Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

We study a class of non-autonomous boundary control and observation linear systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by different fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.

This partial differential equation is also known as port-Hamiltonian systems, and covers the wave equation, the transport equation, beam equations, coupled beam and wave equations as well as certain networks. Autonomous port-Hamiltonian systems, that is when H, P k are time-independent, have been intensively investigated, see e.g., [15,16,3,2,17,22,36,41]. The existence of mild/classical solutions with nonincreasing energy and well-posedness for autonomous port-Hamiltonian systems can in most cases be tested via a simple matrix condition [22,Theorem 4.1]. Well-posedness of linear systems in general is not easy to prove and a necessary condition is that the state operator generates a strongly continuous semigroup. For the class of autonomous port-Hamiltonian systems of first order i.e., N = 1, this condition is even sufficient under some weak assumptions on P 1 H, see [22] or [17,Theorem 13.2.2].
In this paper, we aim to generalize these solvability and well-posedness results to the non-autonomous situation. To our knowledge, in contrast to infinite-dimensional autonomous port-Hamiltonian systems, the non-autonomous counterpart has not been discussed so far. Motivated by this class we start a systematic study of non-autonomous linear boundary control and observation systems, and in particular those of the following formẋ  Evolution families are a generalization of strongly continuous semigroups, and are often used to describe the solution of an abstract non-autonomous Cauchy problem. In Section 2, we therefore review the concept of evolution families and that of C 1 -well posed non-autonomous Cauchy problems. Furthermore, we provide several abstract results which are crucial for the analysis of our non-autonomous boundary control and observation systems.
Fattorini's trick is well known for autonomous boundary control systems [17,12,9]. The basic idea of this approach is to reformulate the state and the control equation into an abstract inhomogeneous Cauchy problem on X. A brief description of the autonomous situation is given in Subsection 3.1. In Subsection 4.1 we provide a generalization to non-autonomous boundary control systems (see Proposition 4.2). This generalization and the results of Section 2 are then used to prove our main classical solvability results: Theorem 4.8 and Theorem 6.5. The second main purpose of this paper is the study of the well-posedness for non-autonomous boundary and observation systems Σ N,M (A, B, C). However, we will restrict ourselves to the case where for every t ≥ 0 the (unperturbed) autonomous system Σ N,id (A(t), B, C) is (R(t), P (t), J(t))-scattering passive i.e., when for all x in an appropriate subspace of X × U where P (t), R(t) and J(t) are bounded linear operators. A precise definition and a characterization of scattering passive autonomous and non-autonomous systems is the subject of Subsection 3.2 and Subsection 4.2, respectively. Under additional conditions we then prove in Theorem 4.8 that the perturbed system Σ N,M (A, B, C) is well-posed. In particular, we deduce in Theorem 6.5 that well-posedness for a large class of non-autonomous port-Hamiltonian systems can be checked via a simples matrix condition.
In the literature most attention has been devoted to autonomous control systems. However, in view of applications, the interest in non-autonomous systems has been rapidly growing in recent years, see e.g., [13,26,7,29,19,6,18,28] and the references therein. In particular, a class of scattering passive linear non-autonomous linear systems of the forṁ (5) has been studied by R. Schnaubelt and G. Weiss in [29]. Here (A, D(A)) generates a strongly continuous semigroup on X, A −1 ∈ L(X, X −1 ) is a bounded extension of (A, D(A)), B ∈ L(U, X −1 ), C ∈ L(Z, Y ) and D ∈ L(U, Y ), where X −1 is the extrapolation space corresponding to A, and Z : The control part (1)-(2) of the nonautonomous boundary control system Σ N,M (A, B, C) can be rewritten in the (standard) abstract formulation (4), however, in the particular case where A(t) = A is constant which for non-autonomous port-Hamiltonian system correspond to the case where the matrices P k , k = 1, · · · , N, are constant with respect to time variable. On the other hand, when A(t) = A the output part (3) could be also written into (5) using the concept of system nodes. Indeed, well-posed autonomous port-Hamiltonian system fit into the framework of compatible system nodes [37,Theorem 10]. This can be also easily generalized for boundary control and observation systems defined in Definition 3.2. Since we do not follow the approach of [29], this topic will not be discussed in this paper and we refer to [31,34] for more details on system nodes.
For the general case, that is when A is not constant, then A −1 , B, C, D and Z will be time dependent. Thus, the abstract results in [29] cannot be immediately applied to deduce classical solvability and wellposedness for (1)- (3). We expect that the results in [29] can be generalized to include this general case. However, for the class of boundary control systems defined in Definition 3.2 we deal directly with (1)- (2) in combination with Fattorini's trick instead of its corresponding system (4)-(5). Our method is indeed much simpler. Moreover, in general it is not clear how the solution of (4)- (5) can be related to that of (1)-(3) even for the special case where A(t) = A is constant. In the autonomous case this relationship is quite simple as we can see in Section 3. The reason is that C 0 -semigroups can be always extended to the extrapolation space. The situation is more delicate for the non-autonomous setting. Indeed, a general extrapolation theory for evolution families is still missing. Moreover, the extrapolation space may also depend on the time variable. In Section 5 we deal with this question by associating a mild solution to the control part (1)-(2) of the nonautonomous boundary control system Σ N,M (A, B, C). Finally, we apply our abstract results to non-autonomous port-Hamiltonian systems, in particular to the time-dependent vibrating string and the time-dependent Timoschenko beam.

