pp. X–XX COMPOSITIONS OF PASSIVE BOUNDARY CONTROL SYSTEMS

We show under mild assumptions that a composition of internally 
 well-posed, impedance passive (or conservative) boundary control 
 systems through Kirchhoff type connections is also an internally 
 well-posed, impedance passive (resp., conservative) boundary 
 control system. The proof is based on results of Malinen and 
 Staffans [21]. We also present an example of such 
 composition involving Webster's equation on a Y-shaped graph.

1. Introduction. We treat the solvability (forward in time) of dynamical boundary control systems that are composed by interconnecting a finite number of more simple boundary control subsystems that are already known to be solvable forward in time. The interconnections are given in terms of algebraic equations involving the boundary control/observation operators of the subsystems. The aggregate formed by the subsystems and their interconnections is called a transmission graph (see Definition 3.1), and it can be seen as a generalisation of mathematical transmission lines and networks. We assume throughout this work that all the subsystems are passive or conservative as described in, e.g., [9], [17], [20], [21], [24], [25], and [26], and they are represented by equations of the form (5) below involving strong boundary nodes. Moreover, the interconnections respect passivity in the sense that they do not create energy. In Theorem 3.3 -the main result of this paper -we give conditions for checking the solvability (i.e., internal well-posedness) and passivity of the transmission graph in terms of simple conditions on the subsystems and interconnections.
To illuminate the purpose of this paper, let us consider the following example from acoustic wave propagation. Given the interconnection graph in Fig. 1, the longitudinal wave propagation on its edges (i.e., wave guides) is governed by x ∈ [0, l j ], and t ∈ R + .
We remark that in acoustics applications the state ψ (j) is chosen to be a velocity potential; then p (j) = ρ ∂ψ (j) ∂t gives the perturbation pressure and v (j) = − ∂ψ (j) ∂x gives the perturbation velocity for each edge. Thus, the first equation in (2) says that the pressure is continuous, and the second equation is a flux conservation law (the weights A j can be understood as the cross-sectional areas of the wave guides). We want to control the pressure at the vertex AC and observe the perturbation flux to the wave guides A and C. Defining the input and output respectively, then equations (1) for j = A, ..., D and (2) define a dynamical system whose solvability and energy conservation we wish to verify using Theorem 3.3.
We must consider first the solvability of the subsystems, that is, equations (1) on the edges with boundary conditions After reducing (1) to a first order equation of formż = Lz with z = ψ (j) p (j) , defining operator G by (4), that is, by Gz(t) = u (j) (t), and K in a similar manner, we obtain an internally well-posed boundary node Ξ (j) = (G, L, K) that is impedance conservative, see Definitions 2.2 and 2.3. As explained after Definition 2.2, the initial value problem has a solution such that ψ (j) in equation (1) satisfies ψ (j) ∈ C 1 (R + , L 2 (0, l j )) ∩ C(R + , H 1 [0, l j ]) for all inputs u (j) ∈ C 2 (R + , C 2 ) and for all initial states z 0 that satisfy the boundary condition Gz 0 = u(0). For technical details, see the (more general) example of Webster's equation presented in Section 5. Now we have boundary nodes Ξ (j) , j = A, ..., D and coupling conditions of the form (2) for all vertices except the one that defines the external input and output through (3). They form a transmission graph as defined in Definition 3.1. Since the components Ξ (j) are solvable and conservative, then by Theorem 3.3, also the resulting composed system is solvable forward in time and conservative in a similar way as any of its components.
Let us review the most relevant literature on compositions of (boundary control) systems. The feedback theory for (regular) well-posed linear systems is treated in [26,Chapter 7] and [28] whose concept of admissibility of the feedback loops is related to the (internal) well-posedness of the composed system, but the theory can be used only when well-posedness of the components is verified by other means.
Transport equation on graphs is studied in [7] by using semigroup techniques. For a study of the non-linear Saint-Venant equations on a star-shaped graph, see [6]. A control algorithm for a string network is developed in [12] using a domain decomposition method. Further practical examples of compositions of PDEs with 1D spatial domains include semiconductor strips and lattice structures constructed of Timoshenko beams. Such systems have also been studied from the spectral point of view: asymptotic spectral properties of the Laplacian are studied in [13] and [23] when its "graph-like" 3D spatial domain collapses to a graph with 1D edges. See also [16] for a study on the spectral properties of the Sturm-Liouville equation on a Y-shaped graph.
Compositions of PDEs on 1D spatial domains are treated in [27] and [30] in terms of port-Hamiltonian framework. Compositions of more general systems are studied in, e.g., [3] and [15] who treat systems that give raise to Dirac structures on their state spaces (see also [5]). These contain impedance conservative, strong boundary control systems (as characterised in Definitions 2.2 and 2.3) as a special case. However, our approach is based on results of [20,21] that are reviewed in Section 2, and we are able to treat couplings of both passive and conservative systems at once.
In Section 5 we present a concrete example of a transmission graph, namely the human vocal tract with nasal cavity, modelled by Webster's equation on a Y-shaped graph. For more concrete examples, we refer to [19].

