# Modeling and boundary control of infinite dimensional systems in the Brayton–Moser framework

Modeling and boundary control of infinite dimensional systems in the Brayton–Moser framework Abstract It is well documented that shaping the energy of finite-dimensional port-Hamiltonian systems by interconnection is severely restricted due to the presence of dissipation. This phenomenon is usually referred to as the dissipation obstacle. In this paper, we show the existence of dissipation obstacle in infinite dimensional systems. Motivated by this, we present the Brayton–Moser formulation, together with its equivalent Dirac structure. Analogous to finite dimensional systems, identifying the underlying gradient structure is crucial in presenting the stability analysis. We elucidate this through an example of Maxwell’s equations with zero energy flows through the boundary. In the case of mixed-finite and infinite-dimensional systems, we find admissible pairs for all the subsystems while preserving the overall structure. We illustrate this using a transmission line system interconnected to finite dimensional systems through its boundary. This ultimately leads to a new passive map, using this we solve a boundary control problem, circumventing the dissipation obstacle. 1. Introduction Boundary control of infinite dimensional systems is a well-studied topic. A significant advance in the port-Hamiltonian setting was presented in (van der Schaft & Maschke, 2002), where the authors extended the classical Hamiltonian formulation of infinite dimensional systems to incorporate boundary energy flow. Most infinite dimensional systems interact with the environment through its boundary, and hence such a formulation has an immediate impact on boundary control of infinite dimensional systems by energy shaping (Pasumarthy, 2006). Port-Hamiltonian systems are passive with storage function as the total energy, which is assumed to be bounded from below, and port variables being power conjugate (e.g. voltage and current). This resulted in the development of so-called ‘Energy Shaping’ methods for control of physical systems. In some cases, the standard power-conjugate port variables do not necessarily help in achieving the control objectives due to the dissipation obstacle (García-Canseco et al., 2010), motivating the search for alternative passive maps. One possible alternative that has been explored extensively in the finite-dimensional case is ‘power shaping’ using Brayton–Moser (BM) framework for modeling electrical RLC1 networks (Ortega et al., 2003). Modeling electrical networks in BM framework is a well-established theory (Brayton & Moser, 1964a,b) and has proven useful in studying the Lyapunov stability of RLC networks. The formulation was extended in (Brayton & Miranker, 1964), to the infinite-dimensional case where the authors developed a pseudo gradient framework to analyze the stability of a transmission line with non-zero boundary conditions. Later control theorists borrowed this framework to generate new passive maps (Jeltsema & Scherpen, 2007, 2009) when usual passive maps with energy as storage function render ineffective due to pervasive dissipation (García-Canseco et al., 2010). The systems in BM framework are modeled as pseudo-gradient systems using a function called mixed potential function which has units of power. Therefore, BM framework is often called as the power based framework. In case of RLC networks, these mixed potential functions are sum of three potential functions called content of all the current controlled resistors, co-content of all the voltage controlled resistors and instantaneous power transfer between storage elements. In energy shaping the total energy is used as storage function (van der Schaft, 2017; Ortega et al., 2002). But in BM formulation, we make use of the mixed potential function as the storage function and the derived passivity will either be with respect to the controlled voltages and the derivatives of currents, or the controlled currents and the derivatives of the voltages. This method has natural advantages over practical drawbacks of energy shaping methods like speeding up the transient response (as derivatives of currents and voltages are used as outputs) and also helps overcome the ‘dissipation obstacle’ (García-Canseco et al., 2010). Even though this theory was well established in finite dimensional systems, it was not fully extended to the infinite dimensional case (Jeltsema et al., 2002; Jeltsema & Scherpen, 2007). The existing literature on boundary control of infinite dimensional systems by energy shaping (in the Hamiltonian case) deals with either lossless systems (Rodríguez et al., 2001) or partially lossless systems as in (Macchelli & Melchiorri, 2005), and thus avoids dissipation obstacle issues. In (Schöberl & Siuka, 2013), the authors present a different port-Hamiltonian formulation for infinite dimensional mechanical systems in which the interconnection matrix is a constant skew-symmetric matrix in contrast to (van der Schaft & Maschke, 2002) where it is a skew-symmetric differential operator. The control strategy relies on finding the Casimirs of the interconnected system, but this methodology does not deal with the dissipation obstacle. BM formulation of Maxwell’s equations and transmission line with zero boundary energy flows is presented in (Jeltsema & van der Schaft, 2007; Pasumarthy et al., 2014) respectively. However, the admissible pairs given impose restrictions on their spatial domain (such as $$\|\frac {\partial }{\partial z}\|\leqslant$$ 1). The main contributions of this paper are as follows: BM formulation: We first motivate the need for BM formulation by proving the existence of dissipation obstacle in infinite-dimensional systems using transmission line system as an example. Thereafter, we begin with BM formulation of port-Hamiltonian system defined using Stokes’ Dirac structure. In the process, we present its Dirac formulation with a non-canonical bilinear form, similar to the finite dimensional case (Blankenstein, 2005). Zero boundary energy flows: Analogous to the finite-dimensional system, identifying the underlying gradient structure of the system is crucial in analyzing the stability. Therefore we identify alternative BM formulations called admissible pairs that help in the stability analysis, with Maxwell’s equations as an example. Non-zero boundary energy flows and passivity: In case of infinite-dimensional systems with non-zero boundary energy flows, to find admissible pairs for the overall interconnected system, we have to find these admissible pairs for all individual subsystems, that is, spatial domain and boundary, while preserving the interconnection between these subsystems. To illustrate this, we use the transmission line system (modeled by telegrapher’s equations) where the boundary is connected to a finite dimensional circuit at both ends. This ultimately leads to a new passive map with controlled current and derivatives of the voltage at boundary as port variables respectively. Boundary control: Using the new passive map, a passivity based controller is constructed to solve a boundary control problem (using control by interconnection), where the original passive maps derived using energy as storage function do not work due to the existence of pervasive dissipation. The control objective is achieved by generating Casimir functions of the overall systems. In our preliminary work, we presented BM formulation of infinite dimensional systems in (Kosaraju et al., 2015; Kosaraju & Pasumarthy, 2015). The remainder of the paper is structured as follows: In Section 2, the need for BM formulation for infinite-dimensional systems is motivated by proving the existence of dissipation obstacle. In Section 3 we present BM formulation of a distributed parameter system expressed in port-Hamiltonian framework and additionally, we express the system using the Dirac formulation. In Section 4 we find alternative BM formulations (called admissible pairs) that aid in proving the stability, with zero boundary conditions. Alternative passive maps for transmission line system with non-zero boundary conditions are presented in Section 5. Along the way, we present Casimirs and conservation laws of Maxwell equation in Section 6. In Section 7, we use the new passive maps (derived in Section 5) to solve boundary control problem for transmission line system. Finally, we conclude in Section 8. Notations and Mathematical Preliminaries Let Z be an n dimensional Riemannian manifold with a smooth (n − 1) dimensional boundary ∂Z. Ωk(Z), k = 0, 1, …, n denotes the space of all exterior k-forms on Z. The dual space $$\left (\Omega ^{k}(Z)\right )^{\ast }$$ of Ωk(Z) can be identified with Ωn−k(Z) with a pairing between α ∈ Ωk(Z) and $$\beta \in \left (\Omega ^{k}(Z)\right )^{\ast }$$ given by $$\left\langle\beta | \alpha \right\rangle=\int _{Z} \beta \wedge \alpha$$. Here ∧ is the usual wedge product of differential forms, resulting in the n-form β ∧ α. Similar pairings can be established between the boundary variables. Further we denote α|∂Z to be k-form α evaluated at boundary ∂Z. Let $$\alpha =(\alpha _{1},\alpha _{2})\in \mathscr {F}:=\Omega ^{k}(Z)\times \Omega ^{l}(\partial Z)$$ and $$\beta =(\beta _{1},\beta _{2})\in \mathscr {F}^{\ast }=\Omega ^{n-k}(Z)\times \Omega ^{n-1-l}(\partial Z)$$, then we define the following pairing between $$\mathscr {F}$$ and $$\mathscr {F}^{\ast }$$ \begin{align} \int_{(Z+ \partial Z)}\alpha \wedge \beta:=\int_{Z}\alpha_{1} \wedge \beta_{1}+\int_{\partial Z}\alpha_{2} \wedge \beta_{2}. \end{align} (1) The operator ‘d′ denotes the exterior derivative and maps k forms on Z to k + 1 forms on Z. The Hodge star operator * (corresponding to Riemannian metric on Z) converts p forms to (n − p) forms. Given α, β ∈ Ωk(Z) and γ ∈ Ωl(Z), the wedge product α ∧ γ ∈ Ωk+l(Z). We additionally have the following properties: \begin{align} \alpha \wedge \gamma = (-1)^{kl}\gamma \wedge \alpha ,\ \ast \ast \alpha = (-1)^{k(n-k)}\alpha , \end{align} (2) \begin{align} \int_{z} \alpha \wedge \ast \beta = \int_{z} \beta \wedge \ast \alpha ,\end{align} (3) \begin{align} \mathrm{d}\left(\alpha \wedge \gamma\right)= \mathrm{d}\alpha \wedge \gamma+(-1)^{k} \alpha \wedge \mathrm{d}\gamma. \end{align} (4) For details on the theory of differential forms, we refer to (Abraham et al., 2012). Given a functional H(αp, αq), we compute its variation as \begin{align} \partial H &= H(\alpha_{p}+\partial \alpha_{p},\alpha_{q}+\partial\alpha_{q})-H(\alpha_{p},\alpha_{q})\nonumber\\ &= \int_{z}\left( \delta_{p}H \wedge \partial \alpha_{p} + \delta_{q}H \wedge \partial \alpha_{q} \right)+ \int_{\partial z}\left( \delta_{ \alpha_{p}|_{\partial z}}H \wedge \partial \alpha_{p} + \delta_{\alpha_{q}|_{\partial z}}H \wedge \partial \alpha_{q} \right)\!, \end{align} (5) where αp, ∂αp ∈ Ωp(Z) and αq, ∂αq ∈ Ωq(Z); and δpH ∈ Ωn−p(Z), δqH ∈ Ωn−q(Z) are variational derivatives of H(αp, αq) with respect to αp and αq; and $$\delta _{\alpha _{p}|_{\partial z}}H \in \varOmega ^{n-p-1}(\partial Z)$$, $$\delta _{\alpha _{q}|_{\partial z}}H \in \varOmega ^{n-q-1}(\partial Z)$$ constitute variations at boundary. Further, the time derivatives of H(αp, αq) are \begin{align*} \dfrac{dH}{dt} = \int_{Z}\left ( \delta_{p}H \wedge \dfrac{\partial \alpha_{p}}{\partial t} +\delta_{q}H \wedge \dfrac{\partial \alpha_{q}}{\partial t} \right )+ \int_{\partial Z}\left( \delta_{ \alpha_{p}|_{\partial z}}H \wedge \dfrac{\partial \alpha_{p}}{\partial t} + \delta_{\alpha_{q}|_{\partial z}}H \wedge \dfrac{\partial \alpha_{q}}{\partial t} \right)\!. \end{align*} Let $$G: \varOmega ^{n-p}(Z)\rightarrow \varOmega ^{n-p}(Z)$$ and $$R: \varOmega ^{n-q}(Z)\rightarrow \varOmega ^{n-q}(Z)$$, we call $$G\geqslant 0$$, if and only if ∀αp ∈ Ωp(Z) \begin{align} \int_{Z} \left ( \alpha_{p} \wedge \ast G \alpha_{p} \right )=\int_{Z} \left\langle\alpha_{p},G\alpha_{p}\right\rangle\text{Vol} \geqslant 0 \end{align} (6) where the inner product is induced by the Riemmanian metric on Z and Vol ∈ Ωn(Z) such that $$\int _{Z} \left ( \text {Vol} \wedge \ast \text {Vol} \right )=1$$. G is said to be symmetric if $$\left\langle\alpha _{p} | G \alpha _{p}\right\rangle=\left\langle G\alpha _{p} |\alpha _{p}\right\rangle$$. Given $$u(z,t):Z \times \mathbb {R}\rightarrow \mathbb {R}$$, we denote $$\frac {\partial u}{\partial t}(z,t)$$ as ut, similarly $$\frac {\partial u}{\partial z}(z,t)$$ as uz and u*(z) represents the value of u(z, t) at equilibrium. Furthermore, for $$P(z,u,u_{z}):Z\times \mathbb {R}\times \mathbb {R}^{n}\rightarrow \mathbb {R}$$, we denote $$\frac {\partial P}{\partial u_{z}}$$ as $$P_{u_{z}}$$. 2. Motivation/examples In standard control by interconnection methodologies (Ortega et al., 2008), we assume that both plant and controller are passive. Plants that extract infinite energy at non-zero equilibrium cannot be stabilized under this assumption (Duindam et al., 2009). A port-Hamiltonian system is said to be stymied with dissipation obstacle if it extracts infinite energy at non-zero equilibrium (Ortega et al., 2001). This is a common phenomenon in finite dimensional RLC circuits (García-Canseco et al., 2010). Next, we show the existence of dissipation obstacle in infinite-dimensional systems, using transmission line system (with non-zero boundary conditions) as an example. Example 1 Let 0 < z < 1 represent the spatial domain of the transmission line with L, C, R and G denoting the specific inductance, capacitance, resistance and conductance respectively. We further assume that these are independent of the spatial variable z. Denote i(z, t) and v(z, t) as the line current and line voltage of transmission line system. Consider the transmission line system (modeled using telegraphers equations) interconnected to the boundary as shown in Fig. 1. The dynamics of this system are \begin{align} \begin{array}{ccc} -Li_{t}&=& v_{z}+Ri\\ Cv_{t}&=& -Gv-i_{z} \end{array}\quad 0<z<1 \end{align} (7) \begin{align} \begin{array}{ccc} I_{0}&=& C_{0}v_{0t}+i_{0}\\ v_{0} &=& v_{C_{0}}-i_{0}R_{0} \end{array}\quad z=0 \end{align} (8) \begin{align} \begin{array}{ccc} i_{1}&=& C_{1}v_{C_{1}t}\\ v_{1}&=& R_{1}i_{1}+v_{C_{1}} \end{array}\quad z=1, \end{align} (9) where vC0 and vC1 denote voltages across the capacitors C0 and C1 respectively and I0 represents the current source at z = 0. Additionally, the boundary voltages and currents are denoted by v0 = v(0, t), i0 = i(0, t), v1 = v(1, t) and i1 = i(1, t). Proposition 1 The transmission line system described by the equations (7–9) cannot be stabilized at any non-trivial equilibrium point with passive maps obtained by using the total energy given by \begin{align} E=\dfrac{1}{2}{\int_{0}^{1}}\left(Li^{2}+Cv^{2}\right)\ \text{d}z+\dfrac{1}{2}C_{0}v_{c_{0}}^{2}+\dfrac{1}{2}C_{1}v_{c_{1}}^{2} \end{align} (10) as the storage function. Proof. Differentiating (10) along the trajectories of (7–9), we arrive at the following inequality \begin{align} \dot E \leqslant I_{0} v_{C_{0}}. \end{align} (11) Equilibrium points: At equilibrium, equations (7–9) evaluate to \begin{align} i^{\ast}_{z} +Gv^{\ast}=0,\quad Ri^{\ast} +v^{\ast}_{z}=0\quad 0<z<1 \end{align} (12) \begin{align} I_{0}^{\ast} = i_{0}^{\ast},\quad v_{0}^{\ast} = v_{C_{0}}^{\ast}-i_{0}^{\ast} R_{0}\quad z=0 \end{align} (13) \begin{align} i_{1}^{\ast} = 0,\quad v_{1}^{\ast}= v_{C_{1}}^{\ast}\quad z=1. \end{align} (14) Finally, solving partial differential equations in (12), using the boundary conditions (13) and (14), the solution for i*(z), v*(z) takes the form \begin{align} i^{\ast}(z)=\dfrac{G}{\omega}v_{C_{1}}^{\ast} \sinh(\omega (1-z)),\quad v^{\ast}(z)= v_{C_{1}}^{\ast} \cosh(\omega (1-z) \end{align} (15) where $$\omega =\sqrt {RG}$$. Using equations (13–15) it can be shown that the supply rate $$I_{0}^{\ast } v_{C_{0}}^{\ast } \neq 0$$ at equilibrium. This implies that at equilibrium, the system extracts infinite energy from the controller, thus proving the existence of dissipation obstacle (Ortega et al., 2001). □ Fig. 1. View largeDownload slide Transmission line system. Fig. 1. View largeDownload slide Transmission line system. This problem can be circumvented either by relaxing the assumption that controller has to be passive (Koopman & Jeltsema, 2012) or by finding new passive maps (García-Canseco et al., 2010). In this note, we make use of the latter. It can be seen from (11) that ‘adding a differentiation’ on the output port variable obviates the dissipation obstacle. One alternative is to search for new passive maps within the BM framework. We start with BM formulation of an infinite-dimensional port-Hamiltonian system and derive their admissible pairs, which aids in establishing stability. 3. BM formulation of infinite-dimensional port-Hamiltonian systems In this section, we present BM formulation of infinite-dimensional port-Hamiltonian system defined using Stokes’ Dirac structure, thereby giving its Dirac formulation with a non-canonical bilinear form (refer (Blankenstein, 2005) for the finite dimensional equivalent). Define the linear space $$\mathscr {F}_{p,q}=\varOmega ^{p}(Z)\times \varOmega ^{q}(Z)\times \varOmega ^{n-p}(\partial Z)$$ called the space of flows and $$\mathscr {E}_{p,q}=\varOmega ^{n-p}(Z)\times \varOmega ^{n-q}(Z)\times \varOmega ^{n-q}(\partial Z)$$, the space of efforts, with integers p, q satisfying p + q = n + 1. Let $$(\,f_{p},f_{q},f_{b})\in \mathscr {F}_{p,q}$$ and $$(e_{p},e_{q},e_{b})\in \mathscr {E}_{p,q}$$. Then, the linear subspace $$\mathscr {D}\subset \mathscr {F}_{p,q}\times \mathscr {E}_{p,q}$$ \begin{align*} \mathscr{D}=\!\left\{\! (\,f_{p},f_{q},f_{b},e_{p},e_{q},e_{b})\in \mathscr{F}_{p,q}\times \mathscr{E}_{p,q}\,\left| \,\left[\begin{array}{@{}c@{}} f_{p}\\ f_{q} \end{array}\right]\right.\!=\! \left[\begin{array}{@{}cc@{}} 0 & (-1)^{r}\mathrm{d}\\\mathrm{d} & 0 \end{array}\right] \left[\begin{array}{@{}c@{}} e_{p}\\ e_{q} \end{array}\right], \left[\begin{array}{@{}c@{}} f_{b}\\ e_{b} \end{array}\right]\!=\! \left[\begin{array}{@{}cc@{}} 1 & 0\\0 & -(-1)^{n-q} \end{array}\right]\left[\begin{array}{@{}c@{}} e_{p}|_{\partial Z}\\ e_{q}|_{\partial Z} \end{array}\right] \right\}\!, \end{align*} with r = pq + 1, is a Stokes–Dirac structure, (van der Schaft & Maschke, 2002) with respect to the bilinear form \begin{align*} \left\langle \left\langle \left({f_{p}^{1}}, {f_{q}^{1}},{f_{b}^{1}},{e_{p}^{1}},{e_{q}^{1}},{e_{b}^{1}}\right),\left({f_{p}^{2}},{f_{q}^{2}},{f_{b}^{2}},{e_{p}^{2}},{e_{q}^{2}},{e_{b}^{2}}\right) \right\rangle \right\rangle&=\left\langle {e_{p}^{2}}\left|\,{f_{p}^{1}}\right.\right\rangle+\left\langle {e_{p}^{1}}\left|\,{f_{p}^{2}}\right.\right\rangle+\left\langle {e_{q}^{2}}\left|\,{f_{q}^{1}}\right.\right\rangle\\ &\quad+\left\langle {e_{q}^{1}}\left|\,{f_{q}^{2}}\right.\right\rangle+\left\langle {e_{b}^{2}}\left|\,{f_{b}^{1}}\right.\right\rangle+\left\langle {e_{b}^{1}}\left|\,{f_{b}^{2}}\right.\right\rangle \end{align*} where for $$i=1,2\ (\,{f_{p}^{i}},{f_{q}^{i}},{f_{b}^{i}}) \in \mathscr F_{p,q}\ \text{and}\ ({e_{p}^{i}},{e_{q}^{i}},{e_{b}^{i}})\in \mathscr E_{p,q}$$. Consider a distributed-parameter port-Hamiltonian system on Ωp(Z) × Ωq(Z) × Ωn−p(∂Z), with energy variables $$\left (\alpha _{p},\alpha _{q}\right ) \in \Omega ^{p}(Z)\times \Omega ^{q}(Z)$$ representing two different physical energy domains interacting with each other. The total stored energy is defined as \begin{align*} H:=\int_{Z} \mathbf{H} \in \mathbb{R}, \end{align*} where H is the Hamiltonian density (energy per volume element). Let $$G\geqslant 0$$ and $$R \geqslant 0$$ (satisfying (6)) represent the dissipative terms in the system. Then, setting fp = −(αp)t and fq = −(αq)t, and ep = δpH and eq = δqH, the system \begin{align} -\frac{\partial}{\partial t}\left[\begin{array}{@{}c@{}} \alpha_{p}\\ \alpha_{q} \end{array}\right]=\left[\begin{array}{@{}cc@{}} \ast G & (-1)^{r}\mathrm{d}\\\mathrm{d} & \ast R \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{p}H\\ \delta_{q}H \end{array}\right] ,\quad \left[\begin{array}{@{}c@{}} f_{b}\\ \ e_{b} \end{array}\right]=\left[\begin{array}{@{}cc@{}} 1 & 0\\0 & -(-1)^{n-q} \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{p}H{|_{\partial Z}}\\ \delta_{q}H{|_{\partial Z}} \end{array}\right]\!, \end{align} (16) with r = pq + 1, represents an infinite-dimensional port-Hamiltonian system with dissipation. The time-derivative of the Hamiltonian is computed as \begin{align*} \frac{dH}{dt} \leqslant \int_{{\partial Z}} e_{b} \wedge f_{b}. \end{align*} This implies that the system is passive with respect to the boundary variables eb, fb and storage function H (where H is assumed to be bounded from below). 