Background on evolution families and preliminary results
Throughout this section (X, · ) is a Banach space. Let A := {A(t) | t ≥ 0} be a family of linear, closed operators with domains {D(A(t)) | t ≥ 0}. Consider the non-autonomous Cauchy problem ∞), X) and u satisfies (6).
For each s ≥ 0 and x s ∈ Y s the Cauchy problem (6) has a unique classical solution u(·, s, x s ) with u(t, s, x s ) ∈ Y t for all t ≥ s. (c) The solutions depend continuously on the initial data s, x s .
If we want to specify the regularity subspaces Y t , t ≥ 0, we also say (6) is C 1 -well posed on Y t .
In the autonomous case, i.e., if A(t) = A is constant, then it is well known that the associated Cauchy problem is well-posed if and only if A generates a C 0 -semigroup (T (t)) t≥0 . In this case, for each x ∈ D(A) the unique classical solution is given by T (·)x. The following definition provides a natural generalization of operator semigroups for non-autonomous evolution equations.

Definition 2.2. A family
The evolution family U is said to be generated by A, if there is a family {Y t | t ≥ 0} of dense subspaces of X with Y t ⊂ D(A(t)) and (iii) For every x s ∈ Y s , the function t → U (t, s)x s is the unique classical solution of (6).
The Cauchy problem (6) is then C 1 -well posed if and only if A(t), t ≥ 0, generates a unique evolution family, see [10,Proposition 9.3] or [24,Proposition 3.10]. Clearly, if (T (t)) t≥0 is a C 0 -semigroup in X with generator (A, D(A)), then U (t, s) := T (t − s) yields an evolution family on X with regularity spaces Y t = D(A).
It is well known that if S is a C 0 -semigroup on X with generator A and Q ∈ L(X) is an isomorphism, then T (·) := Q −1 S(·)Q is again a C 0 -semigroup on X, called similar C 0 -semigroup to S, and its generator is given by The purpose of this section is to generalize the concept of similar semigroups to evolution families.

