Stability and Stabilization of Infinite-dimensional Linear Port-Hamiltonian Systems

Stability and stabilization of linear port-Hamiltonian systems on infinite-dimensional spaces are investigated. This class is general enough to include models of beams and waves as well as transport and Schr\"odinger equations with boundary control and observation. The analysis is based on the frequency domain method which gives new results for second order port-Hamiltonian systems and hybrid systems. Stabilizing controllers with colocated input and output are designed. The obtained results are applied to the Euler-Bernoulli beam.


Introduction
In recent years there has been a growing interest in the stability and stabilization of wave and beam equations. For several of these equations results for structural damping or boundary feedback have been detected using Lyapunov methods, a Riesz basis approach or frequency domain methods. A large class of these equations may be written in the form of port-Hamiltonian systems with suitable boundary conditions. This class covers in particular the wave equation, the transport equation, the Timoshenko beam equation (all N = 1), but also the Schrödinger equation and the Euler-Bernoulli beam equation (both N = 2). For distributed parameter systems as port-Hamiltonian systems see [19] and in particular the Ph.D thesis [21]. We follow this unified approach and employ the rich theory of one-parameter C 0 -semigroups of linear operators (e.g. [7]) and, more specifically, some of the stability theory ( [1], [5], [8], [15], [16], [20]). Our investigation has the following two parts: stability (or stabilization by static feedback, i.e. pure infinite-dimensional systems) and stabilization by dynamical feedback (i.e. hybrid systems). We concentrate only on boundary feedback stabilization, although most of our results naturally extend to situations with structural damping. For the pure infinite-dimensional part already some results for port-Hamiltonian systems have been known, especially for the case N = 1 ( [6], [11], [22]) whereas for the case N = 2 most of the research has been focussed on particular examples of beam equations ( [2], [3], [10]). On the other hand, for beam equations hybrid systems have been investigated for some time now ([9], [13], [14]) and recently for SIP controllers with colocated input-and output map a nice result for the case N = 1 has been established ( [17]). The latter turns out to be a special case of the results presented here. This article is organised as follows. Section 2 is devoted to pure infinite-dimensional port-Hamiltonian systems, where in Subsection 2.1 we derive the contraction semigroup generation theorem for the operator A associated to the evolution equation (1). However, our main objective is to investigate the asymptotic behaviour of port-Hamiltonian systems. We focus on two types of stability concepts. Namely let (T (t)) t≥0 be any C 0 -semigroup on X. We say that (T (t)) t≥0 is asymptotically respectively (uniformly) exponentially stable if there exist M ≥ 1 and ω < 0 with Here (T (t)) t≥0 is the C 0 -semigroup generated by the port-Hamiltonian operator A.
Our approach is based on Stability Theorems 2.5 and 2.6. These results motivate to introduce properties ASP, AIEP and ESP in Subsection 2.2. We then only has to test whether a particular function f : D(A 0 ) → R + has one of these properties to obtain the corresponding stability result. The main advantage of using these properties does not lie in the pure infinite-dimensional case (with static feedback), but in the case of dynamical feedback via (finite-dimensional) controllers which we consider later in Section 3. In the latter case we use the same properties ASP, AIEP and ESP in order to deduce results for interconnected systems without having to reprove the same auxiliary results once again. We start with asymptotic (strong) stability and based on the Stability Theorem 2.5 by Arendt, Batty, Lyubich and Phong give a general asymptotic stability result for port-Hamiltonian systems. Then we continue with exponential stability for the case N = 1 in Subsection 2.4. This class of systems has been extensively studied in the book [11]. Originally in [22] the authors presented an exponential stability result based on some sideways energy estimate (Lemma III.1 in [22]) which goes back to an idea of Cox and Zuazua (Theorem 10.1 in [4]). We establish the same result using a frequency domain method based on Gearthart's Theorem 2.6. It turns out that by this technique we do not only obtain a different proof for exponential stability of first order port-Hamiltonian systems, but the method extends to a proof for second order systems as well, whereas the idea in [22] seems to be restricted to the transport equation-like situation for first order systems. We even present a general exponential stability result for second order port-Hamiltonian systems in Subsection 2.5. Moreover we give a sufficient condition for second order systems with some special structure which applies in particular to Euler-Bernoulli beam equations. Section 3 then constitutes a breach since we leave the pure infinite-dimensional setup and consider hybrid systems which consist of both a infinite-dimensional subsystem (governed by a port-Hamiltonian partial differential equation) and a finitedimensional subsystem which we think of as a controller (modelled by an ordinary differential equation). In applications these situations are characterized by an energy functional which splits into a continuous part and a discrete part. We interpret the total system as an interconnection of two subsystems which interact with each other by means of boundary control and observation. We then depict how the theory for the pure infinite-dimensional case naturally carries over to these hybrid systems. After stating the generation result in Subsection 3.1 we obtain a stability result for hybrid systems in Subsection 3.2 without additional structure conditions. For the special class of strictly input passive (SIP) controllers with colocated input and output we then obtain in Subsection 3.3 a stability result which is much more suitable for applications. As a special case we rediscover the main result of [17] (which has been proved using a Lyapunov method with the same sideways energy estimate mentioned above). Finally, in Section 4 we illustrate how our theoretical results can be used to reobtain some stability results on the Euler-Bernoulli beam equation, namely the situations considered in [3] and [9]. In the latter case we encounter a situation where the finite-dimensional controller naturally appears in the modelling of the problem.