3.1. The BM formulation Next, we aim to write the infinite-dimensional port-Hamiltonian system, defined with respect to a Stokes’ Dirac structure (16) in an equivalent BM form (Blankenstein, 2005; Kosaraju et al., 2015; Kosaraju & Pasumarthy, 2015; Jeltsema et al., 2003). To begin with, we assume that the mapping from the energy variables (αp, αq) to the co-energy variables (ep, eq) = (δpH, δqH) is invertible. This means the inverse transformation from the co-energy variables to the energy variables can be written as $$(\alpha _{p}, \alpha _{q}) =( \delta _{e_{p}}H^{\ast }, \delta _{e_{q}}H^{\ast })$$. H* is the co-energy of H obtained by $$H^{\ast }(e_{p}, e_{q}) = \int _{Z} \left ( (e_{p} \wedge \alpha _{p} + e_{q} \wedge \alpha _{q})\right ) - H(\alpha _{p}, \alpha _{q})$$. Further, assume that the Hamiltonian H splits as H(αp, αq) = Hp(αp) + Hq(αq), with the co-energy variables given by ep = δpHp, eq = δqHq. Consequently the co-Hamiltonian can also be split as $$H^{\ast }(e_{p}, e_{q}) = H_{p}^{\ast }(e_{p}) + H_{q}^{\ast }(e_{q})$$. We can now rewrite the spatial dynamics of the infinite-dimensional port-Hamiltonian system, in terms of the co-energy variables as \begin{align} \left[\begin{array}{@{}cc@{}} \ast \delta^{2}_{e_{p}} H^{\ast} & 0 \\ 0 & \ast \delta^{2}_{e_{q}} H^{\ast} \end{array}\right] \left[\begin{array}{@{}c@{}} -\frac{\partial e_{p}}{\partial t} \\ -\frac{\partial e_{q}}{\partial t} \end{array}\right]=\left[\begin{array}{@{}cc@{}} \ast G & (-1)^{r}\mathrm{d}\\\mathrm{d} & \ast R \end{array}\right]\left[\begin{array}{@{}c@{}} e_{p}\\ e_{q} \end{array}\right]\!. \end{align} (17) For simplicity, we assume that the relation between the energy and co-energy variables is linear and is given as \begin{align} \alpha_{p} = \ast \varepsilon \; e_{p} \; \text{and}\; \alpha_{q} = \ast \mu\; e_ q \end{align} (18) where $$\mu (=\delta ^{2}_{e_{q}} H^{\ast })$$, $$\varepsilon (= \delta ^{2}_{e_{p}} H^{\ast }) \in \mathbb {R}$$. Applying the Hodge star operator to both sides of (17) and arranging terms using (18), we get \begin{align}-\varepsilon \dot{e}_{p} &= \ast \left((-1)^{r} \mathrm{d} e_{q}+G \ast e_{p}\right)(-1)^{(n-p)\times p}\!,\nonumber \\ -\mu \dot{e}_{q} &= \ast \left(\mathrm{d}e_{p}+R \ast e_{q}\right)(-1)^{(n-q)\times q}. \end{align} (19) Next, we find a mixed-potential function $$P=\int _{Z}\text {P}(e_{p},e_{q})$$ (Jeltsema & van der Schaft, 2007) such that (19) can take the pseudo-gradient structure. The lossless case: We first consider the case of a system that is lossless, that is, when R and G are identically equal to zero in (16). To begin with, we also neglect the boundary terms by setting them to zero. Define P to be a functional of the form $$P=\int _{Z}\text {P}(e_{p},e_{q})$$, where \begin{align} \textrm{P}(e_{p},e_{q}):= e_{q}\wedge de_{p}. \end{align} (20) Its variation is given as \begin{align*} \delta P = \int_{Z}\left(\textrm{P}(e_{p}+\partial e_{p}, e_{q}+\partial e_{q})-\text{P}(e_{p},e_{q})\right) = \int_{Z}\left(e_{q} \wedge \mathrm{d}\partial e_{p} +\partial e_{q} \wedge \mathrm{d} e_{p} +\cdots\right)\!. \end{align*} Using the relation $$e_{q}\wedge \mathrm {d} \partial e_{p}=(-1)^{pq}\partial e_{p} \wedge \mathrm {d}e_{q}+(-1)^{n-q}\mathrm {d}\left (e_{q} \wedge \partial e_{p}\right )$$, and the identity (5), we have \begin{align*} \delta_{e_{q}}P=\mathrm{d}e_{p}(-1)^{(n-q)\times q}, \ \ \delta_{e_{p}}P=(-1)^{pq}\mathrm{d}e_{q}(-1)^{(n-p)\times p}, \end{align*} equation (17) can be written in the BM-type as \begin{align}\left[ \begin{array}{@{}lr@{}} -\mu & 0\\ 0 & \varepsilon \end{array}\right]\frac{\partial}{\partial t}\left[\begin{array}{@{}c@{}} e_{q}\\ \ e_{p} \end{array}\right]= \left[\begin{array}{@{}c@{}} \ast \delta_{e_{q}}P\\ \ast \delta_{e_{p}}P \end{array}\right]\!. \end{align} (21) Including dissipation: One may allow for dissipation by defining the content and co-content functions as follows. Consider instead a functional $$P=\int _{Z}\text {P}$$ defined as \begin{align} \textrm{P}(e_{p},e_{q})= e_{q} \wedge \mathrm{d}e_{p}+\underbrace{\textrm{F}(e_{q})\textrm{Vol}}_{\textrm{content}}- \underbrace{\textrm{G}(e_{p})\textrm{Vol}}_{\textrm{co-content}} \end{align} (22) where Vol ∈ Ωn(Z) such that $$\int _{Z}\ \text{Vol}\wedge \ast \text{Vol} =1$$, the content F(eq) and the co-content G(ep) functions are defined respectively as \begin{align} \textrm{F}(e_{q})=\int_{0}^{e_{q}}\left\langle\hat{e}_{p}(e_{q}^{\prime}),\ \text{d}e_{q}^{\prime}\right\rangle,\quad \textrm{G}(e_{p})=\int_{0}^{e_{p}}\left\langle\hat{e}_{q}(e_{p}^{\prime}),\ \text{d}e_{p}^{\prime}\right\rangle \end{align} (23) where the inner product $$\left\langle\cdot ,\cdot \right\rangle$$ is induced by the Riemannian metric defined on Z. In the case of linear dissipation (16), that is $$\hat {e}_{p}(e_{q})=Re_{q}$$ and $$\hat {e}_{q}(e_{p})=Ge_{p}$$ we have \begin{align} \textrm{P}(e_{p},e_{q})&= e_{q} \wedge \mathrm{d}e_{p}+\int_{0}^{e_{q}}\left\langle Re_{q}^{\prime},\ \text{d}e_{q}^{\prime}\right\rangle\text{Vol}-\int_{0}^{e_{p}}\left\langle Ge_{p}^{\prime},\ \text{d}e_{p}^{\prime}\right\rangle\textrm{Vol} \nonumber\\[4pt] &= e_{q} \wedge \mathrm{d}e_{p}+\dfrac{1}{2}\left\langle Re_{q},e_{q}\right\rangle\textrm{Vol}-\dfrac{1}{2}\left\langle Ge_{p},e_{p}\right\rangle\textrm{Vol} \nonumber\\[4pt] &= e_{q} \wedge \mathrm{d}e_{p}+\underbrace{\frac{1}{2}R e_{q} \wedge \ast e_{q}}_{\textrm{content}}- \underbrace{\frac{1}{2}G e_{p} \wedge \ast e_{p}}_{\textrm{co-content}} \end{align} (24) where in the third step we have used (6). The variation in P is computed as \begin{align*} \delta P =& \int_{Z}\left(e_{q} \wedge \mathrm{d}\partial e_{p} +\partial e_{q} \wedge \mathrm{d} e_{p} + \frac{1}{2}(e_{q} \wedge R \ast \partial e_{q}+\partial e_{q} \wedge \ast e_{q})- \frac{1}{2}(e_{p} \wedge G \ast \partial e_{p}+\partial e_{p} \wedge \ast e_{p}\right) \\[5pt] =& \int_{Z}\left( \partial e_{q} \wedge \mathrm{d}e_{p}+\partial e_{p} \wedge (-1)^{pq} \mathrm{d} e_{q}+ \frac{1}{2}(e_{q} \wedge R \ast \partial e_{q}+\partial e_{q} \wedge \ast e_{q})\right. \\[4pt]&\quad\quad\left.- \frac{1}{2}(e_{p} \wedge G \ast \partial e_{p}+\partial e_{p} \wedge \ast e_{p})+(-1)^{n-q}\mathrm{d}\left(e_{q} \wedge \partial e_{p}\right) \right)\\[4pt] =&\int_{Z} \partial e_{q} \wedge \left(\mathrm{d} e_{p}+R \ast e_{q} \right)+\partial e_{p} \wedge \left((-1)^{pq} \mathrm{d} e_{q}-G \ast e_{p}\right)+(-1)^{n-q}\int_{\partial Z}\left(e_{q} \wedge \partial e_{p}\right) \end{align*} where we have used the relation $$e_{q}\wedge \mathrm {d} \partial e_{p}=(-1)^{pq}\partial e_{p} \wedge \mathrm {d}e_{q}+(-1)^{n-q}\mathrm {d}\left (e_{q} \wedge \partial e_{p}\right )$$, together with properties of the wedge and the Hodge star operator defined in (3) and (4). Finally, by making use of (5) we can write \begin{align} \left[\begin{array}{@{}c@{}} \delta_{e_{p}}P\\ \delta_{e_{q}}P\\\delta_{e_{p}|_{\partial z}}P\\ \delta_{e_{q}|_{\partial z}}P \end{array}\right] =\left[\begin{array}{@{}c@{}} \left((-1)^{pq} \mathrm{d} e_{q}-G \ast e_{p}\right)(-1)^{(n-p)\times p}\\(\mathrm{d}e_{p} +R\ast e_{q})(-1)^{(n-q)\times q}\\(-1)^{n-q}e_{q}|_{\partial z}\\0 \end{array}\right]\!. \end{align} (25) The system of equations (17) can be written in a concise way, similar to (21) as \begin{align} A{u_{t}} = \ast \delta_{u} P \end{align} (26) where u = [ep, eq]⊤ and A = diag(ɛ, −μ). Note that if the linearity between energy and co-energy variables is not assumed (18) then A takes the form $$\text {diag}(-\delta ^{2}_{e_{q}} H^{\ast },\delta ^{2}_{e_{p}} H^{\ast })$$. Including boundary energy flow: The system of equations (16) together with boundary terms can be rewritten as \begin{align} \mathscr{A}U_{t}&=\ast \delta_{U}P+B\ast e_{b}\nonumber\\ \dot{f}_{b}&=B^{\top} U_{t}\left(=\dot{e}_{p}|_{\partial z}\right) \end{align} (27) where U = [u;u|∂z], B = [O1I O2]⊤ and $$\mathscr {A}=\text{diag}\{A,O_{3}\}$$ with O1, O2, O3 denoting zero matrices of order (n + 1 × n − q), (n − p × n − q), (n + 1 × n + 1) respectively and I identity matrix of order (n − q). 3.2. The Dirac Formulation In this section, we aim to find an equivalent Dirac structure formalism of the BM equations of the infinite-dimensional system (27), (for an overview of Dirac structure of infinite dimensional systems we refer to (Le Gorrec et al., 2005)). As we shall see such a formulation would result in a non-canonical Dirac structure. Denote by $$f \in \mathscr F:=\varOmega ^{n-p}(Z)\times \varOmega ^{n-q}(Z)\times \varOmega ^{n-p}(\partial Z)\times \varOmega ^{n-q}(\partial Z)$$ as the space of flows and $$e \in \mathscr {E}:= \mathscr {F}^{\ast }$$, as the space of effort variables. Theorem 1 Consider the following subspace \begin{align} \mathscr D = \left \{(\,f,f_{y}, e,e_{u}) \in \mathscr F \times \mathscr Y\times \mathscr E \times \mathscr S : -\mathscr A f =\ast e+Be_{u},\ f_{y}=\ast B^{\top} f\right \} \end{align} (28) where $$\mathscr {S}$$, $$\mathscr {Y}$$ represents space of port variables eu and fy respectively defined on ∂Z. The above defined subspace constitutes a non-canonical Dirac structure, that is $$\mathscr {D}=\mathscr {D}^{\perp }$$, $$\mathscr {D}^{\perp }$$ is the orthogonal complement of $$\mathscr {D}$$ with respect to the bilinear form \begin{align} &\left\langle\left\langle\left(f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}}\right),\left(f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}}\right)\right\rangle\right\rangle\nonumber\\ &\qquad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle+ \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \mathscr A f^{2}+f^{2} \wedge \ast \mathscr A f^{1} \right ) + \left\langle{e_{u}^{1}}\big|\,{f_{y}^{2}}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle \end{align} (29) where $$\mathscr {A}:\mathscr F\rightarrow \mathscr {F}, \text{for}\ i=1,2;\ f^{i} \in \mathscr F, {f_{y}^{i}}\in \mathscr Y, e^{i} \in \mathscr E, {e_{u}^{i}}\in \mathscr S$$. Proof. We follow a similar procedure as in (van der Schaft & Maschke, 2002). We first show that $$\mathscr {D}\subset \mathscr {D}^{\perp }$$, and secondly $$\mathscr {D}^{\perp }\subset \mathscr {D}$$. $$\mathscr {D}\subset \mathscr {D}^{\perp }$$: Consider $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}$$, if we show that $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}^{\perp }$$ then $$\mathscr {D}\subset \mathscr {D}^{\perp }$$. Now consider any $$(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}})\in \mathscr {D}$$ i.e. satisfying (28), substituting in the bilinear form (29) gives \begin{align*} &\left\langle\left\langle\left(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}}\right),\left(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}}\right)\right\rangle\right\rangle\\ &\quad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle+ \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \mathscr A f^{2}+f^{2} \wedge \ast \mathscr A f^{1} \right ) + \left\langle{e_{u}^{1}}\big|\,{f_{y}^{2}}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle\\ &\quad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle- \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast\left(\ast e^{2}+B{e_{u}^{2}}\right)+f^{2} \wedge \ast\left(\ast e^{1}+B{e_{u}^{1}}\right) \right ) \\ &\quad\quad + \left\langle{e_{u}^{1}}\big|\ast B^{\top} f^{2}\right\rangle+\left\langle{e_{u}^{2}}\big|\ast B^{\top} f^{1}\right\rangle\\ &\quad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle-\left\langle e^{1}|\,f^{2}\right\rangle- \left\langle e^{2}|\,f^{1}\right\rangle-\left\langle{e_{u}^{1}}\ast B^{\top} f^{2}\right\rangle-\left\langle{e_{u}^{2}}\big|\ast B^{\top} f^{1}\right\rangle \\ &\quad\quad+\left\langle{e_{u}^{1}}\big|\ast B^{\top} f^{2}\right\rangle+\left\langle{e_{u}^{2}}\big|\ast B^{\top} f^{1}\right\rangle\\ &\quad = 0 \end{align*} where in step 2 we used the properties of wedge product (2) and (4), that is, \begin{align} f^{1}\wedge\ast \ast e^{2}&=e^{2}\wedge f^{1}\ \textrm{and}\ f^{2}\wedge\ast \ast e^{1}=e^{1}\wedge f^{2}\nonumber\\ f^{1}\wedge \ast B {e^{2}_{u}}&=B {e^{2}_{u}}\wedge \ast f^{1}={e^{2}_{u}}\wedge \ast B^{\top} f^{1}\ \textrm{and}\ f^{2}\wedge \ast B {e^{1}_{u}}=B {e^{1}_{u}}\wedge \ast f^{2}={e^{1}_{u}}\wedge \ast B^{\top} f^{2}. \end{align} (30) This implies $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}^{\perp }$$ implying $$\mathscr {D}\subset \mathscr {D}^{\perp }$$. $$\mathscr {D}^{\perp }\subset \mathscr {D}$$: Consider $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}^{\perp }$$ and if we show that $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}$$ then we are through. Now consider any $$(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}})\in \mathscr {D}$$, implies \begin{align} \left\langle\left\langle\left(f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}}\right),\left(f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}}\right)\right\rangle\right\rangle=0 \end{align} (31) now simplifying the right hand side of (31) we get \begin{align*} &= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle+ \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \mathscr A f^{2}+f^{2} \wedge \ast \mathscr A f^{1} \right ) + \left\langle{e_{u}^{1}}\big|\,{f_{y}^{2}}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle\\ &= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle- \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \left(\ast e^{2}+B{e_{u}^{2}}\right) \right ) + \int_{\left(Z+\partial Z\right)} \left (f^{2} \wedge \ast \mathscr A f^{1} \right )\\ &\quad+ \left\langle{e_{u}^{1}}\big|\ast B^{\top} f^{2}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle\\ &= \int_{\left(Z+\partial Z\right)} \left (f^{2} \wedge \ast \left(\mathscr A f^{1} +\ast e^{1}+B{e_{u}^{1}}\right)\right )+\left\langle{e_{u}^{2}}\big|\left({f_{y}^{1}}-\ast B^{\top} f^{1}\right)\right\rangle \end{align*} where in step 2 we used the fact that $$(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}})\in \mathscr {D}$$, in step 3 we used the wedge operator properties in (30). From (31) we get that, for all f2, $${e^{2}_{u}}$$ \begin{align} \int_{\left(Z+\partial Z\right)} \left (f^{2} \wedge \ast \left(\mathscr A f^{1} +\ast e^{1}+B{e_{u}^{1}}\right)\right )+\left\langle{e_{u}^{2}}\big|\left({f_{y}^{1}}-\ast B^{\top} f^{1}\right)\right\rangle=0. \end{align} (32) This clearly implies \begin{align*} \mathscr A f^{1} +\ast e^{1}+B{e_{u}^{1}}&=0\\ {f_{y}^{1}}-\ast B^{\top} f^{1}&=0 \end{align*} proving that $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}$$. □ Proposition 2 The port-Hamiltonian system (16) or BM equations (27) can be equivalently written as a dynamical system with respect to the non-canonical Dirac structure $$\mathscr {D}$$ in Theorem 1 by setting \begin{align} (\,f,f_{y},e,e_{u})=\left ( -U_{t},\,-\ast \dot{f}_{b},\, \delta_{U}\text{P},\, \ast e_{b}\right )\!. \end{align} (33) Moreover, the non-canonical bilinear form (29) evaluates to the ‘power balance equation’ (Blankenstein, 2005) \begin{align} \dfrac{\partial}{\partial t}\mathscr P = \int_{Z} u_{t}\wedge \ast A u_{t} - \int_{\partial Z}\left(e_{b}\wedge \dot{f}_{b}\right)\!. \end{align} (34) Proof. The first part of the Proposition can be verified by using (33) in the Dirac structure (28). For the second part, consider the following. The bilinear form (29) is assumed to be non-degenerate, hence $$\mathscr D=\mathscr {D}^{\perp }$$ implies \begin{align*} \left\langle\left\langle(\,f,f_{y},e,e_{u}),(\,f,f_{y},e,e_{u})\right\rangle\right\rangle=0,\ \quad \forall (\,f,f_{y},e,e_{u})\in \mathscr{D} \end{align*} and can be simplified to \begin{align} \left\langle e|f\right\rangle+ \int_{\left(Z+ \partial Z\right)} \left (\,f \wedge \ast \mathscr A f\right ) + \left\langle e_{u}|\,f_{y}\right\rangle=0 \end{align} (35) finally using (33) we arrive at the power balance equation (34). We can now interconnect (27) to other BM systems defined at the boundary ∂Z using these new port variables eb and $$-\dot {f}_{b}$$ (Blankenstein, 2003). □ 3.3. A Passivity Argument Once we have written down the equations in the BM framework (sometimes also referred to as the pseudo-gradient form (Jeltsema & van der Schaft, 2007)) we can pose the following question: does the mixed potential function serve as a storage function (or a Lyapunov function) to infer passivity (or equivalently stability) properties of the system? A first look at the balance equation (34) might suggest that the system in the BM form (27) is passive with $$\mathscr P$$ serving as the storage function and port variables $$-\dot f_{b}$$ and eb. Unfortunately though (see (Brayton & Miranker, 1964; Jeltsema & van der Schaft, 2007)) this is not the case as the mixed potential function $$\mathscr P$$ and its time derivative (34) are sign in-definite and hence do not serve as a storage function. This motivates our quest for finding a new $$\mathscr {P}\geqslant 0$$ and $$\mathscr {A}\leqslant 0$$, called as admissible pairs, enabling us to derive certain new passivity/stability properties. This work aims to answer these issues. 4. Stability In the case of infinite dimensional systems to prove Lyapunov stability, it is not sufficient enough to show the positive definiteness of the Lyapunov function and the negative definiteness of its time derivative (as in the case of finite dimensional systems). In infinite dimensional systems, one must specify the norm associated with stability argument because stability with respect to a norm does not imply that it is stable with respect to another norm. Let $$\mathscr {U}_{\infty }$$ be the configuration space of a distributed parameter system, and ∥⋅∥ be a norm on $$\mathscr {U}_{\infty }$$. Definition 1 (Luo et al., 2012) Denote by $$\;U^{\ast }\in \mathscr {U}_{\infty }$$ an equilibrium configuration for a distributed parameter system on $$\mathscr {U}_{\infty }$$. Then, U* is said to be stable in the sense of Lyapunov with respect to the norm ∥⋅∥ if, for every $$\varepsilon \geqslant 0$$ there exist $$\delta \geqslant 0$$ such that, \begin{align*} \|U(0)-U^{\ast}\|\leqslant \delta \implies \|U(t)-U^{\ast}\|\leqslant\varepsilon \end{align*} for all $$t\geqslant 0$$, where $$U(0)\in \mathscr {U}_{\infty }$$ is the initial configuration of the system. We state the following stability theorem for infinite-dimensional systems, which is also referred to as Arnolds theorem for the stability of infinite-dimensional systems. Theorem 2 (Stability of an infinite-dimensional system (Swaters, 1999)): Consider a dynamical system $$\dot {U}=f(U)$$ on a linear space $$\mathscr {U}_{\infty }$$, where $$U^{\ast }\in \mathscr {U}_{\infty }$$ is an equilibrium. Assume there exists a solution to the system and suppose there exists function $$P_{d}:\mathscr {U}_{\infty }\rightarrow \mathbb {R}$$ such that \begin{align} \delta_{U}P_{d}(U^{\ast})=0\ \ \textrm{and}\ \ \dfrac{\partial P_{d}}{\partial t} \leqslant 0. \end{align} (36) Denote ΔU = U − U* and $$\mathscr {N}(\Delta U)= P_{d}(U^{\ast } +\Delta U)-P_{d}(U^{\ast })$$. Show that there exist a positive triplet α, γ1 and γ2 satisfying \begin{align} \gamma_{1}\|\Delta U\|^{2} \leqslant \mathscr{N}(\Delta U) \leqslant \gamma_{2}\|\Delta U\|^{\alpha}. \end{align} (37) Then U* is a stable equilibrium. 4.1. Admissible pairs and stability To infer stability properties of the system (26), let us begin with the case of zero energy flow through the boundary of the system. The mixed-potential function (24) is not positive definite. Hence, we cannot use it as Lyapunov or storage functional. Moreover, the rate of change of this function is computed as \begin{align*} \dot{{P}}= \int_{{Z}} \left ( -\mu \dot{e_{p}} \wedge \ast \dot{e_{p}} +\varepsilon \dot{e_{q}} \wedge \ast \dot{e_{q}} \right )\!, \end{align*} which clearly is not sign-definite. We thus need to look for other admissible pairs$$(\tilde {A}$$, $$\tilde {P})$$ like in the case of finite-dimensional systems (Jeltsema et al., 2003) that can be used to prove the stability of the system while preserving the dynamics of (26). Moreover, the admissible pair should be such that the symmetric part of $$\tilde {A}$$ is negative semi-definite. This can be achieved in the following way (Brayton & Miranker, 1964; Jeltsema & van der Schaft, 2007). Consider functional $$\tilde {P}=\int _{Z}\tilde {\text {P}}$$ of the form \begin{align} \tilde{P} = \lambda P+\frac{1}{2}\int_{Z} \left(\delta_{e_{p}}P \wedge M_{1}\ast \delta_{e_{p}}P+\delta_{e_{q}}P \wedge M_{2}\ast \delta_{e_{q}}P\right)\!, \end{align} (38) with $$\lambda \in \mathbb {R}$$ an arbitrary constant and symmetric mappings $$M_{1}:\varOmega ^{p}(Z) \rightarrow \varOmega ^{p}(Z)$$ and $$M_{2}:\varOmega ^{q}(Z) \rightarrow \varOmega ^{q}(Z)$$ are linear maps. Here the aim is to find λ, M1 and M2 such that \begin{align} \dot{\tilde{P}} = u_{t}^{\top} \tilde{A} u_{t} \leqslant -K ||u_{t}||^{2} \leqslant 0, \end{align} (39) where K ≥ 0 represents the magnitude of smallest eigenvalue of $$\tilde { A}$$. If we can find such a pair $$(\tilde { P}, \tilde { A})$$, which satisfies the above condition, then we can conclude stability of the system. Theorem 3 The system of equations (26) has the alternative BM representation $$\tilde {A}u_{t}=\ast \delta _{u} \tilde {P}$$ with $$\tilde {P}$$ defined as in (38) and \begin{align} \tilde{A} \stackrel{\triangle}=\left[ \begin{array}{@{}cc@{}} -\mu\left(\lambda I+ R^{\top} M_{1}\right)& \varepsilon M_{2} \ast \mathrm{d} (-1)^{(n-p)\times p}\\ -\mu (-1)^{q}M_{1}\ast \mathrm{d} & \varepsilon\left(\lambda I-G^{\top} M_{2}\right) \end{array}\right]\!. \end{align} (40) The new mixed potential function satisfies, $$\tilde {P}\geqslant 0$$ for −∥M1R∥s < λ < ∥M2G∥s, where ∥⋅∥s denotes the spectral norm. Additionally, for systems with p = q and ɛM2 = μM1; symmetric part of $$\tilde {A}$$ is negative definite. Proof. We start with finding the variational derivative of $$\tilde {P}$$. Consider the term $$\delta _{e_{p}}P \wedge M_{1}\ast \delta _{e_{p}}P$$ \begin{align*} &=\left((-1)^{pq} \mathrm{d} e_q -\ast G e_p\right) \wedge M_2 \ast \left((-1)^{pq} \mathrm{d} e_q -\ast G e_p\right)\\ &=\mathrm{d} e_q \wedge M_2 \ast \mathrm{d}e_q-(-1)^p\mathrm{d}e_q \wedge M_2 G e_p+e_p\wedge\ast G^\top M_2Ge_p. \end{align*} The variation in first term deq ∧ M2 *deq is \begin{align*} \mathrm{d}(e_q+\partial e_q) \wedge \ast M_2 \mathrm{d}(e_q+\partial e_q)-\mathrm{d} e_q \wedge M_2 \ast \mathrm{d}e_q &=\mathrm{d} \partial e_q \wedge \ast M_2 \mathrm{d} e_q + \mathrm{d} e_q \wedge \ast M_2 \mathrm{d} \partial e_q\textbf{}+ \cdots\\ &= 2 \mathrm{d} \partial e_q \wedge \ast M_2 \mathrm{d} e_q+\cdots \end{align*} the variation in the second term deq ∧ M2Gep is \begin{align*} \mathrm{d}(e_q+\partial e_q) \wedge M_2 G (e_p+\partial e_p)- \mathrm{d} e_q \wedge M_2 G e_p&=\mathrm{d}e_q \wedge M_2 G \partial e_p + \mathrm{d}\partial e_q \wedge M_2 G e_p+\cdots\\ &=\partial e_p \wedge G^ \top M_2 \mathrm{d}e_q (-1)^{(n-p)\times p} + \mathrm{d}\partial e_q \wedge M_2 G e_p+\cdots \end{align*} finally the variation in the last term ep ∧*G⊤M2Gep is given by \begin{align*} (e_p+\partial e_p) \wedge \ast G^\top M_2 G (e_p+\partial e_p)- e_p \wedge \ast G^\top M_2 G e_p =\partial e_p \wedge 2 \ast G^\top M_2 G e_p. \end{align*} By the properties of the exterior derivative: \begin{align*} \mathrm{d}(\partial e_q \wedge \ast M_2 \mathrm{d}e_q) &=\mathrm{d} \partial e_q \wedge \ast M_2 \mathrm{d} e_q+\partial e_q \wedge (-1)^{(n-q)} \mathrm{d} \ast \mathrm{d} M_2 e_q\\ \mathrm{d}(\partial e_q \wedge M_2 G e_p) &= \mathrm{d}\partial e_q \wedge M_2 G e_p + (-1)^{n-q} \partial e_q \wedge M_2 G \mathrm{d} e_p \end{align*} the variation in $$\delta _{e_{p}}P \wedge M_{1}\ast \delta _{e_{p}}P$$ can be simplified to as \begin{align*} &\partial e_q \wedge 2 \left( (-1)^{p} \mathrm{d} \ast \mathrm{d} M_2 e_q -M_2 G\mathrm{d}e_p\right) +\partial e_p \wedge 2 \left( (-1)^{pq+1}G^\top M_2 \mathrm{d}e_q + \ast G^\top M_2 G e_p \right)\\ &\quad=\partial e_q \wedge 2(-1)^{(n-p)\times p}M_2 \mathrm{d} \ast \left( (-1)^{pq} \mathrm{d}e_q - \ast G e_p \right) +\partial e_p \wedge -2G^\top M_2\left( (-1)^{pq} \mathrm{d}e_q - \ast G e_p \right)\!. \end{align*} Similarly the variation in $$\delta _{e_{q}}P \wedge M_{1}\ast \delta _{e_{q}}P$$ is calculated as \begin{align*} &\partial e_{q} \wedge 2 \left( R^{\top} M_{1} \mathrm{d}e_{p} +\ast R^{\top} M_{1} R e_{q} \right) +\partial e_{p} \wedge 2 \left((-1)^{q} \mathrm{d} \ast \mathrm{d} M_{1} e_{p} +(-1)^{pq} M_{1} R \mathrm{d}e_{q} \right)\\ &\quad= \partial e_{q} \wedge 2R^{\top} M_{1} \left( \mathrm{d}e_{p} +\ast R e_{q} \right) +\partial e_{p} \wedge 2(-1)^{q}M_{1}\mathrm{d} \ast \left(\mathrm{d} e_{p} + \ast R \mathrm{d}e_{q} \right)\!. \end{align*} Together the variational derivative of $$\tilde {P}$$ can be computed as \begin{align*} \delta\tilde{P} \stackrel{\triangle}=\left[ \begin{array}{@{}cc@{}} \lambda I+ R^{\top} M_{1} & M_{2} \mathrm{d} \ast (-1)^{(n-p)\times p}\\ (-1)^{q}M_{1}\mathrm{d} \ast & \lambda I-G^{\top} M_{2} \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{e_{p}}P\\ \delta_{e_{q}}P \end{array}\right]\!. \end{align*} Further \begin{align*} \ast \delta\tilde{P} &= \left[\begin{array}{@{}cc@{}} \lambda I+ R^{\top} M_{1} & M_{2} \ast \mathrm{d} (-1)^{(n-p)\times p}\\ (-1)^{q}M_{1}\ast \mathrm{d} & \lambda I-G^{\top} M_{2} \end{array}\right]\left(\ast \left[\begin{array}{@{}c@{}} \delta_{e_{p}}P\\ \delta_{e_{q}}P \end{array}\right]\right)\\ &=\left[ \begin{array}{@{}cc@{}} -\mu\left(\lambda I+ R^{\top} M_{1}\right) & \varepsilon M_{2} \ast \mathrm{d} (-1)^{(n-p)\times p}\\ -\mu (-1)^{q}M_{1}\ast \mathrm{d} & \varepsilon\left(\lambda I-G^{\top} M_{2}\right) \end{array}\right]\left[\begin{array}{@{}c@{}} \dot{e}_{q}\\\dot{e}_{p} \end{array}\right] =\tilde{A}u_{t}. \end{align*} This concludes the first part of the proof. We next to show the positive definiteness of $$\tilde {P}$$. Before that we simplify P in equation (24) as follows: \begin{align*} \textrm{P}(e_p,e_q)&=e_q \wedge \mathrm{d}e_p+\frac{1}{2}R e_q \wedge \ast e_q- \frac{1}{2}G e_p \wedge \ast e_p \\ &=\frac{R^{-1}}{2}\left(\ast R e_q \wedge \ast \ast R e_q +\mathrm{d}e_p \wedge \ast \ast R e_q + \ast Re_q \wedge \ast \mathrm{d}e_p +\mathrm{d}e_p \wedge \ast \mathrm{d}e_p- \mathrm{d}e_p \wedge \ast \mathrm{d}e_p\right)\\ & \quad-\frac{1}{2} Ge_p \wedge \ast e_p \\ &= \frac{R^{-1}}{2}\left(\delta_{e_p}P \wedge \ast \delta_{e_p}P \right) -\frac{R^{-1}}{2}\mathrm{d}e_p \wedge \ast \mathrm{d}e_p -\frac{1}{2} Ge_p \wedge \ast e_p \end{align*} for −∥M1R∥s < λ < 0 we have \begin{align*} \tilde{\text P} = \frac{\lambda R^{-1}+ M_1 }{2} \left(\delta_{e_p}P \wedge \ast \delta_{e_p}P \right) - \frac{\lambda R^{-1}}{2}\mathrm{d}e_p \wedge \ast \mathrm{d}e_p - \frac{\lambda I}{2} Ge_p \wedge \ast e_p + \frac{M_2}{2} \left( \delta_{e_q}P \wedge \ast \delta_{e_q}P \right)> 0 . \end{align*} In a similar way, we can show that \begin{align} \text P(e_{p},e_{q})= -\frac{G^{-1}}{2}\left( \delta_{e_{q}}P \wedge \ast \delta_{e_{q}}P \right) +\frac{G^{-1}}{2}\mathrm{d}e_{q} \wedge \ast \mathrm{d}e_{q} +\frac{1}{2} Re_{q} \wedge \ast e_{q} \end{align} (41) hence for 0 < λ < ∥M2G∥s, we have \begin{align*} \tilde{\text P} = -\frac{\lambda G^{-1} - M_2 }{2}\left( \delta_{e_q}P \wedge \ast \delta_{e_q}P \right) + \frac{\lambda G^{-1}}{2}\mathrm{d}e_q \wedge \ast \mathrm{d}e_q + \frac{\lambda}{2} Re_q \wedge \ast e_q + \frac{M_1}{2} \left( \delta_{e_p}P \wedge \ast \delta_{e_p}P \right)> 0 \end{align*} concluding that $$\tilde P$$ is positive definite for −∥M1R∥s < λ < ∥M2G∥s. Furthermore, with p = q and ɛM2 = μM1 one can easily that symmetric part of $$\tilde {A}$$ is negative definite. □ Remark 1 Note that, if we do not restrict M1 and M2 such that ɛM2 = μM1 in Theorem 3 then for symmetric part of $$\tilde {A}\leqslant 0$$ will lead to constraints on spatial domain like $$\sigma ^{-1}\sqrt {\varepsilon \mu ^{-1}}\|\ast \mathrm {d}\|<1$$, as given in (Jeltsema & van der Schaft, 2007). Example 2 (Maxwell’s equations) Consider an electromagnetic medium with spatial domain $$Z \subset \mathbb R^{3}$$ with a smooth two-dimensional boundary ∂Z. The energy variables (2-form on Z) are the electric field induction $$\mathscr {D}=\frac {1}{2}\mathscr {D}_{ij}z_{i} \wedge z_{j}$$ and the magnetic field induction $$\mathscr {B}= \frac {1}{2}\mathscr {B}_{ij}z_{i} \wedge z_{j}$$ on Z. The associated co-energy variables are electric field intensity $$\mathscr {E}$$ and magnetic field intensity $$\mathscr {H}$$. These co-energy variables (1-form) are linearly related to the energy variables through the constitutive relationships of the medium as \begin{align} \ast \mathscr{D} = \varepsilon \mathscr{E}, \quad \ast \mathscr{B} = \mu \mathscr{H}, \end{align} (42) where ɛ(z, t) and μ(z, t) denote the electric permittivity and the magnetic permeability, respectively. Hamiltonian formulation: The Hamiltonian H can be written as \begin{align} H(\mathscr{D},\mathscr{B})=\int_{Z} \frac{1}{2}\left(\mathscr{E}\wedge \mathscr{D}+ \mathscr{H}\wedge \mathscr{B}\right)\!. \end{align} (43) Therefore, $$\delta _{\mathscr {D}}H=\mathscr {E}$$ and $$\delta _{\mathscr {B}}H=\mathscr {H}$$. Taking into account dissipation in the system, the dynamics can be written in the port-Hamiltonian form as \begin{align} -\frac{\partial }{\partial t}\left[\begin{array}{@{}c@{}} \mathscr{D}\\ \mathscr{B} \end{array}\right]\left[ =\begin{array}{@{}cc@{}} 0& -\mathrm{d}\\ \mathrm{d}& 0 \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{\mathscr{D}}H\\ \delta_{\mathscr{B}}H \end{array}\right]\!+\!\left[\begin{array}{@{}c@{}} J_{d}\\0 \end{array}\right]=\left[\begin{array}{@{}cc@{}} \ast \sigma & -\mathrm{d}\\ \mathrm{d}& 0 \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{\mathscr{D}}H\\ \delta_{\mathscr{B}}H \end{array}\right] \end{align} (44) where $$\ast J_{d}= \sigma \mathscr {E}$$, Jd denotes the current density and σ(z, t) is the specific conductivity of the material. In addition, we define the boundary variables as fb = δDH|∂Z and eb = δBH|∂Z. Hence, we obtain $$\frac {d}{dt} H \leqslant \int _{\partial Z} \mathscr H \wedge \mathscr E$$. For n = 3, p = q = 2 and $$\alpha _{p}=\mathscr {D}$$, $$\alpha _{q}=\mathscr {B}$$ with H given in (43), Maxwell’s equations given in (44) form a Stokes–Dirac structure. The BM form of Maxwell’s equations: In order to write Maxwell’s equations in BM form, we proceed with defining the corresponding mixed-potential functional \begin{align} P=\int_{{Z}} \mathscr{H}\wedge \mathrm{d}\mathscr{E}-\dfrac{1}{2}\sigma \mathscr{E} \wedge \ast \mathscr{E}, \end{align} (45) which yields the following BM form \begin{align} \left[\begin{array}{@{}cc@{}} -\mu I_{3} & 0\\ 0 & \varepsilon I_{3} \end{array}\right]\left[\begin{array}{@{}c@{}} \mathscr{H}_{t}\\ \mathscr{E}_{t} \end{array}\right]=\left[\begin{array}{@{}c@{}} \ast \mathrm{d} \mathscr{E}\\ -\sigma \mathscr{E}+\ast \mathrm{d} \mathscr{H}\end{array}\right] =\left[\begin{array}{@{}c@{}} \ast \delta_{\mathscr{H}}P\\ \ast \delta_{\mathscr{E}} P\end{array}\right]\!. \end{align} (46) Proposition 3 ( Stability analysis) The system of equations (46) constitutes alternate BM formulation $$\tilde {A}\dot {x}=\ast \delta _{u}\tilde {P}$$, where $$\tilde {P}$$ is as defined in (38) and $$\tilde {A}$$ is defined as \begin{align} \tilde{A}= \left[\begin{array}{@{}cc@{}} -\mu \lambda I & \varepsilon M_{2} \ast \mathrm{d} \\ -\mu M_{1} \ast \mathrm{d} & \varepsilon \left(\lambda I-\sigma M_{2}\right) \end{array}\right]\!. \end{align} (47) Additionally, (46) is stable if λ, M1 > 0 and M2 > 0 are selected such that ϵM2 = μM1 and 0 < λ < σ∥M2∥s. Proof. The first part of the proof is straight forward from Theorem 3. The positive definiteness of $$\tilde {P}$$ can be seen by rewriting it as \begin{align*} \tilde{P}= \int_{z} \delta_{\mathscr{E}}P \wedge \frac{\sigma M_{2} -\lambda I}{2 \sigma} \ast \delta_{\mathscr{E}}P +\frac{1}{2 \sigma}\ \text{d}\mathscr{H} \wedge \ast \ \text{d} \mathscr{H} + \frac{1}{2} \left(\delta_{\mathscr{H}}P \wedge M_{1}\ast \delta_{\mathscr{H}}P\right) \geqslant 0. \end{align*} Under the condition 0 < λ < σ∥M2∥s, the time-derivative of $$\tilde P$$ is \begin{align*} \dot{\tilde{P}} =-\int_{Z} \left(\mu \lambda \mathscr{H}_{t} \wedge \ast \mathscr{H}_{t}+\mathscr{E}_{t} \wedge \ast (\sigma M_{2}-\lambda I)\mathscr{E}_{t}\right) \leqslant 0. \end{align*} Denote $$U=(\mathscr {H},\mathscr {E})$$, $$\varDelta U =(\varDelta \mathscr {H}, \varDelta \mathscr {E} )$$ and consider the norm \begin{align} \|\varDelta U\|^{2} =\int_{Z}\left((\varDelta \mathscr{E}-\ast \mathrm{d}\varDelta \mathscr{H})\wedge \ast( \varDelta \mathscr{E}-\ast \mathrm{d}\varDelta \mathscr{H})+\mathrm{d}\varDelta\mathscr{H}\wedge \ast \mathrm{d}\varDelta\mathscr{H}+\mathrm{d}\varDelta\mathscr{E}\wedge \ast \mathrm{d}\varDelta\mathscr{E}\right)\!. \end{align} (48) One can easily show that the system of equations (46) is stable at equilibrium U* = (0, 0) by invoking Theorem 2 with respect to the above-defined norm (48), for α = 2 and \begin{align*} \gamma_{1} = \min\left\{\dfrac{1}{2\sigma},\lambda_{1}^{\min},\lambda_{2}^{\min}\right\}, \gamma_{2} = \max\left\{\dfrac{1}{2\sigma},\lambda_{1}^{\max},\lambda_{2}^{\max}\right\} \end{align*} where $$\lambda _{1}^{\min},\lambda _{1}^{\max}$$ are the minimum and maximum eigen values of $$\frac {\sigma M_{2} -\lambda I}{2 \sigma }$$ respectively and similarly $$\lambda _{2}^{\min},\lambda _{2}^{\max}$$ for $$\frac {1}{2}M_{1}$$. □ 5. Systems with boundary interaction In this section, we present the BM formulation of infinite dimensional port-Hamiltonian systems that interact through the boundary. We derive admissible pairs and present a new passivity property. We derive these results for the transmission line system described in Example 1. 5.1. The BM form: Spatial domain dynamics: The dynamics of the transmission line (7) can be written in an equivalent BM form as follows: define a functional $$P^{a}={\int _{0}^{1}}\text {P}^{a}dz$$ where \begin{align} \textrm{P}^a=-\dfrac{1}{2}Ri\wedge \ast i+\dfrac{1}{2}Gv\wedge \ast v-i\wedge \mathrm{d}v =\left(-\frac{1}{2}Ri^2+\frac{1}{2}Gv^2-iv_z\right)dz. \end{align} (49) In order to simplify the notation, we avoid using the differential geometric notation2 . Using the line voltage and current as the state variables, we can rewrite the dynamics of the spatial domain as follows \begin{align}\left[ \begin{array}{@{}cc@{}} -L & 0\\0 & C \end{array}\right]\left[\begin{array}{@{}c@{}} i_{t}\\ v_{t} \end{array}\right] = \left[\begin{array}{@{}c@{}} \delta_{i} P^{a}\\ \delta_{v} P^{a} \end{array}\right] = \left[\begin{array}{@{}c@{}} -Ri -v_{z}\\ Gv+i_{z} \end{array}\right]\!. \end{align} (50) The above equations, with A diag {−L, C}, denote u = (i(z, t) v(z, t))⊤, can be written in a gradient form \begin{align} Au_{t}=\delta_{u}P^{a} .\end{align} (51) Boundary dynamics: The spatial domain of the transmission line system is represented by a one-dimensional manifold $$Z=(0,1)\in \mathbb {R}$$ with point boundaries ∂Z = {0, 1}. In order to incorporate boundary conditions, we consider the interconnection of the infinite-dimensional system with finite-dimensional systems, via each of the boundary ports. This type of interconnected system is usually referred to as a mixed finite and infinite-dimensional system. Next, we aim to represent the overall system in BM formulation given in equation (27). Consider now a mixed potential function of the form \begin{align} \mathscr{P}(U) = P^{a}(u)+P^{0}(u_{0})+P^{1}(u_{1}) \end{align} (52) where U = [u u0u1]⊤, P0 and P1 are the contributions to the mixed potential function arising from the boundary dynamics at z = 0 and z = 1 respectively. Similar to (27), we represent the overall dynamics of mixed finite and infinite-dimensional system in BM form. The dynamics evolving on the spatial domain (i.e. for 0 < z < 1) are given by (50) (equivalently (51)). At z = 0 the dynamics are \begin{align} A_{0}u_{0t}=\left.\left(\dfrac{\partial P^{0}}{\partial u_{0}}-\textrm{P}^{a}_{u_{z}}\right)\right|_{z=0} +B_{0} I_{0} \end{align} (53) where \begin{align*} u_{0}=[i_{0},v_{0},v_{C0}]^{\top},\quad P^{0}(u_{0})=(v_{C0}-v_{0})i_{0}-\frac{1}{2}R_{0}{i_{0}^{2}},\quad A^{0}=\ \textrm{diag}\ \{ 0,0,-C_{0}\}, \end{align*} with B0 = [0, 0, −1]⊤ as the input matrix, I0 as input, $$\text {P}^{a}_{u_{z}}=\dfrac {\partial \text {P}^{a}}{\partial u_{z}}$$ and $$u_{0t}=\dfrac {du_{0}}{dt}$$. The dynamics at boundary z = 1 are \begin{align} A_{1}u_{1t}=\left.\left(\dfrac{\partial P^{1}}{\partial u_{1}}+\textrm{P}^{a}_{u_{z}}\right)\right|_{z=1} \end{align} (54) where u1 = [i1, v1, vC1]⊤, $$P^{1}= (v_{1}-v_{C1})i_{1}-\frac {1}{2}R_{1}{i_{1}^{2}}$$ and A1 = diag{0, 0, −C1}. Together they can be written compactly in the BM form as \begin{align} \mathscr{A}U_{t}=\delta_{U}\mathscr{P}+BI_{0} \end{align} (55) $$\mathscr {A}=\text{diag}\{A,A_{0},A_{1}\}$$, B = [0 0 B0O3]⊤ and O3 = [0 0 0]. \begin{align*} \delta_{U} \mathscr{P} =\left[ \delta_{u}P \left.\left(\dfrac{\partial P^{0}}{\partial u_{0}}-\textrm{P}_{u_{z}}\right)\right|_{z=0} \left.\left(\dfrac{\partial P^{1}}{\partial u_{1}}+\textrm{P}_{u_{z}}\right)\right|_{z=1} \right]^{\top} \!. \end{align*} Remark 2 Note that the mixed potential functional is not unique. Another choice is $$P^{b}={\int _{0}^{1}} \text {P}^{b}dz$$ where \begin{align} \textrm{P}^{b}=-\frac{1}{2}Ri^{2}+\frac{1}{2}Gv^{2}+i_{z}v. \end{align} (56) This choice of Pa or Pb does not have any effect on spatial domain since it preserves the dynamics (50) and (51), as δuPa = δuPb. If we use Pb as mixed potential function instead of Pa, then we need to change P0 and P1 to $$v_{C_{0}}i_{0}-\frac {1}{2}R_{0}{i_{0}^{2}}$$ and $$-\frac {1}{2}R_{1}{i_{1}^{2}}-v_{C_{1}}i_{1}$$ respectively in (53), (54). Dirac formulation: The transmission line system in BM equations (55) can be equivalently written as \begin{align*} \left ( \left(-u_{t}, -u_{0t}, -u_{1t}\right),\ B_{0}^{\top} u_{0t},\ \left( \delta_{u} \text{P},\ P^{0}_{u_{0}}-\text{P}_{u_{z}}\big|_{z=0}, P^{1}_{u_{1}}+\text{P}_{u_{z}}\big|_{z=1}\right),\ -I_{0}\right ) \in \mathscr{D} \end{align*} with subspace D defined as in Section 3.2. This gives us the ‘balance equation’ \begin{align} \dfrac{d}{dt}\mathscr{P} = {\int^{1}_{0}} \left(Au_{t} \cdot u_{t}\right)\ \text{d}z +A_{0}u_{0t}\cdot u_{0t}+A_{1}u_{1t}\cdot u_{1t}+f^{\top}_{u0} y_{0} \end{align} (57) \begin{align} ={\int_{0}^{1}}\left(-L{i^{2}_{t}}+C{v^{2}_{t}}\right)\ \text{d}z-C_{0}\left(\dfrac{dv_{C0}}{dt}\right)^{2}-C_{1}\left(\dfrac{dv_{C1}}{dt}\right)^{2}+I_{0}\dfrac{dv_{C0}}{dt}. \end{align} (58) 5.2. Admissible pairs To find admissible pairs for transmission line system with non-zero boundary conditions, we need to define $$\tilde A$$ as the following, which will be clear in the subsequent section. In general, new $$\tilde A$$ may contain ∂/∂z in its entries (similar to *d in (40) and Remark 3). In this case, there will be an additional contribution to the terms in the boundary from $$\tilde A$$, which will be clear in Proposition 4. To account this contribution, we split such $$\tilde A$$ as $$\tilde {A}_{nd}+\tilde {A}_{d}\frac {\partial }{\partial z}$$. Definition 2 Admissible pairs: Denote $$\tilde {\mathscr {P}}=\int _{Z}\tilde {\text {P}^{a}}+\tilde {P}^{0}+\tilde {P}^{1}$$ and $$\tilde {\mathscr {A}}=\text {diag}\ \{\tilde {A}, \tilde {A}_{0}, \tilde {A}_{1}\}$$, further $$\tilde {A}$$ is $$\tilde {A}_{nd}+\tilde {A}_{d}\frac {\partial }{\partial z}$$. We call $$\tilde {\mathscr {P}}$$ and $$\tilde {\mathscr {A}}$$ admissible pairs if they satisfy the following: $$\tilde {P}^{a}\geqslant 0$$, $$\tilde A_{d}^{\top }=\tilde A_{d}$$ and $$u_{t}^{\top }\tilde {A}_{nd}u_{t}\leqslant 0$$ such that \begin{align} \tilde{A}u_{t}=\delta_{u}\tilde{\text{P}^{a}} \end{align} (59) $$\tilde {P}^{0}\geqslant 0$$ and $$u_{0t}^{\top }\tilde {A}_{0}u_{0t}\leqslant 0$$ such that \begin{align} \left(\tilde{A}_{0}+\dfrac{1}{2}\tilde{A}_{d}\right)u_{0t}=\left.\left(\dfrac{\partial \tilde{P}}{\partial u_{0}}-\tilde{\text{P}^{a}}_{u_{z}}\right)\right|_{z=0}+\tilde B_{0}I_{0} \end{align} (60) $$\tilde {P}^{1}\geqslant 0$$ and $$u_{1t}^{\top }\tilde {A}_{1}u_{1t}\leqslant 0$$ such that \begin{align} \left(\tilde{A}_{1}-\dfrac{1}{2}\tilde{A}_{d}\right)u_{1t}=\left.\left(\dfrac{\partial \tilde{P}}{\partial u_{1}}+\tilde{\text{P}}_{u_{z}}^{a}\right)\right|_{z=1} \end{align} (61) Together we can write them as \begin{align} \tilde{\mathscr{A}}U_{t} = \delta_{U} \tilde{\mathscr{P}}+ \tilde BI_{0},\quad y_{0} = - \tilde B^{\top}_{0} u_{0t}. \end{align} (62) Proposition 4 If $$\tilde {\mathscr {P}}=\int _{Z}\tilde {\text {P}^{a}}+\tilde {P}^{0}+\tilde {P}^{1}$$ and $$\tilde {\mathscr {A}}= \text {diag}\ \{\tilde {A}, \tilde {A}_{0}, \tilde {A}_{1}\}$$ satisfy the Definition 2 then $$\dot {\tilde {\mathscr {P}}}\leqslant I_{0}^{\top } y_{0}$$, that is the system is passive port variables I0 and y0. Proof. The time derivative of $$\tilde {P}_{d}\geqslant 0$$ along the trajectories of (59–61) is \begin{align*} \dot{\tilde{P}}&= \int_0^1\big(\delta_u\tilde{P^a}.u_t \big)\ \text{d}z+\left.\left(\dfrac{\partial \tilde{P}}{\partial u_0}-\tilde{\text{P}^a}_{u_z}\right)\right|_{z=0}\cdot u_{0t}+\left.\left(\dfrac{\partial \tilde{P}}{\partial u_1}+\tilde{\text{P}^a}_{u_z}\right)\right|_{z=1}\cdot u_{1t}&&\\ &= \int_0^1\big(\tilde{A}u_t.u_t\big)\ \text{d}z+u_{01}^\top\left(\tilde{A}_0+\dfrac{1}{2}\tilde{A}_d\right) u_{0t}+u_{1t}^\top\left(\tilde{A}_1-\dfrac{1}{2}\tilde{A}_d\right) u_{1t}+I_0^\top y_0&&\\ &= \int_0^1\left(u_t^\top\tilde{A}_{nd}u_t\right)\ \text{d}z+\!\dfrac{1}{2}\int_0^1\dfrac{\partial}{\partial z}\left(u_t^\top\tilde{A}_du_t\right)\ \text{d}z+I_0^\top y_0+u_{0t}^\top\left(\tilde{A}_0+\dfrac{1}{2}\tilde{A}_d\right) u_{0t}+u_{1t}^\top\left(\tilde{A}_1-\dfrac{1}{2}\tilde{A}_d\right)u_{1t}&&\\ &= \int_0^1\left(u_t^\top\tilde{A}_{nd}u_t\right)\ \text{d}z+u_{01}^\top\tilde{A}_0 u_{0t}+u_{1t}^\top\tilde{A}_1 u_{1t}+I_0^\top y_0\leqslant I_0^\top y_0.&& \end{align*} □ Admissible pairs for the spatial domain: First, we derive admissible pairs for the spatial domain of the transmission line, that is we find $$(\tilde {P}^{a},\tilde {A})$$ satisfying Definition (59). Next, we find suitable $$(\tilde {P}^{0},\tilde {A}_{0})$$ and $$(\tilde {P}^{1},\tilde {A}_{1})$$ satisfying (60) and (61) respectively so that we achieve the passivity as stated in Proposition 4. We construct a new mixed potential $$\tilde P$$ (for spatial domain) in a similar procedure as followed in (Brayton & Moser, 1964b) \begin{align} \tilde{P}^{a} = \lambda P^{a} +\frac{1}{2} {\int_{0}^{1}} \delta_{u} P^{a\top} M \delta_{u} P^{a} \ \text{d}z .\end{align} (63) We choose $$M= \left[{{\frac{\alpha}{R}}\atop{m_2}}\quad{{m_2}\atop{\frac{\beta}{G}}}\right]$$ where α, β, m2 are positive constants satisfying $$\alpha \frac {L}{R}=\beta \frac {C}{G}$$ and λ is a dimensionless constant. Such a choice will be clear in the following discussions, which will eventually lead to a stability criterion. Note that $$\tilde {P}$$ still have units of power. To simplify the calculations we define new positive constants θ, γ and ζ as follows: \begin{align} \theta\stackrel{\triangle}{=} \alpha\frac{L}{R}=\beta \frac{C}{G},\ \ m_{2}\stackrel{\triangle}{=} \frac{2\gamma}{CR+LG},\ \ \zeta \stackrel{\triangle}{=} \frac{2\gamma}{\sqrt{LC}(\alpha+\beta)} \implies\ \ m_{2}= \frac{\zeta \theta}{\sqrt{LC}} . \end{align} (64) To show that $$\tilde {P}^{a}\geqslant 0$$ we start by simplifying the right-hand side of (63) in the following way. Define \begin{align} \varDelta \stackrel{\triangle}{=} \left( \zeta \sqrt{\frac{C}{2}}(Ri+v_{z})-{\sqrt{\frac{L}{2}}}(Gv+i_{z}) \right)\!. \end{align} (65) Using (64), (65), and after some calculations, we can show that \begin{align*} \frac{1}{2}\left\langle\delta_{u} P,M \delta_{u} P\right\rangle = \varDelta^{2}+\frac{\alpha}{2R}(1-\zeta^{2})(Ri+v_{z})^{2}. \end{align*} With Pa as the mixed potential functional for transmission line, we calculate $$\tilde {P}^{a}$$ using (63) as follows \begin{align*} \tilde{\textrm{P}}^{a}=\lambda \textrm{P}^{a}+\varDelta^{2}+\frac{\alpha}{2R}(1-\zeta^{2})(Ri+v_{z})^{2} =\frac{\alpha(1-\zeta^{2})-\lambda}{2R}(Ri+v_{z})^{2}+\varDelta^{2}+\frac{\lambda}{2R}{v_{z}^{2}}+\frac{\lambda}{2}Gv^{2}. \end{align*} This means $$\tilde {P}^{a} = {\int _{0}^{1}} \textrm {P}^{a}\ dz\ge 0$$, for \begin{align} 0\leq \lambda\leqslant \alpha(1-\zeta^{2}), \ \ 0 \leq \zeta^{2} \leqslant 1. \end{align} (66) Further, if we choose $$\tilde A$$ as \begin{align} \tilde{A}=\left[\begin{array}{@{}cc@{}} L\left(\lambda -\alpha- m_{2} \frac{\partial }{\partial z}\right )& C\left(Rm_{2}+\frac{\beta}{G}\frac{\partial }{\partial z}\right)\\[5pt] L\left(Gm_{2}+\frac{\alpha}{R}\frac{\partial }{\partial z}\right)& -C\left(\lambda+\beta+ m_{2}\frac{\partial }{\partial z}\right) \end{array}\right] \end{align} (67) then, this $$\tilde A$$ together with $$\tilde {P}^{a}$$ will satisfy the gradient form (59). Next we can decompose $$\tilde {A}=\tilde {A}_{nd}+A_{d}\frac {\partial }{\partial z}$$ with \begin{align} \tilde{A}_{nd}=\left[\begin{array}{@{}cc@{}} L(\lambda -\alpha )& CRm_{2}\\ LGm_{2}& -C(\lambda+\beta) \end{array}\right],\quad\tilde{A}_{d}=\left[\begin{array}{@{}cc@{}} - m_{2}L &\beta\frac{C}{G}\\ \alpha\frac{L}{R}& - m_{2}C \end{array}\right] \end{align} (68) and $$\tilde {A}_{nd}$$ is negative semi-definite as long as \begin{align} -\beta \leqslant \lambda \leqslant \alpha,\ \textrm{and}\ (\lambda -\alpha)(\lambda+\beta)+\dfrac{(\alpha+\beta)^{2}}{4}\zeta^{2} \leqslant 0, \end{align} (69) and noting that $$\alpha \frac {L}{R}=\beta \frac {C}{G}$$ from (64), we can show that $$\tilde {A}_{d}$$ is symmetric. Proposition 5 If there exist non-zero constants α, β, λ and ζ satisfying (64), (66) and (69) then $$(\tilde {P^{a}}, \tilde A)$$ is an admissible pair for the transmission line. The transmission line system with zero boundary energy flow is thus stable. Proof. From (64) we define $$\tau \stackrel {\triangle }{=} \dfrac {\alpha }{\beta }=\dfrac {RC}{LG}$$. Given a transmission line, R, C, L and G are fixed. $$\tau \geqslant 0$$ is now related to system parameters and thus can be treated as one. Let $$\lambda ^{\prime }=\dfrac {\lambda }{\beta }$$. Using this in (66) and (69) we get \begin{align} 0\leqslant \lambda^{\prime}\leqslant \tau(1-\zeta^{2}) \end{align} (70) \begin{align} (\lambda^{\prime} -\tau)(\lambda^{\prime}+1)+\frac{(\tau+1)^{2}}{4}\zeta^{2} \leqslant 0. \end{align} (71) Now we have to show that for all $$\tau \geqslant 0$$, there exists a pair of λ′ and ζ that satisfies equation (70) and (71). Given a ζ ∈ (0, 1), we obtain λ′∈ [0, τ(1 − ζ2)] (using equation (70)). Showing that (71) has one positive and one negative root concludes the proof. Using the fact that a quadratic equation with roots r1 and r2 have opposite signs iff $$r_{1}r_{2}\leqslant 0$$, equation (71) leads to \begin{align*} \frac{(\tau+1)^{2}}{4}\zeta^{2}-\tau \leqslant 0 \Rightarrow \zeta^{2} \leqslant \dfrac{4\tau}{(1+\tau)^{2}}. \end{align*} Note that this is a valid condition on ζ since $$\forall \; \tau \geqslant 0$$, $$\dfrac {4\tau }{(1+\tau )^{2}}\leqslant 1$$. Therefore $$\forall \;\zeta \in [0,\frac {4\tau }{(1+\tau )^{2}}]$$ there exists a λ′ which satisfies (70) and (71). Consequently, $$(\tilde {P^{a}}, \tilde A)$$ satisfies the admissible pair’s Definition 2a. This implies stability of transmission line system with zero boundary conditions (Brayton & Miranker, 1964). □ Admissible pairs for boundary dynamics: Assume that m2 and θ satisfy $$m_{2}=\frac {C_{1}{R_{1}^{2}}}{L}=\frac {C_{1}}{C}$$ and θ = C1R1 = C0R0. Next we show that $$(\tilde {P}^{a}, \tilde {A})$$, together with \begin{align*} \tilde{P}^{0} &= \frac{1}{2R_{0}}(v_{0}-v_{C_{0}})^{2} \tilde{P}^{1} = \frac{1}{2R_{1}}(v_{1}-v_{C_{1}})^{2}\\[7pt] \tilde{A}_{0}&= \left[\begin{array}{@{}ccc@{}} -(m_{2}L+{R_{0}^{2}}C_{0}) & 0 & R_{0}C_{0} \\0& -(C_{0}+m_{2}C)& C_{0} \\-R_{0}C_{0} & -C_{0} &0 \end{array}\right],\quad \tilde{A}_{1}=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -C_{1}R_{1} \\0& 0& C_{1} \\C_{1}R_{1} & -C_{1} &0 \end{array}\right] \end{align*} satisfy Definition 2. Now considering the left hand side of (61) with λ = 1 \begin{align*} \left.