 1 ∈ C 1 ([0, ∞); X 1 ). Therefore, We next show the exponential stability. Suppose that ( is a classical solution of system (3.18). Since e A j t is exponentially stable in X j , there exist two positive constants ω j and L j such that Hence, By the admissibility of C 2 for e A 2 t and [27, Proposition 4.3.6, p.124], it follows that We combine [29, Remark 2.6], (3.20), (3.22) and the admissibility of B 1 for e A 1 t to get where M > 0 is a constant independent of t. On the other hand, the solution of the x 1 -subsystem is Combining (3.20), (3.22) and (3.23), for any 0 < θ < 1, we have which, together with (3.21), (3.24) and (3.20), leads to the exponential stability of (x 1 (t), x 2 (t)) in The proof is complete.
Similarly to Lemma 3.2, we obtain immediately Lemma 3.3.
Then, the operator A 1 generates a C 0 -semigroup e A 1 t on X 1 × X 2 . Moreover, if we suppose further that e A i t is exponentially stable in X j , j = 1, 2, then, e A 1 t is exponentially stable in X 1 × X 2 .
Proof. The proof is almost the same as Lemma 3.2 and we omit the details.

Sylvester equations
In view of (2.4), we need extend the Sylvester equation to the infinite-dimensional cases. For this purpose, we first give the definition of the solution of Sylvester equation.
Definition 2. Let X 1 , X 2 and U 1 be Hilbert spaces and A j : . We say that the operator S is a solution of the Sylvester equation if S ∈ L(X 2 , X 1 ) and the following equality holds whereÃ 1 is an extension of A 1 given by (3.2).
Then, X jB j is independent of λ j and can be characterized as Suppose further that Y 2 = U 1 and S ∈ L(X 2 , X 1 ) is a solution of the Sylvester equation (4.1) in the sense of Definition 2. Then, the following assertions hold true: is a regular linear system, then, there exists an extension of S, still denoted Proof. The definition of the space X jB j and its characterization (4.4) can be obtained by [ The proof of (i). Since S solves the Sylvester equation (4.1), for any x 2 ∈ D(A 2 ), we have Since (A 1 , B 1 , C 1 ) is a regular linear system and S ∈ L(X 2 , X 1 ), (4.6) implies that C 1Λ S ∈ The proof of (ii). In terms of the solution S ∈ L(X 2 , X 1 ) of (4.1), we define the operatorS bỹ For any which implies thatS is an extension of S. On the other hand, by the regularity of (A 2 , B 2 , C 2 ) and the definition (4.7), we can conclude that (β −Ã 2 ) −1 B 2 ∈ L(U 2 , D(C 2Λ )) and which implies thatSB 2 ∈ L(U 2 , [D(A * 1 )] ′ ). Moreover, for any u 2 ∈ U 2 , it follows from (4.8) and (4.9) that (4.10) Due to the arbitrariness of u 2 , (4.10) implies thatS solves the Sylvester equation (4.1) on (β − (4.2) and (4.3), we can obtain (4.5) easily with replacement of S bỹ S. The proof is complete.
Lemma 4.2. Let X 1 , X 2 and U 1 be Hilbert spaces and let A 1 : Then, the Sylvester equation (4.1) admits a solution S ∈ L(X 2 , X 1 ) in the sense of Definition 2. Proof. (4.13) By a simple computation, we have which, together with the fact (α −Ã 1 ) −1 ∈ L([D(A * 1 )] ′ , X 1 ), implies that S ∈ L(X 2 , X 1 ). This shows that S is a solution of equation (4.1) in the sense of Definition 2. The proof is complete. Lemma 4.3. Let X 1 , X 2 and U 1 be Hilbert spaces and let and (4.11) holds. Then, the Sylvester equation (4.1) admits a solution S ∈ L(X 2 , X 1 ) in the sense of Definition 2.