Lemma 2.3.
The two parameters family W, defined by (7), defines an evolution family on X.
Proof. Clearly, the evolution law (i) in Definition 2.2 is fulfilled. It remains to prove the strong continuity of W in ∆. Let x ∈ X and T > 0, and set ∆ T := {(t, s) ∈ [0, T ] 2 | t ≥ s}. Let (t, s), (t n , s n ) ∈ ∆ T for n ∈ N such that (t n , s n ) → (t, s). Then {Q −1 (t n ) | n ∈ N} is bounded by the uniform boundedness theorem. Since we deduce that (t, s) → Q −1 (t)U (t, s)x is continuous on ∆ T . Thus, using a similar argument for Q(s) and Q −1 (t)U (t, s) we obtain that (t, s) → W (t, s)x is continuous on ∆ T . Since T > 0 is arbitrary, this proves the assertion.
In contrast to semigroups, the evolution law (i) and the strong continuity (ii) do not guarantee that the given evolution family is generated by some family of linear closed operators.

Proposition 2.4. Assume that Q(·) is in addition strongly
Proof. (i) Assume that U is generated by A with regularity spaces {Y t | t ≥ 0}. We first remark thatỸ t is a dense subspace of X and for every t ≥ 0, where A Q (t) := Q −1 (t)A(t)Q(t) − Q −1 (t)Q(t). Next, let x s ∈Ỹ s . Then Q(s)x s ∈ Y s and by assumption U (·, s)Q(s)x s is the unique classical solution of It follows that W (t, s)x ∈ Y t ⊂ D(A Q (t)) by (8) and Since Q is strongly C 1 -differentiable, it now follows from (10)-(11) that W (·, s)x s ∈ C 1 ((s, ∞), X) and W (·, s)x s solves the non-autonomous problem (12) Clearly, W (·, s)x s is the unique classical solution of (12). We conclude that W is generated by (ii) Conversely, assume that A Q generates the evolution family W with some regularity spaces {Ỹ t | t ≥ 0}. Since Q −1 is C 1 -strongly continuous we obtain by (i) that the family ( This completes the proof. If A : D(A) ⊂ X → X is the generator of a C 0 -semigroup and B ∈ L(X), then the perturbed operator A := A + B is again the generator of a C 0 -semigroup, see e.g., [ and A(t) generates a contractive C 0 -semigroup on X t := (X, · t ) for all t ≥ 0. (2) A family A is said to belong to Kato's class if it is Kato-stable and the operators A(t), t ≥ 0, have a common domain D such that A(·) : [0, ∞) → L(D, X) is strongly C 1 -differentiable.
It is known that Kato-stability is a sufficient condition for C 1 -well posedness of (hyperbolic) nonautonomous evolution equations [21,35,32]. In particular, each non-autonomous evolution equation that is governed by a Kato-class family is C 1 -well posed.
Obviously, A is Kato-stable if each operator A(t) generates a contractive C 0 -semigroup, as one can simply choose · t = · , t ≥ 0. In this case we say that A belongs to the elementary Kato class. Starting from this simple case many less trivial Kato-stable families can be constructed.  (14) x

Proposition 2.7. Let A belong to the Kato-class and let D denote the common domain of
has a unique classical solution given by Proof. It is not difficult to verify that (13) implies that has a unique classical solution x given by (19) u On the other hand, arguing as in the proof of Proposition 2.4 we see that x := Q −1 (·)u is a classical solution of (15). The uniqueness of classical solutions of (15) follows from the uniqueness of classical solutions of (17). Finally, (16) follows from (19) and (7).
Using Example 2.6 and Proposition 2.7 one can formulate the following two corollaries.
has a unique classical solution given by (16).
Proof. For the proof we just have to apply Proposition 2.7 for has a unique classical solution given by (16).
Proof. From Example 2.6 we deduce that the family 2.2. Backward evolution families. Let X be a Hilbert space over K = C or R.
x t solves uniquely the backward non-autonomous problemu Lemma 2.12.
(1) Assume that A = {A(t) | t ≥ 0} belongs to the elementary Kato-class. Then A generates a backward evolution family. (2) Assume that A generates an evolution family U. If the adjoint operators generate a backward evolution family Then, obviously A T belongs to the Kato-class and thus generates an evolution family It is easy to see that S(t, s) := U T (T − s, T − t) for each 0 ≤ s ≤ t ≤ T defines a backward evolution family with generator {A(t) | t ∈ [0, T ]}. This completes the proof since T is arbitrary. (ii) Denote by Y t and Y t, * , t ≥ 0 the regularity spaces corresponding to A and A * , repectively. Let t > s ≥ 0 and let x s ∈ Y s and y t ∈ Y t, * . Then for s ≥ r ≥ t we have Integrating over [s, t] and using that Y s and Y t, * are dense in X yield the desired identity.