\left(\dfrac{\partial \tilde{P^{1}}}{\partial u_{1}}+\tilde{\text{P}}^{a}_{u_{z}}\right)\right|_{z=1} &=\left[\begin{array}{@{}c@{}} m_{2}Li_{1t} -\theta v_{1t}\\ m_{2}Cv_{1t}-\theta i_{1t}\\ -i_{1} \end{array}\right]=\left[\begin{array}{@{}c@{}} m_{2}Li_{1t} -\theta v_{1t}\\ m_{2}Cv_{1t}-\theta i_{1t}\\ -C_{1}v_{C_{1}t} \end{array}\right] =\left[\begin{array}{@{}c@{}} -C_{1}R_{1}v_{C_{1}t}\\ C_{1}v_{C_{1}t}\\ -C_{1}v_{t}+C_{1}R_{1}i_{t} \end{array}\right]\\&=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -C_{1}R_{1} \\0& 0& C_{1} \\C_{1}R_{1} & -C_{1} &0 \end{array}\right]\left[\begin{array}{@{}c@{}} i_{1t}\\v_{1t}\\v_{C_{1}t} \end{array}\right]\!. \end{align*} We can see that $$\tilde {A}^{1}$$ is skew symmetric. Similarly we can show that $$\tilde {P}^{0}$$ and $$\tilde {A}^{0}$$ preserve boundary and satisfy (60). Proposition 6 Transmission line system defined by (7–9) is passive with storage function $$\tilde {P}=\tilde {P}^{a}+\tilde {P}^{0}+\tilde {P^{1}}$$ and port variables I0 and $$\frac {dv_{C_{0}}}{dt}$$. Proof. From definition (2) the time derivative of $$\tilde {P}$$ along the trajectories of (7--9) gives \begin{align} \dot{\tilde{P}} \leqslant I_{0}\frac{dv_{C_{0}}}{dt} \end{align} (72) which concludes the proof. □ 6. Casimirs and conservation laws We obtain conservation laws which are independent of the mixed potential function, as follows: For simplicity, we consider the case of systems without dissipation. We further assume that the energy and the co-energy variables are related via a linear relation, given by \begin{align} \alpha_{p} = \ast \varepsilon \; e_{p} \; \textrm{and}\; \alpha_{q} = \ast \mu\; e_ q \end{align} (73) we can write (17) in the following way \begin{align} \left[\begin{array}{@{}cc@{}} -\mu & 0\\ 0 & \varepsilon \end{array}\right]\left[\begin{array}{@{}c@{}} \dot{e}_{q}\\\dot{e}_{p} \end{array}\right]= \left[\begin{array}{@{}c@{}} \ast \delta_{e_{q}}P\\ \ast \delta_{e_{p}}P \end{array}\right]\!. \end{align} (74) Consider a function $$C : \varOmega ^{n-p}(Z) \times \varOmega ^{n-q}(Z) \times Z \rightarrow \mathrm R$$, which satisfies \begin{align} \mathrm d (\ast \delta_{e_{p}} C) = 0, \ \ \mathrm d (\ast \delta_{e_{q}} C) = 0. \end{align} (75) The time derivative of $$C(e_{p},e_{q})=\int _{Z}\text {C}(e_{p},e_{q})$$ along the trajectories of (74) is \begin{align*} \dfrac{d}{dt} C(e_{q},e_{p}) =& \int_{Z} \left(\delta_{e_{q}}C \wedge \dot e_{q}+\delta_{e_{p}}C \wedge \dot e_{p}\right) \\ =& \int_{Z} \left(-\delta_{e_{q}}C \wedge \ast \dfrac{1}{\mu}\mathrm{d}e_{p}(-1)^{(n-q)\times q}+\delta_{e_{p}}C \wedge \ast \dfrac{1}{\varepsilon}(-1)^{pq}\mathrm{d}e_{q}(-1)^{(n-p)\times p}\right) \\ =& \int_{Z} \left((-1)^{(n-q).q+1}\dfrac{1}{\mu}\mathrm{d}e_{p} \wedge \ast \delta_{e_{q}}C+(-1)^{p} \dfrac{1}{\varepsilon}\mathrm{d}e_{q} \wedge \ast \delta_{e_{p}}C\right) \\ =& \int_{Z} \left((-1)^{(n-q). q +1}\dfrac{1}{\mu}[\mathrm{d}(e_{p} \wedge \ast \delta_{e_{q}}C)+(-1)^{q}e_{p} \wedge \mathrm{d}(\ast \delta_{e_{q}}C)]\right.\\&\left.+(-1)^{p} \dfrac{1}{\varepsilon}\left[\mathrm{d}(e_{q} \wedge \ast \delta_{e_{p}}C)+(-1)^{p}e_{p} \wedge \mathrm{d}(\ast \delta_{e_{p}}C)\right]\right)\\ =& \int_{\partial Z} \left (e_{q} \wedge \ast \delta_{e_{p}}C) \mid_{\partial Z}+(e_{p} \wedge \ast \delta_{e_{q}}C)\! \mid_{\partial Z} \right )\!. \end{align*} This implies that $$\dot { C}$$ is a function of boundary elements, representing a conservation law. Additionally, if $$\ast \delta _{e_{p}}C = \ast \delta _{e_{q}}C =0$$, then dC/dt = 0. C is then called a Casimir function. 6.1. Example: Transmission Line In case of the lossless transmission line, the total current, and voltage \begin{align} C_{I} = {\int_{0}^{1}} i(t,z) \ \text{d}z\qquad C_{v} = {\int_{0}^{1}} v(t,z) \ \text{d}z \end{align} (76) are the systems conservation laws. This can easily be inferred by the following \begin{align*} \frac{d}{dt} C_{I} &= -{\int_{0}^{1}} \frac{1}{L}\frac{\partial v}{\partial z} = \left.\frac{v}{L} \right|_{0} - \left.\frac{v}{L}\right|_{1}\\ \frac{d}{dt} C_{v} &= -{\int_{0}^{1}} \frac{1}{C}\frac{\partial i}{\partial z} = \left.\frac{i}{C}\right|_{0} - \left.\frac{i}{C}\right|_{1}. \end{align*} Lossy Transmission line (R≠0, G≠0): Consider a functional $$C={\int _{0}^{1}} \bar {\text {C}}(i,v)\ \textrm{d}z$$, where $$\bar {\text {C}}(i,v)$$ satisfies \begin{align} \frac{R}{L}\delta_{i} C=\frac{1}{C}\frac{\partial }{\partial z}\delta_{v} C, \;\;\;\ \frac{G}{C}\delta_{v} C=\frac{1}{L}\frac{\partial }{\partial z}\delta_{i} C \end{align} (77) such as: \begin{align*} C(i,v)= {\int_{0}^{1}} \left(\dfrac{\sqrt{G}}{C} \cosh(\omega z)i+\dfrac{\sqrt{R}}{L} \cosh(\omega z)v\right) \ \text{d}z \end{align*} where $$\omega =\sqrt {RG}$$. It can be shown that the above functional satisfying (77) is a conservation law for lossy transmission line system (R≠0, G≠0) by evaluating the time derivative of C, that is \begin{align*} \dfrac{d}{dt} C(i,v) = -\left.\left(\delta_{i}C v+\delta_{v}C i \right)\right|^{1}_{0}. \end{align*} 6.2. Example: Maxwell’s Equations In case of Maxwell’s equations with no dissipation terms, it can easily be checked that the magnetic field intensity $$\int _{Z} \mathscr H$$ and the electric field intensity $$\int _{Z} \mathscr B$$ constitute the conserved quantities. This can be seen via the following expressions: \begin{align*} \int_{Z}\frac{d}{dt} \mathscr H &= -\int_{\partial Z} \frac{1}{\mu} \mathscr E \\ \int_{Z}\frac{d}{dt} \mathscr E &= \int_{\partial Z} \frac{1}{\varepsilon} \mathscr H. \end{align*} Another class of conserved quantities can be identified in the following way: Using (21), the system of equations can be rewritten as (when R = 0, G = 0) \begin{align} \left[\begin{array}{@{}cc@{}} -\mu & 0\\ 0 & \varepsilon \end{array}\right]\left[\begin{array}{@{}c@{}} \dot{e}_{q}\\ \dot{e}_{p} \end{array}\right]=\left[ \begin{array}{@{}c@{}} \ast \mathrm{d}e_{p}(-1)^{(n-q)\times q} \\ \ast (-1)^{pq}\mathrm{d}e_{q}(-1)^{(n-p)\times p} \end{array}\right]\!. \end{align} (78) Note that \begin{align*} \mathrm d \left ({\mu} \ast \dot e_{q}\right ) & = \mathrm d (\ast \ast\mathrm{d}e_{p})(-1)^{(n-q)\times q} = 0\\ \mathrm d \left ({\mu} \ast \dot e_{p}\right ) & = \mathrm d (\ast \ast \mathrm{d}e_{q})(-1)^{(n-p)\times p+pq} = 0. \end{align*} This means that d(μ * eq), d(ɛ * ep) are differential forms which do not vary with time. In terms of Maxwells Equations, this would mean $$\mathrm d (\mu \ast \mathscr H )$$ is a constant three-form representing the charge density and $$\mathrm d ( \varepsilon \ast \mathscr E )$$ is actually zero. In standard electromagnetic texts these would mean $$\nabla \cdot \mathscr D = J$$, and $$\nabla \cdot \mathscr B = 0$$, representing respectively the Gauss’ electric and magnetic law. 7. Boundary control of transmission line system In this section, we consider the stabilization problem of transmission line system in Example 1 at a non-trivial equilibrium point via boundary control. The control objective is to regulate the voltage at the capacitor C1 to $$v_{C1}^{\ast }$$ using the current source I0 connected at z = 0. We use the new passivity property (72) derived in Proposition 5, that is \begin{align} \dfrac{d}{dt}\tilde{\mathscr{P}} \leqslant I_{0}\dfrac{dv_{C0}}{dt} \end{align} (79) in achieving the boundary control objective. Boundary control: The argument used here is same as that presented in (Pasumarthy et al., 2014; Rodríguez et al., 2001), where the authors have presented a boundary control law for a mixed finite and infinite-dimensional system via energy shaping methods. But in this case, the passive maps I0 and $$v_{C_{0}}$$ (obtained using energy as storage function) do not work due to dissipation obstacle as shown in Proposition 1. Therefore we propose a boundary control law via shaping the power of the infinite-dimensional system. Towards achieving this, we adopt control by interconnection methodology using the passivity property (72). As in the finite dimensional case, the method relies on finding Casimir functions for the closed-loop system (Pasumarthy & van der Schaft, 2007). Consider the controller of the form: \begin{align} \dot \eta=u_{c}, \qquad y_{c}=\dfrac{\partial H_{c}(\eta)}{\partial \eta} \end{align} (80) where η, uc and yc are respectively the state, input and output of the controller. Hc(η) denotes the power function of the controller. The interconnection between the system and controller is given by \begin{align} \left[\begin{array}{@{}c@{}} I_{0}\\u_{c} \end{array}\right]= \left[\begin{array}{@{}cc@{}} 0 & 1\\-1 &0 \end{array}\right]\left[\begin{array}{@{}c@{}} \dfrac{dv_{C0}}{dt}\\ y_{c} \end{array}\right] \!.\end{align} (81) Casimirs: It can be easily shown that functions $$C(\eta ,v_{C_{0}})=\eta + v_{C_{0}}$$ are a casimir for the closed loop system. Time differential of $$C(\eta ,v_{C_{0}})$$ is (along (80) and (81)) \begin{align*} \dot C = \dot \eta+\dfrac{dv_{C_{0}}}{dt}= 0. \end{align*} Now the plant state and controller state are related by $$\eta =-v_{C_{0}}+c$$, c is a constant (we can take it to be zero if the initial condition of the plant is known). Using this we choose the Hamiltonian of the controller to be \begin{align*} H_{c}(\eta)=-v_{C_{0}}i_{0}^{\ast}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2} \end{align*} where $$K_{I}\geqslant 0$$ is tuning parameter. We further modify this in the following way (such modification will be useful in power shaping) \begin{align*} H_{c}(\eta)&= -v_{C_{0}}i_{0}^{\ast}\pm v_{0}i_{0}^{\ast}\pm v_{1}i_{1}^{\ast}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2}\\ &= -i_{0}i_{0}^{\ast} R_{0}+{\int_{0}^{1}} \left(v_{z}i^{\ast}+vi^{\ast}_{z} \right)\textrm{d}z-v_{1}i_{1}^{\ast}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2}\!. \end{align*} Using this controller Hamiltonian, we will shape the closed-loop mixed equilibrium points. Let c be a constant. Consider \begin{align*} P_{d} =&\ \tilde{P}^{a}+\tilde{P}^{0}+\tilde{P}^{1}+H_{c}(\eta)+c \\ =&\ {\int_{0}^{1}}\left(\frac{\alpha(1-\zeta^{2})-1}{2R}(Ri+v_{z})^{2}+\varDelta^{2}+\frac{1}{2R}{v_{z}^{2}}+v_{z}i^{\ast}+\frac{1}{2}Gv^{2}+vi^{\ast}_{z}\right)\ \text{d}z+\dfrac{1}{2}R_{0}{i_{0}^{2}}\\ &+\dfrac{1}{2}R_{1}{i_{1}^{2}}-i_{0}i_{0}^{\ast} R_{0}+c{\pm\int_{0}^{1}}\left(Ri^{\ast 2}+\dfrac{i_{z}^{\ast 2}}{2G}\right)\ \text{d}z\pm\dfrac{1}{2}R_{0}i_{0}^{\ast 2} +\frac{1}{2}K_{I}(v_{C_{0}}-v_{C_{0}}^{\ast})^{2}\\ =&\ {\int_{0}^{1}}\left(\frac{\alpha(1-\zeta^{2})-1}{2R}(Ri+v_{z})^{2}+\Delta^{2}+\dfrac{1}{2R}\left({v_{z}^{2}}+2v_{z}Ri^{\ast} +R^{2}i^{\ast 2}\right)+\frac{1}{2G}\left(Gv+i_{z}^{\ast }\right)^{2}\right)\ \text{d}z\\ &+\dfrac{1}{2}R_{0}\left({i_{0}^{2}}-2i_{0}i_{0}^{\ast} +i_{0}^{\ast 2}\right)+\dfrac{1}{2}R_{1}{i_{1}^{2}}-i_{0}i_{0}^{\ast} R_{0}+c-{\int_{0}^{1}}\left(Ri^{\ast 2}+\dfrac{i_{z}^{\ast 2}}{2G}\right)\ \text{d}z-\dfrac{1}{2}R_{0}i_{0}^{\ast 2}\\ &+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2} \!. \end{align*} By choosing $$c=\displaystyle\int _{0}^{1}\left (Ri^{\ast 2}+\dfrac {i_{z}^{\ast 2}}{2G}\right )\ \text {d}z+\dfrac {1}{2}R_{0}i_{0}^{\ast 2}$$, we can see that \begin{align} P_{d}=&\ {\int_{0}^{1}}\left(\frac{\alpha(1-\zeta^{2})-1}{2R}(Ri+v_{z})^{2}+\Delta^{2}+\dfrac{1}{2R}\left(v_{z}+Ri^{\ast} \right)^{2}+\frac{1}{2G}\left(Gv+i_{z}^{\ast} \right)^{2}\right)\ \text{d}z\nonumber\\&+\dfrac{1}{2}R_{0}\left(i_{0}-i_{0}^{\ast} \right)^{2}+\dfrac{1}{2}R_{1}{i_{1}^{2}}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2} \end{align} (82) has a minimum at the equilibrium (12), (13) and (14). The time derivative of Pd along (7–9), (80) and (81) is \begin{align} \dfrac{d}{dt}P_{d}\leqslant\left(I_{s}+K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)-i_{0}^{\ast}\right)\frac{dv_{C_{0}}}{dt}. \end{align} (83) 7.1. Stability analysis Denote $$\varDelta U=(\varDelta i, \varDelta v,\varDelta i_{0},\varDelta v_{0},\varDelta v_{C_{0}},\varDelta i_{1},$$$$\varDelta v_{1},\varDelta v_{C_{1}})$$. Consider the following norm \begin{align} \|\varDelta U\|^{2}={\int_{0}^{1}}\left((R\varDelta i+\varDelta v_{z})^{2}+\varDelta {v_{z}^{2}}+\varDelta v^{2}\right)\ \text{d}z+\varDelta {i_{0}^{2}}+\varDelta {i_{1}^{2}} +\varDelta v_{C_{0}}^{2} .\end{align} (84) Proposition 7 The transmission line system (55) in closed-loop with control \begin{align} I_{0}= i_{0}^{\ast}-K_{P}\frac{dv_{C_{0}}}{dt}-K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right),\;\;\; K_{P},K_{I} \geqslant 0 \end{align} (85) is asymptotically stable at the operating point $$U^{\ast }=(i^{\ast }, v^{\ast },i_{0}^{\ast },v_{0}^{\ast },v_{C_{0}}^{\ast },i_{1}^{\ast },v_{1}^{\ast },v_{C_{1}}^{\ast })$$ as defined in (12), (13) and (14). Proof. From (82), we can show that \begin{align} P_{d}(U)>0\quad \forall \;U\neq U^{\ast}, P_{d}(U)=0\ \textrm{if and only\ if}\ U=U^{\ast},\ \textrm{and}\ \delta_{U}P_{d}(U^{\ast})=0 . \end{align} (86) Moreover, $$\mathscr {N}(\varDelta U)=P_{d}(U^{\ast }+\varDelta U)-P_{d}(U^{\ast })$$ \begin{align*} = \int_0^1\left(\frac{\alpha(1-\zeta^2)-1}{2R}(R\varDelta i+\varDelta v_z)^2\!+\dfrac{1}{2R}\varDelta v_z^2+\frac{1}{2}G\varDelta v^2\right)\ \text{d}z+\dfrac{1}{2}R_0\varDelta i_0^2+\dfrac{1}{2}R_1\varDelta i_1^2+\frac{1}{2}K_I\varDelta v_{C_0}^2. \end{align*} For \begin{align*} \gamma_1&=\min\left\{\frac{\alpha(1-\zeta^2)-1}{2R},\frac{1}{2R},\frac{1}{2}G,\frac{1}{2}R_0,\frac{1}{2}R_1,\frac{1}{2}K_I\right\}\!,\\ \gamma_2&=\max\left\{\frac{\alpha(1-\zeta^2)-1}{2R},\frac{1}{2R},\frac{1}{2}G,\frac{1}{2}R_0,\frac{1}{2}R_1,\frac{1}{2}K_I\right\}\!, \end{align*} we have the following \begin{align} \gamma_{1}\|\Delta U\|^{2}\leqslant P_{d}(U^{\ast}+\Delta U)-P_{d}(U^{\ast})\leqslant \gamma_{2}\|\Delta U\|^{2}. \end{align} (87) Finally, using (83) and (85) the time derivative $$\dot {P}_{d}$$ is \begin{align} \dot{P}_{d}&= {\int_{0}^{1}}\left(u_{t}^{\top}\tilde{A}_{nd}u_{t}\right) \textrm{d}z+u_{01}^{\top}\tilde{A}_{0} u_{0t}+u_{1t}^{\top}\tilde{A}_{1} u_{1t}+ \left(I_{0}+K_{I}(v_{C_{0}}-v_{C_{0}}^{\ast})-i_{0}^{\ast}\right)\frac{\ dv_{C_{0}}}{dt}\nonumber\\ &= -K\left({\int_{0}^{1}}\left({i_{t}^{2}}+{v_{t}^{2}}\right)\ \text{d}z+i_{0t}^{2}+v_{C_{0}t}^{2}\right)\leqslant 0 . \end{align} (88) Arnold’s first stability theorem (Theorem 2) can be proved using (86), (87) and (88). Hence, the transmission line system (56) in closed-loop is Lyapunov stable at U* with respect to the norm ∥⋅∥ defined in (84) . Further from (12), (13) and (14), one can show that $$\dot {P_{d}}=0$$ iff U = U*. Thereby, we conclude the proof by invoking LaSalle’s invariance principle (Pasumarthy, 2006, see Theorem 5.19). □ 8. Conclusions In this paper, we presented a methodology to overcome the dissipation obstacle in the case of infinite-dimensional systems, thus paving way for passivity based control techniques. The basic building block was to write the system equations in the BM form. However, to effectively use the method, we need to construct admissible pairs for a given system, which aids in stability analysis and also in deriving new passivity properties. We present a systematic way to derive these admissible pairs and prove the stability of Maxwell’s equations. Later, we presented boundary control of mixed finite and infinite-dimensional systems. Footnotes 1 Resistive capacitive and inductive circuits. 2 Note that the transmission line system (7) can be written in infinite dimensional port-Hamiltonian formulation (16) with n = p = q = 1, this give rise to real valued (0 − forms) co-energy variable i(z, t) and v(z, t), which are just functions. References Abraham , R. , Marsden , J. E. & Ratiu , T. ( 2012 ) Manifolds, Tensor Analysis, and Applications , vol. 75. Springer Science & Business Media . Blankenstein , G . ( 2003 ) A joined geometric structure for hamiltonian and gradient control systems . IFAC Proc. Vol. , 36 , 51 -- 56 . Google Scholar CrossRef Search ADS Blankenstein , G . ( 2005 ) Geometric modeling of nonlinear RLC circuits . IEEE Trans. Circuits Syst. I: Regular Pap. , 52 , 396 -- 404 . Google Scholar CrossRef Search ADS Brayton , R. & Miranker , W. ( 1964 ) A stability theory for nonlinear mixed initial boundary value problems . Arch. Rational Mechanics Anal. , 17 , 358 -- 376 . 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Jeltsema , D. , Ortega , R. & Scherpen , J. M. ( 2003 ) On passivity and power-balance inequalities of nonlinear rlc circuits . IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. , 50 , 1174 -- 1179 . Google Scholar CrossRef Search ADS Jeltsema , D. & Scherpen , J. M. ( 2007 ) A power-based description of standard mechanical systems . Syst. Control Lett. , 56 , 349 -- 356 . Google Scholar CrossRef Search ADS Jeltsema , D. & Scherpen , J. M. ( 2009 ) Multidomain modeling of nonlinear networks and systems . IEEE Control Syst. , 29 . Jeltsema , D. & van der Schaft , A. J. ( 2007 ) Pseudo-gradient and Lagrangian boundary control system formulation of electromagnetic fields . J. Phys. A: Math. Theor. , 40 , 11627 . Google Scholar CrossRef Search ADS Koopman , J. & Jeltsema , D. ( 2012 ) Casimir-based control beyond the dissipation obstacle . IFAC Proc. Vol. , 45 , 173 -- 177 . Google Scholar CrossRef Search ADS Kosaraju , K. C. , Pasumarthy , R. & Jeltsema , D. ( 2015 ) Alternative passive maps for infinite-dimensional systems using mixed-potential functions . IFAC Workshop Lagrangian Hamiltonian Methods Non Linear Control, Lyon, France , 48 , 1 -- 6 . Kosaraju , K. C. & Pasumarthy , R. ( 2015 ) Power-based methods for infinite-dimensional systems . Math. Control Theory I. Springer , pp . 277 -- 301 . Le Gorrec , Y. , Zwart , H. & Maschke , B. ( 2005 ) Dirac structures and boundary control systems associated with skew-symmetric differential operators . SIAM J. Control Optimization , 44 , 1864 -- 1892 . Google Scholar CrossRef Search ADS Luo , Z.-H. , Guo , B.-Z. & Morgül , Ö . ( 2012 ) Stability and Stabilization of Infinite Dimensional Systems with Applications . Springer Science & Business Media . Macchelli , A. & Melchiorri , C. ( 2005 ) Control by interconnection of mixed port Hamiltonian systems . IEEE Trans. Automatic Control , 50 , 1839 -- 1844 . Google Scholar CrossRef Search ADS Ortega , R. , van der Schaft , A. J. , Mareels , I. & Maschke , B. ( 2001 ) Putting energy back in control . IEEE Control Syst. , 21 , 18 -- 33 . Google Scholar CrossRef Search ADS Ortega , R. , van der Schaft , A. J. , Maschke , B. & Escobar , G. ( 2002 ) Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems . Automatica , 38 , 585 -- 596 . Google Scholar CrossRef Search ADS Ortega , R. , Jeltsema , D. & Scherpen , J. M. ( 2003 ) Power shaping: a new paradigm for stabilization of nonlinear rlc circuits . IEEE Trans. Automatic Control , 48 , 1762 -- 1767 . Google Scholar CrossRef Search ADS Ortega , R. , van der Schaft , A. J. , Castanos , F. & Astolfi , A. ( 2008 ) Control by interconnection and standard passivity-based control of port-Hamiltonian systems . IEEE Trans. Automatic Control , 53 , 2527 -- 2542 . Google Scholar CrossRef Search ADS Pasumarthy , R. ( 2006 ) On analysis and control of interconnected finite-and infinite-dimensional physical systems . Ph.D. Thesis , Twente University Press . Pasumarthy , R. , Kosaraju , K. C. & Chandrasekar , A. ( 2014 ) On power balancing and stabilization for a class of infinite-dimensional systems . Proc. Mathematical Theory of Networks and Systems . Pasumarthy , R. & van der Schaft , A. J. ( 2007 ) Achievable casimirs and its implications on control of port-hamiltonian systems . Int. J. Control , 80 , 1421 -- 1438 . Google Scholar CrossRef Search ADS Rodríguez , H. , van der Schaft , A. J. & Ortega , R. ( 2001 ) On stabilization of nonlinear distributed parameter port-controlled Hamiltonian systems via energy shaping . Decision and Control, 2001. Proceedings of the 40th IEEE Conference on, vol. 1. IEEE , IEEE , pp. 131 -- 136 . Schöberl , M. & Siuka , A. ( 2013 ) On casimir functionals for infinite-dimensional port-hamiltonian control systems . IEEE Trans. Automatic Control , 58 , 1823 -- 1828 . Google Scholar CrossRef Search ADS Swaters , G. E. ( 1999 ) Introduction to Hamiltonian Fluid Dynamics and Stability Theory , vol. 102 . CRC Press . van der Schaft , A. J. ( 2017 ) L2-gain and Passivity Techniques in Nonlinear Control . Springer . Google Scholar CrossRef Search ADS van der Schaft , A. J. & Maschke , B. ( 2002 ) Hamiltonian formulation of distributed-parameter systems with boundary energy flow . J. Geometry Phys. , 42 , 166 -- 194 . Google Scholar CrossRef Search ADS © The Author(s) 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

# Modeling and boundary control of infinite dimensional systems in the Brayton–Moser framework

, Volume Advance Article – Dec 26, 2017
29 pages

/lp/ou_press/modeling-and-boundary-control-of-infinite-dimensional-systems-in-the-0nI9pJnSvV
Publisher
Oxford University Press
ISSN
0265-0754
eISSN
1471-6887
D.O.I.