Proof. Since
That is Since x 1 is arbitrary, the equality Π * A 2 x 2 − A 1 Π * x 2 = −B 1 C 2 x 2 holds in X 1 for any x 2 ∈ D(A 2 ).

Actuator dynamics compensation
This section is devoted to the extension of the results in Section 2 from finite-dimensional systems to the infinite-dimensional ones.
Assumption 5.1. Let X 1 , X 2 , U 1 and U 2 be Hilbert spaces. The operator A j generates a C 0 - is admissible for e A j t and C 2 ∈ L(D(A 2 ), U 1 ) is admissible for e A 2 t , j = 1, 2. In addition, σ(A 1 )∩σ(A 2 ) = ∅ and the semigroup e A 2 t is exponentially stable in X 2 .
Since the stabilization and compensation of the actuator dynamics are two different issues, we assume additionally that the semigroup e A 2 t is exponentially stable, which is just for avoidance of the confusion. Indeed, one just needs to stabilize the system before the actuator dynamics compensation when e A 2 t is not exponentially stable. Since e A 2 t is exponentially stable already, the full state feedback of system (1.1) can be designed, inspired by (2.7), as where the operator S ∈ L(X 2 , X 1 ) is a solution of the Sylvester equation and K 1 ∈ L(D(A 1 ), U 2 ) is selected such that A 1 + SB 2 K 1Λ generates an exponentially stable C 0 -semigroup on X 1 . Under controller (5.1), we obtain the closed-loop system: Then, the closed-loop system (5.3) can be written as In view of (2.10), we define the operator Theorem 5.1. In addition to Assumption 5.1, suppose that A 1 ∈ L(X 1 ) and (A 2 , B 2 , C 2 ) is a regular linear system. Then, the Sylvester equation (5.2) admits a solution S ∈ L(X 2 , X 1 ) in the sense of Definition 2 such that SB 2 ∈ L(U 2 , X 1 ). If we suppose further that there is a K 1 ∈ L(X 1 , U 2 ) such that A 1 + SB 2 K 1 generates an exponentially stable C 0 -semigroup e (A 1 +SB 2 K 1 )t on X 1 , then, the operator A defined by (5.4) generates an exponentially stable C 0 -semigroup e A t on Proof. Since A 1 ∈ L(X 1 ), it follows that B 1 ∈ L(U 1 , X 1 ) and K 1 = K 1Λ ∈ L(X 1 , U 2 ). By Lemmas 4.1 and 4.3, the Sylvester equation (5.2) admits a solution S ∈ L(X 2 , X 1 ) such that SB 2 ∈ L(U 2 , X 1 ) and where X 2B 2 is defined by (4.3) or equivalently by (4.4). Moreover, SB 2 K 1 ∈ L(X 1 ) and thus We claim that A ∼ S A S , i.e., where the transformation S is given by Obviously, S ∈ L(X 1 × X 2 ) is invertible and its inverse is given by For any (x 1 , x 2 ) ⊤ ∈ D(A S ), we haveÃ 2 x 2 + B 2 K 1 x 1 ∈ X 2 and K 1 x 1 ∈ U 2 . It follows from (4.4) and the regularity of (A 2 , B 2 , C 2 ) that x 2 ∈ X 2B 2 ⊂ D(C 2Λ ). As a result, B 1 C 2Λ x 2 ∈ X 1 and thus For any (x 1 , x 2 ) ⊤ ∈ D(A S ), it follows from (5.9), (4.4) and the regularity of (A 2 , B 2 , C 2 ) that x 2 ∈ X 2B 2 . By virtue of (5.8), a straightforward computation shows that SA S −1 (x 1 , x 2 ) ⊤ = A S (x 1 , x 2 ) ⊤ for any (x 1 , x 2 ) ⊤ ∈ D(A S ). Consequently, A and A S are similar each other.
