\mathrm{S}\mathrm{I}\mathrm{A}\mathrm{M} \mathrm{J}. \mathrm{C}\mathrm{O}\mathrm{N}\mathrm{T}\mathrm{R}\mathrm{O}\mathrm{L} \mathrm{O}\mathrm{P}\mathrm{T}\mathrm{I}\mathrm{M}. © 2023 \mathrm{S}\mathrm{o}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{t}\mathrm{y} \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{I}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{A}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{d} \mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{s} \mathrm{V}\mathrm{o}\mathrm{l}. 61, \mathrm{N}\mathrm{o}. 2, \mathrm{p}\mathrm{p}. 511--535 A STRUCTURAL OBSERVATION ON PORT-HAMILTONIAN SYSTEMS\ast RAINER H. PICARD\dagger , SASCHA TROSTORFF\ddagger , BRUCE WATSON\S , AND MARCUS WAURICK\P Abstract. We study port-Hamiltonian systems on a family of intervals and characterize all boundary conditions leading to m-accretive realizations of the port-Hamiltonian operator and thus to generators of contractive semigroups. The proofs are based on a structural observation that the port-Hamiltonian operator can be transformed to the derivative on a familiy of reference intervals by suitable congruence relations, allowing for studying the simpler case of a transport equation. Moreover, we provide well-posedness results for associated control problems without assuming any additional regularity of the operators involved. Key words. port-Hamiltonian systems, m-accretive operators, congruences MSC codes. 35F15, 46N20, 47B44 DOI. 10.1137/21M1441365 1. Introduction. In this paper, we shall revisit port-Hamiltonian differential equations (going back to van der Schaft et al. [20, 21]), that is, a system of first order partial differential equations of the form\Biggl\{ \partial tu+ P1\partial x\scrH u+ P0\scrH u = 0 on ]0,\infty [\times I, u(0, x) = u0(x), x \in I, where I is a real interval, u : ]0,\infty [\times I \rightarrow \BbbR n is a vector field subject to suitable (linear) boundary conditions, \scrH : I \rightarrow \BbbR n\times n is a matrix field attaining values in the symmetric positive definite matrices, P1 = P \ast 1 \in \BbbR n\times n is invertible, and P0 = - P \ast 0 \in \BbbR n\times n.1 There is a vast amount of literature addressing the well-posedness as well as other questions related to the equations at hand (see, e.g., the monograph [9], the survey [8], and the Ph.D. thesis [1] and the references therein). In particular, questions on the theory of boundary control and observation are treated within the framework of port- Hamiltonian systems. Also, higher-dimensional variants of port-Hamiltonian systems or port-Hamiltonian systems of higher order are discussed (see, e.g., [1, 5, 10]). As an intermediate step, several authors have dealt with port-Hamiltonian systems on net- works (see [5, 6, 7], where networks are considered as examples, and [22] for a detailed study). More precisely, the interval I is replaced by a set of intervals. In this case, \ast Received by the editors August 18, 2021; accepted for publication (in revised form) September 12, 2022; published electronically April 11, 2023. https://doi.org/10.1137/21M1441365 \dagger Department of Mathematics, TU Dresden, Dresden, Saxony 01062, Germany (rainer.picard@tu- dresden.de). \ddagger Mathematisches Seminar, CAU Kiel, Kiel 24118, Germany (trostorff@math.uni-kiel.de). \S School of Mathematics, University of the Witwatersrand, Johannesburg 2000, South Africa (Bruce.Watson@wits.ac.za). \P TU Bergakademie Freiberg, Freiberg 09599, Germany (Marcus.waurick@math.tu-freiberg.de). 1The theory developed in this article also works for complex matrices and complex-valued func- tions. However, since the complex case can always be reduced to the real case by considering copies of real spaces, we restrict ourselves to the real case. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 511 D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y https://doi.org/10.1137/21M1441365 mailto:rainer.picard@tu-dresden.de mailto:rainer.picard@tu-dresden.de mailto:trostorff@math.uni-kiel.de mailto:Bruce.Watson@wits.ac.za mailto:Marcus.waurick@math.tu-freiberg.de 512 PICARD, TROSTORFF, WATSON, AND WAURICK the role of boundary conditions becomes more pronounced. In particular, as a re- sult, there is an abundance of descriptions for boundary conditions leading to port- Hamiltonian operators, that is, - P1\partial x\scrH - P0\scrH on a Hilbert space consisting of suitably many copies of L2-type spaces that generate bounded or (quasi-) contractive semigroups (see [6] and [18] for nonlinear boundary conditions). If the operator \scrH satisfies uniform boundedness conditions (from above and below), then \scrH = 1 can be assumed with no loss of generality; the desired boundary conditions result from a subtle interplay of P1 and \partial x. Note that P0 is then dealt with by a standard perturbation argument. It appears to be commonly understood that reducing the port-Hamiltonian operator to - P1\partial x is the optimal way of treating port-Hamiltonian systems. The main tool provided in the paper at hand is the transformation of the latter operator (by suitable congruence transformations) to \partial x, which is arguably the easiest case in which to discuss boundary conditions on networks. We do not rely on semi-group theory as our method to show existence, uniqueness, and continuous dependence on the data, and as such we can address well-posedness of equations of the form (\partial tM0 +M1 + P1\partial x + P0)u = f, on several copies of L2-type spaces by explicit reduction to the case of\bigl( \partial t\widetilde M0 + \widetilde M1 + \partial x \bigr) \widetilde u = \widetilde f. Using the theory of evolutionary equations (see [11, 16] and Chapter 6 of [13]) to study the latter equation, we shall furthermore be able to treat partial differential algebraic equations; that is, we may allow \widetilde M0 to have a proper nullspace, thus generalizing the class of port-Hamiltonian systems significantly. The article is structured as follows. We consider the operator \partial x on networks in section 2 and provide a characterization of all (linear) boundary conditions leading to m-accretive realizations of this operator on a suitable Hilbert space (and hence - \partial x would generate a contraction semigroup). After that, we show in section 3 how the abstract port-Hamiltonian operator P1\partial x\scrH can be reduced to the case treated in section 2, which allows us to provide a new proof for the well-posedness of port- Hamiltonian systems in section 4. Moreover, using the framework of evolutionary equations instead of C0-semigroups, we present a new approach to boundary control problems, which has the benefit that one does not need to assume smooth diag- onalizability of \scrH , which is a standard assumption in the existing literature (see, e.g., [23]). 2. The operator \partial x on networks. In this section, we briefly introduce the main operator of this paper. For this let Ik \subseteq \BbbR be a nonempty interval with nonempty complement (i.e., Ik \not =] - \infty ,\infty [), where k \in \{ 1, . . . , N\} for some N \in \BbbN . