Waves and Diffusion on Metric Graphs with General Vertex Conditions

We prove well-posedness for very general linear wave- and diffusion equations on compact or non-compact metric graphs allowing various different conditions in the vertices. More precisely, using the theory of strongly continuous operator semigroups we show that a large class of (not necessarily self-adjoint) second order differential operators with general (possibly non-local) boundary conditions generate cosine families, hence also analytic semigroups, on ${\mathrm{L}}^p({\mathbb{R}_+},{\mathbb{C}}^{\ell})\times{\mathrm{L}}^p([0,1],{\mathbb{C}}^m)$ for $1\le p<+\infty$.


Introduction
It is well-known (for details see [17,Sect. II.6] and [6,Sect. 3.14]) that first and second order abstract Cauchy problems of the form for a linear (in general unbounded) operator G ∶ D(G) ⊂ X → X on a Banach space X are well-posed if and only if G generates a strongly continuous semigroup and a cosine family on X, respectively. It follows by [6,Thm. 3.14.17] that generators of cosine families generate analytic semigroups of angle π 2 . Hence, well-posedness of (ACP 2 ) always implies well-posedness of (ACP 1 ). In this paper we are concerned with such Cauchy problems for second order elliptic differential operators G acting on spaces of L p -functions defined on a finite union of intervals. Operators of this type appear, e.g., in the modeling of diffusion-and wave equations on metric graphs. In this case the intervals represent the edges of the graph while its structure is encoded in the boundary conditions appearing in the domain D(G) of G. In the simplest case we can take X = (L p [0, 1]) m = L p ([0, 1], C m ) and where λ(s) = diag(λ j (s)) m j=1 for positive, Lipschitz continuous "diffusion" coefficients λ j ( • ) and suitable "boundary" matrices V 0 , V 1 ∈ M k 0 ×m (C) and U 0 , U 1 , W 0 , W 1 ∈ M k 1 ×m (C), for k 0 , k 1 ∈ N satisfying k 0 + k 1 = 2m.
Our main result, Theorem 2.3, gives for such operators a condition implying the generation of a cosine family, hence the well-posedness of (1.1). For example, by Corollary 2.11, the operator G in (1.2) generates a cosine family if where µ(s) ∶= λ(s) = diag( λ j (s)) m j=1 . In particular, (1.3) implies that for G as in (1.2) both Cauchy problems in (1.1) are well-posed.
Motivated by different problems from physics, chemistry, biology, and engineering, the study of dynamical processes on metric graphs (also called networks or one-dimensional ramified spaces) has received much attention in the last decades. Diffusion equations on networks were first considered in the 1980s, the earliest references include [35,39,38,41]. Since then many authors used functional analytic methods to treat such problems, we only mention [20,12,5,30,8,36]. The study of wave equation on networks was initiated about at the same time by [3,4], see also [33,31,13,34,14,30,26,25]. Almost simultaneously, another community of theoretical physicists was mainly interested in the Schrödinger equation on a network structure (calling it a quantum graph), see [18,29,28,32,9,40]. They also considered so-called non-compact graphs, where some edges are allowed to be infinite. All these problems were initially treated in a L 2 -setting using Hilbert-spaces techniques. Then interpolation was used to generalize the results to L p -spaces. Typically, in this context only self-adjoint operators are considered.
On the contrary, we use methods form the theory of operator semigroups and work on L pspaces directly. The novelty of our approach is manifold. In fact, it allows us to • study non-self-adjoint generators G, • treat very general (also non-diagonal) "diffusion coefficient matrices" a( • , • ), cf. (2.1), • treat very general boundary conditions of the form for appropriate boundary functionals Φ 0 , Φ 1 and a bounded operator B, cf. (2.3), • consider state spaces X = L p (R + , C ℓ ) × L p ([0, 1], C m ) with application to non-compact graphs, • treat all cases for p ∈ [1, +∞) simultaneously without using interpolation arguments, • explicitly compute the phase space ker(Φ 0 ) × X of G, cf. Theorem 2.3. Our reasoning is based on a recent result for boundary perturbations of domains of generators developed in [1] (which we recall in Theorem A.3) and the fact that squares of group generators generate cosine families, cf. [6,Expl. 3.14.15]. Roughly speaking, we start from a simple firstorder differential operator A generating a semigroup. Then we perturb its domain to obtain G whose square is closely related to G. Moreover, since we arrange G to be similar to −G, it automatically generates a group. Hence, G 2 and consequently also G generate cosine families. To obtain our main theorem in its most general form we use similarity transformations and bounded perturbations. In this way we are able to generalize the boundary conditions for non-self adjoint and non-compact graphs given in [27,24], see Example 2.12, as well as the general boundary conditions in terms of "boundary subspaces" presented in [36,Sect. 6.5], see Subsection 3.6. We can also treat different non-local boundary conditions (for example those studied in [37], see Example 2.10). This paper is organized as follows. In Section 2 we introduce our setup, state and prove the main generation result (Theorem 2.3) and apply it to two important classes of boundary conditions (Corollary 2.11 and Corollary 2. 16). This facilitates the verification of the generation conditions (2.44) and (2.56) used in Section 3 to show well-posedness of diffusion-and wave equations on (possibly non-compact) graphs for a wide variety of boundary conditions. In the appendix we recall a perturbation result from [1] which is the main tool for our approach. Our notation closely follows [17].