Since the C 0 -semigroups e (A 1 +SB 2 K 1 )t on X 1 and e A 2 t on X 2 are exponentially stable, K 1 ∈ L(X 1 , U 2 ) and B 2 is admissible for e A 2 t , it follows from Lemma 3.3 that the operator A S generates an exponentially stable C 0 -semigroup e A S t on X 1 ×X 2 . By the similarity of A S and A , the operator A generates an exponentially stable C 0 -semigroup e A t on X 1 × X 2 as well. This completes the proof of the theorem.
When X 1 is finite-dimensional, we can characterize the existence of the feedback gain K 1 through the system (1.1) itself.
Corollary 5.1. In addition to Assumption 5.1, suppose that is a regular linear system and system (1.1) is approximately controllable. Then, there exist S ∈ L(X 2 , X 1 ) and K 1 ∈ L(X 1 , U 2 ) such that the operator A defined by (5.4) generates an exponentially Proof. By Lemmas 4.3 and 4.1, the Sylvester equation (5.2) admits a solution S ∈ L(X 2 , X 1 ) such that SB 2 ∈ L(U 2 , X 1 ) and (4.5) holds. Define Appendix that the finite-dimensional system (A 1 , SB 2 ) is controllable. By the pole assignment theorem, there exists a K 1 ∈ L(X 1 , U 2 ) to stabilize system (A 1 , SB 2 ). By Theorem 5.1, A generates an exponentially stable C 0 -semigroup e A t on X 1 × X 2 .
Theorem 5.2. In addition to Assumption 5.1, suppose that A 2 ∈ L(X 2 ), and (A 1 , B 1 , K 1 ) is a regular linear system. Then, the Sylvester equation (5.2) admits a solution S ∈ L(X 2 , X 1 ) in the sense of Definition 2 and SB 2 ∈ L(U 2 , X 1 ). If we suppose further that K 1 stabilizes system (A 1 , SB 2 ) in the sense of [32, Definition 3.1], then, the operator A defined by (5.4) generates an exponentially stable C 0 -semigroup e A t on X 1 × X 2 .
Since K 1Λ S ∈ L(X 2 , U 2 ), B 2 ∈ L(U 2 , X 2 ) and B 2 K 1Λ x 1 ∈ X 2 , we have In view of (5.4) and Consequently, D(A S ) ⊂ SD(A ) due to the arbitrariness of (x 1 , On the other hand, for any (x 1 , x 2 ) ⊤ ∈ D(A ), since K 1Λ S ∈ L(X 2 , U 2 ), S ∈ L(X 2 , X 1 ), As a result, It follows from (5.16) and (5.4) that Since K 1 ∈ L(D(A 1 ), U 2 ) stabilizes system (A 1 , SB 2 ) exponentially, the operatorÃ 1 + SB 2 K 1Λ generates an exponentially stable C 0 -semigroup e (Ã 1 +SB 2 K 1Λ )t on X 1 and K 1 is admissible for Since e A 2 t is exponentially stable and B 2 ∈ L(U 2 , X 2 ), it follows from Lemma 3.3 that the operator A S generates an exponentially stable C 0 -semigroup e A S t on X 1 × X 2 . By the similarity of A S and A , the operator A generates an exponentially stable C 0 -semigroup e A t on X 1 × X 2 as well. This completes the proof of the theorem.
At the end of this section, let us summarize the scheme of the actuator dynamics compensation.
