# Differential-algebraic systems with dissipative Hamiltonian structure

V. Mehrmann33footnotemark: 3 111 Institut für Mathematik, Sekr. MA 4-5, TU
Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany<EMAIL_ADDRESS>berlin.de. Research supported by Deutsche Forschungsgemeinschaft DFG through
priority Programm SPP 1984 within project 790 39-2: Distributed dynamic
control of network security.  A.J. van der Schaft 222Bernoulli institute for
Mathematics, Computer Science and AI, Jan C. Willems Center for Systems and
Control, University of Groningen, Nijenborgh 9, Groningen, the Netherlands

###### Abstract

Different representations of dissipative Hamiltonian and port-Hamiltonian
differential-algebraic equations (DAE) systems are presented and compared.
Using global geometric and algebraic points of view, translations between the
different representations are presented. Characterizations are also derived
when a general DAE system can be transformed into one of these structured
representations. Approaches for computing the structural information and the
described transformations are derived that can be directly implemented as
numerical methods. The results are demonstrated with a large number of
examples.

Keywords. Port-Hamiltonian system, dissipative Hamiltonian system,
differential algebraic equation, Lagrange structure, Dirac structure, matrix
pencil.

AMS subject classification. 15A18, 15A21, 15A22

##  1 Introduction

Since the original introduction of the energy based modeling concept of port-
Hamiltonian (pH) systems in [28, 35], see also [6, 22, 24, 33, 38, 34, 40],
various novel definitions and formulations have been made to incorporate on
the one hand systems defined on manifolds, see [14, 41] and on the other hand
systems with algebraic constraints defined in the form of Differential-
Algebraic Equations (DAEs), see [4, 31]. Constraints in pH systems typically
arise in electrical or transport networks, where Kirchhoff’s laws constrain
the models at network nodes, as balance equations in chemical engineering, or
as holonomic or non-holonomic constraints in mechanical multibody systems.
Furthermore, they arise in the interconnection of pH systems when the
interface conditions are explicitly formulated and enforced via Lagrange
multipliers, see [4, 31, 39, 40, 41, 42] for a variety of examples, or the
recent survey [32].

In this paper we recall different model representations of pHDAEs,
respectively dissipative Hamiltonian DAEs (dHDAEs), and provide a systematic
geometric and algebraic theory that also provides a ’translation’ between the
different representations. We also discuss approaches for explicitly computing
the structural information that can be directly implemented as numerical
methods. We mainly restrict ourselves to finite-dimensional linear time-
invariant systems without inputs and outputs, but indicate in several places
where extensions to general systems are possible. We also discuss only real
systems although most of the results can also be formulated for complex
problems in a similar way.

For general linear time-invariant homogeneous differential-algebraic equations

$\frac{d}{dt}(Ex)=E\dot{x}=Ax,$ (1)

where $E,A\in\mathbb{R}^{n,n}$ (without additional geometric and algebraic
structures), as well as for their extensions to linear time-varying and
nonlinear systems, the theory is well understood [25]. On the other hand, in
contrast to general DAEs (1), the DAEs that arise in energy based modeling
have extra structure and symmetries, and it is natural to identify and exploit
this extra structure for purposes of analysis, simulation and control.

In this paper we will restrict our attention to _regular_ systems, i.e.,
systems with $\det(\lambda E-A)$ not identically zero, although many of the
results are expected to extend to over- and underdetermined systems; cf. the
Conclusions. The structural properties for regular systems are characterized
via the _(real) Weierstraß_ canonical form of the matrix pair $(E,A)$, see
e.g. [20]. To determine this canonical form one computes nonsingular matrices
$U,W$ such that

$UEW=\left[\begin{array}[]{cc}I&0\\\ 0&\mathcal{N}\end{array}\right],\
UAW=\left[\begin{array}[]{cc}\mathcal{J}&0\\\ 0&I\end{array}\right],$

where $\mathcal{J}$ is in real Jordan canonical form and $\mathcal{N}$ is
nilpotent in Jordan canonical form which is associated to the eigenvalue
$\infty$

An important quantity that will be used throughout the paper is the size of
the largest Jordan block in $\mathcal{N}$ (the index of nilpotency), which is
called the _(differentiation) index of the pair $(E,A)$ as well as the
associated DAE_, where, by convention, $\nu=0$ if $E$ is invertible.

The goal of this paper is to study in detail the relationships and differences
between different classes of (extended) Hamiltonian DAEs; without making any a
priori invertibility assumptions. Furthermore, we exploit at the same time a
_geometric_ point of view on Hamiltonian DAEs using Lagrange and Dirac
structures (as well maximally monotone subspaces; cf. Section 4.2), and an
_algebraic_ point of view using (computationally feasible) condensed forms.
From a DAE analysis point of view, similar to the result known already for
Hamiltonian DAEs of the form (2) in [29], we also show that the index of an
(extended) dissipative Hamiltonian DAEs can be at most two. Moreover, we will
show that index two algebraic constraints may only arise from singularity of
$P$, i.e., from the Lagrange structure, and thus are strictly linked to
singular Hamiltonians.

Another important question that we will answer is the characterization when a
general DAE is equivalent to a structured DAE the form ((2) or ((22)).

The paper is organized as follows.

In Section 2 we discuss dissipative Hamiltonian DAEs and present several
examples. The concept of extended Hamiltonian DAEs is discussed in Section 3.
In Section 4 we will present the geometric theory of dissipative Hamiltonian
DAEs, we introduce Dirac and Lagrange structures as well maximally monotone
subspaces to incorporate dissipation. Section 5 presents different coordinate
representations and their relation. To derive algebraic characterizations, in
Section 6 we present different types of equivalence transformations and
corresponding condensed forms for the different classes of (extended
dissipative) Hamiltonian DAEs. In Section 7 it is analyzed when general linear
DAEs can be represented as (extended dissipative) Hamiltonian systems. In all
sections we present examples that illustrate the properties of the different
representations. In several appendices we present proofs that can be
implemented as numerically stable algorithms.

##  2 Dissipative Hamiltonian DAEs

A well established representation of DAE systems with symmetry structure is
that of _dissipative Hamiltonian DAEs (dHDAEs)_ which has been studied in
detail in [4, 29, 30]. These systems have the form

${\frac{d}{dt}(Ez)}=E\dot{z}=(J-R)Qz,$ (2)

where $J=-J^{\top},R=R^{\top}\in\mathbb{R}^{\ell,\ell}$, and
$E,Q\in\mathbb{R}^{\ell,n}$ with $E^{\top}Q=Q^{\top}E$. In this representation
the associated Hamilton function (_Hamiltonian_) is given by a quadratic form

$\mathcal{H}^{z}(z)=\frac{1}{2}z^{\top}E^{\top}Qz,$ (3)

which (as an energy) is typically nonnegative for all $z$, but more general
Hamiltonians also arise in practice. Note that in the ODE case, i.e. if $E=I$,
then this reduces to the standard quadratic Hamiltonian
$\mathcal{H}^{z}(z)=\frac{1}{2}z^{\top}Qz$.

###### Remark 1

The equivalent formulations $\frac{d}{dt}(Ez)=E\dot{z}$ are both used in the
literature and obviously lead to the same results in the linear constant
coefficient case, but not anymore in the linear time-varying or nonlinear
case. Actually, for the structured DAEs considered in this paper $E$ is the
product $E=KP$ of two matrices $K$ and $P$, and for generalizations it is
appropriate to consider $K\frac{d}{dt}(Pz)$.

Furthermore, the dissipation term $R$ is typically positive semidefinite,
denoted as $R\geq 0$, but we will first discuss the case that $R=0$ which we
call _Hamiltonian DAE (HDAE)_.

###### Remark 2

The definition of (dHDAEs) in (2) can be easily extended to systems with ports
in which case it takes the form

$\displaystyle{\frac{d}{dt}(Ez)}=E\dot{z}=$ $\displaystyle=$
$\displaystyle(J-R)Qz+(F-G)u,$ $\displaystyle y$ $\displaystyle=$
$\displaystyle(F+G)^{\top}Qz+Du,$ (4)

with

$\left[\begin{array}[]{cc}Q^{\top}JQ&Q^{\top}F\\\
-F^{\top}Q&\frac{1}{2}(D-D^{\top})\end{array}\right]=-\left[\begin{array}[]{cc}Q^{\top}JQ&Q^{\top}F\\\
-F^{\top}Q&\frac{1}{2}(D-D^{\top})\end{array}\right]^{\top}$

and

$\left[\begin{array}[]{cc}Q^{\top}RQ&Q^{\top}G\\\
G^{\top}Q&\frac{1}{2}(D+D^{\top})\end{array}\right]=\left[\begin{array}[]{cc}Q^{\top}RQ&Q^{\top}G\\\
G^{\top}Q&\frac{1}{2}(D+D^{\top})\end{array}\right]^{\top}\geq 0,$

see [4]. We will call these systems _dissipative pHDAE (dpHDAE) systems_.

###### Remark 3

Note that by the symmetry of $E^{\top}Q$ the Hamiltonian
$\mathcal{H}^{z}(z)=\frac{1}{2}z^{\top}E^{\top}Qz$ can be also written as a
function of $Qz$ or $Ez$.

###### Remark 4

In standard port-Hamiltonian modeling the matrix $R$ in (2) is always assumed
to be symmetric (as well as positive semi-definite). Alternatively, one could
start from a general matrix $R$ only satisfying $R+R^{\top}\geq 0$. Then by
splitting $R$ into its symmetric and skew-symmetric part, one could add the
skew-symmetric part to the skew-symmetric structure matrix $J$ and continue as
in (2) with the symmetric part $\frac{1}{2}(R+R^{\top})$ of $R$.

Structured DAEs of the form (2) arise naturally in all physical domains.

###### Example 5

The subclass of RLC networks as considered in e.g. [4, 12, 19] has the form

$\displaystyle\underbrace{\left[\begin{array}[]{ccc}D_{C}CD_{C}^{\top}&0&0\\\
0&L&0\\\ 0&0&0\end{array}\right]}_{:=E}\left[\begin{array}[]{c}\dot{V}\\\
\dot{I_{L}}\\\
\dot{I_{S}}\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}-D_{R}GD_{R}^{\top}&-D_{L}&-D_{S}\\\
D_{L}^{\top}&0&0\\\
D_{S}^{\top}&0&0\end{array}\right]}_{:=J-R}\left[\begin{array}[]{c}V\\\
I_{L}\\\ I_{S}\end{array}\right],$ (17)

where the real positive definite diagonal matrices $L$, $C$, $G$ describe
inductances, capacitances, and conductances, respectively. The matrices
$D_{C},D_{L},D_{R},D_{S}$ are the parts of the incidence matrix corresponding,
respectively, to the capacitors, inductors, resistors (conductors), and
current sources of the circuit graph, where $D_{S}$ is of full column rank.
Furthermore $V$ are the node potentials and $I_{L},I_{S}$ denote the currents
through the inductors and sources, respectively. This system has the form (2),
with $E=E^{\top}\geq 0$, $Q$ equal to the identity matrix, and where $J$ and
$-R$ are defined to be the skew-symmetric and symmetric part, respectively, of
the matrix on the right hand side of (17). The Hamiltonian is given by
$\mathcal{H}(V,I_{L})=\frac{1}{2}V^{\top}D_{C}CD_{C}^{\top}V+\frac{1}{2}I_{L}^{\top}LI_{L}$
and it does not involve the variables $I_{S}$, which are in the kernel of $E$.

###### Example 6

Space discretization of the Stokes equation in fluid dynamics, see, e.g.,
[18], leads to a dissipative Hamiltonian system

$\left[\begin{array}[]{cc}M&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}\dot{v}_{h}\\\
\dot{p}_{h}\end{array}\right]=\left[\begin{array}[]{cc}A&B\\\
-B^{\top}&0\end{array}\right]\left[\begin{array}[]{c}v_{h}\\\
p_{h}\end{array}\right]+\left[\begin{array}[]{c}f_{h}\\\ 0\end{array}\right],$

where $A=A^{\top}$ is a positive semidefinite discretization of the negative
Laplace operator, $B$ is a discretized gradient, and $M=M^{\top}$ is a
positive definite mass matrix. The homogeneous system has the form (2), with

$R=\left[\begin{array}[]{cc}A&0\\\ 0&0\end{array}\right],\
J=\left[\begin{array}[]{cc}0&B\\\ -B^{\top}&0\end{array}\right],\
E=\left[\begin{array}[]{cc}M&0\\\ 0&0\end{array}\right],\ Q=I.$

The Hamiltonian is given by $\mathcal{H}=\frac{1}{2}v^{\top}_{h}Mv_{h}$; it
does not involve the variables $p_{h}$.

###### Example 7

Space discretization of the Euler equation describing the acoustic wave
propagation in a gas pipeline network [15, 16] leads to a DAE

$\left[\begin{array}[]{ccc}M_{1}&0&0\\\ 0&M_{2}&0\\\
0&0&0\end{array}\right]\left[\begin{array}[]{c}\dot{p}_{h}\\\ \dot{q}_{h}\\\
\dot{\lambda}\end{array}\right]=\left(\left[\begin{array}[]{ccc}0&-G&0\\\
G^{\top}&0&K^{\top}\\\
0&-K&0\end{array}\right]-\left[\begin{array}[]{ccc}0&0&0\\\ 0&D&0\\\
0&0&0\end{array}\right]\right)\left[\begin{array}[]{c}p_{h}\\\ q_{h}\\\
\lambda\end{array}\right]+f,$

where $p_{h}$ is the discretized pressure, $q_{h}$ is a discretized flux, and
$\lambda$ is a Lagrange multiplier that penalizes the violation of the
conservation of mass and momentum at the pipeline nodes.

