Stabilization of port-Hamiltonian systems by nonlinear boundary control in the presence of disturbances

In this paper, we are concerned with the stabilization of linear port-Hamiltonian systems of arbitrary order $N \in \mathbb{N}$ on a bounded $1$-dimensional spatial domain $(a,b)$. In order to achieve stabilization, we couple the system to a dynamic boundary controller, that is, a controller that acts on the system only via the boundary points $a,b$ of the spatial domain. We use a nonlinear controller in order to capture the nonlinear behavior that realistic actuators often exhibit and, moreover, we allow the output of the controller to be corrupted by actuator disturbances before it is fed back into the system. What we show here is that the resulting nonlinear closed-loop system is input-to-state stable w.r.t.~square-integrable disturbance inputs. In particular, we obtain uniform input-to-state stability for systems of order $N=1$ and a special class of nonlinear controllers, and weak input-to-state stability for systems of arbitrary order $N \in \mathbb{N}$ and a more general class of nonlinear controllers. Also, in both cases, we obtain convergence to $0$ of all solutions as $t \to \infty$. Applications are given to vibrating strings and beams.


Introduction
In this paper, we consider linear port-Hamiltonian systems of arbitrary order N ∈ N on a bounded 1dimensional spatial domain (a, b). Such systems are described by a linear partial differential equation of the form ∂ t x(t, ζ) = P N ∂ N ζ (H(ζ)x(t, ζ)) + · · · + P 1 ∂ ζ (H(ζ)x(t, ζ)) + P 0 H(ζ)x(t, ζ) where x(t, ζ) ∈ R m , H(ζ) ∈ R m×m is the energy density at ζ, and P 0 , . . . , P N ∈ R m×m are alternately skewsymmetric and symmetric matrices and P N is invertible. Simple examples of such systems are given by vibrating strings or beams. Also, many of the systems of linear conservation laws considered in [3], namely those of the special form with a ζ-independent invertible matrix P , fall in the above class of linear port-Hamiltonian systems. What we are interested in here is the stabilization of such a system S by means of dynamic boundary control, that is, by coupling the system to a dynamic controller S c that acts on the system only via the boundary points a, b of the spatial domain (a, b). Since realistic controllers often exhibit nonlinear behavior (due to nonquadratic potential energy or nonlinear damping terms, for instance), we want to work with nonlinear controllers -just like [1], [27], [59], [42]. Since, moreover, realistic controllers are typically affected by external disturbances, we -unlike [1], [27], [59], [42] -also want to incorporate such actuator disturbances which corrupt the output of the controller before it is fed back into the system. Coupling such a controller to the system S by standard feedback interconnection In this paper, we establish the input-to-state stability for the closed-loop systemS w.r.t. square-integrable disturbance inputs d. In rough terms, this means thatx * := 0 is an asymptotically stable equilibrium point of the undisturbed closed-loop system with d = 0 and that this stability property is robust w.r.t. disturbances d = 0 in the sense that small disturbances affect the asymptotic stability only slightly (see (1.5) and (1.6) below). In more precise terms, the input-to-state stability ofS w.r.t. square-integrable disturbance inputs d means (i) that S, for every initial statex 0 and every square-integrable disturbance d, has a unique global (generalized) solutioñ x(·,x 0 , d) and (ii) that for all these (generalized) solutions, the following perturbed stability and attractivity estimates hold true: and lim sup t→∞ x(t,x 0 , d) ≤ γ( d 2 ), (1.6) where σ, γ, γ are monotonically increasing comparison functions that are zero at 0. According to whether the limit relation (1.6) holds (locally) uniformly or just pointwise w.r.t.x 0 and d, one speaks of uniform or weak input-to-state stability, respectively. Also, instead of uniform input-to-state stability one usually just speaks of input-to-state stability. In this paper, we show (i) that for a system S of order N = 1 and a special class of nonlinear controllers S c , the resulting closed-loop systemS is uniformly input-to-state stable w.r.t. squareintegrable disturbances d, and (ii) that for a system S of arbitrary order N ∈ N and a more general class of nonlinear controllers S c , the resulting closed-loop systemS is weakly input-to-state stable w.r.t. squareintegrable disturbances d. In particular, we show in both cases that unique global (generalized) solutions x(·,x 0 , d) exist forS. Additionally, we will see that in both cases every such solution converges to zero: for every initial statex 0 and every square-integrable disturbance d. In all these results, we have to impose only mild assumptions on the system S to be stabilized, namely an impedance-passivity condition and an approximate observability condition. We finally apply both our uniform and our weak input-to-state stability result to vibrating strings and beams (modeled according to Timoshenko).
In the literature, the stabilization of port-Hamiltonian systems has been considered so far, to the best of our knowledge, only in the case without actuator disturbances. In [54], [1], [27], [59], [42], no disturbances are considered at all, that is, the situation depicted above is considered in the special case where d = 0. Stabilization of port-Hamiltonian systems of various degrees of generality is achieved by means of linear dynamic boundary controllers in [54] and by means of nonlinear dynamic boundary controllers in [1], [27], [59], [42]. In [51], sensor -instead of actuator -disturbances are considered, that is, disturbances do occur in [51] but they corrupt the input of the controller instead of its output. It is shown in [51] that for a port-Hamiltonian system of the special form (1.3) with negative definite P and a linear dynamic boundary controller, the resulting closed-loop system is uniformly input-to-state stable w.r.t. essentially bounded disturbance inputs d (meaning that the perturbed stability and attractivity estimates (1.5) and (1.6) are satisfied with the 2-norm d 2 replaced by the ∞-norm d ∞ of the disturbance).
In recent years, input-to-state stability of linear and non-linear partial differential equation systems -parabolic and hyperbolic -has attracted a great deal of research, both in the case of distributed inputs (entering in the domain) and in the typically more challenging case of boundary inputs (entering at the boundary of the domain of the considered pde). See [6], [16], [22], [23], [26], [31], [51], [50], [56], [57], [58], [10], [24], for instance. So far, the works establishing input-to-state stability of parabolic pde systems by far outnumber those on hyperbolic pde systems (like the port-Hamiltonian systems considered here). While the aforementioned papers deal with input-to-state stability w.r.t. a single equilibrium point of pde systems, a few papers [8], [49] also establish input-to-state stability w.r.t. attractors of such systems. Apart from the mentioned works on pde systems, also a lot of papers on the input-to-state stability of finite or infinite networks of ordinary or partial differential equation subsystems and on the input-to-state stability of time-delay systems have appeared in recent years. See [33], [7], [34], [35] and, respectively, [40], [21], [36] among many others. And finally, a multitude of papers have recently been published on fundamental characterizations and criteria for input-to-state stability [29], [30], [17], [15], [18], [19] and for versions of input-to-state stability like local [28], integral [15], strong [37], weak [45] and input-to-state practical stability [32], for instance. A nice survey of the state of the art of input-to-state stability for infinite-dimensional systems can be found in [36] -along with an extensive list of references on the topic.
We conclude the introduction with some remarks on the organization of the paper and on notational conventions used throughout the paper. Section 2 provides a detailed description of the setting with the precise assumptions on the system S to be stabilized and the controller S c used for that purpose. In Section 3 we prove the solvability of the closed-loop system -first, in the classical sense for classical data (x 0 , d), and then in the generalized sense for general data (x 0 , d). In Section 4 we establish the main results of this paper, namely the uniform (Section 4.1) and the weak (Section 4.2) input-to-state stability of the closed-loop system. And finally, in Section 5 we present some applications of our general results. In order to get a quick overview of the core results in simplified form, the reader can consult the conference paper [43].
In the entire paper, | · | denotes the standard norm on R k for every k ∈ N. As usual, K, K ∞ , L denote the following classes of comparison functions: , and for d ∈ L 2 ([0, ∞), R k ) we will use the following short-hand notations: And finally, for a semigroup generator A and bounded operators B, C between appropriate spaces, the symbol S(A, B, C) will stand for the state-linear system [5] x = Ax + Bu with y = Cx, where the prime stands for the derivative w.r.t. time.