Review on Autonomous boundary control and observation systems
Many systems governed by linear partial differential equations with inhomogeneous boundary conditions are described by an abstract boundary system of the forṁ , and X, U and Y are complex Hilbert spaces. We shall call X the state space, U the input space and Y the output space of the system. In this section, we recall some well-known results on well-posedness of these system which are needed throughout this paper. (28)-(30), if x is a classical solution of (28)- (29), y ∈ C([0, ∞); Y ) and y satisfies (30).
that for all classical solution of (28)-(30) we have is a classical solution of (28)- (30) if and only if x is a classical solution of (28)-(29).
3.1. Existence of classical solutions. In order to study existence of classical solutions it is often useful to write the boundary control system (28) generates a strongly continuous semigroup on X. (ii) There exists a linear operatorB ∈ L(U, X) such that for all u ∈ U we havẽ In the following Σ(A, B, C) is assumed to be a BCO-system. The following remark will be very useful for non-autonomous boundary control systems.

This is an easily consequence of Definition 3.2.
We denote by X −1 the extrapolation space associated to A, i.e., the completion of X with respect to the norm x → (βI − A) −1 x for some arbitrary β ∈ ρ(A). Let A −1 be the extension of A to X −1 . It is well known that A −1 with domain X generates a C 0 -semigroup (T −1 (t)) t≥0 on X −1 and for all t ≥ 0 the operator T −1 (t) is the unique continuous extension of T (t) to X −1 . We associate with Σ(A, B, C) the linear operator B ∈ L(U, X −1 ) called control operator defined by It turns out, that for sufficiently smooth initial data and inputs the two Cauchy problemṡ and the BCO-system (28)-(30) are equivalent. More precisely, we have . Then (28)-(29) has a unique classical solution x given by Therefore, x is the unique classical solution of (34) and w := x −ũ is the unique classical solution of (33) with initial value w 0 = x 0 −Bu(0).
Proof. The proof follows from a combination of [ Σ(A, B, C) be a BCO-system on (X, U, Y ) and let P = P * ∈ L(X), R = R * ∈ L(U ) and J = J * ∈ L(Y ). The admissible space V ⊂ X × U is defined by Then the BCO-system Σ(A, B, C) is (P, R, J)-energy preserving if and only if equality holds in (39), or equivalently in (40).
Proof. Obviously, the inequalities (39) and (40)  Then (x(t), u(t)) ∈ V and d dt for all t ≥ 0. Inserting this into (38) yields for all t ≥ 0. The previous inequality implies (39) by taking t = 0. The converse implication and the last assertion can be proved similarly.

Non-autonomous boundary and observation systems
In this section, our aim is to extend the results of Section 3 to the more general case where A, B, and C are time dependent. Let X, U and Y be Hilbert spaces over K = C or R. For each t ≥ 0 we consider the linear operators We consider the following abstract non-autonomous boundary systeṁ which we denote by Σ N (A, B, C).  We also consider the time-dependent admissible spaces V(t), t ≥ 0, i.e, Since {A(t) | t ≥ 0} generates an evolution family U on X, for a given f ∈ L 1 Loc ([0, ∞); X) the inhomogeneous non-autonomous Cauchy probleṁ