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 513 Definition 2.1. We define \partial x : N\bigoplus k=1 H1(Ik) \subseteq N\bigoplus k=1 L2(Ik) \rightarrow N\bigoplus k=1 L2(Ik), (\phi k)k \mapsto \rightarrow (\phi k \prime )k, where H1(Ik) is the (standard) Sobolev space of L2(Ik)-functions with weak deriv- ative representable as an L2(Ik)-function. Recall that \bigoplus N k=1 L 2(Ik) is the space \times N k=1 L2(Ik) endowed with the inner product \langle u, v\rangle \bigoplus N k=1 L2(Ik) := n\sum k=1 \langle uk, vk\rangle L2(Ik). With this operator at hand, we can consider the port-Hamiltonian operator P1\partial x for a suitable matrix P1 \in \BbbR N\times N on \bigoplus N k=1 L 2(Ik). In order to get a well-defined object, we have to restrict the class of possible matrices P1. Definition 2.2. A matrix P1 \in \BbbR N\times N is called compatible if P1 leaves the space\bigoplus N k=1 L 2(Ik) invariant. Remark 2.3. A typical example for a compatible matrix P1 is a diagonal matrix. More generally, if we have several copies of one interval, say I1 = \cdot \cdot \cdot = Ij for some j \in \{ 1, . . . , N\} , then P1 could be block-diagonal. Indeed, this block-diagonal structure is equivalent to P1 being compatible. Before we come to a closer analysis of the port-Hamiltonian operator, we shall reduce the operator just introduced to a more managable reference case. For this, we put Nf := \{ k \in \{ 1, . . . , N\} ; Ik bounded\} , M+ := \{ k \in \{ 1, . . . , N\} ; sup Ik = \infty \} , M - := \{ k \in \{ 1, . . . , N\} ; inf Ik = - \infty \} . Moreover, for n,m+,m - \in \BbbN we define the space L2(n,m+,m - ) := \bigl( L2(] - 1/2, 1/2[) \bigr) n\oplus \bigl( L2(] - 1/2,\infty [) \bigr) m+ \oplus \bigl( L2(] - \infty , 1/2[) \bigr) m - . Correspondingly, we introduce H1(n,m+,m - ) := \bigl( H1 ( ] - 1/2, 1/2[ ) \bigr) n \oplus \bigl( H1 ( ] - 1/2,\infty [ ) \bigr) m+ \oplus \bigl( H1 ( ] - \infty , 1/2[ ) \bigr) m - , H1 0 (n,m+,m - ) := \bigl( H1 0 ( ] - 1/2, 1/2[ ) \bigr) n \oplus \bigl( H1 0 ( ] - 1/2,\infty [ ) \bigr) m+ \oplus \bigl( H1 0 ( ] - \infty , 1/2[ ) \bigr) m - , where H1 0 stands for the closure of smooth functions with compact support in the space H1, that is, the space of Sobolev functions that vanish at the boundary. We now provide a congruence allowing us to transform the operator \partial x on \bigoplus N k=1 L 2(Ik) to the standard space L2(\#Nf ,\#M+,\#M - ). Proposition 2.4. Let a, b \in \BbbR , a < b. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 514 PICARD, TROSTORFF, WATSON, AND WAURICK (a) Consider \phi : ] - 1/2, 1/2[ \rightarrow ]a, b[ x \mapsto \rightarrow - \biggl( x - 1 2 \biggr) a+ \biggl( x+ 1 2 \biggr) b. Then \phi is invertible2 and \Phi : L2 (]a, b[) \rightarrow L2 (] - 1/2, 1/2[) , u \mapsto \rightarrow \surd b - a (u \circ \phi ) is unitary. (b) Consider \phi + : ] - 1/2,\infty [ \rightarrow ]a,\infty [, x \mapsto \rightarrow x+ a+ 1 2 . Then \Phi + : L2 (]a,\infty [) \rightarrow L2 (] - 1/2,\infty [) , u \mapsto \rightarrow u \circ \phi + is unitary. (c) Let P1 = P \ast 1 \in \BbbR N\times N be a diagonal (hence compatible) matrix, n := \#Nf ,m+ := \#M+,m - := \#M - , and \partial x,n : H 1(n,m+,m - ) \subseteq L2(n,m+,m - ) \rightarrow L2(n,m+,m - ), (\phi k)k \mapsto \rightarrow (\phi \prime k)k , where the index n serves as a reminder that we are working on the ``normal"" space L2(n,m+,m - ). Then there exists a diagonal, real matrix \~P1 and a unitary operator \Psi : \bigoplus N k=1 L 2(Ik) \rightarrow L2(n,m+,m - ) such that \Psi P1\partial x\Psi \ast = \widetilde P1\partial x,n. Proof. (a) Let u \in L2(]a, b[). Then we compute\int 1/2 - 1/2 \bigm| \bigm| \bigm| \surd b - a u (\phi (x)) \bigm| \bigm| \bigm| 2 dx = \int 1/2 - 1/2 | u (\phi (x))| 2 (b - a) dx = \int b a | u (y)| 2 dy. Since \Phi is also onto by the invertibility of \phi , the assertion follows. (b) The assertion follows with an elementary computation similar to (a). (c) Without loss of generality, we assume that \{ 1, . . . , n\} = Nf and that \{ n + 1, . . . , n +m+\} = M+ as well as \{ n+m+ + 1, . . . , N\} = M - . For k \in \{ 1, . . . , n\} we find ak, bk \in \BbbR such that Ik =]ak, bk[. Let \phi k be as in (a) with ak, bk replacing a, b. By the chain rule we have for all k \in \{ 1, . . . , n\} 2It is easily verified that \phi - 1 : ]a, b[ \rightarrow ] - 1/2, 1/2[, x \mapsto \rightarrow 1 b - a x - b+ a 2 (b - a) is the inverse of \phi . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 515 \partial x (u \circ \phi k) = (bk - ak) (\partial xu) \circ \phi k. Hence, \partial x,n\Phi k = (bk - ak) \Phi k\partial x, where we denoted \Phi k according to \Phi as in (a) replacing \phi by \phi k. Next, let \Phi + n+1, . . . , \Phi + m+ be as in (b) with the lower bound a appropriately replaced in order that \Phi + k : L2(Ik) \rightarrow L2(] - 1/2,\infty [) is unitary, k \in \{ n + 1, . . . , n + m+\} . For k \in \{ n+m++1, . . . , N\} , we find, similar to (b), a unitary \Phi - k : L2(Ik) \rightarrow L2(] - \infty , 1/2[). Thus, for all k \in \{ n+ 1, . . . , N\} , we obtain \partial x,n\Phi +/ - k = \Phi +/ - k \partial x. In consequence, denoting \Psi := \left( \Phi 1 0 \cdot \cdot \cdot 0 0 . . . \Phi n . . . ... ... . . . \Phi + n+1 . . . 0 0 \cdot \cdot \cdot 0 \Phi - N \right) and b - a := \left( b1 - a1 0 \cdot \cdot \cdot 0 0 . . . bn - an . . . ... ... . . . 1 . . . 0 0 \cdot \cdot \cdot 0 1 \right) , we deduce that \Psi is unitary, and since P1 is diagonal and hence it commutes with \Psi , we compute \Psi P1\partial x\Psi \ast = \Psi P1 (b - a)\Psi \ast \partial x,n = P1 (b - a) \partial x,n = \sqrt{} (b - a)P1 \sqrt{} (b - a)\partial x,n. Thus, the assertion follows with \widetilde P1 = \sqrt{} (b - a)P1 \sqrt{} (b - a). Remark 2.5. It can easily be seen from the proof that the statement in (c) remains true if m+ = m - = 0, Ik =]a, b[ for all k \in \{ 1, . . . , N\} , and P1 = P \ast 1 (so P1 need not necessarily be diagonal). The matrix \widetilde P1 claimed to exist then has the form \widetilde P1 = \sqrt{} (b - a)P1 \sqrt{} (b - a) and is not necessarily diagonal either. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 516 PICARD, TROSTORFF, WATSON, AND WAURICK If it is clear from the context, we will also write \partial x instead of \partial x,n (as it is defined in Proposition 2.4(c)). We recall that by the Sobolev embedding theorem, H1(n,m+,m - ) \subseteq (C ([ - 1/2, 1/2])) n \times (C0 ( [ - 1/2,\infty [ )) m+ \times (C0 ( ] - \infty , 1/2] )) m - , where we denote by C0(I) for an interval I \subseteq \BbbR the closure of Cc(I) (continuous functions with compact support) with respect to the supremum-norm. This fact will be used frequently throughout the paper. A consequence of integration by parts and the Sobolev embedding theorem is the next proposition. Note that functions in H1(]a, b[) possess well-defined point evaluations at the boundary points a and b (if a = - \infty or b = \infty , the respective evaluation is 0). Proposition 2.6. Let u = \Bigl( u1/2 u\infty u - \infty \Bigr) , v = \Bigl( v1/2 v\infty v - \infty \Bigr) \in H1(n,m+,m - ). Then we find \langle \partial xu, v\rangle + \langle u, \partial xv\rangle = \biggl\langle u1/2 \biggl( 1 2 \biggr) , v1/2 \biggl( 1 2 \biggr) \biggr\rangle - \biggl\langle u1/2 \biggl( - 1 2 \biggr) , v1/2 \biggl( - 1 2 \biggr) \biggr\rangle - \biggl\langle u\infty \biggl( - 1 2 \biggr) , v\infty \biggl( - 1 2 \biggr) \biggr\rangle + \biggl\langle u - \infty \biggl( 1 2 \biggr) , v - \infty \biggl( 1 2 \biggr) \biggr\rangle = \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( v1/2( 1 2 ) v - \infty ( 12 ) \Biggr) \Biggr\rangle - \Biggl\langle \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) , \Biggl( v1/2( - 1 2 ) v\infty ( - 1 2 ) \Biggr) \Biggr\rangle . Denote by \r \partial x the restriction of \partial x to H1 0 (n,m+,m - ). A straightforward conse- quence of the previous observation is the following. Proposition 2.7. The operators \partial x and \r \partial x are densely defined and closed on L2(n,m+,m - ). Moreover, \partial \ast x = - \r \partial x and - \r \partial \ast x = \partial x, where the adjoints are computed in L2(n,m+,m - ). The next statement summarizes the description of all linear, accretive operator extensions of the minimal operator \r \partial x (see also pages 18--22 of [17] for related ma- terial). We recall that for a symmetric matrix K \in \BbbR n\times n and c \in \BbbR the expression K \leq c means that \langle Kx, x\rangle \leq c\| x\| 2 (x \in \BbbR n). K \geq c is defined analogously. Theorem 2.8. Let \r \partial x \subseteq D \subseteq \partial x be a linear operator on L2(n,m+,m - ). (a) Then the following statements are equivalent: (i) D is accretive; that is, \langle Du, u\rangle \geq 0 (u \in dom(D)). (ii) There exists M \in \BbbR (n+m+)\times (n+m - ) with M\ast M \leq 1 such that dom(D) \subseteq \biggl\{ u = (u1/2, u\infty , u - \infty ) \in H1(n,m+,m - ) ; M \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) + \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) = 0 \biggr\} . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 517 (b) Let M \in \BbbR (n+m+)\times (n+m - ) be such that dom(D) = \biggl\{ u = (u1/2, u\infty , u - \infty ) \in H1(n,m+,m - ) ; M \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) + \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) = 0 \biggr\} . Then3 \r \partial x \subseteq - D\ast \subseteq \partial x and dom(D\ast ) = \biggl\{ v = (v1/2, v\infty , v - \infty ) \in H1(n,m+,m - ) ;\Biggl( v1/2( 1 2 ) v - \infty ( 12 ) \Biggr) +M\ast \Biggl( v1/2( - 1 2 ) v\infty ( - 1 2 ) \Biggr) = 0 \biggr\} . (c) D is maximal accretive; that is, D is accretive and there exists no accretive relation extending D (or, equivalently, D and D\ast are accretive) if and only if there exists M \in \BbbR (n+m+)\times (n+m - ) with M\ast M \leq 1 such that dom(D) = \biggl\{ u = (u1/2, u\infty , u - \infty ) \in H1(n,m+,m - ) ; M \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) + \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) = 0 \biggr\} . Proof. (a) Assume D is accretive. Then, by Proposition 2.6, we deduce that, for all u \in dom(D) \subseteq H1(n,m+,m - ), 0 \leq 2\langle Du, u\rangle = \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) \Biggr\rangle - \Biggl\langle \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) , \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) \Biggr\rangle . Hence, \Biggl\langle \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) , \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) \Biggr\rangle \leq \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) \Biggr\rangle .(2.1) Therefore, for all \biggl( u1/2( 1 2 ) u - \infty ( 1 2 ) \biggr) with u \in dom(D), M \biggl( u1/2( 1 2 ) u - \infty ( 1 2 ) \biggr) := - \biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \biggr) gives rise to a linear mapping, which is well defined by (2.1), defined on R := \Biggl\{ \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) ; u \in dom(D) \Biggr\} \subseteq \BbbR n+m - . We put M = 0 on R\bot \BbbR n+m - . Thus, M \in L(\BbbR n+m - ,\BbbR n+m+) with \| M\| \leq 1 by (2.1). Hence, identifying M with its matrix representation M \in \BbbR (n+m+)\times (n+m - ) we obtain (ii). Next, assume (ii), and let M be as in (ii). Then we compute using Proposition 2.6 for all u \in dom(D) 3Note that skew-self-adjointness of D requires M to be unitary, and so in particular m+ = m - . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 518 PICARD, TROSTORFF, WATSON, AND WAURICK 2\langle Du, u\rangle = \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) \Biggr\rangle - \Biggl\langle \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) , \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) \Biggr\rangle = \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) \Biggr\rangle - \Biggl\langle M \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) ,M \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) \Biggr\rangle \geq 0 as \| M\| \leq 1. (b) Since \r \partial x \subseteq D \subseteq \partial x it follows that - \r \partial x = \partial \ast x \subseteq D\ast \subseteq \r \partial \ast x = - \partial x. Hence, \r \partial x \subseteq - D\ast \subseteq \partial x, and thus H1 0 (n,m+,m - ) \subseteq dom(D\ast ) \subseteq H1(n,m+,m - ). Thus, for u \in dom(D) and v \in H1(n,m+,m - ) we can use Proposition 2.6 and deduce \langle Du, v\rangle + \langle u, \partial xv\rangle = \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( v1/2( 1 2 ) v - \infty ( 12 ) \Biggr) \Biggr\rangle - \Biggl\langle \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) , \Biggl( v1/2( - 1 2 ) v\infty ( - 1 2 ) \Biggr) \Biggr\rangle = \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( v1/2( 1 2 ) v - \infty ( 12 ) \Biggr) \Biggr\rangle + \Biggl\langle M \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( v1/2( - 1 2 ) v\infty ( - 1 2 ) \Biggr) \Biggr\rangle = \Biggl\langle \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) , \Biggl( v1/2( 1 2 ) v - \infty ( 12 ) \Biggr) +M\ast \Biggl( v1/2( - 1 2 ) v\infty ( - 1 2 ) \Biggr) \Biggr\rangle . Next, let (x, y) \in \BbbR n \times \BbbR m - . Then M(x, y) \in \BbbR n \times \BbbR m+ . Using suitable piecewise linear functions, it is not difficult to construct u \in H1(n,m+,m - ) such that\Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) = \biggl( x y \biggr) and - M(x, y) = \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) . Hence, \Biggl\{ \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) ; u \in dom(D) \Biggr\} = \BbbR n \times \BbbR m - . As a consequence of this and the above computation, we deduce that v \in dom(D\ast ) if and only if v \in H1(n,m+,m - ) and\Biggl( v1/2( 1 2 ) v - \infty ( 12 ) \Biggr) +M\ast \Biggl( v1/2( - 1 2 ) v\infty ( - 1 2 ) \Biggr) = 0, which establishes (b). (c) At first we assume that dom(D) can be written as it is given in (c). Then, by (a), D is accretive. Moreover, by (b), Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 519 dom(D\ast ) = \biggl\{ v = (v1/2, v\infty , v - \infty ) \in H1(n,m+,m - ) ;\Biggl( v1/2( 1 2 ) v - \infty ( 12 ) \Biggr) +M\ast \Biggl( v1/2( - 1 2 ) v\infty ( - 1 2 ) \Biggr) = 0 \biggr\} , which means that D\ast is accretive by arguing as in (a) (note that with M\ast M \leq 1 we have MM\ast \leq 1 and also that - \r \partial x \subseteq D\ast \subseteq - \partial x). Since D is closed and densely defined, it follows that D is maximal accretive (see, e.g., [3, Corollary 3.17]). On the other hand, if D is maximal accretive, D is accretive, and therefore by (a), we find M \in \BbbR (n+m+)\times (n+m - ) with M\ast M \leq 1 such that dom(D) \subseteq \biggl\{ u = (u1/2, u\infty , u - \infty ) \in H1(n,m+,m - ) ; M \Biggl( u1/2( 1 2 ) u - \infty ( 12 ) \Biggr) + \Biggl( u1/2( - 1 2 ) u\infty ( - 1 2 ) \Biggr) = 0 \biggr\} . By the first part of the proof of (c), we have that \partial x restricted to the right-hand side of this inclusion is maximal accretive. Hence, by the maximality of D, the inclusion is an equality, which establishes the assertion. Remark 2.9. The latter result is a special case of [22, Theorem 2.1], where a similar result is proved using the concept of boundary systems. However, we decided to provide this more direct and simpler proof above, in order to avoid this general theory and to keep the paper self-contained. We remark here that the results in this section can be generalized also to infinite networks; that is, one considers operators on \oplus j\in JL 2(Ij) for an arbitrary index set J . Note, however, that in this case the mapping \Psi in Proposition 2.4(c) exists as a unitary mapping only if infj\in J | Ij | > 0. For the general case we refer the reader to [22, section 7]. 3. The congruence of \partial x and P1\partial x\scrH . Throughout, let P1 \in \BbbR N\times N be a compatible self-adjoint and invertible matrix, and let \scrH : \BbbR \rightarrow \BbbR N\times N be measurable and bounded such that \scrH (x) is compatible for each x \in \BbbR and \scrH attains values in the symmetric matrices and is uniformly positive definite; i.e., there exists c > 0 such that \scrH (x) \geq c for all x \in \BbbR . Moreover, we identify the function \scrH with its induced multiplication operator on \bigoplus N k=1 L 2(Ik). 