2.
Background. In this work we treat linear boundary control systems described by operator differential equations of the form (5) involving linear mappings G, L, and K: Definition 2.1. Let Ξ := (G, L, K) be a triple of linear mappings.
(i) Ξ is a colligation on the Hilbert spaces (U , X , Y) if G, L, and K have the same domain Z = dom(Ξ) ⊂ X and values in U , X , and Y, respectively; with domain Z, and L is closed with dom(L) = Z. We call L the interior operator, G the input (boundary) operator, and K the output (boundary) operator. The space Z we call the solution space, X the state space, and U and Y the input and output spaces, respectively. In Z we use the graph norm In this paper we use the notations · · · and to represent orthogonal direct sum of (sub)spaces. See also Remark 3 for a discussion on the terms input and output.
The definition of strongness coincides with [21,Definition 4.4]. By [21,Lemma 4.5], Ξ is strong if and only if L is closed with dom(Ξ) and G and K are bounded with respect to the graph norm of L on dom(Ξ). We shall later make use of this fact. Many dynamical systems defined by boundary controlled partial differential equations naturally adopt the form (5) associated with some colligation (G, L, K) on properly chosen spaces (U , X , Y), see the example in Section 5. Equations (5) are solvable forward in time (at least) if Ξ satisfies somewhat stronger assumptions: but we do not consider such nodes in this paper 1 . We remark that also [8], [9], and [15] treat strong colligations (with different names), see [21,Theorem 5.2] and [15,Remark 4.4]. Note that "boundary node" does not refer to the vertices of the underlying graph structure. In fact, boundary nodes are related to the edges of the graph. Therefore, we always talk about vertices when referring to the graph structure.
If Ξ = (G, L, K) is an internally well-posed boundary node, then (5) has a unique solution for sufficiently smooth input functions u and initial states z 0 compatible with u(0). More precisely, as shown in [20,Lemma 2.6], for all z 0 ∈ Z and u ∈ C 2 (R + ; U ) with Gz 0 = u(0) the first, second, and fourth of the equations in (5) have a unique solution z ∈ C 1 (R + ; X )∩C(R + ; Z), and hence we can define y ∈ C(R + ; Y) by the third equation in (5). In the rest of this article, when we say "a smooth solution of (5) on R + " we mean a solution with the above properties.
In a practical application, checking the solvability of (5), that is, verifying the conditions of Definition 2.2 may be difficult. However, in many cases this is not necessary because the system satisfies energy (in)equalities that can be verified using the Green-Lagrange inequality without an a priori knowledge of the well-posedness. Such energy laws make it easier to check the solvability, see Proposition 1 below. First we shall define impedance passivity/conservativity. To keep the notation simple, we assume that U = Y even though it would be enough to assume that U and Y are a dual pair of Hilbert spaces with duality pairing ·, · (Y,U ) ; see [21, Definition 3.6] and the discussion preceding it. (i) Ξ is impedance passive if the following conditions hold: (ii) Impedance passive Ξ is impedance conservative if (6) holds as an equality, and (a) holds also for some α, β ∈ C − . Impedance passivity/conservativity is defined in [21, Definition 3.2] using the external Cayley transform of scattering passivity/conservativity (see also the discussion there). These definitions are equivalent by [21,Theorem 3.4]. We further remark that [21,Theorem 3.4] also states that for an impedance passive Ξ, condition (a) holds for all α, β ∈ C + , and for an impedance conservative Ξ, condition (a) holds also for all α, β ∈ C − . Suppose now that Ξ is an internally well-posed, impedance passive boundary node and z a smooth solution of (5). Then (6) means plainly the energy inequality d dt where the right hand side stands for the instantaneous power entering the system, and the norm of X measures the energy stored in the state.
The following proposition utilising the energy balance laws is needed for checking internal well-posedness and impedance passivity/conservativity. Proposition 1. Let Ξ = (G, L, K) be a strong colligation on Hilbert spaces (U , X , U ).
(i) Suppose that (6) holds for all z ∈ dom(Ξ), and that G α−L is surjective for some α ∈ C with Re(α) ≥ 0. Then Ξ is an internally well-posed, impedance passive boundary node. If, in addition, (6) holds as an equality and G α−L is surjective also for some Re(α) ≤ 0, then the internally well-posed boundary node Ξ is impedance conservative.