2. Generation of cosine families 2.1. The setup. Throughout this section we make the following assumptions. Although the results presented here are abstract, the terminology already suggests that our main motivation arises from the study of dynamical processes on (possibly non-compact) metric graphs. In the sequel we use the notation R + ∶= [0, +∞).
As we will see in Theorem 2.3 below this can be achieved (independently on B) through an invertibility condition on the operator R t 0 defined in (2.12) below. We note that we do not consider the case p = ∞. In fact, this would yield a non-densely defined operator G which cannot be a generator. More generally, it is well-known that on L ∞ -spaces strongly continuous semigroups are uniformly continuous, i.e., have a bounded generator. Hence, an operator G ⊂ a( • , • ) ⋅ d 2 ds 2 will never, independently on the domain, In order to state our main result rigorously we need some more notations.
Next for fixed t 0 > 0 and t ∈ [0, t 0 ] we introduce the bounded linear operators whereû denotes the extension of a function u defined on I ⊂ R to R by the value 0. Observe for µ e (•) and µ i (•) given in (2.4) and (2.6), respectively. Now we are ready to introduce the operator R t 0 as follows. Note that is well-defined and has a unique bounded extension denoted again by The operator R t 0 plays a crucial role in our main result, see Theorem 2.3. As we will see later, in many important cases of boundary conditions involving just the boundary values, cf. Subsection 2.3, the operator R t 0 reduces to a matrix. Before starting the proof, we note that in this section we equip all subspaces Z ⊆ C n with the maximum norm, i.e., we define Hence, it suffices to show that the operator is well-defined and has a bounded extension in In both cases the assumption u(0) = 0 implies that the functionû(t − ϑ( • )) ∈ W 1,p (I, C k ) has compact support and hence Φû(t − ϑ( • )) is well-defined for all t ∈ [0, t 0 ]. Moreover, sincê u (−∞,t 0 ] is uniformly continuous, the map 2 [0, t 0 ] ∋ t ↦û(t − ϑ( • )) ∈ C 0 (I, C k ) is continuous and therefore [0, t 0 ] ∋ t ↦ Φû(t − ϑ( • )) ∈ Y is continuous as well. Summing up, this shows that the operator U t 0 is well-defined.
Next we verify that U t 0 is bounded. Since Φ ∈ L(C 0 (I, C k ), Y ) and Y is finite dimensional, by the Riesz-Markov representation theorem there exists a function η ∶ I → L(C k , Y ) of bounded variation such that Φ is given by the Riemann-Stieltjes integral (2.13) Then by Hölder's inequality and Fubini's theorem we conclude for u ∈ W 1,p 0 ([0, t 0 ], C k ) that where η ∶ I → R + denotes the positive Borel measure defined by the total variation of η and this implies that U t 0 has a unique bounded extension as claimed.
2.2. The main result. We are now ready to state our main generation result.
The proof is split into four parts where in the first three we assume B = 0. We start by showing the result under the hypothesis that the operator matrix G in (2.14) below generates a semigroup. Then, using a series of lemmas we give the proof that G indeed is a generator, first in case q( • , • ) ≡ diag(Id, Id), then for general q( • , • ). Finally, we prove the result for B ≠ 0.