Setting: the system to be stabilized
We consider a linear port-Hamiltonian system S of order N ∈ N on a bounded interval (a, b) with control and observation at the boundary [12], [9], [2]. Such a system evolves according to the following differential equation with boundary control and boundary observation conditions: u(t) = Bx(t) and y(t) = Cx(t) (2.2) and the energy of such a system in the state x is given by In these equations, ζ → H(ζ) ∈ R m×m is a measurable matrix-valued function (the energy density) such that for some positive constants m, m and almost all ζ ∈ (a, b) and P 0 , P 1 , . . . , P N ∈ R m×m are matrices such that P N is invertible and P 1 , . . . , P N are alternately symmetric and skew-symmetric while P 0 is dissipative: . . , N }) and P 0 + P 0 ≤ 0.
(Strictly speaking, such systems should be called port-Hamilitonian only if P 0 is skew-symmetric. If P 0 is only dissipative, they could more precisely be called port-dissipative systems.) As the state space of S one chooses X := L 2 ((a, b), R m ) with norm · X given by the system energy It is clear by (2.4) that the norm · X is equivalent to the standard norm of L 2 ((a, b), R m ) and that it is induced by a scalar product which we denote by ·, · X . As the domain of the linear differential operator In other words, the domain of A incorporates the zero boundary condition which consists of N m−k (scalar) equations and is linear in the boundary values of (Hx), ∂ ζ (Hx), . . . , ∂ N −1 ζ (Hx). Similarly, the boundary control and boundary observation conditions (2.2) consist of k (scalar) equations each and are again linear in (Hx)| ∂ , that is, the boundary control and boundary observation operators B, C : D(A) → R k from (2.2) are linear and of the form with matrices W B,2 , W C ∈ R k×2mN . In all our results, we will impose the following additional impedancepassivity condition on S.
is a contraction semigroup generator on X and that the matrix W B := (W B,1 , W B,2 ) ∈ R mN ×2mN has full row rank. In particular, A and B define a boundary control system [12] (by the same arguments as for Theorem 11.3.2 of [12] and by Lemma A.3 of [25]), so that for every initial state x 0 ∈ D(A) and every control input u ∈ C 2 ([0, ∞), R k ) with u(0) = Bx 0 , the system (2.1) and (2.2) has a unique global classical solution Also, along every such classical solution, the following energy dissipation inequality is satisfied by virtue of (2.8): where E x (t) := E(x(t)) and x(t) := x(t, x 0 , u) and y(t) := y(t, x 0 , u) := Cx(t, x 0 , u).