The homogeneous system has the form (2) with $Q=I$, and Hamiltonian
$\mathcal{H}(p_{h},q_{h})=\frac{1}{2}(p_{h}^{\top}M_{1}p_{h}+q_{h}^{\top}M_{2}q_{h})$.
The Hamiltonian does not involve the Lagrange multiplier $\lambda$.

###### Example 8

Consider a linear mechanical system (with $q$ denoting the vector of position
coordinates) $M\ddot{q}+D\dot{q}+Wq=f$, together with kinematic constraints
$G\dot{q}=0$, see e.g. [17]. The constraints give rise to constraint forces
$G^{\top}\lambda$, with $\lambda$ a vector of Lagrange multipliers. This
yields the dynamics $M\ddot{q}+D\dot{q}+Wq=f+G^{\top}\lambda$, and the
resulting system can be written in first order form as

$\left[\begin{array}[]{ccc}M&0&0\\\ 0&I&0\\\
0&0&0\end{array}\right]\left[\begin{array}[]{c}\ddot{q}\\\ \dot{q}\\\
\dot{\lambda}\end{array}\right]+\left[\begin{array}[]{ccc}D&I&-G^{\top}\\\
-I&0&0\\\ G&0&0\end{array}\right]\left[\begin{array}[]{ccc}I&0&0\\\ 0&W&0\\\
0&0&I\end{array}\right]\left[\begin{array}[]{c}\dot{q}\\\ q\\\
\lambda\end{array}\right]=\left[\begin{array}[]{c}f\\\ 0\\\
0\end{array}\right].$

Here, $E=E^{\top}$ and $Q=Q^{\top}$ are commuting matrices, $E$, $W$, and
$R=R^{\top}$ are positive semidefinite, so the homogeneous system is of the
form (2). The Hamiltonian is given by
$\frac{1}{2}(\dot{q}^{\top}M\dot{q}+q^{\top}Wq)$ (kinetic plus potential
energy) and does not involve the Lagrange multiplier $\lambda$.

As indicated in the presented examples, singularity of $E$, and thus the
presence of algebraic constraints, implies that the Hamiltonian (the total
stored energy) $\mathcal{H}^{z}(z)=\frac{1}{2}z^{\top}E^{\top}Qz$ does not
involve all of the variables contained in the vector $z$. In fact in some of
the examples the variables that do not show up in the Hamiltonian are
_Lagrange multipliers_. Conversely, singularity of $E$ may arise as a limiting
situation of otherwise regular Hamiltonians, as shown by the following simple
example.

###### Example 9

Consider the model of standard mass-spring-damper system with model equation

$\left[\begin{array}[]{c}\dot{q}\\\
\dot{p}\end{array}\right]=\left[\begin{array}[]{cc}0&1\\\
-1&-d\end{array}\right]\left[\begin{array}[]{cc}k&0\\\
0&\frac{1}{m}\end{array}\right]\left[\begin{array}[]{c}q\\\
p\end{array}\right]$

and Hamiltonian $\mathcal{H}(q,p)=\frac{1}{2}kq^{2}+\frac{p^{2}}{2m}$.

To compute the limit $m\to 0$, we rewrite the system in coordinates $q$ and
$v:=\frac{p}{m}$ as

$\left[\begin{array}[]{c}\dot{q}\\\
m\dot{v}\end{array}\right]=\left[\begin{array}[]{cc}0&1\\\
-1&-d\end{array}\right]\left[\begin{array}[]{c}kq\\\ v\end{array}\right]$

with Hamiltonian $\mathcal{H}(q,v)=\frac{1}{2}kq^{2}+\frac{1}{2}mv^{2}$. For
$m\to 0$ this converges to the DAE system

$\left[\begin{array}[]{c}\dot{q}\\\
0\end{array}\right]=\left[\begin{array}[]{cc}0&1\\\
-1&-d\end{array}\right]\left[\begin{array}[]{c}kq\\\ v\end{array}\right],$

which is of (differentiation) index one if $d\neq 0$, and of index two if
$d=0$. The limiting Hamiltonian $\mathcal{H}(q,v)=\frac{1}{2}kq^{2}$ is not a
function of $v$ anymore.

Alternatively, we can compute the limit for $k\to\infty$. For this we rewrite
the system in coordinates $F:=kq$ and $p$ as

$\left[\begin{array}[]{c}\frac{1}{k}\dot{F}\\\
\dot{p}\end{array}\right]=\left[\begin{array}[]{cc}0&1\\\
-1&-d\end{array}\right]\left[\begin{array}[]{c}F\\\
\frac{p}{m}\end{array}\right],$

with Hamiltonian $\mathcal{H}(F,p)=\frac{1}{2k}F^{2}+\frac{1}{2m}p^{2}$. For
$k\to\infty$ this converges to the DAE system

$\left[\begin{array}[]{c}0\\\
\dot{p}\end{array}\right]=\left[\begin{array}[]{cc}0&1\\\
-1&-d\end{array}\right]\left[\begin{array}[]{c}F\\\
\frac{p}{m}\end{array}\right],$

which is index two for any $d$, with Hamiltonian
$\mathcal{H}(F,p)=\frac{1}{2m}p^{2}$, which is not involving $F$.

We may also take the limits $m\to 0$ and $k\to\infty$ simultaneously. By
rewriting the system in the variables $F,v$, with Hamiltonian
$\mathcal{H}(F,p)=\frac{1}{2k}F^{2}+\frac{1}{2}mv^{2}$. This leads to the
purely algebraic system

$\left[\begin{array}[]{c}0\\\
0\end{array}\right]=\left[\begin{array}[]{cc}0&1\\\
-1&-d\end{array}\right]\left[\begin{array}[]{c}F\\\ v\end{array}\right],$

with zero Hamiltonian, and having a single solution $v=0$, $F=0$, irrespective
of the damper.

##  3 Extended Hamiltonian DAEs

Classically, Hamiltonian systems and also their extension to systems with
inputs and outputs, called port-Hamiltonian systems, see e.g. [13, 22, 28, 35,
36, 38, 40], are defined by a _Dirac structure_ , representing the power-
conserving interconnection structure, an _energy-dissipation relation_ , and a
_Hamilton function (Hamiltonian)_ , capturing the total energy storage in the
system. The Dirac structure formalizes the _generalized junction structure_
known from port-based modeling theory. Importantly, the Dirac structure may
entail linear constraints on the so-called effort variables, called _effort
constraints_. Through the gradient vector of the Hamilton function, these
effort constraints induce algebraic constraints on the state variables, see
especially [31, 39] for a discussion of general nonlinear port-Hamiltonian
DAEs. In the special case of a linear autonomous system without energy
dissipation and without inputs and outputs, and after choosing a basis such
that $\mathcal{X}=\mathbb{R}^{n}$, the equations of motion take the DAE form

$K\dot{x}=LQx,$ (18)

where the pair of matrices $K,L\in\mathbb{R}^{n,n}$, satisfying
$KL^{\top}+LK^{\top}=0$ and
$\operatorname{rank}\left[\begin{array}[]{cc}K&L\end{array}\right]=n$,
describes the Dirac structure, and $\mathcal{H}^{x}(x)=\frac{1}{2}x^{\top}Qx$
defines the Hamilton function for some $Q=Q^{\top}$. Denoting by $\ker$ and im
the kernel and image of a linear map or its matrix representation,
geometrically the Dirac structure is defined by the subspace
$\mathcal{D}\subset{\mathcal{X}}\times\mathcal{X}^{*}$ given as
$\ker\left[\begin{array}[]{cc}K&L\end{array}\right]$, where $\mathcal{X}$ is
the linear state space, and $\mathcal{X}^{*}$ its dual space.

###### Remark 10

More precisely $\mathcal{D}\subset\hat{\mathcal{X}}\times\mathcal{X}^{*}$,
where $\hat{\mathcal{X}}$ is the _tangent space_ to $\mathcal{X}$ at
$x\in\mathcal{X}$. However, since $\mathcal{X}$ is linear, tangent spaces at
any $x\in\mathcal{X}$ can be identified with each other and with
$\mathcal{X}$.

###### Remark 11

In this paper we frequently switch between coordinate free representations
with state space $\mathcal{X}$ and coordinate representations that are
obtained for the case $\mathcal{X}=\mathbb{R}^{n}$ after choosing a basis.
Furthermore, in this case it will be tacitly assumed that the dual basis is
chosen for the dual space $\mathcal{X}^{*}$. Although this is not the most
general setup (from an abstract linear algebraic or functional analytic point
of view), it makes the presentation of results much more convenient.

Algebraic constraints occur if the matrix $K$ is singular, and are represented
via

$Qx\in\mbox{\rm im\,}K^{\top}.$

Singularity of $K$ often results from the network structure as the following
example demonstrates.

###### Example 12

Consider a general linear $LC$-electrical circuit. Let the circuit graph be
determined by an incidence matrix $D$, defining Kirchhoff’s current laws
$I\in\ker D$ and voltage laws $V\in\mbox{\rm im\,}D^{\top}$. Split the
currents $I$ into currents $I_{C}$ through the capacitors and currents $I_{L}$
through the inductors, and the voltages $V$ into voltages $V_{C}$ across the
capacitors and voltages $V_{L}$ across the inductors. Furthermore, let
$I_{C}=-\dot{q}_{C}$ and $V_{L}=-\dot{\varphi}$, with $q$ the vector of
charges at the capacitors and $\varphi$ the flux linkages of the inductors.
Split the incidence matrix $D$ accordingly as
$D=\left[\begin{array}[]{cc}D_{C}&D_{L}\end{array}\right]$. Then Kirchhoff’s
current laws take the form

$D_{C}\dot{q}=D_{L}I_{L}.$

Furthermore, let $F$ be a maximal annihilator of $D^{\top}$, i.e., $\ker
F=\mbox{\rm im\,}D^{\top}$. Then Kirchhoff’s voltage laws are given as
$F\left[\begin{array}[]{c}V_{C}\\\ V_{L}\end{array}\right]=0$, and after
splitting $F=\left[\begin{array}[]{cc}F_{C}&F_{L}\end{array}\right]$
accordingly, we have

$F_{L}\dot{\varphi}=F_{C}V_{C}.$

Writing the linear constitutive equations for the capacitors as $q=CV_{C}$ for
some positive definite diagonal capacitance matrix $C$ and those for the
inductors as $\varphi=LI_{L}$ for some positive definite diagonal inductance
matrix $L$, we finally obtain the system of equations

$\left[\begin{array}[]{cc}D_{C}&0\\\
0&F_{L}\end{array}\right]\left[\begin{array}[]{c}\dot{q}\\\
\dot{\varphi}\end{array}\right]=\left[\begin{array}[]{cc}0&D_{L}\\\
F_{C}&0\end{array}\right]\left[\begin{array}[]{cc}C^{-1}&0\\\
0&L^{-1}\end{array}\right]\left[\begin{array}[]{c}q\\\
\varphi\end{array}\right],$

which is in the form (18) with Hamilton function
$\mathcal{H}(q,\varphi)=\frac{1}{2}q^{\top}C^{-1}q+\frac{1}{2}\varphi^{\top}L^{-1}\varphi$.
It is easily checked that singularity of
$K=\left[\begin{array}[]{cc}D_{C}&0\\\ 0&F_{L}\end{array}\right]$ corresponds
to parallel interconnection of capacitors or series interconnection of
inductors. (Note that since $\ker F=\mbox{\rm im\,}D^{\top}$ the linear space
$\mbox{\rm im\,}F^{\top}=\ker D$ is spanned by the cycles of the circuit
graph.) See e.g. [23] for pHDAE modeling of electrical circuits.

Motivated by [4] the port-Hamiltonian point of view on DAE systems was
extended in [41] by replacing the gradient vector $Qx$ of the Hamiltonian
function $\mathcal{H}(x)=\frac{1}{2}x^{\top}Qx$ by a general Lagrangian
subspace, called _Lagrange structure_ in the present paper, since we want to
emphasize the similarity with Dirac structures. As we will see, this extension
allows to bridge the gap with the (dissipative) Hamiltonian formulation (4) of
the previous Section 2.

In the linear homogeneous case without dissipation, one starts with a Dirac
structure $\mathcal{D}\subset{\mathcal{X}}\times\mathcal{X}^{*}$ and a
Lagrange structure $\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$. The
_composition_ of the Dirac structure $\mathcal{D}$ and the Lagrange structure
$\mathcal{L}$, over the _shared variables_ $e\in\mathcal{X}^{*}$, is then
defined as

$\mathcal{D}\circ\mathcal{L}=\\{(f,x)\in{\mathcal{X}}\times\mathcal{X}\mid\mbox{there
exists }e\in\mathcal{X}^{*}\mbox{ such that }(f,e)\in\mathcal{D},\
(e,x)\in\mathcal{L}\\}.$ (19)

Substituting $f=-\dot{x}$ this leads to the coordinate-free definition of the
dynamics

$(-\dot{x},x)\in\mathcal{D}\circ\mathcal{L}.$ (20)

In order to obtain a coordinate representation of the dynamics (20), the
simplest option (later on in Section 5.1 we will discuss another one) is to
start from the _image representation_ of the Lagrange structure $\mathcal{L}$,
defined by a pair of matrices $P,S\in\mathbb{R}^{n,n}$ satisfying
$P^{\top}S=S^{\top}P$ and
$\operatorname{rank}\left[\begin{array}[]{cc}P^{\top}&S^{\top}\end{array}\right]=n$.
Taking coordinates $x$ for $\mathcal{X}=\mathbb{R}^{n}$ and dual coordinates
$e$ for its dual space $\mathcal{X}^{*}=\mathbb{R}^{n}$, the image
representation of $\mathcal{L}$ is given by

$\left[\begin{array}[]{c}x\\\
e\end{array}\right]=\left[\begin{array}[]{c}Pz\\\ Sz\end{array}\right],$ (21)

for some parameterizing vector $z$ in a space $\mathcal{Z}$ (of the same
dimension as $\mathcal{X}$). Analogously, we consider a _kernel
representation_ of the Dirac structure $\mathcal{D}$ given by matrices $K,L$
satisfying $KL^{\top}=-LK^{\top}$ and
$\operatorname{rank}\left[\begin{array}[]{cc}K&L\end{array}\right]=n$, such
that $\mathcal{D}=\\{(f,e)\in\mathcal{X}\times\mathcal{X}^{*}\mid Kf+Le=0\\}$.
(More details will be given in Section 4.)