4 The aim of the present section is to identify the operator P1\partial x\scrH on a suitable Hilbert space and the operator \partial x on L2(n,m+,m - ) as mutually congruent operators, if we choose n,m+,m - \in \BbbN suitably. Since maximal accretivity is preserved by conguences, Theorem 2.8 would then allow us to provide a characterization result for maximal accretivity of the operator P1\partial x\scrH . The reduction to the case \scrH = 1 is standard and well known in the theory of port-Hamiltonian systems (see, e.g., [9, Lemma 7.2.3]). Proposition 3.1. Let H := \bigoplus N k=1 L 2(Ik) equipped with the inner product \langle u, v\rangle H := \langle \scrH u, v\rangle (u, v \in H). 4Note that it suffices to define \scrH on \bigcup N k=1 Ik. However, such a function can easily be extended to \BbbR by setting \scrH (x) = EN for x /\in \bigcup N k=1 Ik. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 520 PICARD, TROSTORFF, WATSON, AND WAURICK Consider the mapping B : H \rightarrow \bigoplus N k=1 L 2(Ik) given by Bu := \scrH u. Then B\ast : N\bigoplus k=1 L2(Ik) \rightarrow H, v \mapsto \rightarrow v, where the adjoint is computed with respect to the inner products on H and \bigoplus N k=1 L 2(Ik). Proof. First note that the inner product on H is well-defined due to the self- adjointness, boundedness, and positive definiteness of \scrH . Moreover, B is obviously linear and bounded, and for u \in H, v \in \bigoplus N k=1 L 2(Ik) we compute \langle Bu, v\rangle = \langle \scrH u, v\rangle = \langle u, v\rangle H , which shows the asserted formula for B\ast . The latter proposition shows that P1\partial x\scrH = B\ast P1\partial xB and hence, P1\partial x\scrH and P1\partial x are congruent as operators on H and \bigoplus N k=1 L 2(Ik), respectively. Next we show that P1\partial x and \partial x are congruent as well (a similar result was obtained in [2, Theorem 3.4] for one interval with finite length). The strategy is as follows. First, we diagonalize P1 and then apply our congruence result Proposition 2.4(c). After that, we use the reflection operators \sigma - 1 given by (\sigma - 1u)(x) := u( - x) to obtain a congruent operator of the form D\partial x on L2(n,m+,m - ), where D is a diagonal matrix with positive diagonal entries. Finally, using \surd D - 1 we obtain the asserted congruence to \partial x. The precise statement is as follows. Theorem 3.2. Let P1 = P \ast 1 \in \BbbR N\times N be invertible and compatible. Then there exist integers n,m+,m - \in \BbbN 0 such that n+m++m - = N and an invertible operator \scrV : \bigoplus k\in \{ 1,...,N\} L2(Ik) \rightarrow L2(n,m+,m - ) such that \scrV P1\partial x\scrV \ast = \partial x. More precisely, \scrV is given as a product of constant matrices, the operator \Psi given in Proposition 2.4(c), and a diagonal operator consisting of identities and reflections on the diagonal. Proof. Since P1 is self-adjoint, we find a unitary matrix K \in \BbbR N\times N such that KP1K \ast = D1, where D1 \in \BbbR N\times N is a diagonal matrix, whose diagonal entries are nonzero thanks to the invertibility of P1. According to Proposition 2.4(c), we find a unitary mapping \Psi : \bigoplus k\in \{ 1,...,N\} L2(Ik) \rightarrow L2(n, \widetilde m+, \widetilde m - ) with suitable n, \widetilde m+, \widetilde m - \in \BbbN 0 such that n+ \widetilde m+ + \widetilde m - = N and a diagonal matrix with real, nonzero entries \widetilde D1 such that \Psi D1\partial x\Psi \ast = \widetilde D1\partial x. Then, \widetilde D1 can be written according to the latter block decomposition with diagonal matrices \widetilde D1,\ell \in \BbbR \ell \times \ell for \ell \in \{ n, \widetilde m+, \widetilde m - \} :\left( \widetilde D1,n 0 0 0 \widetilde D1,\widetilde m+ 0 0 0 \widetilde D1,\widetilde m - \right) . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 521 Let n+ be the number of positive numbers in the diagonal of \widetilde D1,n, n - := n - n+. Similarly we define \widetilde m\pm \pm . Next, we define an operator W : \bigoplus N k=1 L 2(\BbbR ) \rightarrow \bigoplus N k=1 L 2(\BbbR ) acting coordinatewise by (Wku) (x) := \Biggl\{ u(x) if \widetilde D1,kk > 0, u( - x) if \widetilde D1,kk < 0 (x \in \BbbR ) for each u \in L2(\BbbR ) and k \in \{ 1, . . . , N\} . Moreover, it is clear that there exists a permutation matrix Q \in \BbbR N\times N such that QW [L2(n, \widetilde m+, \widetilde m - )] \subseteq L2(n,m+,m - ), where m+ := \widetilde m+ + + \widetilde m - - , m - := \widetilde m - + + \widetilde m+ - . Setting V := QW : L2(n, \widetilde m+, \widetilde m - ) \rightarrow L2(n,m+,m - ), we infer V \Bigl( \widetilde D1\partial x \Bigr) V \ast = D2\partial x on the Hilbert space L2(n,m+,m - ) where D2 \in \BbbR N\times N is a diagonal matrix with strictly positive diagonal entries. Indeed, if \widetilde D1,kk is positive, then Wk is just the identity, and if \widetilde D1,kk is negative, the reflection Wk yields a change of sign by the chain rule. With these transformations at hand, we obtain \partial x = \Bigl( \sqrt{} D2 \Bigr) - 1 D2\partial x \Bigl( \sqrt{} D2 \Bigr) - 1 = \Bigl( \sqrt{} D2 \Bigr) - 1 V \Bigl( \widetilde D1\partial x \Bigr) V \ast \Bigl( \sqrt{} D2 \Bigr) - 1 = \Bigl( \sqrt{} D2 \Bigr) - 1 V\Psi (D1\partial x)\Psi \ast V \ast \Bigl( \sqrt{} D2 \Bigr) - 1 = \Bigl( \sqrt{} D2 \Bigr) - 1 V\Psi K(P1\partial x)K \ast \Psi \ast V \ast \Bigl( \sqrt{} D2 \Bigr) - 1 ; thus the assertion follows with \scrV :=( \surd D2) - 1V\Psi K. Remark 3.3 (Boundary conditions for P1\partial x). Since \scrV P1\partial x\scrV \ast = \partial x we infer that \scrV \ast is a bijection from H1(n,m+,m - ) to \oplus N k=1H 1(Ik). Note that a closer inspection of the proof of Theorem 3.2 reveals that also \scrV P1 \r \partial x\scrV \ast = \r \partial x, where \r \partial x stands for the realization of \partial x restricted to \bigoplus k\in \{ 1,...,N\} H 1 0 (Ik) in the former case and to H1 0 (n,m+,m - ) in the latter case. The reason for this is that the transformation \scrV maps smooth compactly supported functions into smooth compactly supported functions. In particular, this means that any choice of linear boundary conditions for P1\partial x is in one-to-one correspondence to a boundary condition for \partial x (see also Lemma 4.2 below). We have classified all boundary conditions for \partial x leading to an m-accretive operator realization of \partial x. Hence, we obtain a complete description of all boundary conditions for P1\partial x leading to m-accretive operator realizations as a consequence of Theorem 3.2. We shall see in Lemma 4.2 below that the particular form of the congruence is not important. In fact, it turns out that boundary conditions for P1\partial x can be rephrased into boundary conditions for \partial x as long as \scrV P1\partial x\scrV \ast = \partial x for any invertible operator \scrV . Remark 3.4. Using the congruence in Theorem 3.2 and the fact that m-accretivity is preserved by such congruences, we can employ Theorem 2.8 to char- acterize all boundary conditions for the port-Hamiltonian operator P1\partial x\scrH yielding Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 522 PICARD, TROSTORFF, WATSON, AND WAURICK an m-accretive operator. In this way, we can deal with rather general networks with finite and infinite edges in one unified framework. To the best of our knowledge, this has not been achieved in the literature before. In the next section, we will deal with the special case of finite edges with equal length and illustrate what this characteri- zation may look like. In particular, we will recover