. Assume now that x is a classical solution of (42)-(43). Then v(t) ∈ V t ⊂ D(A(t)) for every t ≥ s by Remark 3.3 anḋ
Thus v is a classical solution of (46) with f given by (48). The converse implication can be proved similarly. Finally, (49) follows by the above the remark.
Proof. The equivalence of (ii) and (iii) has been proved in Proposition 3.6. It remains to prove the equivalence of (i) and (ii). Assume that (i) holds and let s ≥ 0 and let (x s , u s ) ∈ V(s). Let u : [s, ∞) → U such that u(s) = u s . If (x, y) is a classical solution of (42)-(44) corresponding to (x s , u) then (x(t), u(t)) ∈ V(t), y(t) = C(t)x(t) and for all t ≥ s. Inserting this into (50) yields for all t ≥ s. The last inequality (ii) by taking t = s. Conversely, assume that (ii) holds and let (x, y) be a classical solution of (42)-(44). Then (x(t), u(t)) ∈ V(t) and (52)-(53) holds for all t ≥ s. This together with (51) imply (50), which completes the proof.
for all t ≥ s ≥ 0. Thus using (54) and that J ≥ 0 we obtain which implies (55). This completes the proof.

Multiplicative perturbed of NBCO-systems.
We will adopt the same notations of the previous sections. The main purpose of this section is the study of some classes of NBCO-systems which are governed by a time-dependent multiplicative perturbation. More precisely, let Σ N (A, B, C) be a NBCOsystem such that the boundary operators are constant, that is C(t) = C and B(t) = B for all t ≥ 0. Thus the domain A(t) should also be constant and we set D(A(t)) = D for all t ≥ 0. Further, throughout this section we assume that the following assumption holds: Assumption 4.6.

The system Σ N,M (A, B, C) is well-posed if in addition
Proof. The first and the second assertion follow from Proposition 4.2 and Corollary 2.8, whereas the last assertion follows from Lemma 4.7, Proposition 4.4 and Lemma 4.5.
Next we consider the case where A(t) = L(t)A with L(t) is as in Assumption 4.6 and such that (A, B, C) is an autonomous BCO-system. This implies that (L(t)A, B, C) is again an autonomous BCO-system for each t ≥ 0 such that the associated operatorB is time-independent. In fact, ifB denotes the operator associated with the autonomous BCO-system Σ(A, B, C), then it is easy to see thatB satisfies all properties listed in Definition 3.2-(ii) corresponding to (L(t)A, B, C). We consider the following perturbed systemẋ   a unique classical solution (x, y) given by