Given an actuator dynamics compensation problem, we can design a compensator through five steps: • Formulate the control plant as the abstract form (1.1); • Find K 2 to stabilize system (A 2 , B 2 ); • Solve the Sylvester equation A 1 S − S(A 2 + B 2 K 2 ) = B 1 C 2 ; • Find K 1 to stabilize system (A 1 , SB 2 ); • With K 1 , K 2 and S at hand, the controller is designed as We need an explicit expression of the solution of the Sylvester operator equation to get the controller. Generally speaking, it is not easy to solve the Sylvester equation. However, under some reasonable additional assumptions, we still can obtain the solution analytically or numerically even for the cascade system involving a multi-dimensional PDE. (see, e.g., [14] and [15]). In particular, when the system (1.1) consists of an ODE and a one-dimensional PDE, the problem becomes quite easy. Indeed, if X 1 is n-dimensional, we can suppose that , · · · , n. Inserting (5.23) into the corresponding Sylvester equation, we will arrive at a vector-valued ODE with respect to the variable Φ. When where Ψ = (Ψ 1 , Ψ 2 , · · · , Ψ m ) ⊤ with Ψ j ∈ X 1 , j = 1, 2, · · · , m. Inserting (5.24) into the corresponding Sylvester equation will lead to a vector-valued ODE as well. Thanks to the ODE theory and numerical analysis theory [26], in both cases, the solution S can be obtained analytically or numerically. In this way, we can stabilize the ODE with actuator dynamics, dominated by the transport equation [9], wave equation [11], heat equation [10] as well as the Schrödinger equation [19], in a unified way. More importantly, the more complicated problem that stabilize the PDEs with ODE actuator dynamics can still be addressed effectively. To validate the effectiveness of the developed method, the proposed scheme of controller design will be applied to stabilization of ODE-transport cascade and heat-ODE cascade in sections 6 and 7, respectively.
Remark 5.1. Another interesting question is to extend Theorems 5.1 and 5.2 to the case where both A 1 and A 2 are unbounded. There are still many difficulties to achieve this problem in the general abstract framework. One of the reasons is that the general Sylvester equation with unbounded operators is hard to be solved. In addition, the proof of well-posedness and exponential stability is also difficult. However, the main idea of the developed approach is still helpful to the actuator dynamics compensation with the unbounded A 1 and A 2 . This will be considered in the third paper [6] of this series works.

ODEs with input delay
In this section, we apply the proposed approach to the input delay compensation for ODEs. Consider the following linear system in the state space X 1 = R n : where A 1 ∈ L(X 1 ) is the system operator, B 1 ∈ L(R, X 1 ) is the control operator, and u : [−τ, ∞) → R is the scalar control that is delayed by τ units of time. It should be pointed out that the input delay compensation problem (6.1) has been considered via many approaches such as the spectrum assignment approach in [8], the "reduction approach" in [1] and the PDE backstepping method in [9]. In this section, we re-consider this problem and show our differences with other approaches. Let Then, system (6.1) can be written as which clearly shows why the time-delay is infinite-dimensional and (6.3) now is delay free. In order to write system (6.3) into the abstract form (1.1), we define and define B 2 q = qδ(·) for any q ∈ R, where δ(·) is the Dirac distribution. System (6.3) can be written as the abstract form: where will be determined later, j = 1, 2, · · · , n. Suppose that the solution of Sylvester equation (5.2) takes the form (5.23). Then, We solve (6.6) to obtain the solution of Sylvester equation (5.2) As a result, If there exists a K ∈ L(X 1 , R) such that A 1 +B 1 K is Hurwitz, then the operator A 1 +e −A 1 τ B 1 Ke A 1 τ is also Hurwitz due to the invertibility of e −A 1 τ . Since e A 2 t is exponentially stable already, by (5.22), the controller is then designed as which leads to the closed-loop system: (6.10) By (6.2), the controller (6.9) can be rewritten as which is the same as those obtained by the spectrum assignment approach in [8], the "reduction approach" in [1] and the PDE backstepping method in [9].
It is seen that in our approach, we never need the target system as that by the backstepping approach. This avoids the possibility that when the target system is not chosen properly, there is no state feedback control and even if the target system is good enough, there is difficulty in solving PDE kernel equation for the backstepping transformation. Another advantage of the proposed approach is that we never construct the Lyapunov function in the stability analysis, which avoids another difficulty of construction of the Lyapunov function. Finally, we point out that the proposed approach is still working for the unbounded operator A 1 , which will be considered in detail in the third paper [6] of this series works.