Substituting $f=-\dot{x}=P\dot{z}$ and $e=Sz$ this leads to the coordinate
representation

$KP\dot{z}=LSz.$ (22)

We will call this class _extended Hamiltonian differential-algebraic systems
(extended HDAEs)_. If dissipation is incorporated, see Section 4.2, then it is
called _extended dHDAEs_.

Importantly, the presence of algebraic constraints in (22) may arise _both_ by
singularity of $K$ (as was already the case for (18)) as well as by
singularity of $P$ (as was the case for (2)). This motivated the introduction
of the notions of _Dirac algebraic constraints_ (corresponding to singularity
of $K$) and of _Lagrange algebraic constraints_ (corresponding to singularity
of $P$) in [41].

Similar to dHDAE systems (2), the Hamiltonian of the extended HDAE system (22)
is specified by the Lagrange structure, and is given by

$\mathcal{H}^{z}(z)=\frac{1}{2}z^{\top}S^{\top}Pz.$ (23)

Indeed, one immediately has the _energy conservation property_

$\frac{d}{dt}\mathcal{H}^{z}(z)=z^{\top}S^{\top}P\dot{z}=-e^{\top}f=0,$

since $(f,e)\in\mathcal{D}$ and thus $e^{\top}f=0$.

While singularity of $K$ in physical systems modeling typically arises from
_interconnection_ due to the network structure, singularity of $P$ often
arises as a limiting situation. An elaborate example will be provided later as
Example 17.

###### Remark 13

Note that if $K$ is _invertible_ , then by multiplying (22) with $K^{-1}$ from
the left, we obtain the lossless version of the dHDAE system (2) in Section 2
with $E=P$, $Q=S$, $J=-J^{\top}=K^{-1}L$ (and $R=0$). Thus the replacement of
the Hamiltonian $\mathcal{H}(x)=\frac{1}{2}x^{\top}Qx$ by a Lagrange subspace
(21) constitutes a first step towards an _overarching_ formulation of
(dissipative) Hamiltonian DAE systems.

##  4 Geometric theory of (dissipative) Hamiltonian DAEs

In this section we take a systematic geometric view on Hamiltonian DAE
systems, extending the existing geometric treatment of extended HDAE systems,
as already discussed in Section 3. We also incorporate the discussed classes
of dHDAE systems from Section 2.

Consider an n-dimensional linear state space $\mathcal{X}$ with elements
denoted by $x$. Let $\hat{\mathcal{X}}$ denote the _tangent space_ to
$\mathcal{X}$ at $x\in\mathcal{X}$, with elements denoted by $f$ and called
_flow_ vectors. As mentioned before, since $\mathcal{X}$ is linear, tangent
spaces at different $x\in\mathcal{X}$ can be identified with each other and
with $\mathcal{X}$; implying that $f\in\mathcal{X}$ as well. Furthermore, let
$\mathcal{X}^{*}$ be the _dual space_ of $\mathcal{X}$, with elements denoted
by $e$ and called _effort_ vectors.

###  4.1 Dirac and Lagrange structures

The product space $\mathcal{X}\times\mathcal{X}^{*}$ is endowed with the two
canonical bilinear forms

$\begin{array}[]{l}\langle(f_{1},e_{1}),(f_{2},e_{2})\rangle_{+}:=f_{1}^{\top}e_{2}+f_{2}^{\top}e_{1},\\\\[5.69054pt]
\langle(f_{1},e_{1}),(f_{2},e_{2})\rangle_{-}:=f_{1}^{\top}e_{2}-f_{2}^{\top}e_{1},\end{array}$
(24)

represented by the two matrices

$\Pi_{+}:=\left[\begin{array}[]{cc}0&I_{n}\\\
I_{n}&0\end{array}\right],\quad\Pi_{-}:=\left[\begin{array}[]{cc}0&I_{n}\\\
-I_{n}&0\end{array}\right],$

where we recognize $\Pi_{-}$ as the standard symplectic form on $\mathcal{X}$.

###### Definition 14

A subspace $\mathcal{D}\subset\mathcal{X}\times\mathcal{X}^{*}$ is called a
_Dirac structure_ if the bilinear form $\langle\cdot,\cdot\rangle_{+}$ is zero
on $\mathcal{D}$ and moreover $\mathcal{D}$ is maximal with respect to this
property. A subspace $\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$ is a
_Lagrange structure_ if the bilinear form $\langle\cdot,\cdot\rangle_{-}$ is
zero on $\mathcal{L}$ and moreover $\mathcal{L}$ is maximal with respect to
this property. A Lagrange structure
$\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$ is called _nonnegative_
if the quadratic form defined by $\Pi_{+}$ is nonnegative on $\mathcal{L}$.

###### Remark 15

In this paper we have chosen the terminology ’Lagrange structure’, instead of
the more common terminology ’Lagrangian subspace’, in order to emphasize the
similarity to Dirac structures. Also note that the definition of Dirac
structures can be extended to manifolds instead of linear state spaces
$\mathcal{X}$; in this context Dirac structures on linear spaces are often
referred to as _constant_ Dirac structures.

We have the following characterizations of Lagrange and Dirac structures, see
e.g. [13, 40].

###### Proposition 16

Consider an $n$-dimensional linear state space $\mathcal{X}$ and its dual
space $\mathcal{X}^{*}$.

  * i)

A subspace $\mathcal{D}\subset\mathcal{X}\times\mathcal{X}^{*}$ is a Dirac
structure if and only if $\mathcal{D}=\mathcal{D}^{\perp\\!\\!\\!\perp_{+}}$,
where ${}^{\perp\\!\\!\\!\perp_{+}}$ denotes the orthogonal complement with
respect to the bilinear form $\langle\cdot,\cdot\rangle_{+}$. Furthermore,
$\mathcal{D}\subset\mathcal{X}\times\mathcal{X}^{*}$ is a Dirac structure if
and only $e^{\top}f=0$ for all $(f,e)\in\mathcal{D}$ and $\dim\mathcal{D}=n$.

  * ii)

A subspace $\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$ is a Lagrange
structure if and only if $\mathcal{L}=\mathcal{L}^{\perp\\!\\!\\!\perp_{-}}$,
where ${}^{\perp\\!\\!\\!\perp_{-}}$ denotes the orthogonal complement with
respect to the bilinear form $\langle\cdot,\cdot\rangle_{-}$. Any Lagrange
structure satisfies $\dim\mathcal{L}=n$.

Dirac and Lagrange structures admit structured _coordinate representations_ ,
see e.g. [40, 41]. For this paper the following representations are most
relevant. Using matrices $K,L\in\mathbb{R}^{n,n}$, any _Dirac structure_
$\mathcal{D}\subset{\mathcal{X}}\times\mathcal{X}^{*}$ admits the kernel/image
representation

$\mathcal{D}=\ker\left[\begin{array}[]{cc}K&L\end{array}\right]=\mbox{\rm
im\,}\left[\begin{array}[]{cc}L^{\top}\\\
K^{\top}\end{array}\right]\subset{\mathcal{X}}\times\mathcal{X}^{*},$ (25)

with $K,L$ satisfying
$\operatorname{rank}\left[\begin{array}[]{cc}K&L\end{array}\right]=n$ and the
generalized _skew-symmetry condition_

$KL^{\top}+LK^{\top}=0.$ (26)

Conversely any such pair $K,L$ defines a Dirac structure.

Analogously, any Lagrange structure
$\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$ can be represented as

$\mathcal{L}=\ker\left[\begin{array}[]{cc}-P^{\top}&S^{\top}\end{array}\right]=\mbox{\rm
im\,}\left[\begin{array}[]{cc}S\\\
P\end{array}\right]\subset\mathcal{X}^{*}\times\mathcal{X},$ (27)

for certain matrices $S,P\in\mathbb{R}^{n,n}$ satisfying
$\operatorname{rank}\left[\begin{array}[]{c}P\\\ S\end{array}\right]=n$ as
well as the generalized _symmetry condition_

$S^{\top}P=P^{\top}S.$ (28)

A Lagrange structure is, furthermore, _nonnegative_ if and only if
$S^{\top}P\geq 0$.

As already described in Section 3, by using the image representation $x=Pz$,
$e=Sz$ of the Lagrange structure $\mathcal{L}$, and the kernel representation
$Kf+Le=0$ of the Dirac structure $\mathcal{D}$ one is led to the
representation (22) of the extended HDAE system defined by $\mathcal{D}$ and
$\mathcal{L}$.

The following is a physical example where both $K$ and $P$ turn out to be
singular. The singularity of $K$ is due to the presence of kinematic
constraints, while the singularity of $P$ is caused by a limiting argument in
the energy expression.

###### Example 17

Consider two masses $m_{1}$ and $m_{2}$ connected by a spring with spring
constant $k$, where the right mass $m_{2}$ is subject to the kinematic
constraint $v_{2}=0$ (velocity is zero). With positions $q_{1},q_{2}$ and
momenta $p_{1},p_{2}$, the Hamiltonian is given by

$\mathcal{H}(q_{1},q_{2},p_{1},p_{2})=\frac{1}{2}k(q_{1}-q_{2})^{2}+\frac{p_{1}^{2}}{2m_{1}}+\frac{p_{2}^{2}}{2m_{2}}.$

Denoting by $e_{q1},e_{q2}$ the spring forces at both ends of the spring, and
by $e_{p1},e_{p2}$ the velocities of the two masses, we obtain the relation

$\begin{bmatrix}q_{1}\\\ q_{2}\\\ p_{1}\\\ p_{2}\\\ \hline\cr e_{q1}\\\
e_{q2}\\\ e_{p1}\\\ e_{p2}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ \hline\cr k&-k&0&0\\\ -k&k&0&0\\\
0&0&\frac{1}{m_{1}}&0\\\ 0&0&0&\frac{1}{m_{2}}\\\
\end{bmatrix}\begin{bmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\ z_{4}\end{bmatrix}$

To consider the limit $k\to\infty$, meaning that the spring is replaced by
rigid connection, we first express the system in different coordinates.

$\begin{bmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\
z_{4}\end{bmatrix}=\begin{bmatrix}\frac{m_{2}}{m_{1}+m_{2}}&1&0&0\\\
-\frac{m_{1}}{m_{1}+m_{2}}&1&0&0\\\ 0&0&1&\frac{m_{1}}{m_{1}+m_{2}}\\\
0&0&-1&\frac{m_{2}}{m_{1}+m_{2}}\end{bmatrix}\begin{bmatrix}\tilde{z}_{1}\\\
\tilde{z}_{2}\\\ \tilde{z}_{3}\\\ \tilde{z}_{4}\end{bmatrix}.$

This yields the transformed representation

$\begin{bmatrix}q_{1}\\\ q_{2}\\\ p_{1}\\\ p_{2}\\\ \hline\cr e_{q1}\\\
e_{q2}\\\ e_{p1}\\\
e_{p2}\end{bmatrix}=\begin{bmatrix}\frac{m_{2}}{m_{1}+m_{2}}&1&0&0\\\
-\frac{m_{1}}{m_{1}+m_{2}}&1&0&0\\\ 0&0&1&\frac{m_{1}}{m_{1}+m_{2}}\\\
0&0&-1&\frac{m_{2}}{m_{1}+m_{2}}\\\ \hline\cr k&0&0&0\\\ -k&0&0&0\\\
0&0&\frac{1}{m_{1}}&\frac{1}{m_{1}+m_{2}}\\\
0&0&-\frac{1}{m_{2}}&\frac{1}{m_{1}+m_{2}}\end{bmatrix}\begin{bmatrix}\tilde{z}_{1}\\\
\tilde{z}_{2}\\\ \tilde{z}_{3}\\\ \tilde{z}_{4}\end{bmatrix}.$

Taking the limit $k\to\infty$ yields the Lagrange structure $\mathcal{L}$ in
image representation

$\begin{bmatrix}q_{1}\\\ q_{2}\\\ p_{1}\\\ p_{2}\\\ \hline\cr e_{q1}\\\
e_{q2}\\\ e_{p1}\\\ e_{p2}\end{bmatrix}=\begin{bmatrix}0&1&0&0\\\ 0&1&0&0\\\
0&0&1&\frac{m_{1}}{m_{1}+m_{2}}\\\ 0&0&-1&\frac{m_{2}}{m_{1}+m_{2}}\\\
\hline\cr 1&0&0&0\\\ -1&0&0&0\\\ 0&0&\frac{1}{m_{1}}&\frac{1}{m_{1}+m_{2}}\\\
0&0&-\frac{1}{m_{2}}&\frac{1}{m_{1}+m_{2}}\end{bmatrix}\begin{bmatrix}\tilde{z}_{1}\\\
\tilde{z}_{2}\\\ \tilde{z}_{3}\\\ \tilde{z}_{4}\end{bmatrix}$

and the limiting Hamiltonian is just the kinetic energy

$\mathcal{H}(\tilde{z}_{1},\tilde{z}_{2},\tilde{z}_{3},\tilde{z}_{4})=\frac{1}{2}\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)\tilde{z}^{2}_{3}+\frac{\tilde{z}^{2}_{4}}{2(m_{1}+m_{2})}=\frac{\tilde{z}^{2}_{4}}{2(m_{1}+m_{2})}.$

The system has a _Lagrange algebraic constraint_ due to the linear dependency
in the rows of $P$. The Dirac structure $\mathcal{D}$ is given as