Setting: the controller
As the controller S c to stabilize S we choose a finite-dimensional nonlinear system which evolves according to the ordinary differential equation (with input u c and output and whose energy in the state v = (v 1 , v 2 ) ∈ R 2mc is given by (potential energy plus kinetic energy). In these equations, K ∈ R mc×mc , B c ∈ R mc×k , S c ∈ R k×k represent a generalized mass matrix, an input matrix, and a direct feedthrough matrix respectively and they are such that Additionally, the potential energy P : R mc → [0, ∞) is differentiable such that ∇P is locally Lipschitz continuous and P(0) = 0 and the damping function R : R mc → R mc is locally Lipschitz continuous such that R(0) = 0.
As the norm on the controller state space V := R 2mc we choose | · | V defined by which is obviously equivalent to the standard norm on R 2mc and is induced by a scalar product ·, · V . In all our results, we will impose the following additional conditions on the controller S c .

Condition 2.2.
(i) P is positive definite and radially unbounded, that is, It follows from Condition 2.2 (ii) and from (2.15) that the controller system S c is passive (in fact, strictly input-passive) w.r.t. the storage function E c .

Setting: the closed-loop system
Coupling S and S c by standard feedback interconnection we obtain the closed-loop systemS described by the following evolution equation and boundary input and output conditions: Its state space is the Hilbert spaceX := X × V = X × R 2mc with norm · = · X defined by V and the energy of the closed-loop system in the statex = (x, v) ∈X is It follows from Condition 2.2 (i) (taking into account Lemma 2.5 of [4]) that the energyẼ is equivalent to the norm · ofX in the following sense: there exist ψ, ψ ∈ K ∞ such that for allx ∈X   and the linear boundary input and output operatorsB,C : D(Ã) → R k are given bỹ In our solvability and stability results, we will need the following semigroup generation and compact resolvent result. A version of this lemma is stated in [42]  Proof. Since assertion (ii) is clear by our assumptions on P and R, we have only to prove assertion (i). We do so in three steps. As a first step, we observe thatÃ is dissipative inX. Indeed, for everyx = (x, v) ∈ D(Ã) we have x ∈ D(A) and B c Kv 2 = −Bx − S c Cx and hence where the last two inequalities follow from (2.8) and (2.15). Consequently,Ã is dissipative inX, as desired.
As a second step, we show thatÃ − λ is surjective ontoX for every λ ∈ (0, ∞) with So let λ ∈ (0, ∞) as in (2.20) and letỹ = (y, w) ∈X. We then have to find anx = (x, v) ∈ D(Ã), that is, añ x ∈X with x ∈ D(A) and such that (Ã − λ)x =ỹ, that is, Since A c − λ is invertible for λ ∈ (0, ∞), finding anx ∈X with (2.21) and (2.22) is equivalent to finding x ∈ X such that where B x := (W B,2 + G λ W C )(Hx)| ∂ and generates a contraction semigroup on X. So let x ∈ D(A ) and u := W B,2 (Hx)| ∂ and y := W C (Hx)| ∂ , then x ∈ D(A) and u + G λ y = 0 and hence by virtue of (2.8). It follows from (2.15) and (2.20) that and thus (2.26) yields the dissipativity of A in X. So, by the characterization of the contraction semigroup generator property for port-Hamiltonian operators from [14] (Corollary 2.3), A is a contraction semigroup generator on X and, moreover, the boundary matrix associated with A has full row rank. In particular, A and B define a boundary control system by the same arguments as for Theorem 11.3.2 from [12] and by Lemma A.3 from [25], and hence there exists an operator B ∈ L(U, X) with U := R k such that B U ⊂ D(A) = D(B ) and AB ∈ L(U, X) and With these preliminary considerations about A we can now finally prove the existence of an x ∈ X with (2.23) and (2.24) and hence the surjectivity ofÃ − λ. Indeed, with the ansatz And since by (2.27) B is surjective and A is a contraction semigroup generator, we really can find h ∈ D(B ) = D(A) and x ∈ D(A ) such that (2.29) and (2.30) are satisfied, as desired.
As a third step, we show thatÃ has compact resolvent. So let λ ∈ (0, ∞) with (2.20). Also, let (ỹ n ) = (y n , w n ) be a bounded sequence inX and write It then follows by (2.28), (2.29), (2.30), (2.27) that x n = x n + h n with Since (ỹ n ) = (y n , w n ) is bounded and AB ∈ L(U, X), the sequences are bounded as well. It follows by the finite-dimensionality of R 2mc and the compactness of (A − λ) −1 (Theorem 2.28 in [53]) that there exists a subsequence (n k ) sucht that (w n k ) and (v n k ) converge in R 2mc and such that (x n k ) converges in X. Consequently, (x n k ) is convergent as well, and we are done.

Solvability of the closed-loop system
In this section, we show that under suitable conditions the initial value problem of the closed-loop system has a global solution for suitable initial valuesx 0 and disturbance inputs d. We will achieve this by applying the standard theory of semilinear evolution equations from [39]. As it stands, however, the closed-loop equation (3.1) is not a semilinear evolution equation in the sense of [39] because, for one thing, the linear partÃ of the differential equation is not a semigroup generator and because, for another thing, in addition to the differential equation the side condition occurs. A way out of this difficulty is to impose the following extra condition, which is easily seen to be satisfied if, for instance, is a matrix of full row rank mN + k.
With the help of this extra condition, we can turn (3.1) into a truly semilinear evolution equation in the sense of [39]. In fact, Condition 3.1 implies the existence of a linear right-inverseR : R k → D(Ã) ofB, that is, which is well-known from linear boundary control problems [11], [12], [5]. Via this transformation, the classical solutionsx of (3.1) are in one-to-one correspondence -for continuously differentiable disturbances d -with the classical solutionsξ of And this is now, in view of Lemma 2.3, a truly semilinear evolution equation (with an explicitly time-dependent nonlinearity). We can therefore apply standard semilinear theory to obtain classical solvability of (3.1) for sufficiently regularx 0 and d (Section 3.1), and, by a suitable density and approximation argument, we then also obtain generalized solvability of (3.1) for sufficiently irregularx 0 and d (Section 3.2).