Mild solutions for NBC-systems
As mentioned in Section 3, for an autonomous BCO-system Σ(A, B, C), for smooth input u and initial data x 0 , the classical solution of the corresponding boundary control system can be formulated as We recall that B ∈ L(U, X −1 ) is given by (32). If x s ∈ X and u ∈ L 2 ([0, ∞); U ), then the above formula makes sense and it is called the mild solution in X −1 of (28)- (30). Moreover, it is well known that the mild solution belongs to C([0, ∞); X) if B is admissible for the semigroup (T (t)) t≥0 , i.e., if for some τ > 0 one has see, e.g., [33,Proposition 4.2.4].
The main purpose of this section is to extend the conceps of mild solutions to non-autonomous boundary control and observation systems Σ N (A, B, C). In contrast to the autonomous case, this is more delicate.
In fact, firstly we remark that the extrapolation spaces X −1,t associated with the family {A(t) | t ≥ 0} of the main operators are in general time-dependent. Secondly, in contrast to semigroups, it is not clear whether the evolution family U generated by {A(t) | t ≥ 0} can be extended to the extrapolation space even if the spaces X −1,t are constant. However, if the latter condition holds, then we can still use the adjoint problem, i.e, A * (t), t ≥ 0, and the associated backward evolution family to extend U to L(X −1 ). The idea to use a duality argument can be found in [7,25,29] to study some classes of non-autonomous systems.
We will adopt here the notations of the previous sections. Let Σ N (A, B, C) be a NBCO-system. Then the main operators {A(t) | t ≥ 0} generate, by definition, an evolution family U = {U (t, s) | (s, t) ∈ ∆} with regularity space Y t , t ≥ 0. We restrict ourselves to case where {A(t) | t ≥ 0} have a common extrapolation space X −1 , i.e., In the following we denote D * := D(A * (0)) equipped with the graph norm and by ·, · the duality between X −1 and D * . Recall from (32) generates a backward evolution family U * . Then U (t, s) has a unique extension U −1 (t, s) ∈ L(X −1 ) for each (t, s) ∈ ∆ and for each T > 0 there is c T > 0 such that Moreover, if the assumptions of Proposition 4.2 hold, then each classical solution x of the boundary control system (42)-(43) satisfies Proof. By [33, Proposition 2.9.3-(b)] we obtain that for each (t, s) ∈ ∆ the operator U (t, s) has a unique extension U −1 (t, s) ∈ L(X −1 ) since [U (t, s)] * D * = U * (t, s)D * ⊂ D * . Next, similar to the proof of [29, Proposition 2.7-(c)] we show the uniform boundedness of U −1 on compact intervals. Next, we claim that for each y ∈ D * , x ∈ X −1 we have In fact, this equality holds for x ∈ X by Lemma 2.12-(ii) since x, U * (t, s)y = (x | U * (t, s)y) = (U (t, s)x | y) = U −1 (t, s)x, y .
Remark that U * (t, s)y ∈ D * , thus the claim follows since X is dense in X −1 .
Using again Lemma 2.12 and (71), we obtain for each y ∈ D * d ds Integrating over [s, t], we obtain Inserting this equality in (49), we obtain that a classical solution x of (42)-(43) satisfies (70).
If the assumptions of Proposition 5.1 hold, then for x s ∈ X and u ∈ L 2 Loc ([0, ∞); U ) we see that (70) is well defined with value in X −1 provided B(·)u(·) ∈ L 1 Loc ([0, ∞); X −1 ). In fact, (69) guaranties that the integral term on the right hand side of (73) is well defined. Thus the following definition makes sense. N (A, B, C) be a NBCO-system and let U and {B(t) | t ≥ 0} the associated evolution family and control operators, respectively. Let (x s , u) ∈ X × L 2 Loc ([0, ∞); U ). If U (t, s) has a unique extension U −1 (t, s) ∈ L(X −1 ) for each (t, s) ∈ ∆ such that U −1 (t, ·)B(·)u(·) ∈ L 1 Loc ([0, ∞); X −1 ), then the function Finally, if Assumption 4.6 holds such that A(t) generates contractive C 0 -semigroup on X for each t ≥ 0 then we can follow [29, Section 2, page 8] to deduce that the extrapolation spaces corresponding to A(t)M (t), t ≥ 0 can be all identified with X −1 and that [A(t)M (t)] −1 = A −1 (t)M (t) for every t ≥ 0. Proof. The proof is an easy consequence of Corollary 5.4 and Lemma 4.7.