Infinite-dimensional Port-Hamiltonian Systems
Throughout this paper we use the following notations. For any Hilbert space X we denote by ·, · its inner product (which is linear in the second component). Moreover B(X, Y ) denotes the space of linear and bounded operators X → Y where as usual B(X) := B(X, X). For any closed linear operator A : D(A) ⊂ X → X we have the resolvent set ρ(A), the spectrum σ(A) and write R(λ, A) := (λI − A) −1 for the resolvent operator and σ p (A) for the point spectrum of A. We investigate port-Hamiltonian systems of order N ∈ N, given by the partial differential equation Here P k ∈ C d×d , k = 0, 1, . . . , N , always denotes some complex matrices satisfying the condition (Note that we do not require P 0 to be skew-adjoint.) Moreover we always assume that P N is invertible. The Hamiltonian density matrix function H : (0, 1) → C d×d is a measurable function such that there exist 0 < m ≤ M such that for almost every ζ ∈ (0, 1) the matrix H(ζ) is self-adjoint and We then say that H is uniformly positive. In this paper we consider the energy state space X = L 2 (0, 1; C d ) with the inner product Note that · H is equivalent to the standard L 2 -norm · L2 . The operator A 0 : D(A 0 ) ⊂ X → X corresponding to equation (4) is given by Thanks to the invertibility of P N the operator A 0 is closed.
Lemma 2.1. The operator A 0 is a closed operator and its graph norm is equivalent to the norm H· H N .
be the boundary trace operator and introduce the boundary port variables f ∂,Hx Note that the boundary port variables do not depend on the matrix P 0 . If P 0 = −P * 0 is skew-adjoint, the boundary port variables determine Re A 0 x, x .

Generation of Contraction Semigroups
Since we did not impose any boundary conditions in equation (8), we could not expect A 0 to generate a C 0 -semigroup (in fact, σ p (A 0 ) = C). However, for suitable boundary conditions, defining a subspace D(A) ⊂ D(A 0 ) the restricted operator A = A 0 | D(A) has the generator property. For this purpose, let W ∈ C N d×2N d be a full rank matrix and define the operator A by Note that thanks to the invertibility of P N , the matrix Q −Q I I is invertible (see Lemma 3.4 in [12]) and thus the condition W f ∂,Hx e ∂,Hx = 0 may be equivalently expressed as W ′ Φ(Hx) = 0 for a suitable matrix W ′ . Using the Lumer-Phillips Theorem II.3.15 in [7] the generators of contraction semigroups have been characterized by a simple matrix condition or alternatively by dissipativity of the operator. Note that usually the hard part of proving that an operator A generates a contraction semigroup is the range condition ran (λI − A) = X for some λ > 0.
Theorem 2.3. The following are equivalent.
In that case A has compact resolvent. Note that this result is a combination of Theorem 7.2.4 in [11] where the authors focus only on the case N = 1 and Theorem 4.1 in [12] where the general case of N -th order Port-Hamiltonian systems is treated for the equivalence of parts 1. and 2. However in both cases the authors only treat the case P 0 = −P * 0 . For the general case where P * 0 = −P 0 is not skew-adjoint we use a perturbation argument.
Proof. Let us first assume that P 0 = −P * 0 is skew-adjoint. The equivalence of conditions 1. and 3. is due to Theorem 4.1 in [12]. The implication 1. ⇒ 2. results from the Lumer-Phillips Theorem II.3.15 in [7]. For the implication 2. ⇒ 1. one only needs to show the range condition ran (I − A) = X (thanks to the Lumer-Phillips result). This can be done similar as in the proof of Theorem 7.2.4 in [11] (with obvious modifications). We leave the details to the interested reader. Let us concentrate on the situation where P 0 = −P * 0 , i.e.
Of course, the implication 1. ⇒ 2. follows by the Lumer-Phillips Theorem II.3.15 in [7]. Next we show that 2. implies 1. Let us writeÃ := A + G 0 H. If we can show thatÃ generates a contractive C 0 -semigroup, then also A generates a C 0 -semigroup by the Bounded Perturbation Theorem III.1.3 in [7] which then is contractive since its generator is dissipative. SinceÃ is a port-Hamiltonian operator with skew-adjointP 0 it suffices to prove dissipativity ofÃ. AssumeÃ were not dissipative. Then by Lemma 2.4 below there exists a X-null sequence ( which leads to a contradiction. HenceÃ generates a contraction semigroup and so does A. Thus 1. and 2. are equivalent also in this case. Further we obtain that if 1. or 2. holds thenÃ generates a C 0 -semigroup, so W ΣW * ≥ 0. Moreover for any by 2. and hence choosing Hx = φξ for φ ∈ C ∞ c (0, 1; C) and ξ ∈ C d it follows Re P 0 ≤ 0, so 3. holds. Finally, from 3. it follows thatÃ (as introduced above) generates a contraction semigroup and hence does A =Ã − G 0 H by Theorem III.2.7 in [7] and the dissipativity of −G 0 H.
In the proof we used the following.
Then for x n := H −1 y n we obtain and so consequently Re Ã x n , x n H = Re Ã x, x H = 1 for all n ∈ N.