Solvability in the classical sense
In this section, we show that for sufficiently regular initial statesx 0 and disturbances d, namely for (set of classical data), the closed-loop equation (3.1) has a classical solution existing globally in time. A classical solution of (3.1) is a continuously differentiable functionx : J →X on an interval J ⊂ [0, ∞) containing 0 such that for every t ∈ J one has (i) thatx(t) ∈ D(Ã) and (ii) that and the output y(·, Proof. As a first step, we show that for every (x 0 , d) ∈ D the initial value problem  [39]). In particular, it follows by variation of constants that for every (x 0 , d) ∈ D the corresponding maximal classical solutionx =x(·,x 0 , d) satisfies the following integral equation: As a second step, we show that for every is continuous. Since, moreover, for some positive constants c, c (Lemma 3.2.3 in [2]), the map is continuous as well. Combining now (3.12) and (3.13), the second step follows. As a third step, we show that there exist σ, γ ∈ K such that for every is a classical solution by the first step, the function is continuously differentiable and its derivative satisfies for all s ∈ [0, Tx 0,d ). With the help of Condition 2.1 and 2.2 we therefore get that for all s ∈ [0, Tx 0,d ) and arbitrary α ∈ (0, ∞), where ς as in (2.15) is the smallest eigenvalue of S c . Choosing now α := 1/(2ς), we see from (3.16) by integration that SinceẼ is equivalent to the norm ofX by (2.18), we further conclude that for every t ∈ [0, Tx 0,d ). So, (3.14) follows because σ, γ defined by obviously belong to the class K.
As a fourth and last step, we show that the maximal existence time Tx 0,d = ∞. Combined with the previous steps, this proves the theorem. So let (x 0 , d) ∈ D and assume that Tx 0 ,d < ∞, that is, the existence interval [0, Tx 0 ,d ) of the maximal classical solutionx(·,x 0 , d) −Rd of (3.4) is bounded. It then follows by the standard blow-up result for semilinear evolution equations (Theorem 6.1.4 of [39]) that the solutionx(·,x 0 , d)−Rd of (3.4) blows up. And thereforex(·,x 0 , d) blows up as well: (recall that d is continuous with compact support). Contradiction to the estimate from the third step! It can be shown that for (x 0 , d) ∈X × W 1,2 loc ([0, ∞), R k ), the initial value problem (3.1) still has a (unique) global mild solution, that is, a continuous functionx : [0, ∞) →X satisfying the variation-of-constants formula (3.9). In order to see this -especially the global existence -one can make a density and approximation argument similar to the one from the next section: one first trivially extends Φ t to a bounded linear operator on W 1,2 ([0, ∞), R k ) and then uses the same arguments as in Theorem 3.4. Since we will not need mild solutions in the sequel, however, we omit the details.

Solvability in the generalized sense
In this section, we show that for sufficiently irregular initial statesx 0 and disturbances d, namely for (set of generalized data), the closed-loop equation (3.1) still has a generalized solution existing globally in time and arising as a suitable limit of classical solutions. In showing the existence of such limits we will make use of the integral equationx for classical solutionsx(·,x 0 , d) with (x 0 , d) ∈ D from (3.21) above. In order to extend this equation to arbitrary Proof. In the entire proof, we use the short-hand notation is continuous and monotonically increasing. Such a choice of Lipschitz constants exists because is monotonically increasing and therefore obviously has a continuous monotonically increasing majorant. (See Lemma 2.5 of [4].) As a first step, we show that the restriction Φ t,0 of Φ t to can be (uniquely) extended to a bounded linear operator Φ t,0 : . It follows by Theorem 3.2 (ii) that is a linear operator that is bounded w.r.t. the norm of L 2 ([0, ∞), R k ) and thus, by the density of C 2 (called composition property in [55]), it further follows that As a second step, we will show that also the non-restricted operator Φ t can be (uniquely) extended to a bounded linear operator Φ t : , which in conjunction with (3.24) proves the lemma. In order to do so, we have only to show that and on the other hand where we used (3.28) to get Φ t/n,0 (d n ) −→ 0 as n → ∞. Combining now (3.32) and (3.33) we finally obtain (3.30), as desired.
It should be noticed that the above lemma means nothing but the (finite-time) admissibility of the linear boundary control systemx With the lemma at hand, we can now prove the following approximation result.
for every such approximating sequence (x 0n , d n ) in D, the corresponding sequence (x(·,x 0n , d n )) of classical solutions of (3.1) is a Cauchy sequence in the locally convex space C([0, ∞),X) and its limit is independent of the particular choice of the approximating sequence (x 0n , d n ).
we see that D is a fortiori dense inX × L 2 loc ([0, ∞), R k ). (ii) As in the proof of Lemma 3.3, choose Lipschitz constants L(ρ) off | Bρ(0) such that ρ → L(ρ) is continuous and monotonically increasing. We show that for arbitrary (x 01 , d 1 ), (x 02 , d 2 ) ∈ D and τ ∈ [0, ∞) one has the estimate for every t ∈ [0, τ ]. So, by Grönwall's lemma, the claimed estimate (3.35) follows. And from this, in turn, assertion (ii) immediately follows because As a consequence of the above theorem, for every generalized datum with (x 0n , d n ) being an arbitrary approximating sequence in D for (x 0 , d), yields a well-defined function. We call this function -following [52] -the generalized solution of the closed-loop system (3.1) corresponding to (x 0 , d) because it obviously coincides with the classical solution for (x 0 , d) ∈ D (and with the mild solution for (x 0 , d) ∈X × W 1,2 loc ([0, ∞), R k )) and because, by the next corollary, it shares many important properties with classical solutions. In particular, the generalized solutions of our closed-loop system satisfy the axioms from [29] (Definition 1).
Proof. Assertion (i) and the causality part of assertion (ii) easily follow by approximation from the cocycle property and the causality of classical solutions. Similarly, the continuity part of assertion (ii) follows by extending the estimate (3.35) to generalized solutions.
It can be shown that the closed-loop system, in addition to having a generalized solution, also has a generalized output y(·, and that the generalized output map is continuous in the respective locally convex topology. In particular, the closed-loop system is well-posed in the spirit of [52] -but we will not need this in the sequel. See [46] and [47] for proofs and general well-posedness results.