Proof of Theorem 2.3, 1 st part. Assume that B = 0 and q( • , • ) ≡ diag(Id, Id). Hence, a( • , • ) = λ( • , • ) and c( • , • ) = µ( • , • ) are diagonal matrices. By λ ′ we denote the derivative of the corresponding diagonal entries, i.e., On X ∶= X × X we consider the operator matrix where . Then G and −G are similar via the similarity transformation induced by diag(Id, −Id). Hence, if for the time being we assume that G generates a C 0 -semigroup, by [17,Sect. II.3.11] it already generates a group. By [6,Expl. 3.14.15] this implies that G 2 generates a cosine family However, G 2 is given by the diagonal matrix with diagonal domain ) induces a bounded multiplication operator on X and therefore P ∶= λ ′ 2 ⋅ d ds ∈ L(V, X). However, by Corollary A.8, D(G) = D(G) and G =G − P , hence by [6, Cor. 3.14.13] it follows that G generates a cosine family with phase space V × X as claimed.
Next we verify the generator property of the matrix G. To do so we proceed in several steps. First we assume again that c( The case of general q( • , • ) as in (2.1) then follows by similarity and bounded perturbation. We start by simplifying G by rearranging the coordinates of X and by normalizing the matrices µ e ( • ), µ i ( • ). Recall thatC is defined in (2.8) and J ϕ e ∈ L(X e ), Jφi ∈ L(X i ) are given by (2.9).
Proof. Consider the invertible transformation We claim thatG

Using this a simple computation shows that
which completes the proof of (2.18).
We now representG as a domain perturbation of a simpler generator A which can be treated by a (slight modification of a) recent perturbation result from [1] (see Theorem A.3). Thanks to Lemma 2.4 we can consider the external and internal part separately.
External Part. We introduce on X e = L p (R + , C ℓ ) the operators and define on X e = X e × X e the operator matrices Note that D e m and −D e 0 generate the strongly continuous left-and right-shift semigroups (T e l (t)) t≥0 and (T e r (t)) t≥0 on X e , respectively, given by wheref denotes the extension of the function f ∶ R + → C ℓ to R by the value 0. This gives immediately the following result.
where δ 0 denotes the point evaluation in s = 0 and ∂X e ∶= C ℓ . Now the following follows easily by inspection.
Lemma 2.6. Let the operators A e m and L e be defined by (2.21) and (2.25), respectively. Then for t 0 > 0 and given is a classical solution of the boundary control system and define on X i = X i × X i the operator matrices

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Then D i 1 and −D i 0 generate the strongly continuous nilpotent left-and right-shift semigroups (T i l (t)) t≥0 and (T i r (t)) t≥0 on X i , respectively, given by . This gives immediately the following result.
As before we observe that in the context of Subsection A.1 we have is a classical solution of the boundary control system (2.37) We are now well-prepared to continue the proof of our main result.
by the right-handside of (2.36) and put , ∂X),X) is strongly continuous. Moreover, by Lemma 2.6 and Lemma 2.8, hence the assertion follows from Lemma A.6. (ii) Using the terminology introduced in Remark A.4, we have to show that Φ in (2.17) is p-admissible for the semigroup (T(t)) t≥0 generated by A. By the representations of T(t) in (2.39) this follows if we verify the p-admissibility of every • Φ ∈ L(C 0 (I, C k ), Y ) for I = R + and k = ℓ with respect to the semigroups (T e l (t)) t≥0 and (T e r (t)) t≥0 given in (2.23), and Then similarly as in the proof of Lemma 2.2 it follows that where η is given by (2.13) and M ∶= c −1 C k . This completes the proof of (ii). (iii)&(iv) By (i), Lemma 2.6, Lemma 2.8 and Lemma A.6 the controllability maps for the problem (2.40) are given by where the last equality is obtained by direct computation using definitions of Φ in (2.17) and R t 0 in (2.12). This combined with Lemma 2.2 implies (iii) and also (iv) follows immediately from the invertibility assumption on R t 0 .
Proof of Theorem 2.3, 3 rd part. Assume that a( • , • ), c( • , • ) are given by (2.1) and (2.11), respectively, where q( • , • ), q( • , • ) −1 are Lipschitz continuous and bounded. Then via the similarity transformation induced by diag(q( • , • ), q( • , • )) we obtain that the operator matrix (2.14), is similar to the operator matrix Since q and q −1 are Lipschitz continuous and bounded, we have P ∈ L(X × X). Moreover, note that Hence, by similarity and bounded perturbation G is a generator iffĜ is. However, by what we proved previously for q( We conclude the proof by considering non-zero boundary operators B ∈ L(X).