Application to non-autonomous Port-Hamiltonian systems
Let N, n ∈ N be fixed and let X := L 2 ([a, b]; K n ) where K = R or C. In this section we investigate the well-posedness of the linear non-autonomous port-Hamiltonian systems of order N ∈ N, given by the boundary control and observation system Here τ denotes the trace operator τ : H N ((a, b) . In this section we assume the following assumptions: Assumption 6.1.
• W B has full rank and W B (t)ΣW * B (t) ≥ 0 for all t ≥ 0. • P N (t) is invertible and P * k (t) = (−1) k−1 P k (t) for all k ≥ 1, t ≥ 0, • P k ∈ C 1 ([0, ∞); L ∞ (a, b; C n×n )) for all t ≥ 0 and k = 0, 1, · · · N . Under these assumptions, the port-Hamiltonian system (76)-(80) can be written as a non-autonomous boundary control and observation system in the sense of Definition 4.1-(ii). In fact, on the Hilbert space X we consider the (maximal) port-Hamiltonian operators Then (A(t), D(A(t))) is a closed and densely defined operator and its graph norm · D(A(t)) is equivalent to the Sobolev norm · H N ((a,b);K n ) as P N (t) is invertible. Moreover, for each t ≥ 0 the operator generates a contractive C 0 -semigroup on X. Further, we define the input operator B and output operator C a follows B : H N ((a, b); K n ) → U, Bx := W B,1 τ (x), and C : H N ((a, b) The operator C is a linear and bounded operator from D(A(t)) to Y , since the trace operator τ is bounded and the norm graph norm of D(A) is equivalent to the H N ((a, b); K n )-norm. Moreover, Lemma 6.2 below shows that there exists an operatorB ∈ L(U, X) which is independent of t ≥ 0 satisfying the assumption (ii) of Definition 3.2. The proof of this fact follows by a minor modification of the proof of [17,Theorem 11.3.2] and that of [2, Lemma 3.2.19] (see also the second step of the proof of [22,Theorem 4.2]). Proof. Since the nN × 2nN -matrix W B has full rank nN there exists a 2nN × nN -matrix S such that In fact, one can choose S as follows Next, let {e j } 2N n j=1 be the standard orthogonal basis in K 2nN . For each j = 1, 2, · · · 2nN we take f j ∈ H N (a, b; K n ) such that τ (f j ) = e j [37,Lemma A.3], and we define the operatorB ∈ L(K m , X) by H N ((a, b); K n )). Furthermore, (84) implies that W B,2S1 = 0 and thus for every u ∈ K m . We deduce thatBK m ⊂ D(A(t)) for all t ≥ 0. It follows that Σ(A(t), B, C) is for each t ≥ 0 a BCO-system on (L 2 ([a, b]; K n ), K m , K d ). Using (84) once more, we obtain that Moreover, if in addition the following assumption holds Proof. Using [2, Lemma 3.2.13] we obtain holds for every x ∈ H N ([a, b]; K n ), since ReP 0 (t, ζ) ≤ 0. Now the claim follows by Lemma 3.6.
Finally, the assumption on H ensures that the family of operators M (t) := H(t)(·) := H(t, ·) as matrix multiplication operators on L 2 (a, b; K n ) satisfies all assumptions of Section 4.3.
Our abstract results in the previous sections hence yield the following main result. Theorem 6.5. If Assumption 6.1 holds, then the port-Hamiltonian system (76)-(80) is a non-autonomous boundary control and observation system. Furthermore, there exists a unique evolution family W in and R ∈ L ∞ Loc ([0, ∞); L(K n )). Finally, we give a result on the existence of mild solution of the non-autonomous port-Hamiltonian system. For that we assume that nN = m = d. Then it is known [22, Lemma A1] (see also [17,Section 7.3]) that there exist a matrix V ∈ K nN ×nN and an invertible matrix S ∈ K nN ×nN such that . For each t ≥ 0, the adjoint operator A * (t) : D(A * (t)) → X of (82)-(83) is given by τ (x) = 0 (91) see e.g., [36,Theorem 2.24], [2,Proposition 3.4.3]. We deduce that the domain of A * (t) are timeindependent if for instance all matrices P k , k = 1, 2, · · · N are constant. Thus using Corollary 5.5 we obtain the followin proposition. Proposition 6.6. Assume that Assumption 6.1 and Assumption 6.3 hold with P k , k = 1, 2, · · · N are constant and J is uniformly coercive and R ∈ L ∞ Loc ([0, ∞); L(K n )). If (86) holds, then the non-autonomous system (76)-(79) has a unique mild solution.
We closed this section by some examples of physical systems which can be modelled as a non-autonomous port-Hamiltonian system. Then the existence of classical and mild solutions as well as well-posedness can be checked by a simple application of the abstracts results presented in this section. Here we will present just two relevant examples, however various other control systems fit into the framework of port-Hamiltonian system and into the general class of NBCO-systems.