10.1093/imamci/dnx057
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### Abstract

Abstract It is well documented that shaping the energy of finite-dimensional port-Hamiltonian systems by interconnection is severely restricted due to the presence of dissipation. This phenomenon is usually referred to as the dissipation obstacle. In this paper, we show the existence of dissipation obstacle in infinite dimensional systems. Motivated by this, we present the Brayton–Moser formulation, together with its equivalent Dirac structure. Analogous to finite dimensional systems, identifying the underlying gradient structure is crucial in presenting the stability analysis. We elucidate this through an example of Maxwell’s equations with zero energy flows through the boundary. In the case of mixed-finite and infinite-dimensional systems, we find admissible pairs for all the subsystems while preserving the overall structure. We illustrate this using a transmission line system interconnected to finite dimensional systems through its boundary. This ultimately leads to a new passive map, using this we solve a boundary control problem, circumventing the dissipation obstacle. 1. Introduction Boundary control of infinite dimensional systems is a well-studied topic. A significant advance in the port-Hamiltonian setting was presented in (van der Schaft & Maschke, 2002), where the authors extended the classical Hamiltonian formulation of infinite dimensional systems to incorporate boundary energy flow. Most infinite dimensional systems interact with the environment through its boundary, and hence such a formulation has an immediate impact on boundary control of infinite dimensional systems by energy shaping (Pasumarthy, 2006). Port-Hamiltonian systems are passive with storage function as the total energy, which is assumed to be bounded from below, and port variables being power conjugate (e.g. voltage and current). This resulted in the development of so-called ‘Energy Shaping’ methods for control of physical systems. In some cases, the standard power-conjugate port variables do not necessarily help in achieving the control objectives due to the dissipation obstacle (García-Canseco et al., 2010), motivating the search for alternative passive maps. One possible alternative that has been explored extensively in the finite-dimensional case is ‘power shaping’ using Brayton–Moser (BM) framework for modeling electrical RLC1 networks (Ortega et al., 2003). Modeling electrical networks in BM framework is a well-established theory (Brayton & Moser, 1964a,b) and has proven useful in studying the Lyapunov stability of RLC networks. The formulation was extended in (Brayton & Miranker, 1964), to the infinite-dimensional case where the authors developed a pseudo gradient framework to analyze the stability of a transmission line with non-zero boundary conditions. Later control theorists borrowed this framework to generate new passive maps (Jeltsema & Scherpen, 2007, 2009) when usual passive maps with energy as storage function render ineffective due to pervasive dissipation (García-Canseco et al., 2010). The systems in BM framework are modeled as pseudo-gradient systems using a function called mixed potential function which has units of power. Therefore, BM framework is often called as the power based framework. In case of RLC networks, these mixed potential functions are sum of three potential functions called content of all the current controlled resistors, co-content of all the voltage controlled resistors and instantaneous power transfer between storage elements. In energy shaping the total energy is used as storage function (van der Schaft, 2017; Ortega et al., 2002). But in BM formulation, we make use of the mixed potential function as the storage function and the derived passivity will either be with respect to the controlled voltages and the derivatives of currents, or the controlled currents and the derivatives of the voltages. This method has natural advantages over practical drawbacks of energy shaping methods like speeding up the transient response (as derivatives of currents and voltages are used as outputs) and also helps overcome the ‘dissipation obstacle’ (García-Canseco et al., 2010). Even though this theory was well established in finite dimensional systems, it was not fully extended to the infinite dimensional case (Jeltsema et al., 2002; Jeltsema & Scherpen, 2007). The existing literature on boundary control of infinite dimensional systems by energy shaping (in the Hamiltonian case) deals with either lossless systems (Rodríguez et al., 2001) or partially lossless systems as in (Macchelli & Melchiorri, 2005), and thus avoids dissipation obstacle issues. In (Schöberl & Siuka, 2013), the authors present a different port-Hamiltonian formulation for infinite dimensional mechanical systems in which the interconnection matrix is a constant skew-symmetric matrix in contrast to (van der Schaft & Maschke, 2002) where it is a skew-symmetric differential operator. The control strategy relies on finding the Casimirs of the interconnected system, but this methodology does not deal with the dissipation obstacle. BM formulation of Maxwell’s equations and transmission line with zero boundary energy flows is presented in (Jeltsema & van der Schaft, 2007; Pasumarthy et al., 2014) respectively. However, the admissible pairs given impose restrictions on their spatial domain (such as $$\|\frac {\partial }{\partial z}\|\leqslant$$ 1). The main contributions of this paper are as follows: BM formulation: We first motivate the need for BM formulation by proving the existence of dissipation obstacle in infinite-dimensional systems using transmission line system as an example. Thereafter, we begin with BM formulation of port-Hamiltonian system defined using Stokes’ Dirac structure. In the process, we present its Dirac formulation with a non-canonical bilinear form, similar to the finite dimensional case (Blankenstein, 2005). Zero boundary energy flows: Analogous to the finite-dimensional system, identifying the underlying gradient structure of the system is crucial in analyzing the stability. Therefore we identify alternative BM formulations called admissible pairs that help in the stability analysis, with Maxwell’s equations as an example. Non-zero boundary energy flows and passivity: In case of infinite-dimensional systems with non-zero boundary energy flows, to find admissible pairs for the overall interconnected system, we have to find these admissible pairs for all individual subsystems, that is, spatial domain and boundary, while preserving the interconnection between these subsystems. To illustrate this, we use the transmission line system (modeled by telegrapher’s equations) where the boundary is connected to a finite dimensional circuit at both ends. This ultimately leads to a new passive map with controlled current and derivatives of the voltage at boundary as port variables respectively. Boundary control: Using the new passive map, a passivity based controller is constructed to solve a boundary control problem (using control by interconnection), where the original passive maps derived using energy as storage function do not work due to the existence of pervasive dissipation. The control objective is achieved by generating Casimir functions of the overall systems. In our preliminary work, we presented BM formulation of infinite dimensional systems in (Kosaraju et al., 2015; Kosaraju & Pasumarthy, 2015). The remainder of the paper is structured as follows: In Section 2, the need for BM formulation for infinite-dimensional systems is motivated by proving the existence of dissipation obstacle. In Section 3 we present BM formulation of a distributed parameter system expressed in port-Hamiltonian framework and additionally, we express the system using the Dirac formulation. In Section 4 we find alternative BM formulations (called admissible pairs) that aid in proving the stability, with zero boundary conditions. Alternative passive maps for transmission line system with non-zero boundary conditions are presented in Section 5. Along the way, we present Casimirs and conservation laws of Maxwell equation in Section 6. In Section 7, we use the new passive maps (derived in Section 5) to solve boundary control problem for transmission line system. Finally, we conclude in Section 8. Notations and Mathematical Preliminaries Let Z be an n dimensional Riemannian manifold with a smooth (n − 1) dimensional boundary ∂Z. Ωk(Z), k = 0, 1, …, n denotes the space of all exterior k-forms on Z. The dual space $$\left (\Omega ^{k}(Z)\right )^{\ast }$$ of Ωk(Z) can be identified with Ωn−k(Z) with a pairing between α ∈ Ωk(Z) and $$\beta \in \left (\Omega ^{k}(Z)\right )^{\ast }$$ given by $$\left\langle\beta | \alpha \right\rangle=\int _{Z} \beta \wedge \alpha$$. Here ∧ is the usual wedge product of differential forms, resulting in the n-form β ∧ α. Similar pairings can be established between the boundary variables. Further we denote α|∂Z to be k-form α evaluated at boundary ∂Z. Let $$\alpha =(\alpha _{1},\alpha _{2})\in \mathscr {F}:=\Omega ^{k}(Z)\times \Omega ^{l}(\partial Z)$$ and $$\beta =(\beta _{1},\beta _{2})\in \mathscr {F}^{\ast }=\Omega ^{n-k}(Z)\times \Omega ^{n-1-l}(\partial Z)$$, then we define the following pairing between $$\mathscr {F}$$ and $$\mathscr {F}^{\ast }$$ \begin{align} \int_{(Z+ \partial Z)}\alpha \wedge \beta:=\int_{Z}\alpha_{1} \wedge \beta_{1}+\int_{\partial Z}\alpha_{2} \wedge \beta_{2}. \end{align} (1) The operator ‘d′ denotes the exterior derivative and maps k forms on Z to k + 1 forms on Z. The Hodge star operator * (corresponding to Riemannian metric on Z) converts p forms to (n − p) forms. Given α, β ∈ Ωk(Z) and γ ∈ Ωl(Z), the wedge product α ∧ γ ∈ Ωk+l(Z). We additionally have the following properties: \begin{align} \alpha \wedge \gamma = (-1)^{kl}\gamma \wedge \alpha ,\ \ast \ast \alpha = (-1)^{k(n-k)}\alpha , \end{align} (2) \begin{align} \int_{z} \alpha \wedge \ast \beta = \int_{z} \beta \wedge \ast \alpha ,\end{align} (3) \begin{align} \mathrm{d}\left(\alpha \wedge \gamma\right)= \mathrm{d}\alpha \wedge \gamma+(-1)^{k} \alpha \wedge \mathrm{d}\gamma. \end{align} (4) For details on the theory of differential forms, we refer to (Abraham et al., 2012). Given a functional H(αp, αq), we compute its variation as \begin{align} \partial H &= H(\alpha_{p}+\partial \alpha_{p},\alpha_{q}+\partial\alpha_{q})-H(\alpha_{p},\alpha_{q})\nonumber\\ &= \int_{z}\left( \delta_{p}H \wedge \partial \alpha_{p} + \delta_{q}H \wedge \partial \alpha_{q} \right)+ \int_{\partial z}\left( \delta_{ \alpha_{p}|_{\partial z}}H \wedge \partial \alpha_{p} + \delta_{\alpha_{q}|_{\partial z}}H \wedge \partial \alpha_{q} \right)\!, \end{align} (5) where αp, ∂αp ∈ Ωp(Z) and αq, ∂αq ∈ Ωq(Z); and δpH ∈ Ωn−p(Z), δqH ∈ Ωn−q(Z) are variational derivatives of H(αp, αq) with respect to αp and αq; and $$\delta _{\alpha _{p}|_{\partial z}}H \in \varOmega ^{n-p-1}(\partial Z)$$, $$\delta _{\alpha _{q}|_{\partial z}}H \in \varOmega ^{n-q-1}(\partial Z)$$ constitute variations at boundary. Further, the time derivatives of H(αp, αq) are \begin{align*} \dfrac{dH}{dt} = \int_{Z}\left ( \delta_{p}H \wedge \dfrac{\partial \alpha_{p}}{\partial t} +\delta_{q}H \wedge \dfrac{\partial \alpha_{q}}{\partial t} \right )+ \int_{\partial Z}\left( \delta_{ \alpha_{p}|_{\partial z}}H \wedge \dfrac{\partial \alpha_{p}}{\partial t} + \delta_{\alpha_{q}|_{\partial z}}H \wedge \dfrac{\partial \alpha_{q}}{\partial t} \right)\!. \end{align*} Let $$G: \varOmega ^{n-p}(Z)\rightarrow \varOmega ^{n-p}(Z)$$ and $$R: \varOmega ^{n-q}(Z)\rightarrow \varOmega ^{n-q}(Z)$$, we call $$G\geqslant 0$$, if and only if ∀αp ∈ Ωp(Z) \begin{align} \int_{Z} \left ( \alpha_{p} \wedge \ast G \alpha_{p} \right )=\int_{Z} \left\langle\alpha_{p},G\alpha_{p}\right\rangle\text{Vol} \geqslant 0 \end{align} (6) where the inner product is induced by the Riemmanian metric on Z and Vol ∈ Ωn(Z) such that $$\int _{Z} \left ( \text {Vol} \wedge \ast \text {Vol} \right )=1$$. G is said to be symmetric if $$\left\langle\alpha _{p} | G \alpha _{p}\right\rangle=\left\langle G\alpha _{p} |\alpha _{p}\right\rangle$$. Given $$u(z,t):Z \times \mathbb {R}\rightarrow \mathbb {R}$$, we denote $$\frac {\partial u}{\partial t}(z,t)$$ as ut, similarly $$\frac {\partial u}{\partial z}(z,t)$$ as uz and u*(z) represents the value of u(z, t) at equilibrium. Furthermore, for $$P(z,u,u_{z}):Z\times \mathbb {R}\times \mathbb {R}^{n}\rightarrow \mathbb {R}$$, we denote $$\frac {\partial P}{\partial u_{z}}$$ as $$P_{u_{z}}$$. 2. Motivation/examples In standard control by interconnection methodologies (Ortega et al., 2008), we assume that both plant and controller are passive. Plants that extract infinite energy at non-zero equilibrium cannot be stabilized under this assumption (Duindam et al., 2009). A port-Hamiltonian system is said to be stymied with dissipation obstacle if it extracts infinite energy at non-zero equilibrium (Ortega et al., 2001). This is a common phenomenon in finite dimensional RLC circuits (García-Canseco et al., 2010). Next, we show the existence of dissipation obstacle in infinite-dimensional systems, using transmission line system (with non-zero boundary conditions) as an example. Example 1 Let 0 < z < 1 represent the spatial domain of the transmission line with L, C, R and G denoting the specific inductance, capacitance, resistance and conductance respectively. We further assume that these are independent of the spatial variable z. Denote i(z, t) and v(z, t) as the line current and line voltage of transmission line system. Consider the transmission line system (modeled using telegraphers equations) interconnected to the boundary as shown in Fig. 1. The dynamics of this system are \begin{align} \begin{array}{ccc} -Li_{t}&=& v_{z}+Ri\\ Cv_{t}&=& -Gv-i_{z} \end{array}\quad 0<z<1 \end{align} (7) \begin{align} \begin{array}{ccc} I_{0}&=& C_{0}v_{0t}+i_{0}\\ v_{0} &=& v_{C_{0}}-i_{0}R_{0} \end{array}\quad z=0 \end{align} (8) \begin{align} \begin{array}{ccc} i_{1}&=& C_{1}v_{C_{1}t}\\ v_{1}&=& R_{1}i_{1}+v_{C_{1}} \end{array}\quad z=1, \end{align} (9) where vC0 and vC1 denote voltages across the capacitors C0 and C1 respectively and I0 represents the current source at z = 0. Additionally, the boundary voltages and currents are denoted by v0 = v(0, t), i0 = i(0, t), v1 = v(1, t) and i1 = i(1, t). Proposition 1 The transmission line system described by the equations (7–9) cannot be stabilized at any non-trivial equilibrium point with passive maps obtained by using the total energy given by \begin{align} E=\dfrac{1}{2}{\int_{0}^{1}}\left(Li^{2}+Cv^{2}\right)\ \text{d}z+\dfrac{1}{2}C_{0}v_{c_{0}}^{2}+\dfrac{1}{2}C_{1}v_{c_{1}}^{2} \end{align} (10) as the storage function. Proof. Differentiating (10) along the trajectories of (7–9), we arrive at the following inequality \begin{align} \dot E \leqslant I_{0} v_{C_{0}}. \end{align} (11) Equilibrium points: At equilibrium, equations (7–9) evaluate to \begin{align} i^{\ast}_{z} +Gv^{\ast}=0,\quad Ri^{\ast} +v^{\ast}_{z}=0\quad 0<z<1 \end{align} (12) \begin{align} I_{0}^{\ast} = i_{0}^{\ast},\quad v_{0}^{\ast} = v_{C_{0}}^{\ast}-i_{0}^{\ast} R_{0}\quad z=0 \end{align} (13) \begin{align} i_{1}^{\ast} = 0,\quad v_{1}^{\ast}= v_{C_{1}}^{\ast}\quad z=1. \end{align} (14) Finally, solving partial differential equations in (12), using the boundary conditions (13) and (14), the solution for i*(z), v*(z) takes the form \begin{align} i^{\ast}(z)=\dfrac{G}{\omega}v_{C_{1}}^{\ast} \sinh(\omega (1-z)),\quad v^{\ast}(z)= v_{C_{1}}^{\ast} \cosh(\omega (1-z) \end{align} (15) where $$\omega =\sqrt {RG}$$. Using equations (13–15) it can be shown that the supply rate $$I_{0}^{\ast } v_{C_{0}}^{\ast } \neq 0$$ at equilibrium. This implies that at equilibrium, the system extracts infinite energy from the controller, thus proving the existence of dissipation obstacle (Ortega et al., 2001). □ Fig. 1. View largeDownload slide Transmission line system. Fig. 1. View largeDownload slide Transmission line system. This problem can be circumvented either by relaxing the assumption that controller has to be passive (Koopman & Jeltsema, 2012) or by finding new passive maps (García-Canseco et al., 2010). In this note, we make use of the latter. It can be seen from (11) that ‘adding a differentiation’ on the output port variable obviates the dissipation obstacle. One alternative is to search for new passive maps within the BM framework. We start with BM formulation of an infinite-dimensional port-Hamiltonian system and derive their admissible pairs, which aids in establishing stability. 3. BM formulation of infinite-dimensional port-Hamiltonian systems In this section, we present BM formulation of infinite-dimensional port-Hamiltonian system defined using Stokes’ Dirac structure, thereby giving its Dirac formulation with a non-canonical bilinear form (refer (Blankenstein, 2005) for the finite dimensional equivalent). Define the linear space $$\mathscr {F}_{p,q}=\varOmega ^{p}(Z)\times \varOmega ^{q}(Z)\times \varOmega ^{n-p}(\partial Z)$$ called the space of flows and $$\mathscr {E}_{p,q}=\varOmega ^{n-p}(Z)\times \varOmega ^{n-q}(Z)\times \varOmega ^{n-q}(\partial Z)$$, the space of efforts, with integers p, q satisfying p + q = n + 1. Let $$(\,f_{p},f_{q},f_{b})\in \mathscr {F}_{p,q}$$ and $$(e_{p},e_{q},e_{b})\in \mathscr {E}_{p,q}$$. Then, the linear subspace $$\mathscr {D}\subset \mathscr {F}_{p,q}\times \mathscr {E}_{p,q}$$ \begin{align*} \mathscr{D}=\!\left\{\! (\,f_{p},f_{q},f_{b},e_{p},e_{q},e_{b})\in \mathscr{F}_{p,q}\times \mathscr{E}_{p,q}\,\left| \,\left[\begin{array}{@{}c@{}} f_{p}\\ f_{q} \end{array}\right]\right.\!=\! \left[\begin{array}{@{}cc@{}} 0 & (-1)^{r}\mathrm{d}\\\mathrm{d} & 0 \end{array}\right] \left[\begin{array}{@{}c@{}} e_{p}\\ e_{q} \end{array}\right], \left[\begin{array}{@{}c@{}} f_{b}\\ e_{b} \end{array}\right]\!=\! \left[\begin{array}{@{}cc@{}} 1 & 0\\0 & -(-1)^{n-q} \end{array}\right]\left[\begin{array}{@{}c@{}} e_{p}|_{\partial Z}\\ e_{q}|_{\partial Z} \end{array}\right] \right\}\!, \end{align*} with r = pq + 1, is a Stokes–Dirac structure, (van der Schaft & Maschke, 2002) with respect to the bilinear form \begin{align*} \left\langle \left\langle \left({f_{p}^{1}}, {f_{q}^{1}},{f_{b}^{1}},{e_{p}^{1}},{e_{q}^{1}},{e_{b}^{1}}\right),\left({f_{p}^{2}},{f_{q}^{2}},{f_{b}^{2}},{e_{p}^{2}},{e_{q}^{2}},{e_{b}^{2}}\right) \right\rangle \right\rangle&=\left\langle {e_{p}^{2}}\left|\,{f_{p}^{1}}\right.\right\rangle+\left\langle {e_{p}^{1}}\left|\,{f_{p}^{2}}\right.\right\rangle+\left\langle {e_{q}^{2}}\left|\,{f_{q}^{1}}\right.\right\rangle\\ &\quad+\left\langle {e_{q}^{1}}\left|\,{f_{q}^{2}}\right.\right\rangle+\left\langle {e_{b}^{2}}\left|\,{f_{b}^{1}}\right.\right\rangle+\left\langle {e_{b}^{1}}\left|\,{f_{b}^{2}}\right.\right\rangle \end{align*} where for $$i=1,2\ (\,{f_{p}^{i}},{f_{q}^{i}},{f_{b}^{i}}) \in \mathscr F_{p,q}\ \text{and}\ ({e_{p}^{i}},{e_{q}^{i}},{e_{b}^{i}})\in \mathscr E_{p,q}$$. Consider a distributed-parameter port-Hamiltonian system on Ωp(Z) × Ωq(Z) × Ωn−p(∂Z), with energy variables $$\left (\alpha _{p},\alpha _{q}\right ) \in \Omega ^{p}(Z)\times \Omega ^{q}(Z)$$ representing two different physical energy domains interacting with each other. The total stored energy is defined as \begin{align*} H:=\int_{Z} \mathbf{H} \in \mathbb{R}, \end{align*} where H is the Hamiltonian density (energy per volume element). Let $$G\geqslant 0$$ and $$R \geqslant 0$$ (satisfying (6)) represent the dissipative terms in the system. Then, setting fp = −(αp)t and fq = −(αq)t, and ep = δpH and eq = δqH, the system \begin{align} -\frac{\partial}{\partial t}\left[\begin{array}{@{}c@{}} \alpha_{p}\\ \alpha_{q} \end{array}\right]=\left[\begin{array}{@{}cc@{}} \ast G & (-1)^{r}\mathrm{d}\\\mathrm{d} & \ast R \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{p}H\\ \delta_{q}H \end{array}\right] ,\quad \left[\begin{array}{@{}c@{}} f_{b}\\ \ e_{b} \end{array}\right]=\left[\begin{array}{@{}cc@{}} 1 & 0\\0 & -(-1)^{n-q} \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{p}H{|_{\partial Z}}\\ \delta_{q}H{|_{\partial Z}} \end{array}\right]\!, \end{align} (16) with r = pq + 1, represents an infinite-dimensional port-Hamiltonian system with dissipation. The time-derivative of the Hamiltonian is computed as \begin{align*} \frac{dH}{dt} \leqslant \int_{{\partial Z}} e_{b} \wedge f_{b}. \end{align*} This implies that the system is passive with respect to the boundary variables eb, fb and storage function H (where H is assumed to be bounded from below). 