Proof of Theorem 2.3, 4 th part. It remains to prove the result for B ≠ 0 satisfying the regularity condition (2.41) To this end we putB ∶= c( • , • ) ⋅ B ∈ L(X) and perturb the matrix G in (2.14) by Then, by Part 3 and the bounded perturbation theorem, G B ∶= G + B generates a group. Now a simple computation using Corollary A.8 shows that X). Moreover, since B ∈ L(X), the regularity property (2.41) combined with Corollary A.8 and the closed graph theorem implyB ∈ L(W 1,p (R + , C ℓ ) × W 1,p ([0, 1], C m )). Hence, Q ∶= c ⋅ d ds ⋅B ∈ L(V, X) and since G =G B − P − Q, the claim follows from [6, Cor. 3.14.13] as in Part 1. (ii) Note that by [6, Cor. 3.14.13] the sum G + P of the generator G of a cosine family with phase space V ×X and a perturbation P ∈ L(V, X) still generates a cosine family with the same phase space. Here in the context of Theorem 2. X). Thus, boundedness and invertibility of R t 0 in (2.12) imply that for arbitrary b( generates a cosine family with the same phase space. (iii) By [6, Thm. 3.14.17] every generator of a cosine family generates an analytic semigroup of angle π 2 . Hence, the previous remark gives also conditions implying that G + P generates an analytic semigroup of angle π 2 . (iv) It is quite remarkable that G in (2.14) might generate a C 0 -semigroup even if none of its entries D Φ 0 and DΦ 1 are generators. For example for ℓ = 0, m = 1 and Y 0 = C ℓ+2m = C 2 , Feller [19] has characterized the boundary conditions in the domain of the generator of the transition semigroup corresponding to one-dimensional diffusion processes. Besides Dirichlet and Neumann boundary conditions (which we discuss in Example 2.13), these include also non-local integral conditions which we discuss next. Note that this also generalizes the wellposednes results in [37]. Example 2.10. We consider a diffusion operator G ⊆ d 2 ds 2 on L p [0, 1] with non-local boundary conditions. More precisely, for h 0 , h 1 ∈ L q [0, 1] where 1 p + 1 q = 1, we define the domain In our setting this corresponds to ℓ = 0, m = 1, the diffusion coefficient a( • ) ≡ 1, the state space X = L p [0, 1], the boundary spaces Y 1 = {0}, Y 0 = C 2 , and the boundary functionals This implies Jφi = Id,c = 1, q( • ) ≡ 1, and for the operators R t , S t defined in (2.10) we obtain for u ∈ L p [0, t 0 ], 0 < t 0 < 1, Moreover, a simple computation shows that Hence, the operator R t 0 in (2.12) is given by where by Young's inequality each convolution operator This implies that R t 0 is invertible for t 0 ∈ (0, 1] sufficiently small and by Theorem 2.3 we conclude that G generates a cosine family on X. We close this section by considering two very common and important classes of boundary conditions. The first one uses a set of "boundary matrices" (2.42) to impose the values in the end points, the second one uses two "boundary spaces" Y 0 , Y 1 instead. As we will see, in both cases our main assumption in Theorem 2.3, the invertibility of the map R t 0 given by (2.12), reduces to a condition which can be easily verified. More precisely, in the first case we obtain the determinant condition (2.44), in the second one the direct sum condition (2.56).
2.3. Boundary conditions via "boundary matrices". For k 0 , k 1 ∈ N 0 satisfying k 0 + k 1 = ℓ + 2m we choose matrices for µ e (•) and µ i (•) given in (2.4) and (2.6), respectively. Next we will use the matrices V e 0 , V i 0 , V i 1 to specify k 0 conditions containing only values at the endpoints, while the matrices W e 0 , W i 0 , W i 1 will determine k 1 (linear independent) conditions regarding derivatives at the endpoints.
Using all this we compute the operator R t 0 given in (2.12) as Since the matrix diag(q e (0), q i (1), q i (0)) ∈ M ℓ+2m (C) is always invertible, the assertion follows from Theorem 2.3.
We give some possible choices for the operators B e , B i appearing in Corollary 2.11. (2.43) gives the mixed boundary condition In particular, this covers the boundary conditions considered in [27,24].
(ii) For arbitrary operators T e ∈ L(L p (R + , C ℓ ), C k 1 ) and Note that by choosing operators T e , T i properly (e.g. as an integral) we thus obtain various non-local boundary conditions.