Stability of the closed-loop system
After having established the global solvability of the closed-loop system, we can now move on to stability. A first very simple result is the following uniform global stability theorem. We recall from [29] that the systemS is called uniformly globally stable iff there exist comparison functions σ, γ ∈ K such that for everyx 0 ∈X and (t ∈ [0, ∞)). Proof. An immediate consequence of Theorem 3.2 (ii) and (3.38).
We are now going to improve this result to a (uniform) input-to-state stability result (Section 4.1) and to a weak input-to-state stability result (Section 4.2), respectively. In particular, we prove that the generalized solutionsx(t,x 0 , d) of the closed-loop system converge to 0 as t → ∞ for every initial statex 0 and every square-integrable disturbance d. In the case of finite-dimensional systems, such a convergence result has been proved in [20] under a strict output passivity and a zero-state detectability assumption. We point out, however, that the theory from [20] cannot be applied in our situation. Indeed, the finite-dimensionality of the systems considered in [20] is essentially used in the proof from [20] (namely to get the relative compactness of orbits needed for the invariance principle).

Input-to-state stability of the closed-loop system
In this section we show that, for systems S of order N = 1 and for a special class of controllers S c , the closed-loop systemS is (uniformly) input-to-state stable. We recall thatS is (uniformly) input-to-state stable w.r.t. inputs from L 2 ([0, ∞), R k ) iff it is uniformly globally stable and of uniform asymptotic gain. See [29] for this and other characterizations of input-to-state stability. In this context, the uniform asymptotic gain property by definition means the following: there is a function γ ∈ K ∪ {0} such that for every ε, r > 0 there is a time τ = τ (ε, r) such that for everyx 0 ∈X with x 0 ≤ r and every d ∈ L 2 ([0, ∞), R k ) A function γ as above is called a uniform (asymptotic) gain (function) forS.
SinceS is uniformly globally stable and of uniform asymptotic gain, there exist σ, γ ∈ K as in (4.1) and a uniform gain function γ ∈ K ∪ {0} along with a time τ 0 = τ 0 (ε, σ( x 0 ) + γ( d 2 )) such that In order to achieve input-to-state stability, we add the following conditions on the system S and the controller S c to the assumptions from the solvability results.  (i) P is quasi-quadratic in the sense that for some constants c 1 , c 1 > 0 (ii) R is quasi-linear in the sense that for some constants c 2 , c 2 > 0  for everyx 0 ∈X and d ∈ L 2 ([0, ∞), R k ).
We now turn to the proof of the theorem and we begin by recording some central ingredients.

Central ingredients of the proof
A first important ingredient is the following estimate for the energy along solutions which essentially uses our assumption thatS be of order N = 1. It is a perturbed version of the respective sideways energy estimate from [42].
Lemma 4.6. Under the assumptions of the above theorem, there exists a function c 0 ∈ L and a t 0 > 0 such that for η = a and η = b one has for every (x 0 , d) ∈ D and for every t ≥ t 0 , where (x, v) :=x(·,x 0 , d) and y := y(·,x 0 , d).
Since F ± τ is absolutely continuous, the differential inequalities (4.11) and (4.12) imply that are monotonically increasing or decreasing, respectively. And from this, in turn, (4.8) follows in a straightforward manner.
As a second step, we show the assertion of the lemma. Choose γ 0 , κ 0 as in (4.9) and let (x 0 , d) ∈ D. In view of the first line of (3.16) we see thatẼx for all t 1 ≤ t 2 . It follows from (4.13) that, for every τ > 2γ 0 (b − a), t |d(s)||y(s)| ds dt
A second important ingredient is the following estimate for the integrated controller energy which shows up in the previous lemma.
Proof. As a first step, we show that E c is equivalent to the auxiliary functions V γ defined by for all sufficiently small γ > 0. In fact, we show that there is a γ 0 > 0 such that for all γ ∈ (0, γ 0 ] and v ∈ R 2mc one has In essence, this is Lemma 16 of [42], but in contrast to that lemma our equivalence constants in (4.18) are independent of γ 0 and our proof is a bit more direct. In order to see (4.18), just observe that by Condition 4.4 one has for all v ∈ R 2mc for C 1 := max{c −1 1 , K −1 }. So, with γ 0 := 1/(2C 1 ) the equivalence relation (4.18) follows. As a second step, we observe that there exists a constant C 0 > 0 and a γ ∈ (0, γ 0 ] such that for every for all t ∈ [0, ∞), where (x, v) :=x(·,x 0 , d) and y := y(·,x 0 , d). Since this follows in the same way as Lemma 17 of [42], we omit the proof. In conjunction with (4.18), the estimate (4.20) proves the lemma.