$\displaystyle\mathcal{D}$ $\displaystyle=$
$\displaystyle\left\\{(f_{q1},f_{q2},f_{p1},f_{p2},e_{q1},e_{q2},e_{p1},e_{p2}=0)\mid\mbox{there
exists }\lambda\right.$ $\displaystyle\mbox{ s.t. }\begin{bmatrix}f_{q1}\\\
f_{q2}\\\ f_{p1}\\\ f_{p2}\end{bmatrix}=\begin{bmatrix}0&0&-1&0\\\ 0&0&0&-1\\\
1&0&0&0\\\ 0&1&0&0\end{bmatrix}\begin{bmatrix}e_{q1}\\\ e_{q2}\\\ e_{p1}\\\
e_{p2}\end{bmatrix}+\begin{bmatrix}0\\\ 0\\\ 0\\\
1\end{bmatrix}\left.\lambda\right\\}.$

After elimination of the Lagrange multiplier $\lambda$ this yields

$K=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&0\end{bmatrix},\quad
L=\begin{bmatrix}0&0&1&0\\\ 0&0&0&1\\\ -1&0&0&0\\\ 0&0&0&0\end{bmatrix},$

and hence

$KP=\begin{bmatrix}0&1&0&0\\\ 0&1&0&0\\\ 0&0&1&\frac{m_{1}}{m_{1}+m_{2}}\\\
0&0&0&0\end{bmatrix},\quad
LS=\begin{bmatrix}0&0&\frac{1}{m_{1}}&\frac{1}{m_{1}+m_{2}}\\\
0&0&-\frac{1}{m_{2}}&\frac{1}{m_{1}+m_{2}}\\\ -1&0&0&0\\\
0&0&-\frac{1}{m_{2}}&\frac{1}{m_{1}+m_{2}}\end{bmatrix}.$

Finally subtracting the second equation from the first equation, we obtain the
HDAE system

$\begin{bmatrix}0&0&0&0\\\ 0&1&0&0\\\ 0&0&1&\frac{m_{1}}{m_{1}+m_{2}}\\\
0&0&0&0\end{bmatrix}\begin{bmatrix}\dot{\tilde{z}}_{1}\\\
\dot{\tilde{z}}_{2}\\\ \dot{\tilde{z}}_{3}\\\
\dot{\tilde{z}}_{4}\end{bmatrix}=\begin{bmatrix}0&0&\frac{1}{m_{1}}+\frac{1}{m_{2}}&0\\\
0&0&-\frac{1}{m_{2}}&\frac{1}{m_{1}+m_{2}}\\\ -1&0&0&0\\\
0&0&-\frac{1}{m_{2}}&\frac{1}{m_{1}+m_{2}}\end{bmatrix}\begin{bmatrix}\tilde{z}_{1}\\\
\tilde{z}_{2}\\\ \tilde{z}_{3}\\\ \tilde{z}_{4}\end{bmatrix}.$

Here the first equation is the Lagrange algebraic constraint $\tilde{z}_{3}=0$
(and eventually $\tilde{z}_{1}=0$) obtained by letting $k\to\infty$
(corresponding to singularity of $P$), and the last equation is the Dirac
algebraic constraint
$-\frac{\tilde{z}_{3}}{m_{2}}+\frac{\tilde{z}_{4}}{m_{1}+m_{2}}=0$, i.e.,
$\tilde{z}_{4}=0$ resulting from the kinematic constraint, leading to
singularity of $K$ and resulting in the trivial dynamics
$\dot{\tilde{z}}_{2}=0$.

###### Remark 18

Instead of using a parametrization of the Lagrange structure $\mathcal{L}$ one
can also use a parameterization of the Dirac structure $\mathcal{D}$,

$\left[\begin{array}[]{c}f\\\
e\end{array}\right]=\left[\begin{array}[]{c}L^{\top}\\\
K^{\top}\end{array}\right]v$ (29)

with $v\in\mathcal{V}$, where $\mathcal{V}$ is an $n$-dimensional parameter
space. This yields an extended dHDAE system (but now in the parameter vector
$v$) given by

$P^{\top}K^{\top}\dot{v}=S^{\top}L^{\top}v,$ (30)

which is the _adjoint system_ of (22). See [26] for a detailed discussion of
adjoint systems of DAEs.

###  4.2 Incorporation of dissipation

As noted in Section 4, extended HDAE systems (22), geometrically defined by a
Dirac and Lagrange structure, already include HDAE systems (2) _without_
dissipation. Conversely, any HDAE system with $K$ invertible can be rewritten
into the form (2) with $R=0$.

In order to complete the geometric viewpoint towards the inclusion of
dissipation (and thus to (2)), we recall the geometric definition of a port-
Hamiltonian system [35, 37, 40]. By replacing the Hamiltonian function by a
Lagrange structure as in [41], and specializing to the case without external
variables (inputs and outputs), such systems will be called _extended dHDAE
systems_.

###### Definition 19

Consider a state space $\mathcal{X}$ with linear coordinates $x$ and a linear
space of resistive flows $\mathcal{F}_{R}$. Furthermore, consider a Dirac
structure $\mathcal{D}$ on $\mathcal{X}\times\mathcal{F}_{R}$, a Lagrange
structure $\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$, and a
nonnegative Lagrange structure
$\mathcal{R}\subset\mathcal{F}_{R}\times\mathcal{F}^{*}_{R}$. Then an
_extended dissipative Hamiltonian DAE (extended dHDAE) system_ is defined as
the tuple $(\mathcal{X},\mathcal{F}_{R},\mathcal{D},\mathcal{L},\mathcal{R})$
with

$\begin{array}[]{rcl}&&\left\\{(\dot{x},x)\mid\mbox{there exist
}e\in\mathcal{X}^{*},f_{R}\in\mathcal{F}_{R},e_{R}\in\mathcal{F}^{*}_{R}\right.\\\\[5.69054pt]
&&\quad\left.\mbox{ such that
}(-\dot{x},-f_{R},e,e_{R})\in\mathcal{D},(f_{R},e_{R})\in\mathcal{R},\,(x,e)\in\mathcal{L}\right\\}.\end{array}$
(31)

If $\mathcal{L}$ is represented as in (27), i.e.

$\mathcal{L}=\left\\{\left[\begin{array}[]{c}x\\\
e\end{array}\right]\mid\left[\begin{array}[]{c}x\\\
e\end{array}\right]=\left[\begin{array}[]{c}P\\\
S\end{array}\right]z,\;z\in\mathbb{R}^{n}\right\\},$ (32)

then it immediately follows from the properties of the Dirac structure
$\mathcal{D}$ and the nonnegative Lagrange structure $\mathcal{R}$ that the
dynamics of the extended dHDAE satisfies

$\frac{d}{dt}(\frac{1}{2}z^{\top}S^{\top}Pz)=e_{R}^{\top}f_{R}\leq 0.$

More generally we will now introduce the notion a _maximally monotone
subspace_ , which is overarching the notions of a Dirac structure
$\mathcal{D}$ and a nonnegative Lagrange structure $\mathcal{R}$.

###### Definition 20

Consider a linear space $\mathcal{X}$. A subspace
$\mathcal{M}\subset\mathcal{X}\times\mathcal{X}^{*}$ is called _monotone_
subspace if

$e^{\top}f\geq 0$ (33)

for all $(f,e)\in\mathcal{M}$, and it is _maximally monotone_ if additionally
$\mathcal{M}$ is maximal with respect to this property (i.e., there does not
exist a monotone subspace
$\mathcal{M}^{\prime}\subset\mathcal{X}\times\mathcal{X}^{*}$ with
$\mathcal{M}\subsetneq\mathcal{M}^{\prime}$).

###### Remark 21

The definition of a monotone subspace is a special case of the notion of a
monotone _relation_ $\widetilde{M}$, which is defined as a _subset_ of
$\mathcal{X}\times\mathcal{X}^{*}$ satisfying

$(e_{1}-e_{2})^{\top}(f_{1}-f_{2})\geq 0$ (34)

for all $(f_{1},e_{1}),(f_{2},e_{2})\in\widetilde{M}$. Clearly if
$\widetilde{M}$ is a subspace then (34) reduces to (33). (Maximally) monotone
subspaces with a sign change were employed before in [21], using the
terminology of ’(maximally) linear dissipative relations’. Nonlinear port-
Hamiltonian systems with respect to a general (maximally) monotone relation
were coined as _incremental port-Hamiltonian systems_ in [10]; see [9] for
further developments.

Obviously, a subspace $\mathcal{M}$ is monotone if and only if the quadratic
form defined by $\Pi_{+}$ is nonnegative on $\mathcal{M}$, since
$\langle(f,e),(f,e)\rangle_{+}=2e^{\top}f$. This yields

###### Proposition 22

Consider a state space $\mathcal{X}$ with $\dim\mathcal{X}=n$. Then any
monotone subspace of $\mathcal{X}\times\mathcal{X}^{*}$ has dimension less
than or equal to $n$, and any maximally monotone subspace of
$\mathcal{X}\times\mathcal{X}^{*}$ has dimension $n$. Any maximally monotone
subspace $\mathcal{M}\subset\mathcal{X}\times\mathcal{X}^{*}$ can be
represented as

$\mathcal{M}=\mbox{\rm im\,}\left[\begin{array}[]{c}N^{\top}\\\
M^{\top}\end{array}\right]$ (35)

for $M,N\in\mathbb{R}^{n,n}$ satisfying
$\operatorname{rank}\left[\begin{array}[]{cc}N&M\end{array}\right]=n$ and

$MN^{\top}+NM^{\top}\geq 0.$ (36)

Conversely, any subspace defined by $M,N$ satisfying (36) is a maximally
monotone subspace.

Proof. The proof follows, since $\Pi_{+}$ has $n$ positive and $n$ negative
eigenvalues.

Obviously any Dirac structure $\mathcal{D}$ given by a pair of matrices $K,L$
is maximally monotone (take $M=K$ and $N=L$). In a similar way any nonnegative
Lagrange structure $\mathcal{R}$ given by a pair of matrices $P,S$ with
$S^{\top}P\geq 0$ is maximally monotone by taking $N^{\top}=P$, $M^{\top}=S$.

Importantly, also the _composition_ of two maximally monotone subspaces is
again maximally monotone. In order to prove this we first state the following
lemma.

###### Lemma 23

Let $A:\mathcal{F}\to\mathcal{G}$ be a linear map between two linear spaces
$\mathcal{F},\mathcal{G}$. Let
$\mathcal{M}_{\mathcal{G}}\subset\mathcal{G}\times\mathcal{G}^{*}$ be a
maximally monotone subspace. Then the _pull-back of
$\mathcal{M}_{\mathcal{G}}$ via $A$_, defined as

$b_{A}(\mathcal{M}_{\mathcal{G}}):=\\{(f,A^{*}g^{*})\in\mathcal{F}\times\mathcal{F}^{*}\mid(Af,g^{*})\in\mathcal{M}_{\mathcal{G}}\\},$

is maximally monotone. Furthermore, let
$\mathcal{M}_{\mathcal{F}}\subset\mathcal{F}\times\mathcal{F}^{*}$ be a
maximally monotone subspace. Then the _push-forward of
$\mathcal{M}_{\mathcal{F}}$ via $A$_, defined as

$f_{A}(\mathcal{M}_{\mathcal{F}}):=\\{(Af,g^{*})\in\mathcal{G}\times\mathcal{G}^{*}\mid(f,A^{*}g^{*})\in\mathcal{M}_{\mathcal{F}}\\},$

is maximally monotone.

Proof. It is immediately checked that $b_{A}(\mathcal{M}_{\mathcal{G}})$ is
monotone. Furthermore

$\begin{array}[]{l}\dim
b_{A}(\mathcal{M}_{\mathcal{G}})=\dim\mathcal{M}_{\mathcal{G}}+\dim\ker
A-\dim\ker A^{*}=\\\\[5.69054pt] \dim\mathcal{G}+\dim\ker A-\dim\ker
A^{*}=\dim\mathcal{F},\end{array}$

and thus $b_{A}(\mathcal{M}_{\mathcal{G}})$ is maximally monotone. The proof
to show that $f_{A}(\mathcal{M}_{\mathcal{F}})$ is maximally monotone is
analogous.

Using Lemma 23 we can show that maximally monotone subspaces satisfy the
following composition property. This same property was recently derived for
maximally monotone relations in [9], assuming additional regularity
conditions.

###### Proposition 24

Consider an extended dHDAE system as in (31) and let $\mathcal{M}_{a}$ and
$\mathcal{M}_{b}$ be maximally monotone subspaces

$\mathcal{M}_{a}\subset\mathcal{F}_{a}\times\mathcal{F}\times\mathcal{E}_{a}\times\mathcal{E},\quad\mathcal{M}_{b}\subset\mathcal{F}_{b}\times\mathcal{F}\times\mathcal{E}_{b}\times\mathcal{E}$

with
$\mathcal{E}=\mathcal{F}^{*},\mathcal{E}_{a}=\mathcal{F}_{a}^{*},\mathcal{E}_{b}=\mathcal{F}_{b}^{*}$.
Define the composition

$\begin{array}[]{rcl}\mathcal{M}_{a}\circ\mathcal{M}_{b}&:=&\\{(f_{a},f_{b},e_{a},e_{b})\mid\mbox{
there exist }f\in\mathcal{F},e\in\mathcal{E}\mbox{ such that }\\\\[5.69054pt]
&&(f_{a},f,e_{a},e)\in\mathcal{M}_{a},(f_{b},-f,e_{b},e)\in\mathcal{M}_{b}\\}.\end{array}$

Then
$\mathcal{M}_{a}\circ\mathcal{M}_{b}\subset\mathcal{F}_{a}\times\mathcal{F}_{b}\times\mathcal{E}_{a}\times\mathcal{E}_{b}$
is again maximally monotone.