(iii) We can also combine the two examples above and obtain the second condition in (2.43) of the form Example 2.13. We consider the second derivative G p and G D with periodic-and Dirichlet boundary conditions, respectively, on In order to write these boundary conditions as in (2.43) we choose ℓ = 0 4 and m = 1. Moreover, in case of G p we take k 0 = k 1 = 1 and scalars V 0 = 1, V 1 = −1, W 0 = W 1 = 1. Then the determinant condition (2.44) is fulfilled, hence G p generates a cosine family. To handle G D one might be tempted to choose again k 0 = k 1 = 1. Then the first boundary condition f (0) = 0 can be implemented by choosing V 0 = 1, V 1 = 0 while the second condition f (1) = 0 follows from (2.51) if we take W 0 = W 1 = U 0 = 0 and U 1 = 1. However, by doing so (2.44) is not fulfilled nevertheless it is well-known that G D generates a cosine family. At a first glance, Corollary 2.11 fails in this simple example, thus only gives a sufficient but in general not necessary generation criterion. However, as pointed out earlier, the matrices W 0 , W 1 should be used to implement k 1 linear independent conditions regarding the derivatives at the endpoints. In case of G D this means that we have to choose k 0 = 2, k 1 = 0 and the two boundary matrices V 0 = 1 0 , V 1 = 0 1 . For this choice (2.44) is fulfilled, yielding the desired generation result. We leave it to the reader to check that also problems with Neumann-or mixed boundary conditions on an interval can be handled in the same way.   (1). Hence, by the previous result G in (2.53) generates a cosine family if a(0) ≠ a(1). We note that in case a(0) = a(1) the operator G never generates a cosine family or even an analytic semigroup. To prove this assertion, we denote the operator obtained by (2.53) for a( • ) ≡ 1 by G 1 . Then for each λ ∈ C we have e λ ∈ ker(λ − G 1 ) where e λ (s) ∶= e Hence, σ(G 1 ) = C implying that G 1 cannot be a generator. For Lipschitz continuous, positive a( • ) ∈ C[0, 1] one can use the similarity transformation induced by Jφi ∈ L(X), Jφif ∶= f ○φ i to show that in case a(0) = a(1) the operators G and G 1 are similar up to the perturbation P ∶= a ′ 2 ⋅ d ds . Since P is relatively bounded with bound 0, this implies the claim. Summing up, in this example Corollary 2.11 gives an optimal result which demonstrates the sharpness of the underlying perturbation argument from Subsection A.1.
We continue with an example on a simple star-shaped non-compact metric graph, cf. Figure 1. Further applications to general metric graphs are presented in Section 3. Example 2.15. We consider a diffusion process described by G ⊂ d 2 ds 2 along the edges of the noncompact star graph presented in Figure 1. The two compact edges e i 1 , e i 2 are parametrized as [0, 1], with 0 in the common vertex, while e e 1 = R + . In the central vertex we assume continuity and an additional boundary condition for the derivatives, i.e., for some α, β, γ ∈ C, while at the remaining endpoints we set the following Neumann-and Robin condition, respectively, that is Then ℓ = 1, m = 2 and if ε ≠ 0 we choose k 0 = 2, k 1 = 3 and boundary matrices Then the determinant in (2.44) equals ε ⋅ (α + β + γ). In case ε = 0 the boundary conditions are essentially different and in order to apply Corollary 2.11 one has to take k 0 = 3, k 1 = 2.
2.4. Boundary conditions via "boundary spaces". We consider another way to impose conditions at the end points using two "boundary spaces" Y 0 , Y 1 ⊂ C ℓ+2m and two operators B e ∈ L(X e , L p (R + , C ℓ+2m )), B i ∈ L(X i , L p ([0, 1], C ℓ+2m )) satisfying Then for f = f e f i ∈ W 2,p (R + , C ℓ ) × W 2,p ([0, 1], C m ) we consider the boundary conditions Applying Theorem 2.3 to this setting yields the following.

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generates a cosine family on X = L p (R + , Proof. Using that for I = [0, 1] or R + we have L p (I, C ℓ+2m ) = L p (I, C ℓ )×L p (I, C m )×L p (I, C m ) we decompose B e and B i accordingly, i.e., we write . Finally, we consider the projection P ∈ L(C ℓ+2m ) associated to the representation (2.56), that is ker(P ) = Then a simple computation using (2.49) and yields that R t 0 in (2.12) is constant and given by Hence, R t 0 is invertible and Theorem 2.3 implies the claim.