then it is an internally well-posed boundary node if and only if its input operator G is surjective.
For a proof, see [21, Theorem 4.3 and Remark 4.6] for part (i) and [21,Theorem 4.7] for part (ii).
Internally well-posed boundary nodes can always be written in terms of more general and complicated system nodes (see [20], [22], and [26]) but they are excluded from state linear systems studied in [4]. A functional analytic setting of boundary control systems, that is independent of the system node setting, was formulated in [8] and significant progress was made in [24] and [25]. See also [10] for a similar presentation.

Transmission graphs as colligations. Assume that we have colligations Ξ
That is, the Hilbert spaces U (j) and Y (j) are represented by an orthogonal direct sum of k j subspaces each, and the corresponding input and output operators are split accordingly.
In order to define the topological structure of the transmission graph, we define control vertices I 1 , ..., I N (where N = 0) and closed vertices J 1 , ..., J M as pairwise disjoint sets of index pairs (j, i) where j refers to the subsystem Ξ (j) and i ∈ {1, ..., k j } refers to the i th component in the splitting (7). We assume that every pair (j, i) for j = 1, ..., m; i = 1, ..., k j belongs to some vertex. This is not a restriction since uncoupled input/output pairs can be included as singleton vertices, as in our example in Section 5.
Each vertex defines a coupling between the subsystems in such a way that all inputs u (j) i whose index pairs (j, i) belong to the same vertex are equal, and the corresponding outputs are summed up. In addition, for closed vertices we require that the outputs sum up to zero. For such coupling to be possible, it is required that the compatibility conditions hold for all (j, i), (p, q) ∈ I k , k = 1, ..., N and for all (j, i), (p, q) ∈ J l , l = 1, ..., M . The couplings are written in terms of input and output operators as follows: (i) for all control and closed vertices, the continuity equations hold, i.e., (9) holds for all (j, i), (p, q) ∈ I k , k = 1, ..., N and for all (j, i), (p, q) ∈ J l , l = 1, ..., M ; and (ii) for closed vertices, also the balance equations hold. Control vertices are exactly those couplings where external signals are applied. If the transfer function (see [20,Section 2]) of each Ξ (j) represents electrical admittance, then the physical dimensions of U (j) and Y (j) are the voltage and current, respectively. Equations (9) and (10) are the classical Kirchhoff laws, namely, the continuity of voltage and the conservation of charge.
Definition 3.1. Assume that Ξ (j) are colligations with splittings as described above in (7). Suppose that sets I 1 , ..., I N and J 1 , ..., J M are defined consistently with this splitting so that the compatibility conditions (8) hold. The ordered triple is called a transmission graph.
A transmission graph is a notion that contains the building blocks and the "assembly instructions" of the composition. Together with coupling conditions (9) and (10), it gives rise to a dynamical system as follows: otherwise.
In order to make the preceding definitions more intuitive, let us return to the example on the wave equation on the graph of Fig. 1, presented in the introduction. We have four boundary nodes Ξ (j) , j = A, ..., D whose input and output spaces are split into two parts, see equation (4). In the graph, there is one control vertex  (9) and (10) and the input and output operators given in Definition 3.2 yield the input/output of equation (3).
The main result of this paper is the following: Then the colligation of Γ is an impedance passive (respectively, conservative), internally well-posed, strong boundary node. This is proved in three steps (Lemmas 4.1, 4.2, and 4.3) presented in the following section. The assumption (11) can be relaxed (see Remark 1) but this condition appears to hold in many applications (as in our example in Section 5).
4. Proof of Theorem 3.3. Suppose we are given a transmission graph Γ. We reconstruct this graph by a finite number of three different kinds of steps, starting from its components Ξ (j) . In step 1, we form a partial parallel connection between two compatible colligations to obtain a new colligation, see

4.1.
Step 1: partial parallel coupling. Assume that we have two colligations Yc with solution spaces The domain of the colligation is Such partial parallel coupling is illustrated in Fig. 2a. We now show that such coupling of two boundary nodes is also a boundary node and the coupling preserves internal well-posedness and passivity/conservativity. Proof. We start by showing that Ξ (AB) is a strong colligation. First, we show that Ξ (AB) is closed. Assume that dom(Ξ (AB) ) ∋ Since ∈ dom(Ξ (AB) ). By a similar computation we can verify    Closedness of L (AB) with domain dom(L (AB) ) = dom(Ξ (AB) ) is shown similarly. Thus, Ξ (AB) is strong colligation. Note that in the preceding computation, we did not need G . We proceed to show that Ξ (AB) is an internally well-posed, impedance passive boundary node with the help of Proposition 1. Surjectivity of (with domain dom(Ξ (AB) )) for some α ∈ C with Re α ≥ 0 follows from the fact that are surjective for the same α. All that is left is to show that the Green-Lagrange identity (6) holds: where the last equation follows from G The conservativity is verified by repeating the latter part of the proof with −α in place of α and replacing the inequality in Green-Lagrange identity by equality.