Proof. Let $\mathcal{V}_{a}:=\mathcal{F}$ and $\mathcal{V}_{b}:=\mathcal{F}$.
Define the linear maps

$\begin{array}[]{lrll}\phi:&\mathcal{F}_{a}\times\mathcal{V}_{a}\times\mathcal{F}_{b}\times\mathcal{V}_{b}&\to&\mathcal{F}_{a}\times\mathcal{V}_{a}\times\mathcal{V}_{a}\times\mathcal{F}_{b}\times\mathcal{V}_{b}\times\mathcal{V}_{b}\\\\[5.69054pt]
&(f_{a},v_{a},f_{b},v_{b})&\mapsto&(f_{a},v_{a},v_{a},f_{b},v_{b},v_{b})\\\\[5.69054pt]
\psi:&\mathcal{F}_{a}\times\mathcal{V}_{a}\times\mathcal{F}_{b}\times\mathcal{V}_{b}&\to&\mathcal{F}_{a}\times\mathcal{F}_{b}\\\\[5.69054pt]
&(f_{a},v_{a},f_{b},v_{b})&\mapsto&(f_{a},f_{b})\\\\[5.69054pt] \end{array}$

Then for the maximally monotone subspace

$\mathcal{M}_{I}=\\{(v_{a},v_{b},v_{a}^{*},v_{b}^{*})\in\mathcal{V}_{a}\times\mathcal{V}_{b}\times\mathcal{V}^{*}_{a}\times\mathcal{V}^{*}_{b}\mid
v_{a}=-v_{b},v^{*}_{a}=v^{*}_{b}\\},$

it can be readily checked that

$\mathcal{M}_{a}\circ\mathcal{M}_{b}=f_{\psi}(b_{\phi}\left(\mathcal{M}_{a}\times\mathcal{M}_{I}\times\mathcal{M}_{b})\right),$

where $\mathcal{M}_{a}\times\mathcal{M}_{I}\times\mathcal{M}_{b}$ is clearly
maximally monotone. Then the proof finishes by applying Lemma 23.

We immediately have the following corollary.

###### Corollary 25

Consider a Dirac structure
$\mathcal{D}\subset\mathcal{X}\times\mathcal{F}_{R}\times\mathcal{X}^{*}\times\mathcal{F}^{*}_{R}$,
together with a nonnegative Lagrangian subspace
$\mathcal{R}\subset\mathcal{F}_{R}\times\mathcal{F}^{*}_{R}$. Then the
composition of $\mathcal{D}$ and $\mathcal{R}$ defined via

$\begin{array}[]{rcl}\mathcal{D}\circ\mathcal{R}:=\\{(f,e)\in\mathcal{X}\times\mathcal{X}^{*}&\mid&\mbox{
there exists }f_{R}\in\mathcal{F}_{R},e_{R}\in\mathcal{F}_{R}^{*}\mbox{ such
that }\\\\[5.69054pt] &&(f,-f_{R},e,e_{R})\in\mathcal{D},\mbox{ and
}(f_{R},e_{R})\in\mathcal{R}\\}\end{array}$

is maximally monotone. In particular, for any
$(f,e)\in\mathcal{D}\circ\mathcal{R}$, one has

$e^{\top}f=e_{R}^{\top}f_{R}\geq 0.$

###### Remark 26

We conjecture that conversely any maximally monotone subspace $\mathcal{M}$
can be generated this way, i.e., as the composition of a certain Dirac
structure $\mathcal{D}$ and a certain nonnegative Lagrangian subspace
$\mathcal{R}$.

The presented analysis of maximally monotone subspaces leads to the following
geometric definition of an _extended dHDAE_ system, covering both dHDAE
systems (2) and extended HDAE systems (22). See [21] for related results
(using the terminology of (maximally) dissipative linear relations).

###### Definition 27

Consider a linear state space $\mathcal{X}$ with coordinates $x$, a maximally
monotone subspace $\mathcal{M}\subset\mathcal{X}\times\mathcal{X}^{*}$, and a
Lagrange structure $\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$. Then
an _extended dHDAE system_ is a system $(\mathcal{X},\mathcal{M},\mathcal{L})$
satisfying

$\\{(\dot{x},x)\mid\mbox{ there exists }e\in\mathcal{X}^{*}\mbox{ such that
}(-\dot{x},e)\in\mathcal{M},(x,e)\in\mathcal{L}\\}.$ (37)

A coordinate representation of an extended dHDAE system is obtained as
follows. Consider a coordinate expression (35) of the maximally monotone
subspace $\mathcal{M}$ (with $M,N$ satisfying (36)). This means that any
element $(f,e)\in\mathcal{M}$ can be represented as

$\left[\begin{array}[]{c}f\\\
e\end{array}\right]=\left[\begin{array}[]{c}N^{\top}\\\
M^{\top}\end{array}\right]v$

for some $v\in\mathbb{R}^{n}$. Furthermore, any $(x,e)\in\mathcal{L}$ can be
represented as in (32). Substituting $-f=\dot{x}=P\dot{z}$ this yields

$\left[\begin{array}[]{c}-P\dot{z}\\\
Sz\end{array}\right]=\left[\begin{array}[]{c}N^{\top}\\\
M^{\top}\end{array}\right]v$

Now construct matrices $C,D$ satisfying

$\ker\left[\begin{array}[]{cc}C&D\end{array}\right]=\mbox{\rm
im\,}\left[\begin{array}[]{c}N^{\top}\\\ M^{\top}\end{array}\right].$ (38)

Then pre-multiplication by such a maximal annihilator
$\left[\begin{array}[]{cc}C&D\end{array}\right]$ eliminates the auxiliary
variables $v$, and one obtains the coordinate representation

$CP\dot{z}=DSz.$ (39)

###### Remark 28

The geometric construction of extended dissipative Hamiltonian system can be
immediately generalized to extended dissipative port-Hamiltonian DAE (dpHDAE)
systems _with external port variables_ (inputs and outputs), by extending the
maximally monotone subspace
$\mathcal{M}\subset\mathcal{X}\times\mathcal{X}^{*}$ to a maximally monotone
subspace
$\mathcal{M}_{e}\subset\mathcal{X}\times\mathcal{X}^{*}\times\mathcal{F}_{P}\times\mathcal{F}_{P}^{*}$,
where $\mathcal{F}_{P}\times\mathcal{F}_{P}^{*}$ is the space of external port
variables.

Two particular cases of Definition 27 are of special interest. The first one
is where the maximally monotone subspace $\mathcal{M}$ is actually a Dirac
structure as in (18) with $K,L$ satisfying (26). In this case one can take
$\left[\begin{array}[]{cc}C&D\end{array}\right]=\left[\begin{array}[]{cc}K&L\end{array}\right]$,
and thus the extended dHDAE system reduces to the extended HDAE system

$KP\dot{z}=LSz.$ (40)

The other special case is where the maximally monotone subspace $\mathcal{M}$
in (35) is such that $M$ is invertible. In this case, without loss of
generality $M^{\top}$ can be taken to be the identity matrix, and the maximal
annihilator $\left[\begin{array}[]{cc}C&D\end{array}\right]$ can be taken to
be of the form $\left[\begin{array}[]{cc}I&D\end{array}\right]$. Hence
$D=-N^{\top}$, and thus (39) reduces to

$P\dot{z}=-N^{\top}Sz.$

Furthermore, $MN^{\top}+NM^{\top}\geq 0$ reduces to $N^{\top}+N\geq 0$, and
hence

$N^{\top}=-\frac{1}{2}(N-N^{\top})+\frac{1}{2}(N+N^{\top})=:-J+R$

with $J=-J^{\top}$ and $R=R^{\top}\geq 0$. Thus in this case the extended
dpHDAE system takes the familiar form (2) with $E=P$, $S=Q$ expressed as

$E\dot{z}=(J-R)Qz,\quad Q^{\top}E=E^{\top}Q,J=-J^{\top},R=R^{\top}\geq 0.$

###### Remark 29

Similar to the theory exposed in [41] for HDAE systems (22), the _algebraic
constraints_ of the dpHDAE system (39) can be split into two classes: one
corresponding to singularity of $P$ (Lagrange algebraic constraints in [41]),
and one corresponding to singularity of $C$. In case of (22) the second class
of algebraic constraints are called Dirac algebraic constraints in [41], but
now they correspond to the maximally monotone subspace.

Furthermore, mimicking the developments in [41], one can transform algebraic
constraints associated with index one belonging to one class into algebraic
constraints in the other, by the use of additional state variables (serving as
Lagrange multipliers).

##  5 Representation of DAE systems generated by Dirac and Lagrange structures
in the state variables $x$

The representation (22) of an extended HDAE system as discussed in the
previous sections does not use the state variables $x$ of the state space
$\mathcal{X}$, but instead an equally dimensioned vector $z\in\mathcal{Z}$
parameterizing the Lagrange structure, cf. (21) and (27). In this section we
show how a _different_ DAE representation involving the original state vector
$x\in\mathcal{X}$ can be obtained. Furthermore we discuss in what sense this
representation in $x$ is equivalent with the representation (22) involving
$z$.

###  5.1 A coordinate representation in the original state variables $x$

Consider a Dirac structure
$\mathcal{D}\subset\mathcal{X}\times\mathcal{X}^{*}$, a Lagrange structure
$\mathcal{L}\subset\mathcal{X}\times\mathcal{X}^{*}$, and the resulting
dynamics specified (in coordinate-free form) as
$\mathcal{D}\circ\mathcal{L}\subset{\mathcal{X}}\times\mathcal{X}$. Let $x$ be
coordinates for the state space $\mathcal{X}$ and let the Dirac structure
represented by a pair of matrices $K,L$ and the Lagrange structure by a pair
of matrices $P,S$. To derive a coordinate representation employing directly
the state vector $x$, we first consider the combined representations of
$\mathcal{D}$ and $\mathcal{L}$, both in kernel representation, i.e.,

$\left[\begin{array}[]{ccc}K&L&0\\\
0&-P^{\top}&S^{\top}\end{array}\right]\left[\begin{array}[]{c}-\dot{x}\\\ e\\\
x\end{array}\right]=0,$ (41)

where $e$ are dual coordinates for $\mathcal{X}^{*}$. In order to obtain a DAE
system only involving $x$ we need to eliminate the variables $e$. This can be
done by considering a _maximal annihilator_ (left null-space)
$\left[\begin{array}[]{cc}M&N\end{array}\right]$ of
$\left[\begin{array}[]{c}L\\\ -P^{\top}\end{array}\right]$, i.e.,

$\ker\left[\begin{array}[]{cc}M&N\end{array}\right]=\mbox{\rm
im\,}\left[\begin{array}[]{c}L\\\ -P^{\top}\end{array}\right],$ (42)

and thus, in particular,

$ML=NP^{\top}.$ (43)

Since

$\left[\begin{array}[]{cc}M&N\end{array}\right]\left[\begin{array}[]{ccc}K&L&0\\\
0&-P^{\top}&S^{\top}\end{array}\right]=\left[\begin{array}[]{ccc}MK&0&NS^{\top}\end{array}\right],$

premultiplication of the equations (41) by
$\left[\begin{array}[]{cc}M&N\end{array}\right]$ thus yields

$\mathcal{D}\circ\mathcal{L}=\ker\left[\begin{array}[]{cc}MK&NS^{\top}\end{array}\right]\subset{\mathcal{X}}\times\mathcal{X}.$

Hence the resulting DAE system is given by

$MK\dot{x}=NS^{\top}x.$ (44)

###### Remark 30

Also for extended dHDAE systems (including _dissipation_) we can consider,
instead of the coordinate representation (39) involving the parametrizing
vector $z$, a representation that is using the original state $x$. In fact,
let as before, cf. (38), $\left[\begin{array}[]{cc}C&D\end{array}\right]$
denote a maximal annihilator of
$\left[\begin{array}[]{cc}N&M\end{array}\right]^{\top}$, i.e.,
$\ker\left[\begin{array}[]{cc}C&D\end{array}\right]=\mbox{\rm
im\,}\left[\begin{array}[]{cc}N&M\end{array}\right]^{\top}$. Then consider,
similarly to (41), the stacked matrix

$\left[\begin{array}[]{ccc}C&D&0\\\ 0&-P^{\top}&S^{\top}\end{array}\right]$

and a maximal annihilator $\left[\begin{array}[]{cc}V&W\end{array}\right]$ to
$\left[\begin{array}[]{cc}D^{\top}&-P\end{array}\right]^{\top}$, that is
$\ker\left[\begin{array}[]{cc}V&W\end{array}\right]=\mbox{\rm
im\,}\left[\begin{array}[]{cc}D^{\top}&-P\end{array}\right]^{\top}$. Then
premultiplication by $\left[\begin{array}[]{cc}V&W\end{array}\right]$ yields
the representation

$VC\dot{x}=WS^{\top}x$

The analysis performed in the current subsection for (44) can be performed,
mutatis mutandis, for (30) as well.

Recall that in the coordinate representation (22) we have the expression
$\mathcal{H}^{z}(z)=\frac{1}{2}z^{\top}S^{\top}Pz$ for the Hamiltonian. In the
representation (44) we do not yet have a Hamiltonian associated with the
extended HDAE system. To define such a Hamiltonian $\mathcal{H}^{x}$ in (44)
we would need that $P$ is invertible, in which case it is given by

$\mathcal{H}^{x}(x)=\frac{1}{2}x^{\top}SP^{-1}x.$ (45)

If $P$ is invertible then there is a direct relation between the Hamiltonians
(23) and (45). In fact, substituting $x=Pz$, we immediately obtain

$\mathcal{H}^{x}(x)=\frac{1}{2}x^{\top}SP^{-1}x=\frac{1}{2}z^{\top}P^{\top}SP^{-1}Pz=\frac{1}{2}z^{\top}P^{\top}Sz=\mathcal{H}^{z}(z).$

Alternatively, if $S$ is invertible then one can use the _co-energy_ (Legendre
transform) of $H^{x}$ given by

$\mathcal{H}^{e}(e)=\frac{1}{2}e^{\top}PS^{-1}e$ (46)

for which $\mathcal{H}^{e}(e)=\mathcal{H}^{z}(z)$ with $e=Sz$.