known results from the literature in this case (see Theorem 4.1). We conclude this section by looking at a variant of the above result under the assumption - \infty < inf k\in \{ 1,...,N\} Ik < sup k\in \{ 1,...,N\} Ik < \infty (3.1) of finiteness of all intervals. Then n = N,m+ = m - = 0 in Theorem 3.2. We first provide another representation of \partial x on L2(N, 0, 0) = L2(] - 1/2, 1/2[)N . It will be instrumental for the proof that the derivative of an odd function is even and that the derivative of an even function is odd. Theorem 3.5 ([12, pp. 61--62] and [18, pp. 2811--2812]). Let \partial x : H 1(] - 1/2, 1/2[)N \subseteq L2(] - 1/2, 1/2[)N \rightarrow L2(] - 1/2, 1/2[)N . We define \iota e : L 2 e(] - 1/2, 1/2[)N \rightarrow L2(] - 1/2, 1/2[)N , \iota o : L 2 o(] - 1/2, 1/2[)N \rightarrow L2(] - 1/2, 1/2[)N , the canonical embeddings from the (componentwise) even and odd functions on L2(] - 1/2, 1/2[) into L2(] - 1/2, 1/2[). Then\biggl( \iota \ast e \iota \ast o \biggr) \partial x \bigl( \iota e \iota o \bigr) = \biggl( 0 \partial x,o \partial x,e 0 \biggr) , where \partial x,e : H 1 e (] - 1/2, 1/2[)N \subseteq L2 e(] - 1/2, 1/2[)N \rightarrow L2 o(] - 1/2, 1/2[)N , \phi \mapsto \rightarrow \phi \prime , and H1 e (] - 1/2, 1/2[)N := H1(] - 1/2, 1/2[)N \cap L2 e(] - 1/2, 1/2[)N ; similarly \partial x,o and H1 o (] - 1/2, 1/2[)N . Remark 3.6. Note that the previous theorem permits us to compute the form of the boundary conditions for \r \partial x \subseteq A \subseteq \partial x m-accretive in terms of restrictions and extensions of \Bigl( 0 \partial x,\mathrm{o} \partial x,\mathrm{e} 0 \Bigr) . Indeed, let \r \partial x \subseteq A \subseteq \partial x be maximal accretive, and let M \in \BbbR N\times N be such that dom(A) = \biggl\{ u \in H1(] - 1/2, 1/2[)N ; Mu \biggl( 1 2 \biggr) + u \biggl( - 1 2 \biggr) = 0 \biggr\} . Note that a small computation (and invariance of smooth compactly supported func- tions) shows that \biggl( \iota \ast e \iota \ast o \biggr) \r \partial x \bigl( \iota e \iota o \bigr) = \biggl( 0 \r \partial x,o \r \partial x,e 0 \biggr) . Also note that by \iota e\iota \ast eu( - 1/2) = \iota e\iota \ast eu(1/2) and \iota o\iota \ast ou( - 1/2) = - \iota o\iota \ast ou(1/2) we obtain Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 523 u \biggl( 1 2 \biggr) = (\iota e\iota \ast eu+ \iota o\iota \ast ou) \biggl( 1 2 \biggr) = (\iota e\iota \ast eu) \biggl( 1 2 \biggr) + (\iota o\iota \ast ou) \biggl( 1 2 \biggr) , u \biggl( - 1 2 \biggr) = (\iota e\iota \ast eu+ \iota o\iota \ast ou) \biggl( - 1 2 \biggr) = (\iota e\iota \ast eu) \biggl( 1 2 \biggr) - (\iota o\iota \ast ou) \biggl( 1 2 \biggr) . Hence, dom(A) = \biggl\{ u \in H1(] - 1/2, 1/2[)N ; M \biggl( (\iota e\iota \ast eu) \biggl( 1 2 \biggr) + (\iota o\iota \ast ou) \biggl( 1 2 \biggr) \biggr) + (\iota e\iota \ast eu) \biggl( 1 2 \biggr) - (\iota o\iota \ast ou) \biggl( 1 2 \biggr) = 0 \biggr\} . Thus, \widetilde A := \Bigl( \iota \ast \mathrm{e} \iota \ast \mathrm{o} \Bigr) A ( \iota \mathrm{e} \iota \mathrm{o} ) is an m-accretive restriction of \Bigl( 0 \partial x,\mathrm{o} \partial x,\mathrm{e} 0 \Bigr) with domain dom( \widetilde A) = \biggl\{ \biggl( ue uo \biggr) \in H1 e (] - 1/2, 1/2[)N \oplus H1 o (] - 1/2, 1/2[)N ; \bigl( M + I M - I \bigr) \biggl( ue(1/2) uo(1/2) \biggr) = 0 \biggr\} . This leads to an alternative proof of one equivalence in [9, Theorem 7.2.4] (also pay attention to [9, Lemma 7.3.1]). 4. Well-posedness of port-Hamiltonian boundary control systems. In this section, we shall have a closer look at port-Hamiltonian systems in the simpler case I1 = \cdot \cdot \cdot = IN =]a, b[ for some a, b \in \BbbR . Note that then n = N,m+ = m - = 0, and L2(N, 0, 0) = L2(] - 1/2, 1/2[)N = L2(] - 1/2, 1/2[;\BbbR N ). In fact, because of the structure theorem, which allows us to represent any port-Hamiltonian system as \partial x, we are in a position to describe a rather general class of port-Hamiltonian systems and discuss related issues like well-posedness of associated boundary control systems. The foundation of this lies in the solution theory for evolutionary equations; see, e.g., [11, 13, 14, 16]. We briefly recall the setting of [8, section 5] (see also [9, section 7]). For this, let a, b \in \BbbR , a < b, P1 = P \ast 1 \in \BbbR N\times N be invertible, P0 = - P \ast 0 \in \BbbR N\times N , \scrH \in L\infty (]a, b[;\BbbR N\times N ). Assume there exists m,M \in \BbbR >0 such that mIN\times N \leq \scrH (x) = \scrH (x)\ast \leq MIN\times N (a.e. x \in ]a, b[). Let A : dom(A) \subseteq L2 \scrH (]a, b[)N \rightarrow L2 \scrH (]a, b[)N , u \mapsto \rightarrow P1 (\scrH u) \prime + P0\scrH u, where dom(\r \partial x\scrH ) \subseteq dom(A) \subseteq dom(\partial x\scrH ) and L2 \scrH (]a, b[)N := (L2(]a, b[)N ; \langle \cdot ,\scrH \cdot \rangle L2(]a,b[)N ). In order to have a meaningful notion of well-posedness, classically, people focus on the generator properties of A. Generating a (C0-)semigroup of contractions can be char- acterized as the closed densely defined operator being m-dissipative, by the Lumer-- Philipps theorem. In the particular case of port-Hamiltonian systems this characteri- zation can be reformulated in terms of the boundary conditions parametrized by some matrix WB ; see, e.g., [6, Theorem 1] or [8, Theorem 5.8]. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 524 PICARD, TROSTORFF, WATSON, AND WAURICK Theorem 4.1. The following conditions are equivalent: (i) - A generates a semigroup of contractions on L2 \scrH (]a, b[)N ; that is, A is max- imal accretive; (ii) A is accretive and there exists WB \in \BbbR N\times 2N such that dom(A) = \biggl\{ u \in L2(]a, b[)N ; \scrH u \in H1(]a, b[)N , WB \biggl( (\scrH u)(b) (\scrH u)(a) \biggr) = 0 \biggr\} ; (iii) there exists WB \in \BbbR N\times 2N such that dom(A) is given as in (ii) and WB has rank N and, in the sense of positive definiteness, WB \biggl( - P1 P1 IN\times N IN\times N \biggr) - 1\biggl( 0 IN\times N IN\times N 0 \biggr) \Biggl( WB \biggl( - P1 P1 IN\times N IN\times N \biggr) - 1 \Biggr) \ast \geq 0. If any of the above holds, then there exist an invertible matrix G \in \BbbR 2N\times 2N and matrices M,L \in \BbbR N\times N with M\ast M \leq 1 and L invertible such that WBG = L \bigl( M 1 \bigr) . We aim to prove this theorem with the help of our congruence result, Theorem 3.2, and the characterization result, Theorem 2.8. For this, we need to inspect how the transformation \scrV of Theorem 3.2 acts on the boundary values of a function u \in H1(]a, b])N . It is remarkable that no knowledge of how \scrV acts on functions with vanishing boundary values is needed in order to obtain that H1 0 (]a, b]) N -functions are mapped onto H1 0 (] - 1/2, 1/2[)N -functions. Lemma 4.2. Let P1 = P \ast 1 \in \BbbR N\times N be invertible. Moreover, let \scrV : L2(]a, b[)N \rightarrow L2(] - 1/2, 1/2[)N be invertible such that \scrV P1\partial x\scrV \ast = \partial x. Then for u \in H1(] - 1/2, 1/2[)N we have that \scrV \ast u \in H1(]a, b[)N , and there exists an invertible matrix G \in \BbbR 2N\times 2N such that\biggl( (\scrV \ast u)(b) (\scrV \ast u)(a) \biggr) = G \biggl( u(1/2) u( - 1/2) \biggr) (u \in H1(] - 1/2, 1/2[)N ). Moreover, G satisfies G\ast \biggl( P1 0 0 - P1 \biggr) G = \biggl( 1 0 0 - 1 \biggr) . Proof. From the congruence we see that \scrV \ast u \in H1(]a, b[)N for functions u \in H1(] - 1/2, 1/2[)N . Moreover, for u,w \in H1(] - 1/2, 1/2[)N we compute by using the integration by parts formula twice\biggl\langle \biggl( (\scrV \ast u)(b) (\scrV \ast u)(a) \biggr) , \biggl( P1(\scrV \ast w)(b) - P1(\scrV \ast w)(a) \biggr) \biggr\rangle \BbbR 2N (4.1) = \langle (\scrV \ast u)(b), P1(\scrV \ast w)(b)\rangle \BbbR N - \langle (\scrV \ast u)(a), P1(\scrV \ast w)(a)\rangle \BbbR N Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 525 = \langle \partial x\scrV \ast u, P1\scrV \ast w\rangle L2(]a,b[)N + \langle \scrV \ast u, \partial xP1\scrV \ast w\rangle L2(]a,b[)N = \langle \scrV P1\partial x\scrV \ast u,w\rangle L2(] - 1/2,1/2[)N + \langle u,\scrV P1\partial x\scrV \ast w\rangle L2(] - 1/2,1/2[)N = \langle \partial xu,w\rangle L2(] - 1/2,1/2[)N + \langle u, \partial xw\rangle L2(] - 1/2,1/2[)N = \biggl\langle \biggl( u(1/2) u( - 1/2) \biggr) , \biggl( w(1/2) - w( - 1/2) \biggr) \biggr\rangle \BbbR 2N . We consider now the binary relation G := \biggl\{ \biggl( \biggl( u(1/2) u( - 1/2) \biggr) , \biggl( (\scrV \ast u)(b) (\scrV \ast u)(a) \biggr) \biggr) ; u \in H1(] - 1/2, 1/2[)N \biggr\} \subseteq \BbbR 2N \times \BbbR 2N . Obviously, G is linear. Moreover, G is a mapping: Indeed, due to the linearity, it suffices to prove that \Bigl( u(1/2) u( - 1/2) \Bigr) = 0 implies \Bigl( (\scrV \ast u)(b) (\scrV \ast u)(a) \Bigr) = 0. For doing so we choose u \in H1(] - 1/2, 1/2[)N with \Bigl( u(1/2) u( - 1/2) \Bigr) = 0. By (4.1) it follows that\biggl\langle \biggl( (\scrV \ast u)(b) (\scrV \ast u)(a) \biggr) , \biggl( P1(\scrV \ast w)(b) - P1(\scrV \ast w)(a) \biggr) \biggr\rangle = 0 (w \in H1(] - 1/2, 1/2[)N ). Let now x, y \in \BbbR N and define f(t) := 1 b - a \bigl( (t - a)P - 1 1 x+ (t - b)P - 1 1 y \bigr) for t \in [a, b]. Then f \in H1(]a, b[)N and we set w := (\scrV \ast ) - 1f \in H1(] - 1/2, 1/2[)N (this follows from \scrV P1\partial x\scrV \ast = \partial x). Then, clearly, P1(\scrV \ast w)(b) = x and - P1(\scrV \ast w)(a) = y, and so, we infer \biggl\langle \biggl( (\scrV \ast u)(b) (\scrV \ast u)(a) \biggr) , \biggl( x y \biggr) \biggr\rangle = 0. Since this holds for all x, y \in \BbbR N , we derive ( (\scrV \ast u)(b) (\scrV \ast u)(a) ) = 0. In the same way one shows that G is one-to-one. Moreover, it is obvious that the domain of G is \BbbR 2N (see also Lemma 4.10), and so G can be represented by an invertible matrix, which we again denote by G \in \BbbR 2N\times 2N . Thus, we have\biggl( (\scrV \ast u)(b) (\scrV \ast u)(a) \biggr) = G \biggl( u(1/2) u( - 1/2) \biggr) (u \in H1(] - 1/2, 1/2[)N ). Plugging this representation into (4.1), we obtain\biggl\langle G \biggl( u(1/2) u( - 1/2) \biggr) , \biggl( P1 0 0 - P1 \biggr) G \biggl( w(1/2) w( - 1/2) \biggr) \biggr\rangle = \biggl\langle \biggl( u(1/2) u( - 1/2) \biggr) , \biggl( 1 0 0 - 1 \biggr) \biggl( w(1/2) w( - 1/2) \biggr) \biggr\rangle for all u,w \in H1(] - 1/2, 1/2[)N , from which we infer G\ast \biggl( P1 0 0 - P1 \biggr) G = \biggl( 1 0 0 - 1 \biggr) . Next, we present our alternative way of characterizing maximal accretivity of A in terms of the boundary conditions. Our approach makes use of the similarity result presented in the previous section. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 526 PICARD, TROSTORFF, WATSON, AND WAURICK Proof of Theorem 4.1. First, we note that we can assume without loss of generality that P0 = 0 (note that P0\scrH is skew-self-adjoint on L2 \scrH (]a, b[)N and thus, does not affect the maximal accretivity of A) and \scrH = 1, by Proposition 3.1. (i) \Rightarrow (ii): Assume that A is maximal accretive. We need to show that there exists WB \in \BbbR N\times 2N with dom(A) = \biggl\{ u \in L2(]a, b[)N ; \scrH u \in H1(]a, b[)N , WB \biggl( (\scrH u)(b) (\scrH u)(a) \biggr) = 0 \biggr\} . By Theorem 3.2 we find \scrV : L2(]a, b[)N \rightarrow L2(] - 1/2, 1/2[)N invertible with \scrV P1\partial x\scrV \ast = \partial x. Hence, D := \scrV A\scrV \ast is maximal accretive and satisfies \r \partial x \subseteq D \subseteq \partial x (note that \scrV leaves H1 0 (]a, b[) N invariant by Lemma 4.2). By Theorem 2.8 we find a matrix M \in \BbbR N\times N such that dom(D) = \{ v \in H1(]a, b[)N ; Mv(1/2) + v( - 1/2) = 0\} , which in turn implies dom(A) = \{ u \in L2(]a, b[)N ; (\scrV \ast ) - 1u \in dom(D)\} = \{ u \in H1(]a, b[)N ; M((\scrV \ast ) - 1u)(1/2) + ((\scrV \ast ) - 1u)( - 1/2) = 0\} = \biggl\{ u \in H1(]a, b[)N ; \bigl( M 1 \bigr) \biggl( ((\scrV \ast ) - 1)u(1/2) ((\scrV \ast ) - 1u)( - 1/2) \biggr) = 0 \biggr\} = \biggl\{ u \in H1(]a, b[)N ; \bigl( M 1 \bigr) G - 1 \biggl( u(b) u(a) \biggr) = 0 \biggr\} , where we have used Lemma 4.2 in the last equality. This establishes the implication (i) \Rightarrow (ii) with WB := \bigl( M 1 \bigr) G - 1 \in \BbbR N\times 2N . (ii) \Rightarrow (iii): Assume that A is accretive and that dom(A) is given as in (ii). Using Theorem 3.2 we find \scrV : L2(]a, b[)N \rightarrow L2(] - 1/2, 1/2[)N invertible, such that \scrV P1\partial x\scrV \ast = \partial x. We set D := \scrV A\scrV \ast and obtain an accretive operator on L2(] - 1/2, 1/2[)N satisfying \circ \partial x \subseteq D \subseteq \partial x. Moreover, by Lemma 4.2 the domain of D is given by dom(D) = \biggl\{ u \in H1(] - 1/2, 1/2[)N ; WBG \biggl( u(1/2) u( - 1/2) \biggr) = 0 \biggr\} with G \in \BbbR 2N\times 2N as in Lemma 4.2. Moreover, by Theorem 2.8(a) there exists M \in \BbbR N with M\ast M \leq 1 such that dom(D) \subseteq \biggl\{ u \in H1(] - 1/2, 1/2[)N ; \bigl( M 1 \bigr) \biggl( u(1/2) u( - 1/2) \biggr) = 0 \biggr\} . Consider now the linear mappings F1 : \BbbR N \times \BbbR N \rightarrow H1(] - 1/2, 1/2[)N , (x, y) \mapsto \rightarrow (t \mapsto \rightarrow (1/2 - t)y + (1/2 + t)x) and F2 : H1(] - 1/2, 1/2[)N \rightarrow \BbbR N \times \BbbR N , u \mapsto \rightarrow (u(1/2), u( - 1/2)). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 527 By definition F1[kerWBG] \subseteq dom(D) and F2[dom(D)] \subseteq ker( M 1 ), and thus, id = F2 \circ F1 : kerWBG \rightarrow ker \bigl( M 1 \bigr) is a well-defined linear injective mapping. Hence, N \leq dimkerWBG \leq dimker \bigl( M 1 \bigr) = N, and so, kerWBG = ker( M 1 ). Thus, there is L \in \BbbR N\times N invertible such that WBG = L \bigl( M 1 \bigr) .(4.2) In particular, WB has rank N . It is left to show that WB \biggl( - P1 P1 IN\times N IN\times N \biggr) - 1\biggl( 0 IN\times N IN\times N 0 \biggr) \Biggl( WB \biggl( - P1 P1 IN\times N IN\times N \biggr) - 1 \Biggr) \ast \geq 0(4.3) and an easy computation shows that (4.3) is equivalent to WB \biggl( - P - 1 1 0 0 P - 1 1 \biggr) W \ast B \geq 0. Using Lemma 4.2, we infer that\biggl( - P - 1 1 0 0 P - 1 1 \biggr) = G \biggl( - 1 0 0 1 \biggr) G\ast , and so WB \biggl( - P - 1 1 0 0 P - 1 1 \biggr) W \ast B = WBG \biggl( - 1 0 0 1 \biggr) G\ast W \ast B = L \bigl( M 1 \bigr) \biggl( - 1 0 0 1 \biggr) \biggl( M\ast 1 \biggr) L\ast = L( - MM\ast + 1)L\ast \geq 0, which shows (ii) \Rightarrow (iii) as well as the formula WBG = L \bigl( M 1 \bigr) . (iii) \Rightarrow (i): We again set D := \scrV A\scrV \ast and prove that D is maximal accretive. We recall that dom(D) = \biggl\{ u \in H1(] - 1/2, 1/2[)N ; WBG \biggl( u(1/2) u( - 1/2) \biggr) = 0 \biggr\} with G \in \BbbR 2N\times 2N from Lemma 4.2, and from the computation above, we see that WBG \biggl( - 1 0 0 1 \biggr) G\ast W \ast B \geq 0. We set K := WBG = ( K1 K2 ) \in \BbbR N\times 2N , which has rank N and satisfies K1K \ast 1 \leq K2K \ast 2 . Since K has rank N , the kernel of \Bigl( K1 K2 \Bigr) is trivial. Let now x \in kerK2. From K1K \ast 1 \leq K2K \ast 2 it follows that x \in kerK1, and hence, x \in ker \Bigl( K1 K2 \Bigr) = \{ 0\} . Thus, K2 is invertible, and we set M := K - 1 2 K1. Then MM\ast = K - 1 2 K1K \ast 1 (K - 1 2 )\ast \leq 1, and hence M\ast M \leq 1. Moreover, kerK = ker( M 1 ), and thus dom(D) = \biggl\{ u \in H1(] - 1/2, 1/2[)N ; \bigl( M 1 \bigr) \biggl( u(1/2) u( - 1/2) \biggr) = 0 \biggr\} . Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 528 PICARD, TROSTORFF, WATSON, AND WAURICK Thus, the maximal accretivity of D, and hence of A, follows from Theorem 2.8(c). The formulation of the generator property as in Theorem 4.1 is instrumental to understanding the well-posedness theorem for boundary control systems in connec- tion with port-Hamiltonian systems, for which we will provide a different perspective below. We recall that in the context of evolutionary equations, the well-posedness of port-Hamiltonian boundary control systems has already been dealt with in [15, section 5.1]. In any case, we need the following notions. Let WB,j \in \BbbR Nj\times 2N and Nj \in \BbbN , j \in \{ 1, 2\} , with N = N1 +N2,K \in \BbbN and WC \in \BbbR K\times 2N . We define \frakA : dom(\frakA ) \subseteq L2(]a, b[)N \rightarrow L2(]a, b[)N , u \mapsto \rightarrow P1 (\scrH u) \prime + P0\scrH u, dom(\frakA ) = \biggl\{ u\in L2(]a, b[)N ; \scrH u \in H1(]a, b[)N ,WB,2 \biggl( \scrH u(b) \scrH u(a) \biggr) = 0 \biggr\} , as well as B : dom(\frakA ) \rightarrow \BbbR N1 , u \mapsto \rightarrow WB,1 \biggl( \scrH u(b) \scrH u(a) \biggr) , and C : dom(\frakA ) \rightarrow \BbbR K , u \mapsto \rightarrow WC \biggl( \scrH u(b) \scrH u(a) \biggr) . Theorem 4.3 ([23, Theorem 2.4] and [8, Theorem 7.7]). Assume that for all x \in ]a, b[ there exist an invertible matrix S(x) and a diagonal matrix \Delta (x) such that P1\scrH (x) = S(x) - 1\Delta (x)S(x), where S and \Delta are continuously differentiable. Moreover, assume that rank \Bigl( WB,1 WB,2 \Bigr) = N , rank \biggl( WB,1 WB,2 WC \biggr) = N + rank(WC) and that - A := - \frakA | ker(B) is a generator of a C0- semigroup on L2(]a, b[)N . Then for each \tau > 0 and all v \in C2([0, \tau ])N1 , u0 \in dom(\frakA ) with Bu0 = v(0) there exists a unique classical solution u \in C1([0, \tau ])N of \.u(t) = - \frakA u(t) = - P1 (\scrH u) \prime - P0\scrH u, u(0) = u0, v(t) = Bu(t) = WB,1 \biggl( \scrH u(t, b) \scrH u(t, a) \biggr) , y(t) = Cu(t) = WC \biggl( \scrH u(t, b) \scrH u(t, a) \biggr) . Moreover, there exists a constant m\tau \geq 0 (just depending on \tau ) such that \| u(\tau )\| 2L2(]a,b[)N + \int \tau 0 \| y(t)\| 2\BbbR K dt \leq m\tau \biggl( \| u0\| 2L2(]a,b[)N + \int \tau 0 \| v(t)\| 2\BbbR N1 dt \biggr) . Owing to the flexibility of evolutionary equations, we are able to significantly improve the well-posedness result in as much as we do not need to impose any regu- larity conditions on \scrH . Further, we can address systems which are more general than Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 529 the Cauchy problems of the previous theorem. In particular, we consider differential- algebraic equations. For this we consider equations of the following form. Definition 4.4. Let P1 = P \ast 1 \in \BbbR N\times N be invertible and \scrH \in L\infty (]a, b[;\BbbR N\times N ). Assume there exist m,M \in \BbbR >0 such that mIN\times N \leq \scrH (x) = \scrH (x)\ast \leq MIN\times N (a.e. x \in ]a, b[). Let M0 = M\ast 0 ,M1 \in L(L2(]a, b[)N ) such that M0 \geq 0. An equation of the form (\partial tM0 +M1 + P1\partial x)\scrH U = F, where F : \BbbR \times ]a, b[\rightarrow \BbbR N is given and U : \BbbR \times ]a, b[\rightarrow \BbbR N is the unknown, is called a differential-algebraic port-Hamiltonian equation; here \partial t is the derivative with respect to the \BbbR -variable (``time""), and \partial x is the coordinatewise derivative with respect to the spatial variable in ]a, b[. Remark 4.5. The classical port-Hamiltonian operator is then covered by choosing M1 = P0 and M0 = \scrH - 1. We emphasize that in the literature different types of differential-algebraic port- Hamiltonian systems are treated which are not covered by the class above. Indeed, in [4] a port-Hamiltonian system modeling a viscoelastic nanorod is studied, and the resulting system is differential-algebraic in the sense that P1 is nonregular. This is not covered by the class above. However, to deal with this example, the authors needed to restrict the state space to obtain a well-posed system, which is not needed for the class considered here. We provide the counterpart of the generation property first. We will restrict ourselves to maximal accretive restrictions of P1\partial x (corresponding to the case of a generator of a contraction semigroup) and note that the additional generality enters the problem via the operators M0 and M1. In order to formulate the well-posedness result, we need some notation from the theory of evolutionary equations. Definition 4.6. For a Hilbert space H and \rho > 0 we define L2 \rho (\BbbR ;H) := \biggl\{ u : \BbbR \rightarrow H ; u Bochner-measurable, \int \BbbR \| u(t)\| 2e - 2\rho t dt < \infty \biggr\} equipped with the natural inner product. Moreover, we define the operator \partial t on L2 \rho (\BbbR ;H) as the closure of C1 c (\BbbR ;H) \subseteq L2 \rho (\BbbR ;H) \rightarrow L2 \rho (\BbbR ;H), \phi \mapsto \rightarrow \phi \prime , where C1 c (\BbbR ;H) denotes the space of continuously differentiable functions with com- pact support on \BbbR taking values in H. The domain of \partial t is given by all functions u \in L2 \rho (\BbbR ;H) such that there exists v \in L2 \rho (\BbbR ;H) with \int \BbbR u\phi \prime = - \int \BbbR v\phi (\phi \in C\infty c (\BbbR )) and in the latter case we have v = \partial tu (see [16, Proposition 4.1.1]). We remark that the so-defined operator \partial t is continuously invertible on L2 \rho (\BbbR ;H) (see, e.g., [13, 16]) and thus allows for the following definition. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 530 PICARD, TROSTORFF, WATSON, AND WAURICK Definition 4.7. For k \in \BbbN and \rho > 0 we define the space H - k \rho (\BbbR ;H) as the completion of L2 \rho (\BbbR ;H) with respect to the norm \| u\| \rho , - k := \| \partial - k t u\| L2 \rho . Remark 4.8. The space H - k \rho (\BbbR ;H) is a so-called extrapolation space for \partial t. We refer the reader to[13, Chapter 2] for the general concept of Sobolev chains/lattices or to [3, Chapter II.5] for extrapolation spaces in the framework of C0-semigroups. It is easy to see that \partial t can be extended to a continuously invertible operator on H - k \rho (\BbbR ;H) and that each closed densely defined operator between two Hilbert spaces H0, H1 can be canonically extended to a closed and densely defined operator between H - k \rho (\BbbR ;H0) and H - k \rho (\BbbR ;H1). We recall the well-posedness theorem for evolutionary equations in the form needed here; see also [16, Chapter 6]. Theorem 4.9 ([11, Solution Theory], [13, Theorem 6.2.5]). Let H be a Hilbert space, M0,M1 \in L(H) with M0 = M\ast 0 \geq 0, and assume there exist \rho 0 > 0 and c > 0 such that \rho 0\langle M0y, y\rangle + \langle M1y, y\rangle \geq c\| y\| 2 for all y \in H. Moreover, let A : dom(A) \subseteq H \rightarrow H be maximal accretive. Then for each \rho \geq \rho 0 the operator \bigl( \partial tM0 +M1 +A \bigr) is continuously invertible on H - k \rho (\BbbR ;H) for each k \in \BbbN 0. We give a short sketch of the proof of the above theorem. By the accretivity of A and the conditions on M0,M1 one confirms that \partial tM0 +M1 +A - c is accretive in L2 \rho (\BbbR ;H) for \rho large enough. Using the spectral representation of \partial t one can prove that the same holds for its adjoint. Thus, \partial tM0 +M1 +A is boundedly invertible in L2 \rho (\BbbR ;H), and invoking Remark 4.8, one obtains the assertion. Before we can state the well-posedness result for boundary control problems for differential-algebraic port-Hamiltonian equations, we need the following prerequisite. Lemma 4.10. Let N \in \BbbN . Then there exists \eta : \BbbR 2N \rightarrow H1(]a, b[)N continuous such that \gamma (\eta (v)) = v for all v \in \BbbR 2N , where \gamma : H1(]a, b[)N \ni (uk)k\in \{ 1,...,N\} \mapsto \rightarrow \biggl( (uk(b))k\in \{ 1,...,N\} (uk(a))k\in \{ 1,...,N\} \biggr) . Proof. Let v = (v1, v2) \in \BbbR N \times \BbbR N . Then \eta (v)(x) := 1 b - a ((x - a)v1 + (b - x)v2) is a valid choice for \eta . The continuity properties are easily checked. The next result puts Theorem 4.3 into the perspective of evolutionary equations. Note that we do not assume any regularity condition on \scrH . As our main assumption, we shall assume the accretivity of the `derivative part' of the port-Hamiltonian. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 531 Theorem 4.11. Consider a differential-algebraic port-Hamiltonian equation as in Definition 4.4. Assume that there exists \rho 0 \geq 0 such that for all y \in L2(]a, b[)N \rho 0\langle M0y, y\rangle + \langle M1y, y\rangle \geq c\langle y, y\rangle for some c > 0. Let \gamma : H1(]a, b[)N \rightarrow \BbbR 2N be given by \gamma (uk)k\in \{ 1,...,N\} = \biggl( (uk(b))k\in \{ 1,...,N\} (uk(a))k\in \{ 1,...,N\} \biggr) . Let W \in \BbbR N\times 2N be such that A : dom(A) \subseteq L2(]a, b[)N \rightarrow L2(]a, b[)N , u \mapsto \rightarrow P1\partial xu with dom(A) = \{ u \in H1(]a, b[)N ; W\gamma u = 0\} being maximal accretive (cf. Theorem 4.1). Furthermore, let v \in H - k \rho (\BbbR ;\BbbR N ) for some k \in \BbbN and \rho \geq \rho 0. Then there exists a unique u \in H - k - 1 \rho (\BbbR ;L2 \scrH (]a, b[)N ) such that (\partial tM0 +M1 + P1\partial x)\scrH u = 0, W\gamma \scrH u = v. Moreover, the mapping H - k \rho (\BbbR ;\BbbR N ) \ni v \mapsto \rightarrow \scrH u \in H - k - 1 \rho (\BbbR ;L2(]a, b[)N ) \cap H - k - 2 \rho (\BbbR ;H1(]a, b[)N ) is continuous. In particular, v \mapsto \rightarrow C\gamma \scrH v \in H - k - 2 \rho (\BbbR ;\BbbR K) is continuous for each C \in \BbbR K\times 2N . Remark 4.12. The continuity statements in the previous theorem are the proper replacements for the inequality asserted to hold in Theorem 4.3. However, Theo- rem 4.3 seems not to be a direct corollary of Theorem 4.11, since the continuous differentiability of u in Theorem 4.3 cannot be guaranteed in the general setting of differential-algebraic port-Hamiltonian equations and needs more assumptions on the operators involved (see, e.g., [19] for the interplay between evolutionary equations and C0-semigroups). We emphasize that the previous theorem also deals with differential- algebraic equations as well as with rough \scrH . The price we have to pay is the regularity loss of the solution u and the observation y = C\gamma \scrH u. So far, we were not able to provide either an example or a theorem showing that the asserted regularity loss is really occurring. This is postponed to future research. Proof of Theorem 4.11. We consider the equation (\partial tM0 +M1 + P1\partial x)\scrH u = 0, W\gamma \scrH u = v. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 532 PICARD, TROSTORFF, WATSON, AND WAURICK Using that A is m-accretive, we apply Theorem 4.1 to find an invertible matrix G \in \BbbR 2N\times 2N and two matrices L,M \in \BbbR N\times N such that L is invertible and M\ast M \leq 1 with WG = L \bigl( M 1 \bigr) . Then W\gamma \scrH u = v is equivalent to 0 = L \bigl( M 1 \bigr) G - 1\gamma \scrH u - v = L \bigl( M 1 \bigr) \biggl( G - 1\gamma \scrH u - \biggl( 0 L - 1v \biggr) \biggr) = W \biggl( \gamma \scrH u - G \biggl( 0 L - 1v \biggr) \biggr) . Let \eta be as in Lemma 4.10. Then the latter can equivalently be formulated by 0 = W \biggl( \gamma \scrH u - G \biggl( 0 L - 1v \biggr) \biggr) = W \biggl( \gamma \scrH u - \gamma \eta G \biggl( 0 L - 1v \biggr) \biggr) = W\gamma \biggl( \scrH u - \eta G \biggl( 0 L - 1v \biggr) \biggr) , which in turn is equivalent to \scrH u - \widetilde v \in dom(A), where \widetilde v = \eta G \bigl( 0 L - 1v \bigr) \in H - k \rho (\BbbR ;H1(]a, b[)N ). Thus, we obtain (\partial tM0 +M1 + P1\partial x)\scrH u = 0, W\gamma \scrH u = v, which amounts to asking for (\partial tM0 +M1 +A)(\scrH u - \widetilde v) = - (\partial tM0 +M1 + P1\partial x) \widetilde v \in H - k - 1 \rho (\BbbR ;L2(]a, b[)N ). (4.4) Next, Theorem 4.9 leads to unique existence of \scrH u - \widetilde v \in H - k - 1 \rho (\BbbR ;L2(]a, b[)N ), which shows the unique existence of u \in H - k - 1 \rho (\BbbR ;L2 \scrH (]a, b[)N ) (the computation above shows existence, and performing the steps backwards, we obtain uniqueness), solving the problem. Moreover, since the mapping H - k \rho (\BbbR ;\BbbR N ) \ni v \mapsto \rightarrow \widetilde v \in H - k \rho (\BbbR ;H1(]a, b[)N ) is continuous by Lemma 4.10 and H - k \rho (\BbbR ;H1(]a, b[)N ) \ni \widetilde v \mapsto \rightarrow (\partial tM0 +M1 + P1\partial x) \widetilde v \in H - k - 1 \rho (\BbbR ;L2(]a, b[)N ) Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y PORT-HAMILTONIAN SYSTEMS 533 is easily seen to be continuous, Theorem 4.9 yields the continuity of H - k \rho (\BbbR ;\BbbR N ) \ni v \mapsto \rightarrow \scrH u \in H - k - 1 \rho (\BbbR ;L2(]a, b[)N ). Moreover, by (4.4) we have that A(\scrH u - \widetilde v) = - (\partial tM0 +M1 + P1\partial x) \widetilde v - (\partial tM0+M1)(\scrH u - \widetilde v) \in H - k - 2 \rho (\BbbR ;L2(]a, b[)N ), which yields the continuity of the mapping H - k \rho (\BbbR ;\BbbR N ) \ni v \mapsto \rightarrow \scrH u \in H - k - 2 \rho (\BbbR ;H1(]a, b[)N ) since \widetilde v \in H - k \rho (\BbbR ;H1(]a, b[)N ) \subseteq H - k - 2 \rho (\BbbR ;H1(]a, b[)N ). We illustrate the versatility of differential-algebraic port-Hamiltonian equations considered here in the following example. Example 4.13. We consider a differential-algebraic port-Hamiltonian equation with P1 = ( 0 1 1 0 ) and M0 and M1 given by M0 := \biggl( 1Ih\cup Ip 0 0 1Ih \biggr) , M1 = \biggl( 1Ie 0 0 1Ip\cup Ie \biggr) , where Ie, Ip, Ih \subseteq ]a, b[ are pairwise disjoint measurable subsets (not necessarily inter- vals) with Ie \cup Ip \cup Ih =]a, b[. The resulting equation then takes the form\biggl( \partial t \biggl( 1Ih\cup Ip 0 0 1Ih \biggr) + \biggl( 1Ie 0 0 1Ip\cup Ie \biggr) + \biggl( 0 \partial x \partial x 0 \biggr) \biggr) \scrH U = F. This problem satisfies the assumption of Theorem 4.11 for \rho 0 = c = 1. Hence, we can study boundary control problems like that in Theorem 4.11 with a boundary control of the form W\gamma \scrH u = v, where \gamma is the point evaluation at the boundaries b and a and W \in \BbbR 4\times 4 satisfies W \ast W \leq 1. We note that the equation is of hyperbolic type on Ih, of parabolic type on Ip, and of elliptic type on Ie. Indeed, assuming for simplicity that \scrH (x) = I2 and F = ( f 0 ) and decomposing U into U = ( u1 u2 ), we end up with the equation u1 + \partial xu2 = f, u2 + \partial xu1 = 0, on Ie, which can be written as the second order problem u1 - \partial 2 xu1 = f , which is an equation of elliptic type. On Ip we obtain \partial tu1 + \partial xu2 = f, u2 + \partial xu1 = 0, which results in the parabolic problem \partial tu1 - \partial 2 xu1 = f . Finally, on Ih, we have \partial tu1 + \partial xu2 = f, \partial tu2 + \partial xu1 = 0, Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. D ow nl oa de d 02 /1 8/ 24 to 1 46 .1 41 .6 3. 27 . R ed is tr ib ut io n su bj ec t t o SI A M li ce ns e or c op yr ig ht ; s ee h ttp s: //e pu bs .s ia m .o rg /te rm s- pr iv ac y 534 PICARD, TROSTORFF, WATSON, AND WAURICK which yields after applying the temporal derivative to the first equation and then substituting the second equation \partial 2 t u1 - \partial 2 xu1 = \partial tf, which is a hyperbolic equation. Remark 4.14. We emphasize that in the above equation no transmission con- ditions need to be imposed in order to apply Theorem 4.11. To the best of our knowledge, such boundary control systems have not been considered before in the literature. Acknowledgment. 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