Conclusion of the proof
With the above lemmas at hand, we can now conclude the proof of our input-to-state stability result in two steps. Since we already know that the closed-loop system is uniformly globally stable (Theorem 4.1), we have only to show that it is of uniform asymptotic gain γ := ψ −1 (2C · 2 ) for every C > 1/(4ς).
As a first step, we show that for every C > 1/(4ς) there exist a constant β ∈ (0, 1) and a time τ ∈ (0, ∞) such that for every (x 0 , d) ∈ DẼ (4.21) So let C > 1/(4ς) and let η be the endpoint of (a, b) for which (4.6) is satisfied. Also, let (x 0 , d) ∈ D and, as usual, write (x, v) :=x(·,x 0 , d) and u := Bx, y := Cx. We know from the first line of (3.16) that for all t ∈ [0, ∞) and ε ∈ (0, 1), where Condition 4.4 and (2.15) have been used. Also, from (4.6) it follows that for all s ∈ [0, ∞). And from Lemma 4.6 and 4.7 it follows that for all t ∈ [t 0 , ∞), where C 0 , t 0 , c 0 are the constants and the function from the above lemmas. Inserting now (4.23) and (4.24) in the last integral on the right-hand side of (4.22), we obtaiñ for all t ∈ [t 0 , ∞). So, by regrouping terms, we see that for all t ∈ [t 0 , ∞) and arbitrary α ∈ (0, ∞). In particular, this holds true for α chosen such that (which choice is possible because C > 1/(4ς)). We choose now τ ∈ [t 0 , ∞) so large that which is possible because c 0 ∈ L. We also choose ε ∈ (0, 1) so small that which last two choices are possible because of (4.26). Combining now (4.25) with (4.28), we conclude that where by virtue of (4.27) the constant β is smaller than 1, as desired: As a second step, we show that for every C > 1/(4ς) and for every constant β and time τ as in the first step one hasẼ for every t ∈ [0, ∞) and every (x 0 , d) ∈ D. So let C > 1/(4ς) and let β and τ be as in the first step. It then follows from (4.21) by the cocycle property and induction that for every n ∈ N 0 and every (x 0 , d) ∈ D. If now t ∈ [0, ∞) is arbitrary, then for a unique n = n t ∈ N 0 and thus it follows by the cocycle property and by (4.32) and (3.17) that every (x 0 , d) ∈ D, from which (4.31) and hence the second step follows. We conclude by density and continuity that for every C > 1/(4ς) there is a constant β ∈ (0, 1) and a time τ ∈ (0, ∞) such that (4.31) also holds for arbitrary data (x 0 , d) ∈X × L 2 ([0, ∞), R k ) and t ∈ [0, ∞). So, by (2.18) for every t ∈ [0, ∞) and every (x 0 , d) ∈X × L 2 ([0, ∞), R k ). In particular,S is of uniform asymptotic gain γ := ψ −1 (2C · 2 ) for every C > 1/(4ς), as desired.