Note that if $P$ is invertible, then also $M$ is invertible. This follows,
since then $ML=NP^{\top}$ implies that the columns of $N$ are in $\mbox{\rm
im\,}M$, and since
$\operatorname{rank}\left[\begin{array}[]{cc}M&N\end{array}\right]=n$ this
means $M$ is invertible. The converse that $M$ invertible implies $P$
invertible follows analogously. In a similar fashion, it follows that $L$ is
invertible if and only if $N$ is invertible. These observations imply the
following simplifications of the representation (44) under additional
assumptions.

  * 1.

    * a)

If $P$ is invertible then $N=MLP^{-\top}$ and by multiplying (44) from the
left by $M^{-1}$ we obtain the system

$K\dot{x}=LP^{-\top}S^{\top}x=L\left(SP^{-1}\right)^{\top}x=LSP^{-1}x,$ (47)

where the last equality follows from $S^{\top}P=P^{\top}S$. This is exactly
the form of a Hamiltonian DAE system in case of a general Dirac structure and
a Lagrange structure that is given as the graph of a symmetric matrix
$Q:=SP^{-1}$, see [35, 38, 39, 40]. Indeed, the Lagrange structure simplifies
to the gradient of the Hamiltonian function
$\mathcal{H}^{x}(x)=\frac{1}{2}x^{\top}SP^{-1}x$.

    * b)

If in this case additionally $K$ is invertible, then we obtain the _Poisson
formulation_ of Hamiltonian systems, see e.g. [2],

$\dot{x}=\left(K^{-1}L\right)\left(SP^{-1}\right)x=JQx,$ (48)

with $J=-J^{\top}=K^{-1}L$, and $Q=Q^{\top}$.

  * 2.

    * a)

If $L$ and thus also $N$ is invertible, then by (43) we have
$M=NP^{\top}L^{-1}$ and multiplying with $N^{-1}$ from the left we get the DAE

$P^{\top}J\dot{x}=P^{\top}L^{-1}K\dot{x}=S^{\top}x$ (49)

with $J:=\left(L^{-1}K\right)^{\top}=-L^{-1}K$.

    * b)

If additionally $P$ is invertible, then with $Q=SP^{-1}=P^{-\top}S^{\top}$
this may be rewritten as

$J\dot{x}=Qx,$ (50)

which is the standard _symplectic formulation_ of a Hamiltonian system in case
additionally $J$ is invertible, see e.g. [2].

###  5.2 Relation between the representations (22) and (44)

An immediate question that arises is how the representations (22) and (44) are
related. We have already seen that if $P$ is _invertible_ then the
relationship is obvious, since in this case $x=Pz$ defines an ordinary state
space transformation. However, if $P$ is not invertible then the
representations are _not_ state space equivalent, as the following simple
example demonstrates.

###### Example 31

For $P=[0]$, $S=[1]$, $K=[1]$, $L=[0]$, we have that (22) is the singular
system $0\cdot\dot{z}=0\cdot z$. On the other hand

$\operatorname{rank}\left[\begin{array}[]{c}L\\\
-P^{\top}\end{array}\right]=0,$

and $\mathcal{D}\circ\mathcal{L}$ is the origin in
${\mathcal{X}}\times\mathcal{X}$, defining the degenerate DAE system
$\dot{x}=x=0$.

However, representations (22) (in the parameterizing $z$ variables) and (44)
(in the original state variables $x$) can be shown to be equivalent in the
following generalized sense. First note that for any representation $P,S$ of a
Lagrange structure there exist nonsingular matrices $V,W$ such that

$\displaystyle V^{-1}PW=V^{\top}PW=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&0\end{array}\right],\ V^{\top}SW=\left[\begin{array}[]{cc}S_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ (55) $\displaystyle
KV=\left[\begin{array}[]{cc}K_{11}&K_{12}\\\ K_{21}&K_{22}\end{array}\right],\
LV^{-T}=LV=\left[\begin{array}[]{cc}L_{11}&L_{12}\\\
L_{21}&L_{22}\end{array}\right].$ (60)

This is a direct consequence of Lemma 37 that will be presented in the next
section. Setting $\left[\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right]=W^{-1}z$ and $\left[\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right]=V^{-1}x$, it follows that $z_{1}=x_{1}$ and
$z_{2}=e_{2}$. After such a transformation the system $KP\dot{z}=LSz$ takes
the form

$\left[\begin{array}[]{cc}K_{11}&K_{12}\\\
K_{21}&K_{22}\end{array}\right]\left[\begin{array}[]{cc}I&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}\dot{x}_{1}\\\
\dot{e}_{2}\end{array}\right]=\left[\begin{array}[]{cc}L_{11}&L_{12}\\\
L_{21}&L_{22}\end{array}\right]\left[\begin{array}[]{cc}S_{11}&0\\\
0&I\end{array}\right]\left[\begin{array}[]{c}x_{1}\\\
e_{2}\end{array}\right].$ (61)

If we _add_ to the vector $\left[\begin{array}[]{cc}x_{1}\\\
e_{2}\end{array}\right]$ the subvector $x_{2}$, and if we consider the
equations (61) together with the original Lagrange algebraic constraint
$x_{2}=0$, then the so extended system can be rewritten as

$\left[\begin{array}[]{cc}K_{11}&K_{12}\\\ K_{21}&K_{22}\\\
0&0\end{array}\right]\left[\begin{array}[]{cc}I&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}\dot{x}_{1}\\\
\dot{x}_{2}\end{array}\right]=\left[\begin{array}[]{cc}L_{11}S_{11}x_{1}\\\
L_{21}S_{11}x_{1}\\\ x_{2}\end{array}\right]+\left[\begin{array}[]{c}L_{12}\\\
L_{22}\\\ 0\end{array}\right]e_{2}.$ (62)

On the other hand, as shown in Subsection 5.1, the extended dHDAE system
defined by the Dirac structure $\mathcal{D}$ and the Lagrangian structure
$\mathcal{L}$ in the state space variables $x$ can be expressed as

$\left[\begin{array}[]{cccccc}K_{11}&K_{12}&L_{11}&L_{12}&0&0\\\
K_{21}&K_{22}&L_{21}&L_{22}&0&0\\\ 0&0&-I&0&S_{11}&0\\\
0&0&0&0&0&I\end{array}\right]\left[\begin{array}[]{c}-\dot{x}_{1}\\\
-\dot{x}_{2}\\\ e_{1}\\\ e_{2}\\\ x_{1}\\\ x_{2}\end{array}\right]=0,$ (63)

with $e_{1},e_{2}$ serving as _auxiliary variables_. Instead of eliminating
$e_{1},e_{2}$ from these equations, as discussed in Subsection 4.1, we can
only eliminate $e_{1}$ by premultiplication of (63) by the full row rank
matrix

$\left[\begin{array}[]{cc}M&N\end{array}\right]=\left[\begin{array}[]{cccc}I&0&L_{11}&0\\\
0&I&L_{21}&0\\\ 0&0&0&I\end{array}\right],$

which directly leads to the system (62). This extended equivalence between
(22) and (44) is summarized in the following proposition.

###### Proposition 32

Consider the pHDAE representations (22) and (44) defined by the same Lagrange
structure $\mathcal{L}$ represented by matrices $P,S$, and by the same Dirac
structure $\mathcal{D}$ represented by $K,L$. Consider a transformation such
that $P,S$ and $K,L$ are transformed into the form (60) with corresponding
partitioning

$x=\left[\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right],\;z=\left[\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right],$

where $x_{1}=z_{1}$. Adding to (22) the Lagrange algebraic constraint
$x_{2}=0$ corresponding to $x=Pz$, the resulting dHDAE system is given by
(62). This system is equivalent to the representation (41) of (44) after
elimination of the variables $e_{1}$.

Note that the subvector $e_{2}=z_{2}$ can be regarded as the _Lagrange
multiplier vector_ corresponding to the constraint $x_{2}=0$. As such,
$e_{2}=z_{2}$ does not contribute to the expression of the Hamiltonian
$\mathcal{H}^{z}(z)$.

Let us illustrate the previous discussion with some further examples.

###### Example 33

Consider a system in the form (22) with $K=\left[\begin{array}[]{cc}I&0\\\
0&I\end{array}\right]$, $L=\left[\begin{array}[]{cc}0&I\\\
-I&0\end{array}\right]$, $P=\left[\begin{array}[]{cc}I&0\\\
0&0\end{array}\right]$, $S=\left[\begin{array}[]{cc}I&0\\\
0&I\end{array}\right]$ which is the general DAE

$\left[\begin{array}[]{cc}I&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}\dot{z}_{1}\\\
\dot{z}_{2}\end{array}\right]=\left[\begin{array}[]{cc}0&I\\\
-I&0\end{array}\right]\left[\begin{array}[]{c}z_{1}\\\
z_{2}\end{array}\right]$

In order to compute the representation (44), we solve

$0=\left[\begin{array}[]{cc}M&N\end{array}\right]\left[\begin{array}[]{cc}0&I\\\
-I&0\\\ -I&0\\\ 0&0\end{array}\right],$

and with

$\left[\begin{array}[]{cc}M&N\end{array}\right]=\left[\begin{array}[]{cccc}0&0&0&I\\\
0&I&-I&0\end{array}\right]$

we get the system

$\left[\begin{array}[]{cc}0&0\\\
0&I\end{array}\right]\left[\begin{array}[]{c}\dot{x}_{1}\\\
\dot{x}_{2}\end{array}\right]=\left[\begin{array}[]{cc}0&I\\\
-I&0\end{array}\right]\left[\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right].$

###### Example 34

Consider the system of the form (22) with $K=0$, $L=I$, $P=I$, $S=I$, i.e.,
$0\cdot\dot{z}=z$. Solving

$0=\left[\begin{array}[]{cc}M&N\end{array}\right]\left[\begin{array}[]{c}I\\\
-I\end{array}\right],$

yields $M=I,N=I$, and we obtain the system (44) given by $0\cdot\dot{x}=x$.

###### Example 35

Consider the system of the form (22) with $K=0$, $L=I$, $P=0$, $S=I$, i.e.
$0\cdot\dot{z}=z$. Solving

$0=\left[\begin{array}[]{cc}M&N\end{array}\right]\left[\begin{array}[]{c}I\\\
0\end{array}\right]$

yields $M=0,N=I$, and thus the representation (44) is $0\cdot\dot{x}=x$.

###### Example 36

Consider the system of the form (22) with $K=0$, $L=\begin{bmatrix}0&I\\\
-I&0\end{bmatrix}$, $P=I$, $S=I$, i.e., $0\cdot\dot{z}=\begin{bmatrix}0&I\\\
-I&0\end{bmatrix}z$. Solving

$0=\left[\begin{array}[]{cc}M&N\end{array}\right]\left[\begin{array}[]{cc}0&I\\\
-I&0\\\ -I&0\\\ 0&-I\end{array}\right]$

yields $M=I,N=\left[\begin{array}[]{cc}0&I\\\ -I&0\end{array}\right]$, and
thus the representation (44) is given by

$0\cdot\dot{x}=\left[\begin{array}[]{cc}0&I\\\ -I&0\end{array}\right]x.$

##  6 Equivalence transformations and condensed forms

To characterize the properties of extended dHDAEs we use transformations to
condensed forms from which the properties can be read off.

For general DAEs (1) given by matrix pairs $(E,A)$, $E,A\mathbb{R}^{\ell,n}$(
or the representation via matrix pencils $\lambda E-A$) we can perform
equivalence transformations of the coefficients of the form

$(\tilde{E},\tilde{A})=(U^{\top}EW,U^{\top}AW),$ (64)

with $U\in\mathbb{R}^{\ell,ell}$, $W\in\mathbb{R}^{n,n}$ nonsingular. This
corresponds to a scaling of the equation with $U^{\top}$ and a change of
variables $z=W\tilde{z}$. Under such transformations there is a one-to-one
relationship between the solution spaces, see [25] and the canonical form is
the Weierstraß canonical form.

For structured systems of the form (2), the associated equivalence
transformation that preserves the structure is of the form

$(\tilde{E},(\tilde{J}-\tilde{R})\tilde{Q})=(U^{\top}EW,(U^{\top}(J-R)U)U^{-1}QW),$

with $U\in\mathbb{R}^{\ell,\ell}$, $W\in\mathbb{R}^{n,n}$ nonsingular. A
condensed form for this case has been presented in [29].

Finally for systems of the form (22), the equivalence transformations have the
form

$\tilde{K}=U^{\top}KV,\ \tilde{L}=U^{\top}LV^{-T},\ \tilde{P}=V^{-1}PW,\
\tilde{S}=V^{\top}SW,$ (65)

where $U\in\mathbb{R}^{\ell,\ell}$, $V\in\mathbb{R}^{n,n}$,
$W\in\mathbb{R}^{m,m}$ are nonsingular.

The geometric interpretation of the set of transformations in (65) is clear:
$V$ defines a coordinate transformation on the state space $\mathcal{X}$ while
$V^{-\top}$ is the corresponding dual transformation on the dual state space
$\mathcal{X}^{*}$. Also note that the combination of $V$ and $V^{-\top}$ on
the product space $\mathcal{X}\times\mathcal{X}^{*}$ leaves the canonical
bilinear forms defined by the matrices $\Pi_{-}$ and $\Pi_{+}$ invariant. (In
fact, it can be shown that any transformation on
$\mathcal{X}\times\mathcal{X}^{*}$ that leaves both canonical bilinear forms
invariant is necessarily of this form for some invertible $V$.) Finally,
$U^{\top}$ is an invertible transformation on the equation space for the
kernel representation of the Dirac structure $\mathcal{D}$, while $W$ is an
invertible transformation on the parametrization space $\mathcal{Z}$ for the
Lagrange structure $\mathcal{L}$.