Heat equation with ODE dynamics
In this section, we consider the stabilization of an unstable heat equation with m-dimensional ODE actuator dynamics as follows: where w(·, t) is the state of the heat system, µ > 0, A 2 ∈ R m×m is the system matrix of the actuator dynamics, C 2 ∈ L(R m , R) represents the connection, B 2 ∈ L(R, R m ) is the control operator and u(t) is the control. We assume without loss of the generality that A 2 is Hurwitz. Compared with the stabilization of finite-dimensional systems through infinite-dimensional dynamics in existing literature, stabilization of infinite-dimensional unstable system through finite-dimensional dynamics is a difficult problem and the corresponding result is very scarce.
By a simple computation, the operator S given by (7.6) and (7.5) solves the Sylvester equation Thanks to the block-diagonal structure of SAS −1 and Lemma 10.2 in Appendix, system (A 1 , SB 2 ) is approximately controllable. By (7.8), system (A 1 , b(·)) is approximately controllable as well.

Numerical simulations
In this section, we carry out some simulations for system (7.19) to validate our theoretical results.
We choose show that the control free system is indeed unstable. The trajectory of state feedback law u(t) is plotted in Figure 1 (c). The state w(·, t) of the closed-loop system (7.19) is plotted in Figure 2 (a) and the state x 2 (t) is plotted in Figure 2 (b). Comparing Figure 1 with Figure 2, it is found that the proposed approach is very effective and the controller is smooth.

Conclusions
In this paper, we develop a systematic method to compensate the actuator dynamics dominated by general abstract linear systems. A scheme of full state feedback law design is proposed. As a result, a sufficient condition of the existence of compensator for ODE with PDE actuator dynamics is obtained and the existing results about stabilization of ODE with actuator dynamics dominated by the transport equation [9], wave equation [11], heat equation [10] as well as the Schrödinger equation [19] can be treated in a unified way. More importantly, the more complicated problem that stabilize the infinite-dimensional system through finite-dimensional actuator dynamics can still be addressed effectively. We present two examples to demonstrate the effectiveness of the proposed approach. One is on input delay compensation for ODE system and another is for unstable heat equation with ODE actuator dynamics.
It should be pointed out that the proposed approach in Theorems 5.1 and 5.2 is not limited to the examples considered in Sections 6 and 7. In [34], it has been applied to the stabilization of ODEs with actuator dynamics dominated by Euler-Bernoulli beam equation. More importantly, the approach opens up a new road leading to the stabilization of cascade systems particularly for those systems which consist of ODE and multi-dimensional PDE.
Furthermore, the main idea of the approach is still applicable to the stabilization of PDE-PDE cascade systems like those arising from PDEs with input delay. This will be considered in the third paper [6] of this series works. The present paper focuses only on the full state feedback. After being investigated in the next paper [5] of this series studies for the state observer design through sensor dynamics, the output feedback will become straightforward by the separation principle of the linear systems.

Appendix
Lemma 10.1. Let the operator A 1 be given by (7.2), SB 2 , b(·) be given by (7.8) and φ n (·), λ n be given by (7.9). Suppose that the integer N satisfies (7.14), Λ N , B N is given by (7.16) and there exists an L N = (l 1 , l 2 , · · · , l N ) ∈ L(R N , R) such that Λ N +B N L N is Hurwitz. Then, the operator A 1 + SB 2 K N generates an exponentially stable C 0 -semigroup on L 2 [0, 1], where K N is given by (7.17).
The proof will be accomplished if we can show that σ(A 1 + SB 2 K N ) ⊂ {s | Re(s) < 0}. For any λ ∈ σ(A 1 + SB 2 K N ), we consider the characteristic equation (A 1 + SB 2 K N )f = λf with f = 0.