Weak input-to-state stability of the closed-loop system
In this section we show that for systems S of arbitrary order N ∈ N and for a general class of controllers S c , the closed-loop systemS is weakly input-to-state stable. We callS weakly input-to-state stable w.r.t. inputs from L 2 ([0, ∞), R k ) iff it is uniformly globally stable and of weak asymptotic gain. See [45] for other characterizations of weak input-to-state stability. In this context, the weak asymptotic gain property by definition means the following: there is a function γ ∈ K ∪ {0} such that for every ε > 0 and everyx 0 ∈X and d ∈ L 2 ([0, ∞), R k ) there is a time τ = τ (ε,x 0 , d) such that A function γ as above is called a weak (asymptotic) gain (function) forS. We observe that the only difference to the uniform asymptotic gain property is that the time τ is allowed to depend on the initial statex 0 (instead of only on its norm) and on the disturbance d. In [29] the weak asymptotic gain property is called just asymptotic gain property. Also, the weak asymptotic gain property with weak gain 0 is often just termed global attractivity [29], [30].
Proof. Completely analogous to the proof of Lemma 4.2, the only difference being that there the convergence (4.3) was even locally uniform w.r.t.x 0 .
In order to achieve weak input-to-state stability, we add the following conditions on the system S and the controller S c to the assumptions from the solvability results.  (i) R is strictly damping, that is, for some constants c, c, δ > 0, (ii) B c , the input operator of the controller, is injective. is the pointx * = 0. Then the closed-loop systemS is weakly input-to-state stable and the function γ = 0 is a weak asymptotic gain forS. In particular, for everyx 0 ∈X and d ∈ L 2 ([0, ∞), R k ).
We now turn to the proof of the theorem and, for that purpose, it will be most helpful to write the nonlinear part of (3.1) asf whereB : R mc →X,C :X → R mc and g, h : R mc → R mc are such that In particular,B * x = Kv 2 for everyx = (x, v 1 , v 2 ) ∈X. We begin by rewriting the closed-loop equation (3.1) as that is, as a perturbation of the respective linear boundary control system where λ > 0. It follows from (4.39) by the transformation (3.3) and variation of constants that classical solutions of (3.1) satisfy the following integral equation:  for n ∈ N, we have of course that and in view of (4.43) we also have that where we used the notation (x n (t), v 1n (t), v 2n (t)) :=x n (t) := eÃ tx n0 . We will also use the abbreviations u n (t) := Bx n (t) and y n (t) := Cx n (t) in the following. Choose and fix now n ∈ N. We know from (4.44.a) thatx n = eÃ ·x n0 is continuously differentiable withx for all t ∈ [0, ∞). So, by (4.45) it follows that v 1n = v 1n0 is constant and that And from this, in turn, it follows by the injectivity of B c (Condition 4.10 (ii)) that where L c is an arbitrary left-inverse of B c . We also know from (4.44.a) thatx n (t) = eÃ tx n0 ∈ D(Ã) ⊂ kerB for all t and so by (4.45) and (4.48) it follows that Since S is impedance-passive (Condition 2.1) and since x n = x(·, x n0 , u n ) by the uniqueness statement made around (2.10), it further follows using (4.48) and (4.49) that for all t ∈ [0, ∞). Since now S is classically approximately observable in infinite time (Condition 4.9) and since y n = y(·, x n0 , u n ), we conclude by (4.50.a) and (4.50.c) that x n0 = 0. A second important ingredient is the following stabilization result for the collocated linear system S(Ã,B,B * ), which hinges on the approximate observability property just established (Lemma 4.12) and on the compactness of the resolvent ofÃ (Lemma 2.3).
A third important ingredient is the following lemma which says that the linear boundary control system Proof. Choose and fix λ ∈ (0, ∞). We adopt the short-hand notations ρ(x 0 , d) and L g (ρ), L h (ρ) from the proof of Lemma 4.14.
As a first step, we show that the restriction Φ λ t,0 of Φ λ t to can be (uniquely) extended to a bounded linear operator Φ λ t,0 : So let t ∈ [0, ∞). We see by (4.41) that for all d ∈ C 2 c,0 ([0, ∞), R k ). And finally, it follows by integration by parts (which is allowed by Lemma 4.14 (ii)) and by Lemma 4.13 (i) together with (4.59), (4.52) that Combining now (4.63) and (4.64), (4.65), (4.66) we see that is a continuous monotonically increasing function. It follows that Φ λ t,0 : is a linear operator that is bounded w.r.t. the norm of L 2 ([0, ∞), R k ) and thus, by the density of C 2 c,0 ([0, ∞), R k ) in L 2 ([0, ∞), R k ), can be uniquely extended to a bounded linear operator Φ λ t,0 : As a second step, we observe that also the non-restricted operator Φ λ t can be (uniquely) extended to a bounded linear operator Φ λ t : L 2 ([0, ∞), R k ) →X and that Φ λ t = Φ λ t,0 for every t ∈ [0, ∞). In order to do so, we have only to show that . And this, in turn, can be achieved in the same way as in the second step of the proof of Lemma 3.3. (Instead of (3.28) the essential ingredient now is the estimate (4.67), for it again allows us to conclude Φ λ t/n,0 (d n ) −→ 0 as n → ∞ for every sequence (d n ) as in (3.31).) In conjunction with (4.62) the second step proves the lemma. We finally remark that the mere bounded extendability of Φ λ t (without the finiteness condition (4.61)) could alternatively also be concluded from Lemma 3.3 by a perturbation argument (for which the approximate observability of S(Ã,B * ,B) would not be needed). See [48] for a collection of positive and negative perturbation results for (infinite-time) admissibility.

Conclusion of the proof
With the above lemmas at hand, we can now conclude the proof of weak input-to-state stability with weak asymptotic gain γ = 0 in three simple steps. Since we already know that the closed-loop system is uniformly globally stable (Theorem 4.1), we have only to show that for everyx 0 ∈X and d ∈ L 2 ([0, ∞), R k ). So let (x 0 , d) ∈X × L 2 ([0, ∞), R k ). It then follows from (4.41) by a simple approximation argument (using (3.38) and Lemma 4.15) that for all t ∈ [0, ∞) and λ ∈ (0, ∞). We now show in three steps that all four terms on the right-hand side of (4.70) converge to 0 as t → ∞.
As a first step, we observe from Lemma 4.13 (ii) and from Lemma 4.13 (iii) in conjunction with Lemma 4.14 (i) that the first term and the second term (the first integral) on the right-hand side of (4.70) converge to 0 as t → ∞ for every λ ∈ (0, ∞).
As a second step, we observe from Lemma 4.15 that the last term on the right-hand side of (4.70) converges to 0 as t → ∞ for every λ ∈ (0, ∞). Indeed, for d 0 ∈ C 2 c ([0, ∞), R k ) the convergence Φ λ t (d 0 ) −→ 0 as t → ∞ immediately follows from (4.42) by virtue of Lemma 4.13 (ii). And from this, in turn, the desired convergence Φ λ t (d) −→ 0 follows by density and Lemma 4.15. As a third and last step, we show that also the third term (the second integral) on the right-hand side of (4.70) converges to 0 as t → ∞ for every λ ∈ (0, ∞). Choose an arbitrary sequence (t n ) with t n −→ ∞ as n → ∞. Then by Lemma 4.13 (iv) in conjunction with Lemma 4.14 (ii) there exists a subsequence (t n l ) such that for both λ = 1 and λ = 2 one has: (4.71) Combining now (4.70) with the first and second step and with (4.71), we see thatx * = lim l→∞x (t n l ,x 0 , d) exists inX and thatx * = −(Ã − λBB * ) −1B h(Cx * ) (4.72) for both λ = 1 and λ = 2. So,x * ∈ D(Ã) and It follows from this that on the one hand and on the other handÃx * +f (x * ) =Ãx * +Bg(B * x * ) +Bh(Cx * ) =Ãx * +Bh(Cx * ) = 0. In other words, (4.74) and (4.75) say thatx * is an equilibrium point of (4.37) and thusx * = 0 by assumption. So, summarizing we have shown that for every sequence (t n ) with t n −→ ∞ there exists a subsequence (t n l ) such that lim l→∞x (t n l ,x 0 , d) =x * = 0. And from this, in turn, the asserted convergence (4.69) follows (and therefore, in view of the first two steps, also the third step follows).