In all three cases, in view of an implementation of the transformations as
numerically stable procedures, we are also interested in the case that $U,V,W$
are real orthogonal matrices. We then have that $V^{-1}=V^{\top}$ and for both
pairs $(K,L)$ and $(P,S)$ this is a classical orthogonal equivalence
transformation.

Using the described equivalence transformations we can derive condensed forms
for pencils $\lambda P-S$ with $P^{\top}S=S^{\top}P$ associated with Lagrange
subspaces (or _isotropic subspaces_ if the dimension is not $n$, see e.g.
[29]). Here we slightly modify the representation and also give a constructive
proof that can be implemented as numerically stable algorithm in Appendix A.

###### Lemma 37

Let $P,S\in\mathbb{R}^{n,m}$ be such that $P^{\top}S=S^{\top}P$. Then there
exist invertible matrices $V\in\mathbb{R}^{n,n}$, $W\in\mathbb{R}^{m,m}$ such
that

$V^{-1}PW=\left[\begin{array}[]{ccccc}I_{m_{1}}&0&0&0&0\\\
0&I_{m_{2}}&0&0&0\\\ 0&0&I_{m_{3}}&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right],V^{\top}SW=\left[\begin{array}[]{cccccc}I_{m_{1}}&0&0&0&0\\\
0&-I_{m_{2}}&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&I_{m_{4}}&0\\\
S_{51}&S_{5,2}&S_{5,3}&0&0\\\ 0&0&0&0&0\end{array}\right],$ (66)

with $\left[\begin{array}[]{ccc}S_{51}&S_{5,2}&S_{5,3}\end{array}\right]$ of
full row rank $n_{5}$. (Note that block sizes may be zero). Moreover, if the
pencil $\lambda P-S$ is regular then the condensed form is unique, except for
the order of blocks, and just contains the first four block rows and columns.

Proof. See Appendix A.

Note that the condensed form is in general not unique in the fifth block row,
but the block sizes $m_{1},m_{2},m_{3},m_{4}$ and the row dimension $n_{5}$
are.

###### Corollary 38

Let $P,S\in\mathbb{R}^{n,m}$ be such that $P^{\top}S=S^{\top}P$. Then there
exist real orthogonal matrices $V\in\mathbb{R}^{n,n}$, $W\in\mathbb{R}^{m,m}$
such that

$V^{\top}PW=\left[\begin{array}[]{cccc}P_{11}&0&0&0\\\ P_{21}&P_{22}&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right],V^{\top}SW=\left[\begin{array}[]{cccccc}S_{11}&0&0&0\\\
0&0&0&0\\\ S_{31}&S_{32}&S_{33}&0\\\ S_{41}&S_{42}&0&0\\\
0&0&0&0\end{array}\right],$ (67)

with $P_{11},S_{11}\in\mathbb{R}^{m_{1}+m_{2},m_{1}+m_{2}}$,
$P_{22}\in\mathbb{R}^{m_{3},m_{3}}$, $S_{33}\in\mathbb{R}^{m_{4},m_{4}}$
invertible, $\left[\begin{array}[]{ccc}S_{41}&S_{42}\end{array}\right]$ of
full row rank $n_{5}$, and $P_{11}^{\top}S_{11}=S_{11}^{\top}P_{11}$. Here the
block sizes $m_{1}+m_{2}$, $m_{3}$, $m_{4}$, and $m_{5}$ are as in (66).

Proof. The proof follows by performing Steps 1. and 2. of the proof of Lemma
37, see Appendix B, which yields

$V_{2}^{\top}V_{1}^{\top}PW_{1}W_{2}=\left[\begin{array}[]{ccc}\hat{P}_{11}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right],\
V_{2}^{\top}V_{1}^{\top}SW_{1}W_{2}=\left[\begin{array}[]{ccc}\hat{S}_{11}&0&0\\\
\hat{S}_{21}&\hat{S}_{22}&0\\\ \hat{S}_{31}&0&0\end{array}\right],$

followed by a singular value decomposition
$\hat{V}^{\top}_{3}\hat{S}_{11}\hat{W}_{3}=\left[\begin{array}[]{cc}\check{S}_{11}&0\\\
0&0\end{array}\right]$ with $\check{S}_{11}$ nonsingular diagonal and a full
rank decomposition
$\check{V}_{4}^{\top}\hat{S}_{31}\hat{W}_{3}=\left[\begin{array}[]{cc}S_{41}&S_{42}\end{array}\right]$.

Corollary 38 shows that the characteristic quantities $m_{1}+m_{2}$, $m_{3}$
and $m_{4}$, as well as $n_{5}$ can be obtained by purely real orthogonal
transformations. The quantities $m_{1},m_{2}$ can then be determined from the
real orthogonal staircase form of the symmetric pencil $\lambda
P_{11}^{\top}S_{11}-S_{11}^{\top}P_{11}$ which has been presented in [8] and
implemented as production software in [7].

There is an analogous condensed form for pencils of the form $\lambda K-L$
satisfying $KL^{\top}=-LK^{\top}$. For the case of regular pairs this directly
follows from the canonical form presented in [11], but again we present the
construction so that it can be directly implemented as a numerical method, see
Appendix B.

###### Lemma 39

Let $K,L\in\mathbb{R}^{\ell,n}$ be such that $KL^{\top}=-LK^{\top}$. Then
there exist invertible matrices $U\in\mathbb{R}^{\ell,\ell}$,
$V\in\mathbb{R}^{n,n}$ such that

$U^{\top}KV=\left[\begin{array}[]{cccccc}I_{\ell_{1}}&0&0&0&0&0\\\
0&I_{\ell_{1}}&0&0&0&0\\\ 0&0&I_{\ell_{3}}&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\end{array}\right],U^{\top}LV^{-\top}=\left[\begin{array}[]{cccccc}0&I_{\ell_{1}}&0&0&L_{15}&0\\\
-I_{\ell_{1}}&0&0&0&L_{25}&0\\\ 0&0&0&0&L_{35}&0\\\ 0&0&0&I_{\ell_{4}}&0&0\\\
0&0&0&0&0&0\end{array}\right],$ (68)

with $\left[\begin{array}[]{c}L_{15}\\\ L_{2,5}\\\ L_{3,5}\end{array}\right]$
of full column rank $n_{5}$. (Note that block sizes may be zero). Moreover, if
the pencil $\lambda K-L$ is regular then the condensed form is unique except
for the order of blocks and just contains the first four block rows and
columns.

Proof. See Appendix B.

Note again that the form (68) is not unique in general but the block sizes
$\ell_{1},\ell_{2},\ell_{3},\ell_{4}$ and the column dimension $n_{5}$ are.

###### Corollary 40

Let $K,L\in\mathbb{R}^{\ell,n}$ be such that $LK^{\top}=-KL^{\top}$. Then
there exist real orthogonal matrices $U\in\mathbb{R}^{\ell,\ell}$,
$V\in\mathbb{R}^{n,n}$ such that

$U^{\top}KV=\left[\begin{array}[]{ccccc}K_{11}&K_{12}&0&0&0\\\
0&K_{22}&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right],U^{\top}LV=\left[\begin{array}[]{cccccc}L_{11}&0&L_{13}&L_{14}&0\\\
0&0&L_{23}&L_{24}&0\\\ 0&0&L_{33}&0&0\\\ 0&0&0&0&0\end{array}\right].$ (69)

with $K_{11},L_{11}\in\mathbb{R}^{2\ell_{1},2\ell_{1}}$,
$K_{22}\in\mathbb{R}^{\ell_{3},\ell_{3}}$,
$L_{33}\in\mathbb{R}^{\ell_{4},\ell_{4}}$ invertible,
$\left[\begin{array}[]{c}L_{14}\\\ L_{2,4}\end{array}\right]$ is of full
column rank $n_{5}$, and $K_{11}^{\top}L_{11}=-L_{11}^{\top}K_{11}$. Here the
block-sizes $\ell_{1}$, $\ell_{3}$, $\ell_{4}$, and $n_{5}$ are as in (68).

Proof. The proof follows by performing Steps 1. and 2. of the proof of Lemma
39, which yields

$U_{2}^{\top}U_{1}^{\top}KV_{1}V_{2}=\left[\begin{array}[]{ccc}\hat{K}_{11}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right],\
U_{2}^{\top}U_{1}^{\top}LV_{1}V_{2}=\left[\begin{array}[]{ccc}\hat{L}_{11}&\hat{L}_{12}&\hat{L}_{13}\\\
0&\hat{L}_{22}&0\\\ 0&0&0\end{array}\right],$

followed by a singular value decomposition
$\hat{U}^{\top}_{3}\hat{L}_{11}\hat{V}_{4}=\left[\begin{array}[]{cc}\check{L}_{11}&0\\\
0&0\end{array}\right]$ with $\check{L}_{11}$ nonsingular diagonal and a full
rank decomposition
$\hat{U}_{3}^{\top}\hat{L}_{13}\hat{V}_{3}=\left[\begin{array}[]{c}L_{14}\\\
L_{24}\end{array}\right]$.

Corollary 40 shows that the characteristic quantities $\ell_{1}$, $\ell_{3}$,
$\ell_{4}$, as well as $n_{5}$ can be obtained by purely real orthogonal
transformations.

The presented condensed forms can now be used in generating a condensed form
for systems of the form (22).

###### Lemma 41

Consider a system of the form (22) with $K,P,L,S\in\mathbb{R}^{n,n}$ and
regular pencil $\lambda KP-LS$. Then there exists invertible matrices $U,V,W$
as in (65) such that

$\displaystyle\hat{K}$ $\displaystyle=$ $\displaystyle
U^{\top}KV=\left[\begin{array}[]{cc}K_{11}&K_{12}\\\
K_{21}&K_{22}\end{array}\right],\
\hat{L}=U^{\top}LV^{-\top}=\left[\begin{array}[]{cc}L_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ (74) $\displaystyle\hat{P}$ $\displaystyle=$
$\displaystyle V^{-1}PW=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&0\end{array}\right],\
\hat{S}=V^{\top}SW=\left[\begin{array}[]{cc}S_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ (79)

where $S_{11}=S_{11}^{\top}$ and $K_{11}L_{11}^{\top}=-L_{11}K_{11}^{\top}$.

Proof. Since the pencil $\lambda KP-LS$ is square and regular, it is square,
and also the pencil $\lambda P-S$ is regular, otherwise by Lemma 37 there
would be common kernel of $P$ and $S$ which would imply the pencil $\lambda
KP-LS$ to be singular.

Thus, by Lemma 37 there exist nonsingular matrices
$W_{1},V_{1}\in\mathbb{R}^{n,n}$ such that

$\displaystyle\tilde{K}$ $\displaystyle=$ $\displaystyle
KV_{1}=\left[\begin{array}[]{cc}\tilde{K}_{11}&\tilde{K}_{12}\\\
\tilde{K}_{21}&\tilde{K}_{22}\end{array}\right],\
\tilde{L}=LV_{1}^{-\top}=\left[\begin{array}[]{cc}\tilde{L}_{11}&\tilde{L}_{12}\\\
\tilde{L}_{21}&\tilde{L}_{22}\end{array}\right],$ $\displaystyle\tilde{P}$
$\displaystyle=$ $\displaystyle
V_{1}^{-1}PW_{1}=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&0\end{array}\right],\
\tilde{S}=V_{1}^{\top}SW_{1}=\left[\begin{array}[]{cc}\tilde{S}_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$

with $n_{1}=m_{1}+m_{2}+m_{3}$, $n_{2}=m_{4}$ and $\tilde{S}_{11}$ symmetric.
The regularity of the pencil $\lambda KP-LS$ implies that

$\left[\begin{array}[]{c}\tilde{L}_{12}\\\ \tilde{L}_{22}\end{array}\right]$

has full column rank and hence there exist invertible matrices
$U_{2}\in\mathbb{R}^{n,n}$, $\tilde{V}_{2}\in\mathbb{R}^{n_{2},n_{2}}$, and

$V_{2}=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&\tilde{V}_{2}\end{array}\right]$

such that

$U_{2}^{\top}\tilde{L}V_{2}=\left[\begin{array}[]{cc}\hat{L}_{11}&0\\\
\hat{L}_{21}&I_{n_{2}}\end{array}\right].$

With

$V_{3}=\left[\begin{array}[]{cc}I_{n_{1}}&\hat{L}_{21}^{\top}\\\
0&I_{n_{2}}\end{array}\right],$

we then get that

$\displaystyle\hat{K}$ $\displaystyle=$ $\displaystyle
U_{2}^{\top}KV_{1}V_{2}V_{3}=\left[\begin{array}[]{cc}K_{11}&K_{12}\\\
K_{21}&K_{22}\end{array}\right],\
\hat{L}=U_{2}^{\top}LV_{1}^{-\top}V_{2}^{-\top}V_{3}^{-\top}=\left[\begin{array}[]{cc}L_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ $\displaystyle\hat{P}$ $\displaystyle=$
$\displaystyle
V_{3}^{-1}V_{2}^{-1}V_{1}^{-1}PW_{1}=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&I_{n_{2}}\end{array}\right],\
\hat{S}=V_{3}^{\top}V_{2}^{\top}V_{1}^{\top}SW_{1}=\left[\begin{array}[]{cc}S_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$

has the desired form with $U=U_{2}$, $V=V_{1}V_{2}V_{3}$, $W=W_{1}$, and where
$S_{11}=S_{11}^{\top}$ and $K_{11}L_{11}^{\top}=-L_{11}K_{11}^{\top}$.