4.2.
Step 2: loop coupling. Now assume that we have a colligation Ξ = (G, L, K) on the Hilbert spaces . The block diagram of such coupling is shown in Fig. 2b. As in step 1, we show that if the original colligation Ξ is an internally well-posed, impedance passive (conservative), strong boundary node, then Ξ is one as well.
Lemma 4.2. Let Ξ and Ξ be as defined above. If the colligation Ξ is an internally well-posed, impedance passive (conservative), strong boundary node such that [ G K ] is surjective, then also Ξ is an internally well-posed, impedance passive (resp., conservative), strong boundary node with the property that G K is surjective.
Proof. Strongness of Ξ is shown as before in Lemma 4.1.
Surjectivity of G α− L for some α ∈ C with Re α ≥ 0 is easy to see, and also Green-Lagrange identity holds in dom( Ξ): where the second equality follows from G 2 z = G 3 z and the last from the definitions of G and K. Surjectivity of G K follows from surjectivity of [ G K ]. If Ξ is conservative, then to show conservativity of Ξ, just repeat the proof with −α in place of α and replace the inequality in the Green-Lagrange identity with equality.

4.3.
Step 3: closing the vertices. In this step, we single out some vertices as control/observation vertices and permanently "close" all others with respect to additional external signals. Note that after steps 1 and 2, under the assumptions of Lemmas 4.1 and 4.2, the resulting colligation is an internally well-posed boundary node, such that (9) is satisfied. This closing means that we require also (10) to be satisfied, and we now show that this can be done without sacrificing the internal well-posedness or passivity/conservativity.  is surjective, then also Ξ is an internally well-posed, impedance passive (resp., conservative), strong boundary node.
Proof. We carry out a partial flow inversion and interchange the roles of G 2 and K 2 . More precisely, we shall prove that Ξ := G, L, K on Hilbert spaces U1 Y2 , X , Y1

U2
where G := G1 K2 , K := K1 G2 , and dom( Ξ) := dom(Ξ), is an internally well-posed, impedance passive (conservative), strong boundary node. Colligation Ξ is then obtained from Ξ by restricting the solution space to N (K 2 ), and it clearly has all the properties as claimed, see Definition 2.2 and the comment after Definition 2.1 concerning the strongness of Ξ.
It is trivial that Ξ is a strong colligation. One way to see the interchangeability of G 2 and K 2 is directly by Definition 2.3 with β = 1: The surjectivity of this operator follows from impedance passivity of Ξ. Similarly for the conservative system we also need the operator to be surjective which holds by the conservativity of Ξ, see Definition 2.3 with β = −1. The Green-Lagrange (in)equality trivially holds, and it follows that Ξ is an impedance passive (conservative), strong colligation. Finally, by Proposition 1, the surjectivity of G1 K2 implies that Ξ is an internally well-posed boundary node. Remark 1. Assumption (11) is actually stronger than what was needed in Theorem 3.3. Indeed, it was only used in the last lines of the proof of Lemma 4.3. However, the minimal sufficient conditions are impossible to formulate in terms of the control/observation operators of the subsystems. Instead, we would have to consider the whole composed system. The requirement is that the control operator of the composed system has to remain surjective despite the couplings in the closed vertices.
Remark 2. The partial parallel coupling could be constructed by first forming a cross product of systems Ξ (A) and Ξ (B) , see [26,Example 2.3.10]. It is easy to see that the cross product preserves all the desired properties of the colligations. The partial parallel coupling can then be formed by applying a loop coupling to the product system. This means that Lemma 4.1 actually follows from Lemma 4.2.
Remark 3. Using the words input and output for Gz and Kz is somewhat misleading. In fact, since our coupling equations (9) and (10) include conditions involving both Gz and Kz, we have to assume that also the flow-inverted system is solvable; that is, solvable if G and K are interchanged. For many systems this is not a serious restriction and, in fact, the whole concept of abstract boundary spaces (introduced in [9]) is based on the existence of such interchangeable pair of possible boundary conditions. See also [5] for a study of compositions of systems using such abstract boundary spaces and [14] for an introduction of state/signal systems that are based on equal treatment of inputs and outputs. . Thus, as described in Section 4.1, we obtain a system with Again, the colligation Ξ is induced by a transmission graph In block operators G and K, before closing any vertices, each column corresponds to one system (an edge of the graph) and each row corresponds to a coupling (a vertex of the graph). Thus, in phase 2, the block operators G (AB) , K (AB) , G (CD) , and K (CD) have three rows and two columns. In phase 3, G and K have five rows and four columns. And finally, when connecting vertex {(B, 2)} to {(C, 2), (D, 1)} and {(A, 1)} to {(C, 1)}, two rows are lost.