Some remarks on the assumptions
In this section, we discuss specializations and generalizations of the assumptions from the weak input-to-state stability result (Theorem 4.11). In particular, we clarify their relation to the assumptions from the uniform input-to-state stability result (Theorem 4.5). As the following two lemmas show, the assumption on S from Theorem 4.5 (Condition 4.3) is more or less -apart from some minor extra conditions which are often satisfied in applications -a sufficient condition both for the approximate observability assumption (Condition 4.9) and for the equilibrium point assumption from Theorem 4.11. See the conference paper [43] for versions of the stability theorems above that are simplified accordingly. In our applied examples (Section 5), we will make ample use of this. Lemma 4.16. Suppose that the assumptions (Conditions 2.1, 2.2, 3.1) of the solvability theorems are satisfied and, in addition, that S is even impedance-energy-preserving meaning that (2.8) holds with equality. If then Condition 4.3 is satisfied, then so is Condition 4.9.
Proof. Suppose that Condition 4.3 is satisfied and that x 0 ∈ D(A) is such that y(t, x 0 , 0) = Cx(t, x 0 , 0) = 0 (t ∈ [0, ∞)). Since x(·, x 0 , 0) = e A· x 0 is a classical solution of x = Ax with t → x(t, x 0 , 0) X being monotonically decreasing and since Condition 4.3 is satisfied, there exist positive constants C 0 , t 0 such that for all t ∈ [t 0 , ∞) (Lemma 9.1.2 of [12] or, more precisely, the third step in the proof of Theorem 3.5 of [44]). So, as Bx(·, x 0 , 0) ≡ 0 and Cx(·, x 0 , 0) ≡ 0 by (4.77), we conclude that at least for all t ∈ [t 0 , ∞). Since by the assumed impedance-energy-preservation of S and by (4.77) the energy t → 1/2 x(t, x 0 , 0) 2 X is constant on the whole of [0, ∞), it follows from (4.79) that x 0 = 0 as desired. Lemma 4.17. Suppose that the assumptions (Conditions 2.1, 2.2, 3.1) of the solvability theorems are satisfied and, in addition, that 0 is the only critical point of the potential energy P of S c . If then Condition 4.3 is satisfied, then the only equilibrium pointx * of (4.37) is 0.

Some applications
In this section, we apply the uniform and the weak input-to-state stability results to a vibrating string and a Timoshenko beam.
In these equations, ρ, T are the mass density and the Young modulus of the string and they are assumed to be absolutely continuous and to be bounded below and above by positive finite constants. Also, assume that the string is clamped at its left end, that is, ∂ t w(t, a) = 0 (t ∈ [0, ∞)) (5.2) and that the control input u(t) and observation output y(t) are given respectively by the force and by the velocity at the right end of the string, that is, u(t) = T (b)∂ ζ w(t, b) and y(t) = ∂ t w(t, b) and P 0 := 0 ∈ R 2×2 , the pde (5.1) takes the form (2.1) of a port-Hamiltonian system of order N = 1 and, moreover, the boundary condition (5.2) and the in-and output conditions (5.3) take the desired form (2.6) and (2.2), (2.7) with matrices W B,1 , W B,2 , W C ∈ R 1×4 . It is straightforward to verify that this system S is impedance-energy-preserving, that the matrix W ∈ R 3×4 from (3.2) has full rank, and that H is absolutely continuous and (4.6) holds true. In particular, Conditions 2.1, 3.1, 4.3 are satisfied. So, as soon as the controller S c is chosen such that • Conditions 2.2 and 4.4 are satisfied, or • Conditions 2.2 and 4.10 are satisfied and 0 is the only critical point of P, respectively, the resulting closed-loop systemS is input-to-state stable or weakly input-to-state stable, respectively (Lemma 4.16 and 4.17).
In these equations, ρ, E, I, I r , K are respectively the mass density, the Young modulus, the moment of inertia, the rotatory moment of inertia, and the shear modulus of the beam and they are assumed to be absolutely continuous and to be bounded below and above by positive finite constants. Also, assume that the beam is clamped at its left end, that is, ∂ t w(t, a) = 0 and ∂ t ϕ(t, a) = 0 (t ∈ [0, ∞)) (5.6) (velocity and angular velocity at the left endpoint a are zero), and that the control input u(t) is given by the force and the torsional moment at the right end of the beam and the observation output y(t) is given by the velocity and angular velocity at the right end of the beam, that is, for all t ∈ [0, ∞). and an appropriate choice of P 1 , P 0 ∈ R 4×4 , the pde (5.4) take the form (2.1) of a port-Hamiltonian system of order N = 1 and, moreover, the boundary condition (5.6) and the in-and output conditions (5.7) take the desired form (2.6) and (2.2), (2.7) with matrices W B,1 , W B,2 , W C ∈ R 2×8 . It is straightforward to verify that this system S is impedance-energy-preserving, that the matrix W ∈ R 6×8 from (3.2) has full rank, and that H is absolutely continuous and (4.6) holds true. In particular, Conditions 2.1, 3.1, 4.3 are satisfied. So, as soon as the controller S c is chosen such that respectively, the resulting closed-loop systemS is input-to-state stable or weakly input-to-state stable, respectively (Lemma 4. 16 and 4.17).