We give two possible choices for the operators B e , B i appearing in the boundary condition (2.55).
This generalizes for example the boundary conditions considered in [36,Sect. 6.5], see also Subsection 3.6.
(ii) For operators T e ∈ L(L p (R + , C ℓ ), C ℓ+2m ) and Then again B e , B i satisfy (2.54) and for f = f e f i ∈ W 2,p (R + , C ℓ ) × W 2,p ([0, 1], C m ) the second boundary condition in (2.55) simplifies to 3. Applications to waves and diffusion on metric graphs 3.1. Introduction. In this section we use our abstract results to show the well-posedness of wave-and diffusion equations on networks. That is, we study first and second order abstract initial-boundary value problems of the form (1.1). The structure of the graph is encoded in the boundary conditions contained in the domain D(G). We consider a finite metric graph (network) with n vertices v 1 , . . . , v n , m internal edges e i 1 , . . . , e i m , which we parametrize on the unit interval [0, 1], and ℓ external edges e e 1 , . . . , e e ℓ , parametrized on the half-line R + . A graph without external edges is called compact. The structure of the graph is given by the n × m internal incidence matrices Φ i,− ∶= (ϕ i,− rs ), and By using the incidence matrices we obtain the diagonal matrices with in-and out-degrees of all vertices on the diagonal as The diffusion-and wave equation on a metric graph is defined by considering on each edge the heat equation or the wave equation respectively, for some Lipschitz continuous functions λ e k ( • ) ∈ C(R + ), λ i j ( • ) ∈ C[0, 1] for k = 1, . . . , ℓ, j = 1, . . . , m, satisfying (2.2). Additionally, one needs to impose some transmission conditions in the vertices. These types of problems for compact graphs were studied for example in [30]. Next we present some types of these transmission conditions and show how our results apply in these examples.

3.2.
Standard conditions. The most natural assumption for the solutions to either heat or wave equations on a metric graph is continuity in the vertices. We say that a function f = f e f i ∈ C(R + , C ℓ ) × C([0, 1], C m ), defined on the edges of the graph, is continuous on the graph if its values at the endpoints of the contiguous edges coincide, i.e., whenever two edges e j and e k (both internal, both external or mixed) have a common vertex v then for the appropriate functions it holds A direct computation shows that the continuity property of f can be easily expressed using the incidence matrices as which is equivalent to Furthermore, in each of the vertices v r , r = 1, . . . , n, we infer the standard Kirchhoff (also called Neumann) conditions where Γ(v r ) denotes the set of all edges incident to the vertex v r and ∂f j ∂s j (v r ) is the normal derivative of f j computed at the appropriate endpoint of the edge e j . Using incidence matrices we can express this condition more accurately as
3.3. δ-type conditions. This condition appears in the literature on quantum graphs, see [9]. It consists of the continuity condition (3.5) and the condition e j ∈Γ(vr) in every vertex v r , r = 1, . . . , n. Here f (v r ) denotes the common value of the functions f j corresponding to the edges e j ∈ Γ(v r ) that meet in vertex v r , and α r are some fixed complex coefficients. Again we can rewrite this using incidence matrices as . . . , c n ) ⊺ is the vector appearing in the continuity condition (3.5).
In order to obtain the appropriate matrix form first note that by (3.5), (3.1) and (3.2) we have Let L ∶= diag(α r ) n r=1 ∈ M n (C). Since every column of an incidence matrix consists of exactly one nonzero entry corresponding to the appropriate endpoint of an edge, there are m × m and ℓ × ℓ diagonal matricesD † , such that Hence we can rewrite δ-type conditions in the matrix form as (3.10) Defining Y 0 and Y 1 as in Subsection 3.2 and the operators 3.4. Non-local boundary conditions. We now further generalize the standard boundary conditions, taking the continuity condition (3.6) together with the condition for some matrices M e ∈ M ℓ (C) and M i,− , M i,+ ∈ M m (C). Note that in this way the Kirchhoff conditions in a vertex are supplemented with a linear combination of values in some other -even non-adjacent -vertices. This models, for example, a network, in which some nodes are able to communicate instantly and directly via another network, atop of the one under consideration. To treat this case we may again define Y 0 and Y 1 as in Subsection 3.2, take the boundary operators and apply Corollary 2.16.