Sufficient Conditions for Stability
Our main tools to deduce stability results are the following two theorems.
Here σ r (B) := {λ ∈ C : ran (λI − B) not dense in Y } denotes the residual spectrum of B which coincides with the point spectrum of the adjoint operator B ′ .
Note that in particular for generators B with compact resolvent we have asymptotic stability if and only if σ p (B) ⊂ C − 0 := {λ ∈ C : Re λ < 0}. The second result requires Hilbert space structure.
Remark 2.7. The uniform boundedness of the resolvent on iR in Theorem 2.6 is equivalent to the condition For the moment let Y be any Hilbert space. The following definition enables us to lift stability results to hybrid systems which we investigate later on.
• ESP (for the operator B) if it has properties ASP and AIEP.
Note the following property which easily may be verified using the above definition. The abbreviations ASP, AIEP and ESP stand for asymptotic stability property, asymptotic implies exponential stability property and exponential stability property, where a typical choice of f are functions of the form for some non-negative constants α j,k ≥ 0. That the above terminology is indeed appropriate is the statement of the following lemma.
Lemma 2.10. Let B have compact resolvent and generate a C 0 -semigroup (S(t)) t≥0 on Y and assume that for some function f : Then 1. If f has property ASP then (S(t)) t≥0 is asymptotically (strongly) stable.
Proof. 1.) If f has property ASP and iβx = Bx for some x ∈ D(B) and β ∈ R then and by the property ASP it follows x = 0, so iR ∩ σ p (B) = ∅ and asymptotic stability follows from Stability Theorem 2.5.
i.e. f (x n ) → 0 and by property AIEP this leads to x n → 0 so exponential stability follows from Stability Theorem 2.6. 3. is a direct consequence of 1. and 2.

Asymptotic Stability of Port-Hamiltonian Systems
An example for a function f : D(A 0 ) → R + which has property ASP is the square of the Euclidean norm of Hx(ζ) and its derivatives at position ζ = 0. (Of course, the choice ζ = 1 is possible as well.) The asymptotic stability result reads as follows.
Proposition 2.11. Assume that A satisfies for some positive κ > 0. Then (T (t)) t≥0 is an asymptotically stable and contractive C 0 -semigroup.
Proof. We prove that has property ASP and use Lemma 2.9. Let β ∈ R and x ∈ D(A 0 ) with which is a system of ordinary differential equations with boundary conditions Since P N is invertible the unique solution of this initial value problem is x = 0, so f has property ASP and the result follows from Lemma 2.10.