3.1. The BM formulation Next, we aim to write the infinite-dimensional port-Hamiltonian system, defined with respect to a Stokes’ Dirac structure (16) in an equivalent BM form (Blankenstein, 2005; Kosaraju et al., 2015; Kosaraju & Pasumarthy, 2015; Jeltsema et al., 2003). To begin with, we assume that the mapping from the energy variables (αp, αq) to the co-energy variables (ep, eq) = (δpH, δqH) is invertible. This means the inverse transformation from the co-energy variables to the energy variables can be written as $$(\alpha _{p}, \alpha _{q}) =( \delta _{e_{p}}H^{\ast }, \delta _{e_{q}}H^{\ast })$$. H* is the co-energy of H obtained by $$H^{\ast }(e_{p}, e_{q}) = \int _{Z} \left ( (e_{p} \wedge \alpha _{p} + e_{q} \wedge \alpha _{q})\right ) - H(\alpha _{p}, \alpha _{q})$$. Further, assume that the Hamiltonian H splits as H(αp, αq) = Hp(αp) + Hq(αq), with the co-energy variables given by ep = δpHp, eq = δqHq. Consequently the co-Hamiltonian can also be split as $$H^{\ast }(e_{p}, e_{q}) = H_{p}^{\ast }(e_{p}) + H_{q}^{\ast }(e_{q})$$. We can now rewrite the spatial dynamics of the infinite-dimensional port-Hamiltonian system, in terms of the co-energy variables as \begin{align} \left[\begin{array}{@{}cc@{}} \ast \delta^{2}_{e_{p}} H^{\ast} & 0 \\ 0 & \ast \delta^{2}_{e_{q}} H^{\ast} \end{array}\right] \left[\begin{array}{@{}c@{}} -\frac{\partial e_{p}}{\partial t} \\ -\frac{\partial e_{q}}{\partial t} \end{array}\right]=\left[\begin{array}{@{}cc@{}} \ast G & (-1)^{r}\mathrm{d}\\\mathrm{d} & \ast R \end{array}\right]\left[\begin{array}{@{}c@{}} e_{p}\\ e_{q} \end{array}\right]\!. \end{align} (17) For simplicity, we assume that the relation between the energy and co-energy variables is linear and is given as \begin{align} \alpha_{p} = \ast \varepsilon \; e_{p} \; \text{and}\; \alpha_{q} = \ast \mu\; e_ q \end{align} (18) where $$\mu (=\delta ^{2}_{e_{q}} H^{\ast })$$, $$\varepsilon (= \delta ^{2}_{e_{p}} H^{\ast }) \in \mathbb {R}$$. Applying the Hodge star operator to both sides of (17) and arranging terms using (18), we get \begin{align}-\varepsilon \dot{e}_{p} &= \ast \left((-1)^{r} \mathrm{d} e_{q}+G \ast e_{p}\right)(-1)^{(n-p)\times p}\!,\nonumber \\ -\mu \dot{e}_{q} &= \ast \left(\mathrm{d}e_{p}+R \ast e_{q}\right)(-1)^{(n-q)\times q}. \end{align} (19) Next, we find a mixed-potential function $$P=\int _{Z}\text {P}(e_{p},e_{q})$$ (Jeltsema & van der Schaft, 2007) such that (19) can take the pseudo-gradient structure. The lossless case: We first consider the case of a system that is lossless, that is, when R and G are identically equal to zero in (16). To begin with, we also neglect the boundary terms by setting them to zero. Define P to be a functional of the form $$P=\int _{Z}\text {P}(e_{p},e_{q})$$, where \begin{align} \textrm{P}(e_{p},e_{q}):= e_{q}\wedge de_{p}. \end{align} (20) Its variation is given as \begin{align*} \delta P = \int_{Z}\left(\textrm{P}(e_{p}+\partial e_{p}, e_{q}+\partial e_{q})-\text{P}(e_{p},e_{q})\right) = \int_{Z}\left(e_{q} \wedge \mathrm{d}\partial e_{p} +\partial e_{q} \wedge \mathrm{d} e_{p} +\cdots\right)\!. \end{align*} Using the relation $$e_{q}\wedge \mathrm {d} \partial e_{p}=(-1)^{pq}\partial e_{p} \wedge \mathrm {d}e_{q}+(-1)^{n-q}\mathrm {d}\left (e_{q} \wedge \partial e_{p}\right )$$, and the identity (5), we have \begin{align*} \delta_{e_{q}}P=\mathrm{d}e_{p}(-1)^{(n-q)\times q}, \ \ \delta_{e_{p}}P=(-1)^{pq}\mathrm{d}e_{q}(-1)^{(n-p)\times p}, \end{align*} equation (17) can be written in the BM-type as \begin{align}\left[ \begin{array}{@{}lr@{}} -\mu & 0\\ 0 & \varepsilon \end{array}\right]\frac{\partial}{\partial t}\left[\begin{array}{@{}c@{}} e_{q}\\ \ e_{p} \end{array}\right]= \left[\begin{array}{@{}c@{}} \ast \delta_{e_{q}}P\\ \ast \delta_{e_{p}}P \end{array}\right]\!. \end{align} (21) Including dissipation: One may allow for dissipation by defining the content and co-content functions as follows. Consider instead a functional $$P=\int _{Z}\text {P}$$ defined as \begin{align} \textrm{P}(e_{p},e_{q})= e_{q} \wedge \mathrm{d}e_{p}+\underbrace{\textrm{F}(e_{q})\textrm{Vol}}_{\textrm{content}}- \underbrace{\textrm{G}(e_{p})\textrm{Vol}}_{\textrm{co-content}} \end{align} (22) where Vol ∈ Ωn(Z) such that $$\int _{Z}\ \text{Vol}\wedge \ast \text{Vol} =1$$, the content F(eq) and the co-content G(ep) functions are defined respectively as \begin{align} \textrm{F}(e_{q})=\int_{0}^{e_{q}}\left\langle\hat{e}_{p}(e_{q}^{\prime}),\ \text{d}e_{q}^{\prime}\right\rangle,\quad \textrm{G}(e_{p})=\int_{0}^{e_{p}}\left\langle\hat{e}_{q}(e_{p}^{\prime}),\ \text{d}e_{p}^{\prime}\right\rangle \end{align} (23) where the inner product $$\left\langle\cdot ,\cdot \right\rangle$$ is induced by the Riemannian metric defined on Z. In the case of linear dissipation (16), that is $$\hat {e}_{p}(e_{q})=Re_{q}$$ and $$\hat {e}_{q}(e_{p})=Ge_{p}$$ we have \begin{align} \textrm{P}(e_{p},e_{q})&= e_{q} \wedge \mathrm{d}e_{p}+\int_{0}^{e_{q}}\left\langle Re_{q}^{\prime},\ \text{d}e_{q}^{\prime}\right\rangle\text{Vol}-\int_{0}^{e_{p}}\left\langle Ge_{p}^{\prime},\ \text{d}e_{p}^{\prime}\right\rangle\textrm{Vol} \nonumber\\[4pt] &= e_{q} \wedge \mathrm{d}e_{p}+\dfrac{1}{2}\left\langle Re_{q},e_{q}\right\rangle\textrm{Vol}-\dfrac{1}{2}\left\langle Ge_{p},e_{p}\right\rangle\textrm{Vol} \nonumber\\[4pt] &= e_{q} \wedge \mathrm{d}e_{p}+\underbrace{\frac{1}{2}R e_{q} \wedge \ast e_{q}}_{\textrm{content}}- \underbrace{\frac{1}{2}G e_{p} \wedge \ast e_{p}}_{\textrm{co-content}} \end{align} (24) where in the third step we have used (6). The variation in P is computed as \begin{align*} \delta P =& \int_{Z}\left(e_{q} \wedge \mathrm{d}\partial e_{p} +\partial e_{q} \wedge \mathrm{d} e_{p} + \frac{1}{2}(e_{q} \wedge R \ast \partial e_{q}+\partial e_{q} \wedge \ast e_{q})- \frac{1}{2}(e_{p} \wedge G \ast \partial e_{p}+\partial e_{p} \wedge \ast e_{p}\right) \\[5pt] =& \int_{Z}\left( \partial e_{q} \wedge \mathrm{d}e_{p}+\partial e_{p} \wedge (-1)^{pq} \mathrm{d} e_{q}+ \frac{1}{2}(e_{q} \wedge R \ast \partial e_{q}+\partial e_{q} \wedge \ast e_{q})\right. \\[4pt]&\quad\quad\left.- \frac{1}{2}(e_{p} \wedge G \ast \partial e_{p}+\partial e_{p} \wedge \ast e_{p})+(-1)^{n-q}\mathrm{d}\left(e_{q} \wedge \partial e_{p}\right) \right)\\[4pt] =&\int_{Z} \partial e_{q} \wedge \left(\mathrm{d} e_{p}+R \ast e_{q} \right)+\partial e_{p} \wedge \left((-1)^{pq} \mathrm{d} e_{q}-G \ast e_{p}\right)+(-1)^{n-q}\int_{\partial Z}\left(e_{q} \wedge \partial e_{p}\right) \end{align*} where we have used the relation $$e_{q}\wedge \mathrm {d} \partial e_{p}=(-1)^{pq}\partial e_{p} \wedge \mathrm {d}e_{q}+(-1)^{n-q}\mathrm {d}\left (e_{q} \wedge \partial e_{p}\right )$$, together with properties of the wedge and the Hodge star operator defined in (3) and (4). Finally, by making use of (5) we can write \begin{align} \left[\begin{array}{@{}c@{}} \delta_{e_{p}}P\\ \delta_{e_{q}}P\\\delta_{e_{p}|_{\partial z}}P\\ \delta_{e_{q}|_{\partial z}}P \end{array}\right] =\left[\begin{array}{@{}c@{}} \left((-1)^{pq} \mathrm{d} e_{q}-G \ast e_{p}\right)(-1)^{(n-p)\times p}\\(\mathrm{d}e_{p} +R\ast e_{q})(-1)^{(n-q)\times q}\\(-1)^{n-q}e_{q}|_{\partial z}\\0 \end{array}\right]\!. \end{align} (25) The system of equations (17) can be written in a concise way, similar to (21) as \begin{align} A{u_{t}} = \ast \delta_{u} P \end{align} (26) where u = [ep, eq]⊤ and A = diag(ɛ, −μ). Note that if the linearity between energy and co-energy variables is not assumed (18) then A takes the form $$\text {diag}(-\delta ^{2}_{e_{q}} H^{\ast },\delta ^{2}_{e_{p}} H^{\ast })$$. Including boundary energy flow: The system of equations (16) together with boundary terms can be rewritten as \begin{align} \mathscr{A}U_{t}&=\ast \delta_{U}P+B\ast e_{b}\nonumber\\ \dot{f}_{b}&=B^{\top} U_{t}\left(=\dot{e}_{p}|_{\partial z}\right) \end{align} (27) where U = [u;u|∂z], B = [O1I O2]⊤ and $$\mathscr {A}=\text{diag}\{A,O_{3}\}$$ with O1, O2, O3 denoting zero matrices of order (n + 1 × n − q), (n − p × n − q), (n + 1 × n + 1) respectively and I identity matrix of order (n − q). 3.2. The Dirac Formulation In this section, we aim to find an equivalent Dirac structure formalism of the BM equations of the infinite-dimensional system (27), (for an overview of Dirac structure of infinite dimensional systems we refer to (Le Gorrec et al., 2005)). As we shall see such a formulation would result in a non-canonical Dirac structure. Denote by $$f \in \mathscr F:=\varOmega ^{n-p}(Z)\times \varOmega ^{n-q}(Z)\times \varOmega ^{n-p}(\partial Z)\times \varOmega ^{n-q}(\partial Z)$$ as the space of flows and $$e \in \mathscr {E}:= \mathscr {F}^{\ast }$$, as the space of effort variables. Theorem 1 Consider the following subspace \begin{align} \mathscr D = \left \{(\,f,f_{y}, e,e_{u}) \in \mathscr F \times \mathscr Y\times \mathscr E \times \mathscr S : -\mathscr A f =\ast e+Be_{u},\ f_{y}=\ast B^{\top} f\right \} \end{align} (28) where $$\mathscr {S}$$, $$\mathscr {Y}$$ represents space of port variables eu and fy respectively defined on ∂Z. The above defined subspace constitutes a non-canonical Dirac structure, that is $$\mathscr {D}=\mathscr {D}^{\perp }$$, $$\mathscr {D}^{\perp }$$ is the orthogonal complement of $$\mathscr {D}$$ with respect to the bilinear form \begin{align} &\left\langle\left\langle\left(f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}}\right),\left(f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}}\right)\right\rangle\right\rangle\nonumber\\ &\qquad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle+ \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \mathscr A f^{2}+f^{2} \wedge \ast \mathscr A f^{1} \right ) + \left\langle{e_{u}^{1}}\big|\,{f_{y}^{2}}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle \end{align} (29) where $$\mathscr {A}:\mathscr F\rightarrow \mathscr {F}, \text{for}\ i=1,2;\ f^{i} \in \mathscr F, {f_{y}^{i}}\in \mathscr Y, e^{i} \in \mathscr E, {e_{u}^{i}}\in \mathscr S$$. Proof. We follow a similar procedure as in (van der Schaft & Maschke, 2002). We first show that $$\mathscr {D}\subset \mathscr {D}^{\perp }$$, and secondly $$\mathscr {D}^{\perp }\subset \mathscr {D}$$. $$\mathscr {D}\subset \mathscr {D}^{\perp }$$: Consider $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}$$, if we show that $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}^{\perp }$$ then $$\mathscr {D}\subset \mathscr {D}^{\perp }$$. Now consider any $$(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}})\in \mathscr {D}$$ i.e. satisfying (28), substituting in the bilinear form (29) gives \begin{align*} &\left\langle\left\langle\left(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}}\right),\left(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}}\right)\right\rangle\right\rangle\\ &\quad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle+ \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \mathscr A f^{2}+f^{2} \wedge \ast \mathscr A f^{1} \right ) + \left\langle{e_{u}^{1}}\big|\,{f_{y}^{2}}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle\\ &\quad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle- \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast\left(\ast e^{2}+B{e_{u}^{2}}\right)+f^{2} \wedge \ast\left(\ast e^{1}+B{e_{u}^{1}}\right) \right ) \\ &\quad\quad + \left\langle{e_{u}^{1}}\big|\ast B^{\top} f^{2}\right\rangle+\left\langle{e_{u}^{2}}\big|\ast B^{\top} f^{1}\right\rangle\\ &\quad= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle-\left\langle e^{1}|\,f^{2}\right\rangle- \left\langle e^{2}|\,f^{1}\right\rangle-\left\langle{e_{u}^{1}}\ast B^{\top} f^{2}\right\rangle-\left\langle{e_{u}^{2}}\big|\ast B^{\top} f^{1}\right\rangle \\ &\quad\quad+\left\langle{e_{u}^{1}}\big|\ast B^{\top} f^{2}\right\rangle+\left\langle{e_{u}^{2}}\big|\ast B^{\top} f^{1}\right\rangle\\ &\quad = 0 \end{align*} where in step 2 we used the properties of wedge product (2) and (4), that is, \begin{align} f^{1}\wedge\ast \ast e^{2}&=e^{2}\wedge f^{1}\ \textrm{and}\ f^{2}\wedge\ast \ast e^{1}=e^{1}\wedge f^{2}\nonumber\\ f^{1}\wedge \ast B {e^{2}_{u}}&=B {e^{2}_{u}}\wedge \ast f^{1}={e^{2}_{u}}\wedge \ast B^{\top} f^{1}\ \textrm{and}\ f^{2}\wedge \ast B {e^{1}_{u}}=B {e^{1}_{u}}\wedge \ast f^{2}={e^{1}_{u}}\wedge \ast B^{\top} f^{2}. \end{align} (30) This implies $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}^{\perp }$$ implying $$\mathscr {D}\subset \mathscr {D}^{\perp }$$. $$\mathscr {D}^{\perp }\subset \mathscr {D}$$: Consider $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}^{\perp }$$ and if we show that $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}$$ then we are through. Now consider any $$(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}})\in \mathscr {D}$$, implies \begin{align} \left\langle\left\langle\left(f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}}\right),\left(f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}}\right)\right\rangle\right\rangle=0 \end{align} (31) now simplifying the right hand side of (31) we get \begin{align*} &= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle+ \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \mathscr A f^{2}+f^{2} \wedge \ast \mathscr A f^{1} \right ) + \left\langle{e_{u}^{1}}\big|\,{f_{y}^{2}}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle\\ &= \left\langle e^{1}|\,f^{2}\right\rangle+ \left\langle e^{2}|\,f^{1}\right\rangle- \int_{\left(Z+\partial Z\right)} \left (f^{1} \wedge \ast \left(\ast e^{2}+B{e_{u}^{2}}\right) \right ) + \int_{\left(Z+\partial Z\right)} \left (f^{2} \wedge \ast \mathscr A f^{1} \right )\\ &\quad+ \left\langle{e_{u}^{1}}\big|\ast B^{\top} f^{2}\right\rangle+\left\langle{e_{u}^{2}}\big|\,{f_{y}^{1}}\right\rangle\\ &= \int_{\left(Z+\partial Z\right)} \left (f^{2} \wedge \ast \left(\mathscr A f^{1} +\ast e^{1}+B{e_{u}^{1}}\right)\right )+\left\langle{e_{u}^{2}}\big|\left({f_{y}^{1}}-\ast B^{\top} f^{1}\right)\right\rangle \end{align*} where in step 2 we used the fact that $$(\,f^{2},{f_{y}^{2}},e^{2},{e_{u}^{2}})\in \mathscr {D}$$, in step 3 we used the wedge operator properties in (30). From (31) we get that, for all f2, $${e^{2}_{u}}$$ \begin{align} \int_{\left(Z+\partial Z\right)} \left (f^{2} \wedge \ast \left(\mathscr A f^{1} +\ast e^{1}+B{e_{u}^{1}}\right)\right )+\left\langle{e_{u}^{2}}\big|\left({f_{y}^{1}}-\ast B^{\top} f^{1}\right)\right\rangle=0. \end{align} (32) This clearly implies \begin{align*} \mathscr A f^{1} +\ast e^{1}+B{e_{u}^{1}}&=0\\ {f_{y}^{1}}-\ast B^{\top} f^{1}&=0 \end{align*} proving that $$(\,f^{1},{f_{y}^{1}},e^{1},{e_{u}^{1}})\in \mathscr {D}$$. □ Proposition 2 The port-Hamiltonian system (16) or BM equations (27) can be equivalently written as a dynamical system with respect to the non-canonical Dirac structure $$\mathscr {D}$$ in Theorem 1 by setting \begin{align} (\,f,f_{y},e,e_{u})=\left ( -U_{t},\,-\ast \dot{f}_{b},\, \delta_{U}\text{P},\, \ast e_{b}\right )\!. \end{align} (33) Moreover, the non-canonical bilinear form (29) evaluates to the ‘power balance equation’ (Blankenstein, 2005) \begin{align} \dfrac{\partial}{\partial t}\mathscr P = \int_{Z} u_{t}\wedge \ast A u_{t} - \int_{\partial Z}\left(e_{b}\wedge \dot{f}_{b}\right)\!. \end{align} (34) Proof. The first part of the Proposition can be verified by using (33) in the Dirac structure (28). For the second part, consider the following. The bilinear form (29) is assumed to be non-degenerate, hence $$\mathscr D=\mathscr {D}^{\perp }$$ implies \begin{align*} \left\langle\left\langle(\,f,f_{y},e,e_{u}),(\,f,f_{y},e,e_{u})\right\rangle\right\rangle=0,\ \quad \forall (\,f,f_{y},e,e_{u})\in \mathscr{D} \end{align*} and can be simplified to \begin{align} \left\langle e|f\right\rangle+ \int_{\left(Z+ \partial Z\right)} \left (\,f \wedge \ast \mathscr A f\right ) + \left\langle e_{u}|\,f_{y}\right\rangle=0 \end{align} (35) finally using (33) we arrive at the power balance equation (34). We can now interconnect (27) to other BM systems defined at the boundary ∂Z using these new port variables eb and $$-\dot {f}_{b}$$ (Blankenstein, 2003). □ 3.3. A Passivity Argument Once we have written down the equations in the BM framework (sometimes also referred to as the pseudo-gradient form (Jeltsema & van der Schaft, 2007)) we can pose the following question: does the mixed potential function serve as a storage function (or a Lyapunov function) to infer passivity (or equivalently stability) properties of the system? A first look at the balance equation (34) might suggest that the system in the BM form (27) is passive with $$\mathscr P$$ serving as the storage function and port variables $$-\dot f_{b}$$ and eb. Unfortunately though (see (Brayton & Miranker, 1964; Jeltsema & van der Schaft, 2007)) this is not the case as the mixed potential function $$\mathscr P$$ and its time derivative (34) are sign in-definite and hence do not serve as a storage function. This motivates our quest for finding a new $$\mathscr {P}\geqslant 0$$ and $$\mathscr {A}\leqslant 0$$, called as admissible pairs, enabling us to derive certain new passivity/stability properties. This work aims to answer these issues. 4. Stability In the case of infinite dimensional systems to prove Lyapunov stability, it is not sufficient enough to show the positive definiteness of the Lyapunov function and the negative definiteness of its time derivative (as in the case of finite dimensional systems). In infinite dimensional systems, one must specify the norm associated with stability argument because stability with respect to a norm does not imply that it is stable with respect to another norm. Let $$\mathscr {U}_{\infty }$$ be the configuration space of a distributed parameter system, and ∥⋅∥ be a norm on $$\mathscr {U}_{\infty }$$. Definition 1 (Luo et al., 2012) Denote by $$\;U^{\ast }\in \mathscr {U}_{\infty }$$ an equilibrium configuration for a distributed parameter system on $$\mathscr {U}_{\infty }$$. Then, U* is said to be stable in the sense of Lyapunov with respect to the norm ∥⋅∥ if, for every $$\varepsilon \geqslant 0$$ there exist $$\delta \geqslant 0$$ such that, \begin{align*} \|U(0)-U^{\ast}\|\leqslant \delta \implies \|U(t)-U^{\ast}\|\leqslant\varepsilon \end{align*} for all $$t\geqslant 0$$, where $$U(0)\in \mathscr {U}_{\infty }$$ is the initial configuration of the system. We state the following stability theorem for infinite-dimensional systems, which is also referred to as Arnolds theorem for the stability of infinite-dimensional systems. Theorem 2 (Stability of an infinite-dimensional system (Swaters, 1999)): Consider a dynamical system $$\dot {U}=f(U)$$ on a linear space $$\mathscr {U}_{\infty }$$, where $$U^{\ast }\in \mathscr {U}_{\infty }$$ is an equilibrium. Assume there exists a solution to the system and suppose there exists function $$P_{d}:\mathscr {U}_{\infty }\rightarrow \mathbb {R}$$ such that \begin{align} \delta_{U}P_{d}(U^{\ast})=0\ \ \textrm{and}\ \ \dfrac{\partial P_{d}}{\partial t} \leqslant 0. \end{align} (36) Denote ΔU = U − U* and $$\mathscr {N}(\Delta U)= P_{d}(U^{\ast } +\Delta U)-P_{d}(U^{\ast })$$. Show that there exist a positive triplet α, γ1 and γ2 satisfying \begin{align} \gamma_{1}\|\Delta U\|^{2} \leqslant \mathscr{N}(\Delta U) \leqslant \gamma_{2}\|\Delta U\|^{\alpha}. \end{align} (37) Then U* is a stable equilibrium. 