3.5. Matrix mixed conditions. Motivated by applications in population dynamics, in [7,8] a diffusion problem on a compact network with the general boundary condition for a matrix K ∈ M 2m (C) is considered. In this case Corollary 2.16 applies directly by choosing 3.6. Generalized node conditions. In [36,Sect. 6.5] the boundary condition appears for compact graphs, where Y ⊆ C 2m and W ∈ L(Y ). We show that also this condition fits in the setting of Corollary 2.16 for ℓ = 0. To this end we define . Then a simple computation shows that for these choices (3.12) is equivalent to (2.55). Next, the representation Y 0 ∶= C Y 1 for positive definite C implies by the same reasoning as at the end of Subsection 3.2 condition (2.56). Hence, Corollary 2.16 applies to the operator G = a( • ) ⋅ d 2 ds 2 satisfying the boundary conditions (3.12). Moreover, this condition can be easily generalized to the non-compact metric graphs.
Appendix A.
A.1. Domain perturbation for generators of C 0 -semigroups. In this appendix we briefly recall a perturbation result from [1,Sect. 4.3] which is our main tool to prove Theorem 2.3 (similar see also [22,23]). Moreover, we give an admissibility criterion which significantly simplifies the computation of the so-called controllability-and input-output maps. To explain the general setup we consider • two Banach spaces X and ∂X, called "state" and "boundary" spaces, respectively; • a closed, densely defined "maximal" operator 5  Hence, one can consider G with boundary condition Lf = Cf as a perturbation of the operator A with abstract "Dirichlet type" boundary condition Lf = 0. In order to proceed we make the following Assumption A.1.
5 "maximal" concerns the size of the domain, e.g., a differential operator without boundary conditions.
(i) The operator A generates a C 0 -semigroup (T (t)) t≥0 on X; (ii) the boundary operator L ∶ D(A m ) → ∂X is surjective.
Under these assumptions the following lemma is shown in [21,Lem. 1.2].
In what follows, the extrapolated space X −1 associated with A is the completion of X with respect to the norm x −1 ∶= R(λ 0 , A)x , x ∈ X, for some fixed λ 0 ∈ ρ(A), T −1 (t) ∈ L(X −1 ) is the unique bounded extension of the operator T (t) to X −1 , and A −1 is the generator of the extrapolated semigroup (T −1 (t)) t≥0 with domain D(A −1 ) = X. For more details on extrapolated spaces and semigroups we refer to [17, Sect. II.5.a]. Now one can verify that the operator L A ∶= (λ − A −1 )L λ ∈ L(∂X, X −1 ) is independent of λ ∈ ρ(A) and that G = (A −1 + L A ⋅ C) X . Before stating the perturbation result [1,Cor. 22], we note that from the assumptions (i)-(iii) in Theorem A.3 below it follows that there exists a bounded "input-output map" F t 0 ∈ L(L p ([0, t 0 ], ∂X)) such that (F t 0 u)( • ) = C Then G ⊂ A m , D(G) ∶= ker(Φ), generates a C 0 -semigroup on the Banach space X.
Proof. Since the assumptions (i) are the same, it suffices to show that the hypotheses (ii)-(iv) imply the corresponding assumptions in Theorem A.3.
(ii) This is clear since LT (s)x = 0 for all x ∈ D(A) = ker(L) and s ≥ 0.
(iii) Using integration by parts twice, one sees that for all v ∈ W 2,p 0 ([0, t 0 ], ∂X) which implies (iii) in the previous result.
(iv) By the previous point it also follows that Id − F t 0 = Q t 0 which implies the corresponding assumption in Theorem A.3.
In [16] we showed two versions of variation of parameters formula for the solutions to boundary control problems. By using them we obtain the following equivalence which is quite helpful to verify the first assumption in the previous two results. In this case for t ∈ (0, t 0 ] the operator B t coincides with the "controllability map", i.e., For more details and examples regarding the above perturbation results we refer to [1,2,10].
A.2. Two auxiliary results. We state and prove two results concerning the inverse and derivative of matrix-valued functions.
Lemma A.7. Let I ⊆ R be an interval and d( • ) ∶ I → M n (C) be Lipschitz continuous and bounded, such that σ(d(s)) ⊂ (0, +∞) for all s ∈ I. If I is not compact assume in addition that there exists ε > 0 such that σ(d(s)) ⊆ [ε, 1 ε ] for all s ∈ I. Then d −1 ( • ) ∶ I → M n (C) is Lipschitz continuous and bounded as well.