First Order Port-Hamiltonian Systems
The following exponential stability result can already be found as Theorem III.2 in [22]. Here we present a different proof using a frequency domain method.
for some κ > 0, then A generates an exponentially stable and contractive C 0semigroup on the Hilbert space X.
We remark that in (28) we could alternatively choose −κ |(Hx)(1)| 2 for the right hand side. For the proof we need the following lemma.
Lemma 2.13. Let Q ∈ W 1 ∞ (0, 1; C d×d ) be a function of self-adjoint operators and x ∈ H 1 (0, 1; C d ). Then Proof of Proposition 2.12. Theorem 2.3 implies that A generates a contraction C 0semigroup Let f : We show that f has the ESP property. By Proposition 2.11 property ASP holds and thus we only need to prove the property AIEP. Let ((x n , β n )) n≥1 ⊂ D(A 0 ) × R be any sequence with x n L2 ≤ c and |β n | → ∞ such that Then we obtain the definition of f that Moreover xn βn is bounded in the graph norm · A0 and by Lemma 2.1 we get Letting q ∈ C 1 ([0, 1]; R) with q(1) = 0 and having Lemma 2.13 in mind we find since (Hx n )(0) → 0, q(1) = 0 and |β n | → ∞, using integration by parts and P 1 = P * 1 . In particular we may choose q ≤ 0 such that where H(ζ) ≥ mI and ±H ′ (ζ) ≤ λI for a.e. ζ ∈ [0, 1], so qH ′ − q ′ H is uniformly positive. This implies Hence property AIEP holds and exponential stability follows with Lemma 2.10.

Second Order Port-Hamiltonian Systems
As we have seen in the preceding subsection for first order (N = 1) port-Hamiltonian systems the sufficient criterion for asymptotic stability in Proposition 2.11 even guarantees exponential stability (Proposition 2.12). We now consider second order port-Hamiltonian systems, i.e.
Adding an additional term |(Hx)(1)| 2 (or, |(Hx) ′ (1)| 2 ) in the dissipativity relation (24) we again obtain exponential stability. By means of the example of the onedimensional Schrödinger equation we show that the sufficient criterion for asymptotic stability as in Proposition 2.11 is not sufficient for exponential stability in the case N = 2.
Remark that again one may interchange 0 and 1 in equation (34). For the proof, let us first state an auxiliary embedding-and-interpolation result.
Lemma 2.15. Let 0 ≤ k < N ∈ N 0 and θ ∈ (0, 1) such that η := θN ∈ (k+ 1 2 , k+1). Then there exist a constant c θ > 0 such that for all f ∈ H N (0, 1; C d ) Further for σ := k N there exists a constant c σ > 0 such that for all f ∈ H N (0, 1; C d ) Proof. Let p ∈ (1, ∞) such that η − 1 2 > k + 1 − 1 p > k. Then by the Sobolev-Morrey Embedding Theorem is continuously embedded. Further, using the notation of [18], we have by the theorems of Subsections 3.3.1 and 3.3.6 in [18] that and the first assertion follows by the interpolation inequality. The second assertion is a special case of the Gagliardo-Nirenberg inequality. In the language and with the theory of [18] it results from Proof of Proposition 2.14. Theorem 2.3 implies that A generates a contraction C 0semigroup. We show that has the property ESP. By Proposition 2.11 and Lemma 2.9 it remains to verify the AIEP property. Let (x n , β n ) n≥1 ⊂ D(A 0 ) × R be a sequence with x n L2 ≤ c for all n ∈ N and |β n | → +∞ as n → +∞ such that ⊆ H 2 (0, 1; C d ) is bounded and by Lemma

2.15
Hxn βn converges to zero in C 1 ([0, 1]; C d ) (since |β n | → ∞). Let q ∈ C 2 ([0, 1]; R) be some real function. Integrating by parts and employing the assumptions on the matrices P 1 and P 2 and Lemma 2.13 we conclude and Subtracting (42) from two times (41) this implies Choosing q ∈ C 2 ([0, 1]; R) such that q(1) = 0 and 2q ′ H − qH ′ is uniformly positive this leads in the case that also f (x n ) → 0 to and thus f also has property AIEP.
Without proof we remark that using the same proof technique as for Proposition 2.14 one obtains the following generalization to port-Hamiltonian systems of even order.
Remark 2.17. One could hope to relax the dissipativity condition in Proposition 2.14 to However, the following example shows that even in the case d = 1 and H ≡ 1 one generally only has asymptotic (strong) stability.