4.1. Admissible pairs and stability To infer stability properties of the system (26), let us begin with the case of zero energy flow through the boundary of the system. The mixed-potential function (24) is not positive definite. Hence, we cannot use it as Lyapunov or storage functional. Moreover, the rate of change of this function is computed as \begin{align*} \dot{{P}}= \int_{{Z}} \left ( -\mu \dot{e_{p}} \wedge \ast \dot{e_{p}} +\varepsilon \dot{e_{q}} \wedge \ast \dot{e_{q}} \right )\!, \end{align*} which clearly is not sign-definite. We thus need to look for other admissible pairs$$(\tilde {A}$$, $$\tilde {P})$$ like in the case of finite-dimensional systems (Jeltsema et al., 2003) that can be used to prove the stability of the system while preserving the dynamics of (26). Moreover, the admissible pair should be such that the symmetric part of $$\tilde {A}$$ is negative semi-definite. This can be achieved in the following way (Brayton & Miranker, 1964; Jeltsema & van der Schaft, 2007). Consider functional $$\tilde {P}=\int _{Z}\tilde {\text {P}}$$ of the form \begin{align} \tilde{P} = \lambda P+\frac{1}{2}\int_{Z} \left(\delta_{e_{p}}P \wedge M_{1}\ast \delta_{e_{p}}P+\delta_{e_{q}}P \wedge M_{2}\ast \delta_{e_{q}}P\right)\!, \end{align} (38) with $$\lambda \in \mathbb {R}$$ an arbitrary constant and symmetric mappings $$M_{1}:\varOmega ^{p}(Z) \rightarrow \varOmega ^{p}(Z)$$ and $$M_{2}:\varOmega ^{q}(Z) \rightarrow \varOmega ^{q}(Z)$$ are linear maps. Here the aim is to find λ, M1 and M2 such that \begin{align} \dot{\tilde{P}} = u_{t}^{\top} \tilde{A} u_{t} \leqslant -K ||u_{t}||^{2} \leqslant 0, \end{align} (39) where K ≥ 0 represents the magnitude of smallest eigenvalue of $$\tilde { A}$$. If we can find such a pair $$(\tilde { P}, \tilde { A})$$, which satisfies the above condition, then we can conclude stability of the system. Theorem 3 The system of equations (26) has the alternative BM representation $$\tilde {A}u_{t}=\ast \delta _{u} \tilde {P}$$ with $$\tilde {P}$$ defined as in (38) and \begin{align} \tilde{A} \stackrel{\triangle}=\left[ \begin{array}{@{}cc@{}} -\mu\left(\lambda I+ R^{\top} M_{1}\right)& \varepsilon M_{2} \ast \mathrm{d} (-1)^{(n-p)\times p}\\ -\mu (-1)^{q}M_{1}\ast \mathrm{d} & \varepsilon\left(\lambda I-G^{\top} M_{2}\right) \end{array}\right]\!. \end{align} (40) The new mixed potential function satisfies, $$\tilde {P}\geqslant 0$$ for −∥M1R∥s < λ < ∥M2G∥s, where ∥⋅∥s denotes the spectral norm. Additionally, for systems with p = q and ɛM2 = μM1; symmetric part of $$\tilde {A}$$ is negative definite. Proof. We start with finding the variational derivative of $$\tilde {P}$$. Consider the term $$\delta _{e_{p}}P \wedge M_{1}\ast \delta _{e_{p}}P$$ \begin{align*} &=\left((-1)^{pq} \mathrm{d} e_q -\ast G e_p\right) \wedge M_2 \ast \left((-1)^{pq} \mathrm{d} e_q -\ast G e_p\right)\\ &=\mathrm{d} e_q \wedge M_2 \ast \mathrm{d}e_q-(-1)^p\mathrm{d}e_q \wedge M_2 G e_p+e_p\wedge\ast G^\top M_2Ge_p. \end{align*} The variation in first term deq ∧ M2 *deq is \begin{align*} \mathrm{d}(e_q+\partial e_q) \wedge \ast M_2 \mathrm{d}(e_q+\partial e_q)-\mathrm{d} e_q \wedge M_2 \ast \mathrm{d}e_q &=\mathrm{d} \partial e_q \wedge \ast M_2 \mathrm{d} e_q + \mathrm{d} e_q \wedge \ast M_2 \mathrm{d} \partial e_q\textbf{}+ \cdots\\ &= 2 \mathrm{d} \partial e_q \wedge \ast M_2 \mathrm{d} e_q+\cdots \end{align*} the variation in the second term deq ∧ M2Gep is \begin{align*} \mathrm{d}(e_q+\partial e_q) \wedge M_2 G (e_p+\partial e_p)- \mathrm{d} e_q \wedge M_2 G e_p&=\mathrm{d}e_q \wedge M_2 G \partial e_p + \mathrm{d}\partial e_q \wedge M_2 G e_p+\cdots\\ &=\partial e_p \wedge G^ \top M_2 \mathrm{d}e_q (-1)^{(n-p)\times p} + \mathrm{d}\partial e_q \wedge M_2 G e_p+\cdots \end{align*} finally the variation in the last term ep ∧*G⊤M2Gep is given by \begin{align*} (e_p+\partial e_p) \wedge \ast G^\top M_2 G (e_p+\partial e_p)- e_p \wedge \ast G^\top M_2 G e_p =\partial e_p \wedge 2 \ast G^\top M_2 G e_p. \end{align*} By the properties of the exterior derivative: \begin{align*} \mathrm{d}(\partial e_q \wedge \ast M_2 \mathrm{d}e_q) &=\mathrm{d} \partial e_q \wedge \ast M_2 \mathrm{d} e_q+\partial e_q \wedge (-1)^{(n-q)} \mathrm{d} \ast \mathrm{d} M_2 e_q\\ \mathrm{d}(\partial e_q \wedge M_2 G e_p) &= \mathrm{d}\partial e_q \wedge M_2 G e_p + (-1)^{n-q} \partial e_q \wedge M_2 G \mathrm{d} e_p \end{align*} the variation in $$\delta _{e_{p}}P \wedge M_{1}\ast \delta _{e_{p}}P$$ can be simplified to as \begin{align*} &\partial e_q \wedge 2 \left( (-1)^{p} \mathrm{d} \ast \mathrm{d} M_2 e_q -M_2 G\mathrm{d}e_p\right) +\partial e_p \wedge 2 \left( (-1)^{pq+1}G^\top M_2 \mathrm{d}e_q + \ast G^\top M_2 G e_p \right)\\ &\quad=\partial e_q \wedge 2(-1)^{(n-p)\times p}M_2 \mathrm{d} \ast \left( (-1)^{pq} \mathrm{d}e_q - \ast G e_p \right) +\partial e_p \wedge -2G^\top M_2\left( (-1)^{pq} \mathrm{d}e_q - \ast G e_p \right)\!. \end{align*} Similarly the variation in $$\delta _{e_{q}}P \wedge M_{1}\ast \delta _{e_{q}}P$$ is calculated as \begin{align*} &\partial e_{q} \wedge 2 \left( R^{\top} M_{1} \mathrm{d}e_{p} +\ast R^{\top} M_{1} R e_{q} \right) +\partial e_{p} \wedge 2 \left((-1)^{q} \mathrm{d} \ast \mathrm{d} M_{1} e_{p} +(-1)^{pq} M_{1} R \mathrm{d}e_{q} \right)\\ &\quad= \partial e_{q} \wedge 2R^{\top} M_{1} \left( \mathrm{d}e_{p} +\ast R e_{q} \right) +\partial e_{p} \wedge 2(-1)^{q}M_{1}\mathrm{d} \ast \left(\mathrm{d} e_{p} + \ast R \mathrm{d}e_{q} \right)\!. \end{align*} Together the variational derivative of $$\tilde {P}$$ can be computed as \begin{align*} \delta\tilde{P} \stackrel{\triangle}=\left[ \begin{array}{@{}cc@{}} \lambda I+ R^{\top} M_{1} & M_{2} \mathrm{d} \ast (-1)^{(n-p)\times p}\\ (-1)^{q}M_{1}\mathrm{d} \ast & \lambda I-G^{\top} M_{2} \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{e_{p}}P\\ \delta_{e_{q}}P \end{array}\right]\!. \end{align*} Further \begin{align*} \ast \delta\tilde{P} &= \left[\begin{array}{@{}cc@{}} \lambda I+ R^{\top} M_{1} & M_{2} \ast \mathrm{d} (-1)^{(n-p)\times p}\\ (-1)^{q}M_{1}\ast \mathrm{d} & \lambda I-G^{\top} M_{2} \end{array}\right]\left(\ast \left[\begin{array}{@{}c@{}} \delta_{e_{p}}P\\ \delta_{e_{q}}P \end{array}\right]\right)\\ &=\left[ \begin{array}{@{}cc@{}} -\mu\left(\lambda I+ R^{\top} M_{1}\right) & \varepsilon M_{2} \ast \mathrm{d} (-1)^{(n-p)\times p}\\ -\mu (-1)^{q}M_{1}\ast \mathrm{d} & \varepsilon\left(\lambda I-G^{\top} M_{2}\right) \end{array}\right]\left[\begin{array}{@{}c@{}} \dot{e}_{q}\\\dot{e}_{p} \end{array}\right] =\tilde{A}u_{t}. \end{align*} This concludes the first part of the proof. We next to show the positive definiteness of $$\tilde {P}$$. Before that we simplify P in equation (24) as follows: \begin{align*} \textrm{P}(e_p,e_q)&=e_q \wedge \mathrm{d}e_p+\frac{1}{2}R e_q \wedge \ast e_q- \frac{1}{2}G e_p \wedge \ast e_p \\ &=\frac{R^{-1}}{2}\left(\ast R e_q \wedge \ast \ast R e_q +\mathrm{d}e_p \wedge \ast \ast R e_q + \ast Re_q \wedge \ast \mathrm{d}e_p +\mathrm{d}e_p \wedge \ast \mathrm{d}e_p- \mathrm{d}e_p \wedge \ast \mathrm{d}e_p\right)\\ & \quad-\frac{1}{2} Ge_p \wedge \ast e_p \\ &= \frac{R^{-1}}{2}\left(\delta_{e_p}P \wedge \ast \delta_{e_p}P \right) -\frac{R^{-1}}{2}\mathrm{d}e_p \wedge \ast \mathrm{d}e_p -\frac{1}{2} Ge_p \wedge \ast e_p \end{align*} for −∥M1R∥s < λ < 0 we have \begin{align*} \tilde{\text P} = \frac{\lambda R^{-1}+ M_1 }{2} \left(\delta_{e_p}P \wedge \ast \delta_{e_p}P \right) - \frac{\lambda R^{-1}}{2}\mathrm{d}e_p \wedge \ast \mathrm{d}e_p - \frac{\lambda I}{2} Ge_p \wedge \ast e_p + \frac{M_2}{2} \left( \delta_{e_q}P \wedge \ast \delta_{e_q}P \right)> 0 . \end{align*} In a similar way, we can show that \begin{align} \text P(e_{p},e_{q})= -\frac{G^{-1}}{2}\left( \delta_{e_{q}}P \wedge \ast \delta_{e_{q}}P \right) +\frac{G^{-1}}{2}\mathrm{d}e_{q} \wedge \ast \mathrm{d}e_{q} +\frac{1}{2} Re_{q} \wedge \ast e_{q} \end{align} (41) hence for 0 < λ < ∥M2G∥s, we have \begin{align*} \tilde{\text P} = -\frac{\lambda G^{-1} - M_2 }{2}\left( \delta_{e_q}P \wedge \ast \delta_{e_q}P \right) + \frac{\lambda G^{-1}}{2}\mathrm{d}e_q \wedge \ast \mathrm{d}e_q + \frac{\lambda}{2} Re_q \wedge \ast e_q + \frac{M_1}{2} \left( \delta_{e_p}P \wedge \ast \delta_{e_p}P \right)> 0 \end{align*} concluding that $$\tilde P$$ is positive definite for −∥M1R∥s < λ < ∥M2G∥s. Furthermore, with p = q and ɛM2 = μM1 one can easily that symmetric part of $$\tilde {A}$$ is negative definite. □ Remark 1 Note that, if we do not restrict M1 and M2 such that ɛM2 = μM1 in Theorem 3 then for symmetric part of $$\tilde {A}\leqslant 0$$ will lead to constraints on spatial domain like $$\sigma ^{-1}\sqrt {\varepsilon \mu ^{-1}}\|\ast \mathrm {d}\|<1$$, as given in (Jeltsema & van der Schaft, 2007). Example 2 (Maxwell’s equations) Consider an electromagnetic medium with spatial domain $$Z \subset \mathbb R^{3}$$ with a smooth two-dimensional boundary ∂Z. The energy variables (2-form on Z) are the electric field induction $$\mathscr {D}=\frac {1}{2}\mathscr {D}_{ij}z_{i} \wedge z_{j}$$ and the magnetic field induction $$\mathscr {B}= \frac {1}{2}\mathscr {B}_{ij}z_{i} \wedge z_{j}$$ on Z. The associated co-energy variables are electric field intensity $$\mathscr {E}$$ and magnetic field intensity $$\mathscr {H}$$. These co-energy variables (1-form) are linearly related to the energy variables through the constitutive relationships of the medium as \begin{align} \ast \mathscr{D} = \varepsilon \mathscr{E}, \quad \ast \mathscr{B} = \mu \mathscr{H}, \end{align} (42) where ɛ(z, t) and μ(z, t) denote the electric permittivity and the magnetic permeability, respectively. Hamiltonian formulation: The Hamiltonian H can be written as \begin{align} H(\mathscr{D},\mathscr{B})=\int_{Z} \frac{1}{2}\left(\mathscr{E}\wedge \mathscr{D}+ \mathscr{H}\wedge \mathscr{B}\right)\!. \end{align} (43) Therefore, $$\delta _{\mathscr {D}}H=\mathscr {E}$$ and $$\delta _{\mathscr {B}}H=\mathscr {H}$$. Taking into account dissipation in the system, the dynamics can be written in the port-Hamiltonian form as \begin{align} -\frac{\partial }{\partial t}\left[\begin{array}{@{}c@{}} \mathscr{D}\\ \mathscr{B} \end{array}\right]\left[ =\begin{array}{@{}cc@{}} 0& -\mathrm{d}\\ \mathrm{d}& 0 \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{\mathscr{D}}H\\ \delta_{\mathscr{B}}H \end{array}\right]\!+\!\left[\begin{array}{@{}c@{}} J_{d}\\0 \end{array}\right]=\left[\begin{array}{@{}cc@{}} \ast \sigma & -\mathrm{d}\\ \mathrm{d}& 0 \end{array}\right]\left[\begin{array}{@{}c@{}} \delta_{\mathscr{D}}H\\ \delta_{\mathscr{B}}H \end{array}\right] \end{align} (44) where $$\ast J_{d}= \sigma \mathscr {E}$$, Jd denotes the current density and σ(z, t) is the specific conductivity of the material. In addition, we define the boundary variables as fb = δDH|∂Z and eb = δBH|∂Z. Hence, we obtain $$\frac {d}{dt} H \leqslant \int _{\partial Z} \mathscr H \wedge \mathscr E$$. For n = 3, p = q = 2 and $$\alpha _{p}=\mathscr {D}$$, $$\alpha _{q}=\mathscr {B}$$ with H given in (43), Maxwell’s equations given in (44) form a Stokes–Dirac structure. The BM form of Maxwell’s equations: In order to write Maxwell’s equations in BM form, we proceed with defining the corresponding mixed-potential functional \begin{align} P=\int_{{Z}} \mathscr{H}\wedge \mathrm{d}\mathscr{E}-\dfrac{1}{2}\sigma \mathscr{E} \wedge \ast \mathscr{E}, \end{align} (45) which yields the following BM form \begin{align} \left[\begin{array}{@{}cc@{}} -\mu I_{3} & 0\\ 0 & \varepsilon I_{3} \end{array}\right]\left[\begin{array}{@{}c@{}} \mathscr{H}_{t}\\ \mathscr{E}_{t} \end{array}\right]=\left[\begin{array}{@{}c@{}} \ast \mathrm{d} \mathscr{E}\\ -\sigma \mathscr{E}+\ast \mathrm{d} \mathscr{H}\end{array}\right] =\left[\begin{array}{@{}c@{}} \ast \delta_{\mathscr{H}}P\\ \ast \delta_{\mathscr{E}} P\end{array}\right]\!. \end{align} (46) Proposition 3 ( Stability analysis) The system of equations (46) constitutes alternate BM formulation $$\tilde {A}\dot {x}=\ast \delta _{u}\tilde {P}$$, where $$\tilde {P}$$ is as defined in (38) and $$\tilde {A}$$ is defined as \begin{align} \tilde{A}= \left[\begin{array}{@{}cc@{}} -\mu \lambda I & \varepsilon M_{2} \ast \mathrm{d} \\ -\mu M_{1} \ast \mathrm{d} & \varepsilon \left(\lambda I-\sigma M_{2}\right) \end{array}\right]\!. \end{align} (47) Additionally, (46) is stable if λ, M1 > 0 and M2 > 0 are selected such that ϵM2 = μM1 and 0 < λ < σ∥M2∥s. Proof. The first part of the proof is straight forward from Theorem 3. The positive definiteness of $$\tilde {P}$$ can be seen by rewriting it as \begin{align*} \tilde{P}= \int_{z} \delta_{\mathscr{E}}P \wedge \frac{\sigma M_{2} -\lambda I}{2 \sigma} \ast \delta_{\mathscr{E}}P +\frac{1}{2 \sigma}\ \text{d}\mathscr{H} \wedge \ast \ \text{d} \mathscr{H} + \frac{1}{2} \left(\delta_{\mathscr{H}}P \wedge M_{1}\ast \delta_{\mathscr{H}}P\right) \geqslant 0. \end{align*} Under the condition 0 < λ < σ∥M2∥s, the time-derivative of $$\tilde P$$ is \begin{align*} \dot{\tilde{P}} =-\int_{Z} \left(\mu \lambda \mathscr{H}_{t} \wedge \ast \mathscr{H}_{t}+\mathscr{E}_{t} \wedge \ast (\sigma M_{2}-\lambda I)\mathscr{E}_{t}\right) \leqslant 0. \end{align*} Denote $$U=(\mathscr {H},\mathscr {E})$$, $$\varDelta U =(\varDelta \mathscr {H}, \varDelta \mathscr {E} )$$ and consider the norm \begin{align} \|\varDelta U\|^{2} =\int_{Z}\left((\varDelta \mathscr{E}-\ast \mathrm{d}\varDelta \mathscr{H})\wedge \ast( \varDelta \mathscr{E}-\ast \mathrm{d}\varDelta \mathscr{H})+\mathrm{d}\varDelta\mathscr{H}\wedge \ast \mathrm{d}\varDelta\mathscr{H}+\mathrm{d}\varDelta\mathscr{E}\wedge \ast \mathrm{d}\varDelta\mathscr{E}\right)\!. \end{align} (48) One can easily show that the system of equations (46) is stable at equilibrium U* = (0, 0) by invoking Theorem 2 with respect to the above-defined norm (48), for α = 2 and \begin{align*} \gamma_{1} = \min\left\{\dfrac{1}{2\sigma},\lambda_{1}^{\min},\lambda_{2}^{\min}\right\}, \gamma_{2} = \max\left\{\dfrac{1}{2\sigma},\lambda_{1}^{\max},\lambda_{2}^{\max}\right\} \end{align*} where $$\lambda _{1}^{\min},\lambda _{1}^{\max}$$ are the minimum and maximum eigen values of $$\frac {\sigma M_{2} -\lambda I}{2 \sigma }$$ respectively and similarly $$\lambda _{2}^{\min},\lambda _{2}^{\max}$$ for $$\frac {1}{2}M_{1}$$. □ 5. Systems with boundary interaction In this section, we present the BM formulation of infinite dimensional port-Hamiltonian systems that interact through the boundary. We derive admissible pairs and present a new passivity property. We derive these results for the transmission line system described in Example 1. 5.1. The BM form: Spatial domain dynamics: The dynamics of the transmission line (7) can be written in an equivalent BM form as follows: define a functional $$P^{a}={\int _{0}^{1}}\text {P}^{a}dz$$ where \begin{align} \textrm{P}^a=-\dfrac{1}{2}Ri\wedge \ast i+\dfrac{1}{2}Gv\wedge \ast v-i\wedge \mathrm{d}v =\left(-\frac{1}{2}Ri^2+\frac{1}{2}Gv^2-iv_z\right)dz. \end{align} (49) In order to simplify the notation, we avoid using the differential geometric notation2 . Using the line voltage and current as the state variables, we can rewrite the dynamics of the spatial domain as follows \begin{align}\left[ \begin{array}{@{}cc@{}} -L & 0\\0 & C \end{array}\right]\left[\begin{array}{@{}c@{}} i_{t}\\ v_{t} \end{array}\right] = \left[\begin{array}{@{}c@{}} \delta_{i} P^{a}\\ \delta_{v} P^{a} \end{array}\right] = \left[\begin{array}{@{}c@{}} -Ri -v_{z}\\ Gv+i_{z} \end{array}\right]\!. \end{align} (50) The above equations, with A diag {−L, C}, denote u = (i(z, t) v(z, t))⊤, can be written in a gradient form \begin{align} Au_{t}=\delta_{u}P^{a} .\end{align} (51) Boundary dynamics: The spatial domain of the transmission line system is represented by a one-dimensional manifold $$Z=(0,1)\in \mathbb {R}$$ with point boundaries ∂Z = {0, 1}. In order to incorporate boundary conditions, we consider the interconnection of the infinite-dimensional system with finite-dimensional systems, via each of the boundary ports. This type of interconnected system is usually referred to as a mixed finite and infinite-dimensional system. Next, we aim to represent the overall system in BM formulation given in equation (27). Consider now a mixed potential function of the form \begin{align} \mathscr{P}(U) = P^{a}(u)+P^{0}(u_{0})+P^{1}(u_{1}) \end{align} (52) where U = [u u0u1]⊤, P0 and P1 are the contributions to the mixed potential function arising from the boundary dynamics at z = 0 and z = 1 respectively. Similar to (27), we represent the overall dynamics of mixed finite and infinite-dimensional system in BM form. The dynamics evolving on the spatial domain (i.e. for 0 < z < 1) are given by (50) (equivalently (51)). At z = 0 the dynamics are \begin{align} A_{0}u_{0t}=\left.\left(\dfrac{\partial P^{0}}{\partial u_{0}}-\textrm{P}^{a}_{u_{z}}\right)\right|_{z=0} +B_{0} I_{0} \end{align} (53) where \begin{align*} u_{0}=[i_{0},v_{0},v_{C0}]^{\top},\quad P^{0}(u_{0})=(v_{C0}-v_{0})i_{0}-\frac{1}{2}R_{0}{i_{0}^{2}},\quad A^{0}=\ \textrm{diag}\ \{ 0,0,-C_{0}\}, \end{align*} with B0 = [0, 0, −1]⊤ as the input matrix, I0 as input, $$\text {P}^{a}_{u_{z}}=\dfrac {\partial \text {P}^{a}}{\partial u_{z}}$$ and $$u_{0t}=\dfrac {du_{0}}{dt}$$. The dynamics at boundary z = 1 are \begin{align} A_{1}u_{1t}=\left.\left(\dfrac{\partial P^{1}}{\partial u_{1}}+\textrm{P}^{a}_{u_{z}}\right)\right|_{z=1} \end{align} (54) where u1 = [i1, v1, vC1]⊤, $$P^{1}= (v_{1}-v_{C1})i_{1}-\frac {1}{2}R_{1}{i_{1}^{2}}$$ and A1 = diag{0, 0, −C1}. Together they can be written compactly in the BM form as \begin{align} \mathscr{A}U_{t}=\delta_{U}\mathscr{P}+BI_{0} \end{align} (55) $$\mathscr {A}=\text{diag}\{A,A_{0},A_{1}\}$$, B = [0 0 B0O3]⊤ and O3 = [0 0 0]. \begin{align*} \delta_{U} \mathscr{P} =\left[ \delta_{u}P \left.\left(\dfrac{\partial P^{0}}{\partial u_{0}}-\textrm{P}_{u_{z}}\right)\right|_{z=0} \left.\left(\dfrac{\partial P^{1}}{\partial u_{1}}+\textrm{P}_{u_{z}}\right)\right|_{z=1} \right]^{\top} \!. \end{align*} Remark 2 Note that the mixed potential functional is not unique. Another choice is $$P^{b}={\int _{0}^{1}} \text {P}^{b}dz$$ where \begin{align} \textrm{P}^{b}=-\frac{1}{2}Ri^{2}+\frac{1}{2}Gv^{2}+i_{z}v. \end{align} (56) This choice of Pa or Pb does not have any effect on spatial domain since it preserves the dynamics (50) and (51), as δuPa = δuPb. If we use Pb as mixed potential function instead of Pa, then we need to change P0 and P1 to $$v_{C_{0}}i_{0}-\frac {1}{2}R_{0}{i_{0}^{2}}$$ and $$-\frac {1}{2}R_{1}{i_{1}^{2}}-v_{C_{1}}i_{1}$$ respectively in (53), (54). Dirac formulation: The transmission line system in BM equations (55) can be equivalently written as \begin{align*} \left ( \left(-u_{t}, -u_{0t}, -u_{1t}\right),\ B_{0}^{\top} u_{0t},\ \left( \delta_{u} \text{P},\ P^{0}_{u_{0}}-\text{P}_{u_{z}}\big|_{z=0}, P^{1}_{u_{1}}+\text{P}_{u_{z}}\big|_{z=1}\right),\ -I_{0}\right ) \in \mathscr{D} \end{align*} with subspace D defined as in Section 3.2. This gives us the ‘balance equation’ \begin{align} \dfrac{d}{dt}\mathscr{P} = {\int^{1}_{0}} \left(Au_{t} \cdot u_{t}\right)\ \text{d}z +A_{0}u_{0t}\cdot u_{0t}+A_{1}u_{1t}\cdot u_{1t}+f^{\top}_{u0} y_{0} \end{align} (57) \begin{align} ={\int_{0}^{1}}\left(-L{i^{2}_{t}}+C{v^{2}_{t}}\right)\ \text{d}z-C_{0}\left(\dfrac{dv_{C0}}{dt}\right)^{2}-C_{1}\left(\dfrac{dv_{C1}}{dt}\right)^{2}+I_{0}\dfrac{dv_{C0}}{dt}. \end{align} (58) 5.2. Admissible pairs To find admissible pairs for transmission line system with non-zero boundary conditions, we need to define $$\tilde A$$ as the following, which will be clear in the subsequent section. In general, new $$\tilde A$$ may contain ∂/∂z in its entries (similar to *d in (40) and Remark 3). In this case, there will be an additional contribution to the terms in the boundary from $$\tilde A$$, which will be clear in Proposition 4. To account this contribution, we split such $$\tilde A$$ as $$\tilde {A}_{nd}+\tilde {A}_{d}\frac {\partial }{\partial z}$$. Definition 2 Admissible pairs: Denote $$\tilde {\mathscr {P}}=\int _{Z}\tilde {\text {P}^{a}}+\tilde {P}^{0}+\tilde {P}^{1}$$ and $$\tilde {\mathscr {A}}=\text {diag}\ \{\tilde {A}, \tilde {A}_{0}, \tilde {A}_{1}\}$$, further $$\tilde {A}$$ is $$\tilde {A}_{nd}+\tilde {A}_{d}\frac {\partial }{\partial z}$$. We call $$\tilde {\mathscr {P}}$$ and $$\tilde {\mathscr {A}}$$ admissible pairs if they satisfy the following: $$\tilde {P}^{a}\geqslant 0$$, $$\tilde A_{d}^{\top }=\tilde A_{d}$$ and $$u_{t}^{\top }\tilde {A}_{nd}u_{t}\leqslant 0$$ such that \begin{align} \tilde{A}u_{t}=\delta_{u}\tilde{\text{P}^{a}} \end{align} (59) $$\tilde {P}^{0}\geqslant 0$$ and $$u_{0t}^{\top }\tilde {A}_{0}u_{0t}\leqslant 0$$ such that \begin{align} \left(\tilde{A}_{0}+\dfrac{1}{2}\tilde{A}_{d}\right)u_{0t}=\left.\left(\dfrac{\partial \tilde{P}}{\partial u_{0}}-\tilde{\text{P}^{a}}_{u_{z}}\right)\right|_{z=0}+\tilde B_{0}I_{0} \end{align} (60) $$\tilde {P}^{1}\geqslant 0$$ and $$u_{1t}^{\top }\tilde {A}_{1}u_{1t}\leqslant 0$$ such that \begin{align} \left(\tilde{A}_{1}-\dfrac{1}{2}\tilde{A}_{d}\right)u_{1t}=\left.\left(\dfrac{\partial \tilde{P}}{\partial u_{1}}+\tilde{\text{P}}_{u_{z}}^{a}\right)\right|_{z=1} \end{align} (61) Together we can write them as \begin{align} \tilde{\mathscr{A}}U_{t} = \delta_{U} \tilde{\mathscr{P}}+ \tilde BI_{0},\quad y_{0} = - \tilde B^{\top}_{0} u_{0t}. \end{align} (62) Proposition 4 If $$\tilde {\mathscr {P}}=\int _{Z}\tilde {\text {P}^{a}}+\tilde {P}^{0}+\tilde {P}^{1}$$ and $$\tilde {\mathscr {A}}= \text {diag}\ \{\tilde {A}, \tilde {A}_{0}, \tilde {A}_{1}\}$$ satisfy the Definition 2 then $$\dot {\tilde {\mathscr {P}}}\leqslant I_{0}^{\top } y_{0}$$, that is the system is passive port variables I0 and y0. Proof. The time derivative of $$\tilde {P}_{d}\geqslant 0$$ along the trajectories of (59–61) is \begin{align*} \dot{\tilde{P}}&= \int_0^1\big(\delta_u\tilde{P^a}.u_t \big)\ \text{d}z+\left.\left(\dfrac{\partial \tilde{P}}{\partial u_0}-\tilde{\text{P}^a}_{u_z}\right)\right|_{z=0}\cdot u_{0t}+\left.\left(\dfrac{\partial \tilde{P}}{\partial u_1}+\tilde{\text{P}^a}_{u_z}\right)\right|_{z=1}\cdot u_{1t}&&\\ &= \int_0^1\big(\tilde{A}u_t.u_t\big)\ \text{d}z+u_{01}^\top\left(\tilde{A}_0+\dfrac{1}{2}\tilde{A}_d\right) u_{0t}+u_{1t}^\top\left(\tilde{A}_1-\dfrac{1}{2}\tilde{A}_d\right) u_{1t}+I_0^\top y_0&&\\ &= \int_0^1\left(u_t^\top\tilde{A}_{nd}u_t\right)\ \text{d}z+\!\dfrac{1}{2}\int_0^1\dfrac{\partial}{\partial z}\left(u_t^\top\tilde{A}_du_t\right)\ \text{d}z+I_0^\top y_0+u_{0t}^\top\left(\tilde{A}_0+\dfrac{1}{2}\tilde{A}_d\right) u_{0t}+u_{1t}^\top\left(\tilde{A}_1-\dfrac{1}{2}\tilde{A}_d\right)u_{1t}&&\\ &= \int_0^1\left(u_t^\top\tilde{A}_{nd}u_t\right)\ \text{d}z+u_{01}^\top\tilde{A}_0 u_{0t}+u_{1t}^\top\tilde{A}_1 u_{1t}+I_0^\top y_0\leqslant I_0^\top y_0.&& \end{align*} □ Admissible pairs for the spatial domain: First, we derive admissible pairs for the spatial domain of the transmission line, that is we find $$(\tilde {P}^{a},\tilde {A})$$ satisfying Definition (59). Next, we find suitable $$(\tilde {P}^{0},\tilde {A}_{0})$$ and $$(\tilde {P}^{1},\tilde {A}_{1})$$ satisfying (60) and (61) respectively so that we achieve the passivity as stated in Proposition 4. We construct a new mixed potential $$\tilde P$$ (for spatial domain) in a similar procedure as followed in (Brayton & Moser, 1964b) \begin{align} \tilde{P}^{a} = \lambda P^{a} +\frac{1}{2} {\int_{0}^{1}} \delta_{u} P^{a\top} M \delta_{u} P^{a} \ \text{d}z .