5.
Webster's equation with dissipation on a graph. An MR-image of the human vocal tract is shown in Fig. 4. The vocal tract can be considered as a Y-shaped graph whose three free vertices are at the vocal folds, mouth, and nose (in Fig. 4, the nasal cavity is only partially visible). The closed vertex with three outgoing edges is located in the pharynx. Wave propagation in such domain can be computed by Webster's equation up to frequencies of about 4 kHz where the effect of the transversal resonances becomes significant, see [11, Section 5 and Fig. 1].
The generalised Webster's equation is derived in [18], and it is given by Mouth Nose Pharynx Vocal folds The solution ψ is Webster's velocity potential that approximates the wave equation velocity potential when averaged over a transversal cross-section at distance x ∈ [0, l] from the tube end. Functions A(·), S(·), and c(·) are the cross-sectional area of the tube, the surface area factor, and the corrected sound velocity, respectively. The coefficient θ ≥ 0 regulates the dissipation at the tube walls. The classical Webster's equation is obtained by setting θ = 0 and c(·) = c.
As explained above, the model for the vocal tract is divided into three parts. In each of these parts we have velocity potentials ψ (j) : [0, l j ] × R + → C, j = A, B, C that satisfy (12) with respective functions A j ∈ C 1 [0, l j ] such that A j (x) > ǫ > 0, S j ∈ L 2 (0, l j ) such that S j (x) ≥ 0, and c j such that ∞ > c j (x) > ǫ > 0 and c −2 j (x) ∈ L 2 (0, l j ). The potentials are connected through Kirchhoff conditions The system is controlled by the flow u through the vocal folds, and there is an acoustic resistance at the mouth and nose openings: where θ B and θ C are the dimensionless normalised acoustic resistances. We proceed to formulate this model as a transmission graph. First, we write Webster's equation as a first order system by choosing the state vector as z = ψ ψt . The state and solution spaces are where ρ is the fluid density. The induced X (j) -norm corresponds to the physical energy -the first term gives the kinetic energy of the fluid and the second term gives the potential energy (recall that acoustic pressure is obtained from the velocity potential through p(x, t) = ρψ t (x, t)). In the solution spaces we use norms Here also the vertices corresponding to the mouth and nose are also chosen to be control vertices which does not correspond to boundary conditions (14). It can be shown that an impedance passive internally well-posed system remains as one with such resistive termination but we do not do it here.
6. Remarks and conclusions. Many kinds of passive boundary control systems can be interconnected with each other so that the composed system is also a passive and internally well-posed boundary control system. The presented Kirchhoff couplings are natural when connecting impedance passive systems. We remark that it is also possible to form partial couplings using the presented techniques. This is needed, e.g., when beams are connected to each other by a hinge that does not transmit all the degrees of freedom between the subsystems. This can be done by splitting the input and output spaces using orthogonal projections and then treating these as independent inputs and outputs. However, if the junctions themselves have (finite-dimensional) dynamics then these methods are not (directly) applicable -consider, for example, a hinge junction between two beams with a spring or a damper. In such case the resulting system is not necessarily of boundary control form, and instead, these systems should be treated in the more general system node setting. See [29] for this kind of ideas.
All results in this paper require the colligations to be strong in the sense of Definition 2.1. As mentioned before, there are internally well-posed boundary nodes (in the sense of [21, Definition 2.2]) that are even impedance conservative and satisfy U = Y but are not strong. One such example is given in [21,Proposition 6.3] in terms of the boundary controlled wave equation on Ω ⊂ R n with smooth boundary ∂Ω. However, the same PDE with the same boundary control can be written as a strong node at the cost of U = Y; these spaces are still a dual pair. Note that Theorem 3.3 can be applied also in this case even though the smoothness assumption on ∂Ω seriously restricts the possible couplings of this kind of systems.