Example 2.18 (Schrödinger Equation)
. Let us investigate the one-dimensional Schrödinger equation on the unit interval with boundary conditions for some constants k > 0 and α ∈ R \ {0}. The energy functional is given as and the corresponding port-Hamiltonian operator is Integrating by parts and using the boundary conditions we deduce We claim that the semigroup is not exponentially stable, though it is asymptotically (strongly) stable. For this end we apply Stability Theorem 2.6 and prove sup iR R(·, A) = ∞.
Let β > 0 be arbitrary, hence iβ ∈ ρ(A). For f ∈ L 2 (0, 1) we solve (iβ − A)x = f and obtain the solution (51) with the value x(0) = x β,f (0) given by Now we choose f = 1 ∈ L 2 (0, 1) and get Thus for all ζ ∈ (0, 1) in particular Thus the resolvents cannot be uniformly bounded on the imaginary axis and hence A does not generate an exponentially stable C 0 -semigroup.
However, for a special class of port-Hamiltonian systems which have some antidiagonal structure we can weaken the assumptions on the boundary dissipation.
Again one may interchange 0 and 1 in the dissipativity estimate.
Proof. The result may be proved in similar fashion as Proposition 2.14.

Hybrid Systems
In this section we study stability of hybrid systems. The preconditions for the infinite-dimensional part of the interconnected system stay the same, except for input and output variables which we utilize for interconnection with the finitedimensional controller. So, instead of a static boundary condition W f ∂,Hx e ∂,Hx = 0 we use (part of) W f ∂,Hx e ∂,Hx to define the input function for the interconnection with a finite-dimensional system and on the other hand use the remaining information from f ∂,Hx e ∂,Hx to define the output map for the interconnection structure. So let W,W ∈ C N d×2N d be two full rank matrices and such that the matrix W W is invertible. Let 1 ≤ m,m ≤ N d ∈ N and decompose W,W as where W 1 ∈ C m×2nd andW 1 ∈ Cm ×2N d . The infinite-dimensional subsystem may then be written as (Further we use the notation B := (B 1 , B 2 ) and C = (C 1 , C 2 ).) Additionally we consider the space Ξ = C n with inner product ξ, η Qc := ξ * Q c η, η, ξ ∈ Ξ, for some positive n× n-matrix Q c = Q * c > 0. We assume that the finite-dimensional controller has the form for some matrices A c , B c , C c , D c of suitable dimension. We are interested in situations without external input signal and interconnect the two subsystems by standard feedback interconnection u c = y 1 , Then we obtain an operator A on the product space X × Ξ which we equip with the canonical inner product (x, ξ), (y, η) H,Qc = x, y H + ξ, η Qc , (x, ξ), (y, η) ∈ X × Ξ. Namely, on the domain with the matrix W cl given by

Semigroup Generation
Similar to the pure infinite-dimensional case we have the following generation result which includes the case of strictly passive controllers as in Theorem 4 of [17].
Remark 3.2. Similar to the pure infinite-dimensional case one sees that the condition Re P 0 := 1 2 (P 0 + P * 0 ) ≤ 0 is necessary for A to generate a contraction C 0 -semigroup.
For the proof we need the following results which follow from step 2 in the proof of Theorem 4.2 in [12].
. . , N . Then there exists an operator B ∈ B(C N d ; H N (0, 1; C d )) such that Proof of Theorem 3.1. The operator A is densely defined. Namely let (x, ξ) ∈ X ×Ξ be arbitrary. Observe that the matrix so D(A) is densely defined. Thanks to the Lumer-Phillips Theorem II.3.15 in [7] and the dissipativity of A, it remains to check that ran (λI − A) = X × Ξ for some λ > 0. To this end let λ > max(0, s(A c )) where s(A c ) := sup{Re λ : λ ∈ σ(A c )} denotes the spectral bound of A c . Further let (y, η) ∈ X × Ξ be given. We are looking for some (x, ξ) ∈ D(A) such that Solving (67) for ξ and substitution lead to Using the operatorB ∈ B(Ξ, D(A 0 )) from Corollary 3.4 forW cl we set x new := x −Bη and get the equivalent system Let us consider the operatorÃ cl = A 0 | D(Ã cl ) with domain For any thus (x, ξ) ∈ D(A) and we have for all x ∈ D(Ã cl ). HenceÃ cl generates a contractive C 0 -semigroup on X by Theorem 2.3. Consequently, (λI −Ã cl ) −1 ∈ B(X) exists and we then get a unique solution x new of (70) which implies the existence of (x, ξ) ∈ D(A), such that (λI −A)(x, ξ) = (y, η). It follows ran (λI −A) = X and the Lumer-Phillips Theorem II.3.15 in [7] yields the result.