\end{align} (63) We choose $$M= \left[{{\frac{\alpha}{R}}\atop{m_2}}\quad{{m_2}\atop{\frac{\beta}{G}}}\right]$$ where α, β, m2 are positive constants satisfying $$\alpha \frac {L}{R}=\beta \frac {C}{G}$$ and λ is a dimensionless constant. Such a choice will be clear in the following discussions, which will eventually lead to a stability criterion. Note that $$\tilde {P}$$ still have units of power. To simplify the calculations we define new positive constants θ, γ and ζ as follows: \begin{align} \theta\stackrel{\triangle}{=} \alpha\frac{L}{R}=\beta \frac{C}{G},\ \ m_{2}\stackrel{\triangle}{=} \frac{2\gamma}{CR+LG},\ \ \zeta \stackrel{\triangle}{=} \frac{2\gamma}{\sqrt{LC}(\alpha+\beta)} \implies\ \ m_{2}= \frac{\zeta \theta}{\sqrt{LC}} . \end{align} (64) To show that $$\tilde {P}^{a}\geqslant 0$$ we start by simplifying the right-hand side of (63) in the following way. Define \begin{align} \varDelta \stackrel{\triangle}{=} \left( \zeta \sqrt{\frac{C}{2}}(Ri+v_{z})-{\sqrt{\frac{L}{2}}}(Gv+i_{z}) \right)\!. \end{align} (65) Using (64), (65), and after some calculations, we can show that \begin{align*} \frac{1}{2}\left\langle\delta_{u} P,M \delta_{u} P\right\rangle = \varDelta^{2}+\frac{\alpha}{2R}(1-\zeta^{2})(Ri+v_{z})^{2}. \end{align*} With Pa as the mixed potential functional for transmission line, we calculate $$\tilde {P}^{a}$$ using (63) as follows \begin{align*} \tilde{\textrm{P}}^{a}=\lambda \textrm{P}^{a}+\varDelta^{2}+\frac{\alpha}{2R}(1-\zeta^{2})(Ri+v_{z})^{2} =\frac{\alpha(1-\zeta^{2})-\lambda}{2R}(Ri+v_{z})^{2}+\varDelta^{2}+\frac{\lambda}{2R}{v_{z}^{2}}+\frac{\lambda}{2}Gv^{2}. \end{align*} This means $$\tilde {P}^{a} = {\int _{0}^{1}} \textrm {P}^{a}\ dz\ge 0$$, for \begin{align} 0\leq \lambda\leqslant \alpha(1-\zeta^{2}), \ \ 0 \leq \zeta^{2} \leqslant 1. \end{align} (66) Further, if we choose $$\tilde A$$ as \begin{align} \tilde{A}=\left[\begin{array}{@{}cc@{}} L\left(\lambda -\alpha- m_{2} \frac{\partial }{\partial z}\right )& C\left(Rm_{2}+\frac{\beta}{G}\frac{\partial }{\partial z}\right)\\[5pt] L\left(Gm_{2}+\frac{\alpha}{R}\frac{\partial }{\partial z}\right)& -C\left(\lambda+\beta+ m_{2}\frac{\partial }{\partial z}\right) \end{array}\right] \end{align} (67) then, this $$\tilde A$$ together with $$\tilde {P}^{a}$$ will satisfy the gradient form (59). Next we can decompose $$\tilde {A}=\tilde {A}_{nd}+A_{d}\frac {\partial }{\partial z}$$ with \begin{align} \tilde{A}_{nd}=\left[\begin{array}{@{}cc@{}} L(\lambda -\alpha )& CRm_{2}\\ LGm_{2}& -C(\lambda+\beta) \end{array}\right],\quad\tilde{A}_{d}=\left[\begin{array}{@{}cc@{}} - m_{2}L &\beta\frac{C}{G}\\ \alpha\frac{L}{R}& - m_{2}C \end{array}\right] \end{align} (68) and $$\tilde {A}_{nd}$$ is negative semi-definite as long as \begin{align} -\beta \leqslant \lambda \leqslant \alpha,\ \textrm{and}\ (\lambda -\alpha)(\lambda+\beta)+\dfrac{(\alpha+\beta)^{2}}{4}\zeta^{2} \leqslant 0, \end{align} (69) and noting that $$\alpha \frac {L}{R}=\beta \frac {C}{G}$$ from (64), we can show that $$\tilde {A}_{d}$$ is symmetric. Proposition 5 If there exist non-zero constants α, β, λ and ζ satisfying (64), (66) and (69) then $$(\tilde {P^{a}}, \tilde A)$$ is an admissible pair for the transmission line. The transmission line system with zero boundary energy flow is thus stable. Proof. From (64) we define $$\tau \stackrel {\triangle }{=} \dfrac {\alpha }{\beta }=\dfrac {RC}{LG}$$. Given a transmission line, R, C, L and G are fixed. $$\tau \geqslant 0$$ is now related to system parameters and thus can be treated as one. Let $$\lambda ^{\prime }=\dfrac {\lambda }{\beta }$$. Using this in (66) and (69) we get \begin{align} 0\leqslant \lambda^{\prime}\leqslant \tau(1-\zeta^{2}) \end{align} (70) \begin{align} (\lambda^{\prime} -\tau)(\lambda^{\prime}+1)+\frac{(\tau+1)^{2}}{4}\zeta^{2} \leqslant 0. \end{align} (71) Now we have to show that for all $$\tau \geqslant 0$$, there exists a pair of λ′ and ζ that satisfies equation (70) and (71). Given a ζ ∈ (0, 1), we obtain λ′∈ [0, τ(1 − ζ2)] (using equation (70)). Showing that (71) has one positive and one negative root concludes the proof. Using the fact that a quadratic equation with roots r1 and r2 have opposite signs iff $$r_{1}r_{2}\leqslant 0$$, equation (71) leads to \begin{align*} \frac{(\tau+1)^{2}}{4}\zeta^{2}-\tau \leqslant 0 \Rightarrow \zeta^{2} \leqslant \dfrac{4\tau}{(1+\tau)^{2}}. \end{align*} Note that this is a valid condition on ζ since $$\forall \; \tau \geqslant 0$$, $$\dfrac {4\tau }{(1+\tau )^{2}}\leqslant 1$$. Therefore $$\forall \;\zeta \in [0,\frac {4\tau }{(1+\tau )^{2}}]$$ there exists a λ′ which satisfies (70) and (71). Consequently, $$(\tilde {P^{a}}, \tilde A)$$ satisfies the admissible pair’s Definition 2a. This implies stability of transmission line system with zero boundary conditions (Brayton & Miranker, 1964). □ Admissible pairs for boundary dynamics: Assume that m2 and θ satisfy $$m_{2}=\frac {C_{1}{R_{1}^{2}}}{L}=\frac {C_{1}}{C}$$ and θ = C1R1 = C0R0. Next we show that $$(\tilde {P}^{a}, \tilde {A})$$, together with \begin{align*} \tilde{P}^{0} &= \frac{1}{2R_{0}}(v_{0}-v_{C_{0}})^{2} \tilde{P}^{1} = \frac{1}{2R_{1}}(v_{1}-v_{C_{1}})^{2}\\[7pt] \tilde{A}_{0}&= \left[\begin{array}{@{}ccc@{}} -(m_{2}L+{R_{0}^{2}}C_{0}) & 0 & R_{0}C_{0} \\0& -(C_{0}+m_{2}C)& C_{0} \\-R_{0}C_{0} & -C_{0} &0 \end{array}\right],\quad \tilde{A}_{1}=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -C_{1}R_{1} \\0& 0& C_{1} \\C_{1}R_{1} & -C_{1} &0 \end{array}\right] \end{align*} satisfy Definition 2. Now considering the left hand side of (61) with λ = 1 \begin{align*} \left.\left(\dfrac{\partial \tilde{P^{1}}}{\partial u_{1}}+\tilde{\text{P}}^{a}_{u_{z}}\right)\right|_{z=1} &=\left[\begin{array}{@{}c@{}} m_{2}Li_{1t} -\theta v_{1t}\\ m_{2}Cv_{1t}-\theta i_{1t}\\ -i_{1} \end{array}\right]=\left[\begin{array}{@{}c@{}} m_{2}Li_{1t} -\theta v_{1t}\\ m_{2}Cv_{1t}-\theta i_{1t}\\ -C_{1}v_{C_{1}t} \end{array}\right] =\left[\begin{array}{@{}c@{}} -C_{1}R_{1}v_{C_{1}t}\\ C_{1}v_{C_{1}t}\\ -C_{1}v_{t}+C_{1}R_{1}i_{t} \end{array}\right]\\&=\left[\begin{array}{@{}ccc@{}} 0 & 0 & -C_{1}R_{1} \\0& 0& C_{1} \\C_{1}R_{1} & -C_{1} &0 \end{array}\right]\left[\begin{array}{@{}c@{}} i_{1t}\\v_{1t}\\v_{C_{1}t} \end{array}\right]\!. \end{align*} We can see that $$\tilde {A}^{1}$$ is skew symmetric. Similarly we can show that $$\tilde {P}^{0}$$ and $$\tilde {A}^{0}$$ preserve boundary and satisfy (60). Proposition 6 Transmission line system defined by (7–9) is passive with storage function $$\tilde {P}=\tilde {P}^{a}+\tilde {P}^{0}+\tilde {P^{1}}$$ and port variables I0 and $$\frac {dv_{C_{0}}}{dt}$$. Proof. From definition (2) the time derivative of $$\tilde {P}$$ along the trajectories of (7--9) gives \begin{align} \dot{\tilde{P}} \leqslant I_{0}\frac{dv_{C_{0}}}{dt} \end{align} (72) which concludes the proof. □ 6. Casimirs and conservation laws We obtain conservation laws which are independent of the mixed potential function, as follows: For simplicity, we consider the case of systems without dissipation. We further assume that the energy and the co-energy variables are related via a linear relation, given by \begin{align} \alpha_{p} = \ast \varepsilon \; e_{p} \; \textrm{and}\; \alpha_{q} = \ast \mu\; e_ q \end{align} (73) we can write (17) in the following way \begin{align} \left[\begin{array}{@{}cc@{}} -\mu & 0\\ 0 & \varepsilon \end{array}\right]\left[\begin{array}{@{}c@{}} \dot{e}_{q}\\\dot{e}_{p} \end{array}\right]= \left[\begin{array}{@{}c@{}} \ast \delta_{e_{q}}P\\ \ast \delta_{e_{p}}P \end{array}\right]\!. \end{align} (74) Consider a function $$C : \varOmega ^{n-p}(Z) \times \varOmega ^{n-q}(Z) \times Z \rightarrow \mathrm R$$, which satisfies \begin{align} \mathrm d (\ast \delta_{e_{p}} C) = 0, \ \ \mathrm d (\ast \delta_{e_{q}} C) = 0. \end{align} (75) The time derivative of $$C(e_{p},e_{q})=\int _{Z}\text {C}(e_{p},e_{q})$$ along the trajectories of (74) is \begin{align*} \dfrac{d}{dt} C(e_{q},e_{p}) =& \int_{Z} \left(\delta_{e_{q}}C \wedge \dot e_{q}+\delta_{e_{p}}C \wedge \dot e_{p}\right) \\ =& \int_{Z} \left(-\delta_{e_{q}}C \wedge \ast \dfrac{1}{\mu}\mathrm{d}e_{p}(-1)^{(n-q)\times q}+\delta_{e_{p}}C \wedge \ast \dfrac{1}{\varepsilon}(-1)^{pq}\mathrm{d}e_{q}(-1)^{(n-p)\times p}\right) \\ =& \int_{Z} \left((-1)^{(n-q).q+1}\dfrac{1}{\mu}\mathrm{d}e_{p} \wedge \ast \delta_{e_{q}}C+(-1)^{p} \dfrac{1}{\varepsilon}\mathrm{d}e_{q} \wedge \ast \delta_{e_{p}}C\right) \\ =& \int_{Z} \left((-1)^{(n-q). q +1}\dfrac{1}{\mu}[\mathrm{d}(e_{p} \wedge \ast \delta_{e_{q}}C)+(-1)^{q}e_{p} \wedge \mathrm{d}(\ast \delta_{e_{q}}C)]\right.\\&\left.+(-1)^{p} \dfrac{1}{\varepsilon}\left[\mathrm{d}(e_{q} \wedge \ast \delta_{e_{p}}C)+(-1)^{p}e_{p} \wedge \mathrm{d}(\ast \delta_{e_{p}}C)\right]\right)\\ =& \int_{\partial Z} \left (e_{q} \wedge \ast \delta_{e_{p}}C) \mid_{\partial Z}+(e_{p} \wedge \ast \delta_{e_{q}}C)\! \mid_{\partial Z} \right )\!. \end{align*} This implies that $$\dot { C}$$ is a function of boundary elements, representing a conservation law. Additionally, if $$\ast \delta _{e_{p}}C = \ast \delta _{e_{q}}C =0$$, then dC/dt = 0. C is then called a Casimir function. 6.1. Example: Transmission Line In case of the lossless transmission line, the total current, and voltage \begin{align} C_{I} = {\int_{0}^{1}} i(t,z) \ \text{d}z\qquad C_{v} = {\int_{0}^{1}} v(t,z) \ \text{d}z \end{align} (76) are the systems conservation laws. This can easily be inferred by the following \begin{align*} \frac{d}{dt} C_{I} &= -{\int_{0}^{1}} \frac{1}{L}\frac{\partial v}{\partial z} = \left.\frac{v}{L} \right|_{0} - \left.\frac{v}{L}\right|_{1}\\ \frac{d}{dt} C_{v} &= -{\int_{0}^{1}} \frac{1}{C}\frac{\partial i}{\partial z} = \left.\frac{i}{C}\right|_{0} - \left.\frac{i}{C}\right|_{1}. \end{align*} Lossy Transmission line (R≠0, G≠0): Consider a functional $$C={\int _{0}^{1}} \bar {\text {C}}(i,v)\ \textrm{d}z$$, where $$\bar {\text {C}}(i,v)$$ satisfies \begin{align} \frac{R}{L}\delta_{i} C=\frac{1}{C}\frac{\partial }{\partial z}\delta_{v} C, \;\;\;\ \frac{G}{C}\delta_{v} C=\frac{1}{L}\frac{\partial }{\partial z}\delta_{i} C \end{align} (77) such as: \begin{align*} C(i,v)= {\int_{0}^{1}} \left(\dfrac{\sqrt{G}}{C} \cosh(\omega z)i+\dfrac{\sqrt{R}}{L} \cosh(\omega z)v\right) \ \text{d}z \end{align*} where $$\omega =\sqrt {RG}$$. It can be shown that the above functional satisfying (77) is a conservation law for lossy transmission line system (R≠0, G≠0) by evaluating the time derivative of C, that is \begin{align*} \dfrac{d}{dt} C(i,v) = -\left.\left(\delta_{i}C v+\delta_{v}C i \right)\right|^{1}_{0}. \end{align*} 6.2. Example: Maxwell’s Equations In case of Maxwell’s equations with no dissipation terms, it can easily be checked that the magnetic field intensity $$\int _{Z} \mathscr H$$ and the electric field intensity $$\int _{Z} \mathscr B$$ constitute the conserved quantities. This can be seen via the following expressions: \begin{align*} \int_{Z}\frac{d}{dt} \mathscr H &= -\int_{\partial Z} \frac{1}{\mu} \mathscr E \\ \int_{Z}\frac{d}{dt} \mathscr E &= \int_{\partial Z} \frac{1}{\varepsilon} \mathscr H. \end{align*} Another class of conserved quantities can be identified in the following way: Using (21), the system of equations can be rewritten as (when R = 0, G = 0) \begin{align} \left[\begin{array}{@{}cc@{}} -\mu & 0\\ 0 & \varepsilon \end{array}\right]\left[\begin{array}{@{}c@{}} \dot{e}_{q}\\ \dot{e}_{p} \end{array}\right]=\left[ \begin{array}{@{}c@{}} \ast \mathrm{d}e_{p}(-1)^{(n-q)\times q} \\ \ast (-1)^{pq}\mathrm{d}e_{q}(-1)^{(n-p)\times p} \end{array}\right]\!. \end{align} (78) Note that \begin{align*} \mathrm d \left ({\mu} \ast \dot e_{q}\right ) & = \mathrm d (\ast \ast\mathrm{d}e_{p})(-1)^{(n-q)\times q} = 0\\ \mathrm d \left ({\mu} \ast \dot e_{p}\right ) & = \mathrm d (\ast \ast \mathrm{d}e_{q})(-1)^{(n-p)\times p+pq} = 0. \end{align*} This means that d(μ * eq), d(ɛ * ep) are differential forms which do not vary with time. In terms of Maxwells Equations, this would mean $$\mathrm d (\mu \ast \mathscr H )$$ is a constant three-form representing the charge density and $$\mathrm d ( \varepsilon \ast \mathscr E )$$ is actually zero. In standard electromagnetic texts these would mean $$\nabla \cdot \mathscr D = J$$, and $$\nabla \cdot \mathscr B = 0$$, representing respectively the Gauss’ electric and magnetic law. 7. Boundary control of transmission line system In this section, we consider the stabilization problem of transmission line system in Example 1 at a non-trivial equilibrium point via boundary control. The control objective is to regulate the voltage at the capacitor C1 to $$v_{C1}^{\ast }$$ using the current source I0 connected at z = 0. We use the new passivity property (72) derived in Proposition 5, that is \begin{align} \dfrac{d}{dt}\tilde{\mathscr{P}} \leqslant I_{0}\dfrac{dv_{C0}}{dt} \end{align} (79) in achieving the boundary control objective. Boundary control: The argument used here is same as that presented in (Pasumarthy et al., 2014; Rodríguez et al., 2001), where the authors have presented a boundary control law for a mixed finite and infinite-dimensional system via energy shaping methods. But in this case, the passive maps I0 and $$v_{C_{0}}$$ (obtained using energy as storage function) do not work due to dissipation obstacle as shown in Proposition 1. Therefore we propose a boundary control law via shaping the power of the infinite-dimensional system. Towards achieving this, we adopt control by interconnection methodology using the passivity property (72). As in the finite dimensional case, the method relies on finding Casimir functions for the closed-loop system (Pasumarthy & van der Schaft, 2007). Consider the controller of the form: \begin{align} \dot \eta=u_{c}, \qquad y_{c}=\dfrac{\partial H_{c}(\eta)}{\partial \eta} \end{align} (80) where η, uc and yc are respectively the state, input and output of the controller. Hc(η) denotes the power function of the controller. The interconnection between the system and controller is given by \begin{align} \left[\begin{array}{@{}c@{}} I_{0}\\u_{c} \end{array}\right]= \left[\begin{array}{@{}cc@{}} 0 & 1\\-1 &0 \end{array}\right]\left[\begin{array}{@{}c@{}} \dfrac{dv_{C0}}{dt}\\ y_{c} \end{array}\right] \!.\end{align} (81) Casimirs: It can be easily shown that functions $$C(\eta ,v_{C_{0}})=\eta + v_{C_{0}}$$ are a casimir for the closed loop system. Time differential of $$C(\eta ,v_{C_{0}})$$ is (along (80) and (81)) \begin{align*} \dot C = \dot \eta+\dfrac{dv_{C_{0}}}{dt}= 0. \end{align*} Now the plant state and controller state are related by $$\eta =-v_{C_{0}}+c$$, c is a constant (we can take it to be zero if the initial condition of the plant is known). Using this we choose the Hamiltonian of the controller to be \begin{align*} H_{c}(\eta)=-v_{C_{0}}i_{0}^{\ast}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2} \end{align*} where $$K_{I}\geqslant 0$$ is tuning parameter. We further modify this in the following way (such modification will be useful in power shaping) \begin{align*} H_{c}(\eta)&= -v_{C_{0}}i_{0}^{\ast}\pm v_{0}i_{0}^{\ast}\pm v_{1}i_{1}^{\ast}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2}\\ &= -i_{0}i_{0}^{\ast} R_{0}+{\int_{0}^{1}} \left(v_{z}i^{\ast}+vi^{\ast}_{z} \right)\textrm{d}z-v_{1}i_{1}^{\ast}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2}\!. \end{align*} Using this controller Hamiltonian, we will shape the closed-loop mixed equilibrium points. Let c be a constant. Consider \begin{align*} P_{d} =&\ \tilde{P}^{a}+\tilde{P}^{0}+\tilde{P}^{1}+H_{c}(\eta)+c \\ =&\ {\int_{0}^{1}}\left(\frac{\alpha(1-\zeta^{2})-1}{2R}(Ri+v_{z})^{2}+\varDelta^{2}+\frac{1}{2R}{v_{z}^{2}}+v_{z}i^{\ast}+\frac{1}{2}Gv^{2}+vi^{\ast}_{z}\right)\ \text{d}z+\dfrac{1}{2}R_{0}{i_{0}^{2}}\\ &+\dfrac{1}{2}R_{1}{i_{1}^{2}}-i_{0}i_{0}^{\ast} R_{0}+c{\pm\int_{0}^{1}}\left(Ri^{\ast 2}+\dfrac{i_{z}^{\ast 2}}{2G}\right)\ \text{d}z\pm\dfrac{1}{2}R_{0}i_{0}^{\ast 2} +\frac{1}{2}K_{I}(v_{C_{0}}-v_{C_{0}}^{\ast})^{2}\\ =&\ {\int_{0}^{1}}\left(\frac{\alpha(1-\zeta^{2})-1}{2R}(Ri+v_{z})^{2}+\Delta^{2}+\dfrac{1}{2R}\left({v_{z}^{2}}+2v_{z}Ri^{\ast} +R^{2}i^{\ast 2}\right)+\frac{1}{2G}\left(Gv+i_{z}^{\ast }\right)^{2}\right)\ \text{d}z\\ &+\dfrac{1}{2}R_{0}\left({i_{0}^{2}}-2i_{0}i_{0}^{\ast} +i_{0}^{\ast 2}\right)+\dfrac{1}{2}R_{1}{i_{1}^{2}}-i_{0}i_{0}^{\ast} R_{0}+c-{\int_{0}^{1}}\left(Ri^{\ast 2}+\dfrac{i_{z}^{\ast 2}}{2G}\right)\ \text{d}z-\dfrac{1}{2}R_{0}i_{0}^{\ast 2}\\ &+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2} \!. \end{align*} By choosing $$c=\displaystyle\int _{0}^{1}\left (Ri^{\ast 2}+\dfrac {i_{z}^{\ast 2}}{2G}\right )\ \text {d}z+\dfrac {1}{2}R_{0}i_{0}^{\ast 2}$$, we can see that \begin{align} P_{d}=&\ {\int_{0}^{1}}\left(\frac{\alpha(1-\zeta^{2})-1}{2R}(Ri+v_{z})^{2}+\Delta^{2}+\dfrac{1}{2R}\left(v_{z}+Ri^{\ast} \right)^{2}+\frac{1}{2G}\left(Gv+i_{z}^{\ast} \right)^{2}\right)\ \text{d}z\nonumber\\&+\dfrac{1}{2}R_{0}\left(i_{0}-i_{0}^{\ast} \right)^{2}+\dfrac{1}{2}R_{1}{i_{1}^{2}}+\frac{1}{2}K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)^{2} \end{align} (82) has a minimum at the equilibrium (12), (13) and (14). The time derivative of Pd along (7–9), (80) and (81) is \begin{align} \dfrac{d}{dt}P_{d}\leqslant\left(I_{s}+K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right)-i_{0}^{\ast}\right)\frac{dv_{C_{0}}}{dt}. \end{align} (83) 7.1. Stability analysis Denote $$\varDelta U=(\varDelta i, \varDelta v,\varDelta i_{0},\varDelta v_{0},\varDelta v_{C_{0}},\varDelta i_{1},$$$$\varDelta v_{1},\varDelta v_{C_{1}})$$. Consider the following norm \begin{align} \|\varDelta U\|^{2}={\int_{0}^{1}}\left((R\varDelta i+\varDelta v_{z})^{2}+\varDelta {v_{z}^{2}}+\varDelta v^{2}\right)\ \text{d}z+\varDelta {i_{0}^{2}}+\varDelta {i_{1}^{2}} +\varDelta v_{C_{0}}^{2} .\end{align} (84) Proposition 7 The transmission line system (55) in closed-loop with control \begin{align} I_{0}= i_{0}^{\ast}-K_{P}\frac{dv_{C_{0}}}{dt}-K_{I}\left(v_{C_{0}}-v_{C_{0}}^{\ast}\right),\;\;\; K_{P},K_{I} \geqslant 0 \end{align} (85) is asymptotically stable at the operating point $$U^{\ast }=(i^{\ast }, v^{\ast },i_{0}^{\ast },v_{0}^{\ast },v_{C_{0}}^{\ast },i_{1}^{\ast },v_{1}^{\ast },v_{C_{1}}^{\ast })$$ as defined in (12), (13) and (14). Proof. From (82), we can show that \begin{align} P_{d}(U)>0\quad \forall \;U\neq U^{\ast}, P_{d}(U)=0\ \textrm{if and only\ if}\ U=U^{\ast},\ \textrm{and}\ \delta_{U}P_{d}(U^{\ast})=0 . \end{align} (86) Moreover, $$\mathscr {N}(\varDelta U)=P_{d}(U^{\ast }+\varDelta U)-P_{d}(U^{\ast })$$ \begin{align*} = \int_0^1\left(\frac{\alpha(1-\zeta^2)-1}{2R}(R\varDelta i+\varDelta v_z)^2\!+\dfrac{1}{2R}\varDelta v_z^2+\frac{1}{2}G\varDelta v^2\right)\ \text{d}z+\dfrac{1}{2}R_0\varDelta i_0^2+\dfrac{1}{2}R_1\varDelta i_1^2+\frac{1}{2}K_I\varDelta v_{C_0}^2. \end{align*} For \begin{align*} \gamma_1&=\min\left\{\frac{\alpha(1-\zeta^2)-1}{2R},\frac{1}{2R},\frac{1}{2}G,\frac{1}{2}R_0,\frac{1}{2}R_1,\frac{1}{2}K_I\right\}\!,\\ \gamma_2&=\max\left\{\frac{\alpha(1-\zeta^2)-1}{2R},\frac{1}{2R},\frac{1}{2}G,\frac{1}{2}R_0,\frac{1}{2}R_1,\frac{1}{2}K_I\right\}\!, \end{align*} we have the following \begin{align} \gamma_{1}\|\Delta U\|^{2}\leqslant P_{d}(U^{\ast}+\Delta U)-P_{d}(U^{\ast})\leqslant \gamma_{2}\|\Delta U\|^{2}. \end{align} (87) Finally, using (83) and (85) the time derivative $$\dot {P}_{d}$$ is \begin{align} \dot{P}_{d}&= {\int_{0}^{1}}\left(u_{t}^{\top}\tilde{A}_{nd}u_{t}\right) \textrm{d}z+u_{01}^{\top}\tilde{A}_{0} u_{0t}+u_{1t}^{\top}\tilde{A}_{1} u_{1t}+ \left(I_{0}+K_{I}(v_{C_{0}}-v_{C_{0}}^{\ast})-i_{0}^{\ast}\right)\frac{\ dv_{C_{0}}}{dt}\nonumber\\ &= -K\left({\int_{0}^{1}}\left({i_{t}^{2}}+{v_{t}^{2}}\right)\ \text{d}z+i_{0t}^{2}+v_{C_{0}t}^{2}\right)\leqslant 0 . \end{align} (88) Arnold’s first stability theorem (Theorem 2) can be proved using (86), (87) and (88). Hence, the transmission line system (56) in closed-loop is Lyapunov stable at U* with respect to the norm ∥⋅∥ defined in (84) . Further from (12), (13) and (14), one can show that $$\dot {P_{d}}=0$$ iff U = U*. Thereby, we conclude the proof by invoking LaSalle’s invariance principle (Pasumarthy, 2006, see Theorem 5.19). □ 8. Conclusions In this paper, we presented a methodology to overcome the dissipation obstacle in the case of infinite-dimensional systems, thus paving way for passivity based control techniques. The basic building block was to write the system equations in the BM form. However, to effectively use the method, we need to construct admissible pairs for a given system, which aids in stability analysis and also in deriving new passivity properties. We present a systematic way to derive these admissible pairs and prove the stability of Maxwell’s equations. Later, we presented boundary control of mixed finite and infinite-dimensional systems. 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( 2015 ) Alternative passive maps for infinite-dimensional systems using mixed-potential functions . IFAC Workshop Lagrangian Hamiltonian Methods Non Linear Control, Lyon, France , 48 , 1 -- 6 . Kosaraju , K. C. & Pasumarthy , R. ( 2015 ) Power-based methods for infinite-dimensional systems . Math. Control Theory I. Springer , pp . 277 -- 301 . Le Gorrec , Y. , Zwart , H. & Maschke , B. ( 2005 ) Dirac structures and boundary control systems associated with skew-symmetric differential operators . SIAM J. Control Optimization , 44 , 1864 -- 1892 . Google Scholar CrossRef Search ADS Luo , Z.-H. , Guo , B.-Z. & Morgül , Ö . ( 2012 ) Stability and Stabilization of Infinite Dimensional Systems with Applications . Springer Science & Business Media . Macchelli , A. & Melchiorri , C. ( 2005 ) Control by interconnection of mixed port Hamiltonian systems . IEEE Trans. Automatic Control , 50 , 1839 -- 1844 . Google Scholar CrossRef Search ADS Ortega , R. , van der Schaft , A. 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( 1999 ) Introduction to Hamiltonian Fluid Dynamics and Stability Theory , vol. 102 . CRC Press . van der Schaft , A. J. ( 2017 ) L2-gain and Passivity Techniques in Nonlinear Control . Springer . Google Scholar CrossRef Search ADS van der Schaft , A. J. & Maschke , B. ( 2002 ) Hamiltonian formulation of distributed-parameter systems with boundary energy flow . J. Geometry Phys. , 42 , 166 -- 194 . Google Scholar CrossRef Search ADS © The Author(s) 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Dec 26, 2017

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