Asymptotic Behaviour
For dissipative hybrid systems we obtain essentially the same stability results as in the pure infinite-dimensional case.
Proposition 3.5. Assume that s(A c ) < 0 and for a function f : 1. If f has property ASP then (T (t)) t≥0 is asymptotically (strongly) stable.
Proof. 1.) Asymptotic stability: By Theorem 3.1 A generates a contractive C 0semigroup and has compact resolvent, so σ(A) = σ p (A). We want to use Stability Theorem 2.5 and thus prove that iR ∩ σ p (A) = ∅. Let β ∈ R and (x, ξ) ∈ D(A) such that iβ(x, ξ) = A(x, ξ), and by property ASP x = 0. The finite-dimensional component reads then also ξ = (iβ−A c ) −1 B c C 1 x = 0. As a result, σ(A) = σ p (A) ⊂ C − 0 and (T (t)) t≥0 is asymptotically stable due to Stability Theorem 2.5. 2.) Exponential stability: Let a sequence ((x n , ξ n , β n )) n≥1 ⊂ D(A) × R with (x n , ξ n ) X×Ξ ≤ c, |β n | n→∞ − −−− → +∞ such that be given. Since (x n , ξ n ) X×Ξ ≤ c it especially follows that Since and f has the property AIEP this implies Let us now consider (ξ n ) n ⊂ Ξ. We have by assumption and dividing by β n = 0 (for n sufficiently large) we get Moreover A(xn,ξn) βn is bounded and using Lemma 2.1 we have Hence Hxn βn is a bounded sequence in H N (0, 1; C d ) and thus by Lemma 2.15 it is a null sequence in C N −1 ([0, 1]; C d ). Since B c C 1 x n continuously depends on Hx n ∈ C N −1 ([0, 1]; C d ) this implies ξ n → 0, so From Stability Theorem 2.6 we deduce exponential stability. 3. is a direct consequence of 1. and 2.

SIP Controllers with Colocated Input/Output
We make the following assumption on the infinite-dimensional part.
Assumption 3.6. Assume that the infinite-dimensional port-Hamiltonian system is passive, i.e. for all x ∈ D(A 0 ) it satisfies the balance equation (In particular the corresponding operator on X for Bx = 0 generates a contraction C 0 -semigroup.) Further we concentrate on finite-dimensional controllers with colocated input and output which are strictly input passive.
1. We say that input and output are colocated ifC ∈ B(X,Ũ ) is the adjoint operator ofB ∈ B(Ũ ,X).

2.
The system is called strictly input passive (SIP) if for some σ > 0 and any solution x one has the estimate In our case we assume m =m and the controller has the forṁ where ξ ∈ Ξ = C n with inner product ξ, η Qc = ξ * Q c η for the n × n-matrix Q c = Q * c > 0 and J c = −J * c , R c = R * c ≥ 0, B c , D c matrices of suitable dimension. For the system to be SIP we demand D c = D * c ≥ σI > 0. We then have the following generation result for the operator.
Theorem 3.8. The operator A generates a contractive C 0 -semigroup on X × Ξ and has compact resolvent.
Proof. We already know that A generates a (contraction) C 0 -semigroup and has compact resolvent. Remark that for any (x, ξ) ∈ D(A) we get 1.) Assume that f has property ASP. We prove that A has no eigenvalues on the imaginary axis. Let β ∈ R and (x, ξ) ∈ D(A) with be arbitrary. Then and from σ(A c ) ⊂ C − 0 we then deduce ξ = 0 and this also implies y c = 0. So and hence f (x) = 0 and A 0 x = iβx. From property ASP we also conclude x = 0 and hence A has no eigenvalues on the imaginary axis. The result follows from Theorem 2.5. 2.) Let us assume (T (t)) t≥0 is asymptotically stable and f has property AIEP. Let ((x n , ξ n ), β n ) n≥1 ⊂ D(A) × Ξ be any sequence with x n ≤ c, |β n | → ∞ and A(x n , ξ n ) − iβ n (x n , ξ n ) n→∞ − −−− → 0 in X × Ξ.
As a result, Theorem 14 of [17] follows directly from Proposition 2.12 and Theorem 3.9.
For the standard port-Hamiltonian formulation we thus set For the related operator A 0 we obtain the balance equation Example 4.1 (Clamped Left End). In [2] the authors consider clamped left end boundary conditions and static feedback at the right end.