Transforming the system as in (79) and setting

$z=W\left[\begin{array}[]{c}z_{1}\\\ z_{2}\end{array}\right],$

partitioned accordingly, from the first block row of the coefficient matrices
we obtain a reduced system given by

$\bar{K}\bar{P}\dot{z}_{1}=\bar{L}\bar{S}z_{1},$ (84)

with $\bar{P}=I_{n_{1}}$, $\bar{S}=\bar{S}^{\top}=S_{11}$, $\bar{K}=K_{11}$
and $\bar{L}=L_{11}$, together with an equation $z_{2}=K_{21}\dot{z}_{1}$,
where $z_{2}$ does not contribute to the Hamiltonian
$\mathcal{H}^{z}(z)=\frac{1}{2}z^{\top}P^{\top}Sz$. Note that the second
equation is an index two constraint, because it uses the derivative of
$z_{1}$, [25]. It arises from the Lagrange structure due to the singularity of
$P$.

An analogous representation can be constructed from the condensed form of
Lemma 39.

###### Lemma 42

Consider a system of the form (22) with $K,P,L,S\in\mathbb{R}^{n,n}$ and
regular pencil $\lambda KP-LS$. Then there exist invertible matrices $U,V,W$
as in (65) such that

$\displaystyle\hat{K}$ $\displaystyle=$ $\displaystyle
U^{\top}KV=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\ 0&0\end{array}\right],\
\hat{L}=U^{\top}LV^{-\top}=\left[\begin{array}[]{cc}L_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ (89) $\displaystyle\hat{P}$ $\displaystyle=$
$\displaystyle V^{-1}PW=\left[\begin{array}[]{cc}P_{11}&P_{12}\\\
P_{21}&P_{22}\end{array}\right],\
\hat{S}=V^{\top}SW=\left[\begin{array}[]{cc}S_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ (94)

where $L_{11}=-L_{11}^{\top}$ and $P_{11}^{\top}S_{11}=S_{11}^{\top}P_{11}$.

Proof. Since the pencil $\lambda KP-LS$ is square and regular, also the pencil
$\lambda K-L$ is regular, otherwise by Lemma 39 there would be a common left
nullspace of $K$ and $L$ which would imply the pencil $\lambda KP-LS$ to be
singular.

Thus by Lemma 39, there exist nonsingular matrices
$U_{1},V_{1}\in\mathbb{R}^{n,n}$ such that

$\displaystyle\tilde{K}$ $\displaystyle=$ $\displaystyle
U_{1}KV_{1}=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\ 0&0\end{array}\right],\
\tilde{L}=U_{1}LV_{1}^{-\top}=\left[\begin{array}[]{cc}\tilde{L}_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ $\displaystyle\tilde{P}$ $\displaystyle=$
$\displaystyle
V_{1}^{-1}P=\left[\begin{array}[]{cc}\tilde{P}_{11}&\tilde{P}_{12}\\\
\tilde{P}_{21}&\tilde{P}_{22}\end{array}\right],\
\tilde{S}=V_{1}^{\top}S=\left[\begin{array}[]{cc}\tilde{S}_{11}&\tilde{S}_{12}\\\
\tilde{S}_{21}&\tilde{S}_{22}\end{array}\right],$

with $n_{1}=2\ell_{1}+\ell_{3}$, $n_{2}=\ell_{4}$ and $\tilde{L}_{11}$ skew-
symmetric. The regularity of the pencil $\lambda KP-LS$ implies that

$\left[\begin{array}[]{cc}\tilde{S}_{21}&\tilde{S}_{22}\end{array}\right]$

has full row rank and hence there exist invertible matrices
$W_{2}\in\mathbb{R}^{n,n}$, $\tilde{V}_{2}\in\mathbb{R}^{n_{2},n_{2}}$, and

$V_{2}=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&\tilde{V}_{2}\end{array}\right]$

such that

$V_{2}^{\top}\tilde{S}W_{2}=\left[\begin{array}[]{cc}\hat{S}_{11}&\hat{S}_{12}\\\
0&I_{n_{2}}\end{array}\right].$

With

$V_{3}=\left[\begin{array}[]{cc}I_{n_{1}}&-\hat{S}_{12}\\\
0&I_{n_{2}}\end{array}\right],$

we then get that

$\displaystyle\hat{K}$ $\displaystyle=$ $\displaystyle
U_{1}^{\top}KV_{1}V_{2}V_{3}=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&0\end{array}\right],\
\hat{L}=U_{1}^{\top}LV_{1}^{-\top}V_{2}^{-\top}V_{3}^{-\top}=\left[\begin{array}[]{cc}L_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$ $\displaystyle\hat{P}$ $\displaystyle=$
$\displaystyle
V_{3}^{-1}V_{2}^{-1}V_{1}^{-1}PW_{2}=\left[\begin{array}[]{cc}P_{11}&P_{12}\\\
P_{21}&P_{22}\end{array}\right],\
\hat{S}=V_{3}^{\top}V_{2}^{\top}V_{1}^{\top}SW_{2}=\left[\begin{array}[]{cc}S_{11}&0\\\
0&I_{n_{2}}\end{array}\right],$

has the desired form with $W=W_{2}$, $V=V_{1}V_{2}V_{3}$, $U=U_{1}$, and where
$L_{11}=-L_{11}^{\top}$ and $P_{11}^{\top}S_{11}=S_{11}^{\top}P_{11}$.

Transforming $KP\dot{z}=LSz$ as in (94) and setting

$z=W\left[\begin{array}[]{c}z_{1}\\\ z_{2}\end{array}\right],$

partitioned accordingly, from the first block row of the coefficient matrices
we obtain a reduced system given by

$\bar{K}\bar{P}\dot{z}_{1}=\bar{L}\bar{S}z_{1},$ (99)

with $\bar{P}=P_{11}$, $\bar{S}=S_{11}$, $\bar{K}=I_{n_{1}}$ and
$\bar{L}=L_{11}=-L_{11}^{\top}$, together with a differential algebraic
equation $0=z_{2}$, so that $z_{2}$ does not contribute to the Hamiltonian.

###### Remark 43

System (84) is a dHDAE of the form (47) in which the Lagrange structure is
spanned by the columns of

$\left[\begin{array}[]{c}I_{n_{1}}\\\ \bar{S}_{11}\end{array}\right].$

See also [30, 32] for similar constructions in the context of removing the
factor $Q$ in systems of the form (2).

Similarly, System (99) is a pHDAE of the form (2) (with $R=0$) and the Dirac
structure is spanned by the columns of

$\left[\begin{array}[]{c}I_{n_{1}}\\\ \bar{L}_{11}\end{array}\right].$

We also perform a similar construction for systems of the form (39). Since $C$
and $D$ are chosen to be a maximal annihilator such that
$CN^{\top}+DM^{\top}=0$ in (38) and $MN^{\top}+NM^{\top}\geq 0$ with
$\operatorname{rank}\left[\begin{array}[]{cc}N&M\end{array}\right]=n$ we can
use the same construction as in the proof of Lemma 39 to first transform $N$
and $M$ in such a way that

$N=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\ 0&0\end{array}\right],\
M=\left[\begin{array}[]{cc}M_{11}&0\\\ 0&I_{n_{2}}\end{array}\right].$

This implies that we may choose $C$ and $D$ such that

$C=\left[\begin{array}[]{cc}-M_{11}^{\top}&0\\\ \\\
0&I_{n_{2}}\end{array}\right],\ D=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&0\end{array}\right].$

If $\lambda CP-DS$ is regular, then it follows that the last $n_{2}$ rows of
$CP$ have full row rank and hence altogether we have the following condensed
form.

###### Lemma 44

Consider a system of the form (39) with $C,P,L,S\in\mathbb{R}^{n,n}$ and
regular pencil $\lambda CP-DS$ and
$\left[\begin{array}[]{cc}C&D\end{array}\right]$ a maximal annihilator as in
(38). Then there exist invertible matrices $U,V,W$ as in (65) such that

$\displaystyle\hat{C}$ $\displaystyle=$ $\displaystyle
U^{\top}CV=\left[\begin{array}[]{cc}C_{11}&0\\\
0&I_{n_{2}}\end{array}\right],\
\hat{D}=U^{\top}DV^{-\top}=\left[\begin{array}[]{cc}I_{n_{1}}&0\\\
0&0\end{array}\right],$ (104) $\displaystyle\hat{P}$ $\displaystyle=$
$\displaystyle V^{-1}PW=\left[\begin{array}[]{cc}P_{11}&P_{12}\\\
0&I_{n_{2}}\end{array}\right],\
\hat{S}=V^{\top}SW=\left[\begin{array}[]{cc}S_{11}&S_{12}\\\
S_{21}&S_{22}\end{array}\right],$ (109)

where $C_{11}=-C_{11}^{\top}$ and $P_{11}^{\top}S_{11}=S_{11}^{\top}P_{11}$.

As a consequence, by transforming $CP\dot{z}=DSz$ as in (109) and setting

$z=W\left[\begin{array}[]{c}z_{1}\\\ z_{2}\end{array}\right],$

partitioned accordingly, from the second block row of the coefficient matrices
we obtain $\dot{z}_{2}=0$, i.e. $z_{2}$ is a constant function and the first
block row gives an inhomogeneous reduced system

$\bar{C}\bar{P}\dot{z}_{1}=\bar{D}\bar{S}z_{1}+S_{12}z_{2},$ (110)

with $\bar{P}=P_{11}$, $\bar{S}=S_{11}$,
$\bar{P}^{\top}\bar{S}=\bar{S}^{\top}\bar{P}$, $\bar{C}=-M_{11}^{T}$ and
$\bar{D}=I_{n_{1}}$. It will then depend on the initial condition for $z_{2}$
whether $z_{2}=0$ in which case it does not contribute to the Hamiltonian,
otherwise the Hamiltonian is still a quadratic function in $z_{1}$ plus some
linear and constant terms.

###### Remark 45

The condensed forms in this section require rank decisions. Even if they are
done in a numerically stable way using singular value decompositions, they can
give wrong decisions in finite precision arithmetic. It is a common strategy
to use in the case of doubt the worst case scenario. In the case of condensed
forms this would be to assume that the problem is a DAE of index two.

In this section we have derived structured condensed forms and shown that
these can also be used to identify a subsystem which is of one of the well-
established forms plus an algebraic constraint whose solution does not
contribute to the Hamiltonian. In the next section we analyze, when general
DAEs can be transformed to the forms (2) or (22).

##  7 Representation of DAEs into the form $KP\dot{z}=LSz$ or
$\frac{d}{dt}(Ez)=(J-R)Qz$

For general DAE systems $E\dot{x}=Ax$ it has been characterized in [30] when
they are equivalent to a dHDAE system of the form (2). We present here a
simplified result for the regular case.

###### Theorem 46

i) A regular pencil $L(\lambda)=\lambda\hat{E}-\hat{A}$ is equivalent to a
pencil of the form $\lambda E-(J-R)Q$ as in (2) with $\lambda E-Q$ being
regular if and only if the following conditions are satisfied:

  1. 1.

The spectrum of $L(\lambda)$ is contained in the closed left half plane.

  2. 2.

The finite nonzero eigenvalues on the imaginary axis are semisimple and the
partial multiplicities of the eigenvalue zero are at most two.

  3. 3.

The index of $L(\lambda)$ is at most two.

ii) A regular pencil $L(\lambda)=\lambda\hat{E}-\hat{A}$ is equivalent to a
pencil of the form $\lambda E-(J-R)$ as in (2) (i.e., with $Q=I$) if and only
if the following conditions are satisfied:

  1. 1.

The spectrum of $L(\lambda)$ is contained in the closed left half plane.

  2. 2.

The finite eigenvalues on the imaginary axis (including zero) are semisimple.

  3. 3.

The index of $L(\lambda)$ is at most two.

As a Corollary for the case without dissipation we have the following result.

###### Corollary 47

A regular pencil $L(\lambda)=\lambda\hat{E}-\hat{A}$ is equivalent to a pencil
of the form $\lambda E-J$ as in (2) (with $Q=I,R=0$) if and only if the
following conditions are satisfied:

  1. 1.

All finite eigenvalues are on the imaginary axis and semisimple.

  2. 2.

The index of $L(\lambda)$ is at most two.

To study when general regular DAEs of the form (1) can be expressed as
extended dHDAEs of the form (22) we first consider a condensed form under
orthogonal equivalence.

###### Theorem 48

Consider a regular pencil $\lambda E-A$ with $E,A\in\mathbb{R}^{n,n}$ of index
at most two. Then there exist real orthogonal matrices $U\in\mathbb{R}^{n,n}$
and $V\in\mathbb{R}^{n,n}$ such that

$U^{\top}EV=\left[\begin{array}[]{cccc}E_{11}&E_{12}&0&0\\\
E_{21}&E_{22}&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right],\
U^{\top}AV=\left[\begin{array}[]{ccccc}A_{11}&A_{12}&A_{13}&A_{14}\\\
A_{21}&A_{22}&A_{23}&0\\\ A_{31}&A_{32}&A_{33}&0\\\
A_{41}&0&0&0\end{array}\right],$ (111)

with $A_{14}\in\mathbb{R}^{n_{1},n_{1}}$, $A_{41}\in\mathbb{R}^{n_{1},n_{1}}$,
$E_{22}\in\mathbb{R}^{n_{2},n_{2}}$, and $A_{33}\in\mathbb{R}^{n_{3},n_{3}}$
invertible.

Proof. The proof is presented in Appendix C.

Transforming the DAE (1) as $U^{\top}EVV^{\top}\dot{x}=U^{\top}AVV^{\top}x$
and setting $V^{\top}x=[x_{1}^{\top},\ldots,x_{4}^{\top}]^{\top}$, it follows
that $x_{1}=0$, $x_{2}$ is determined form the implicit ordinary differential
equation (note that $E_{22}$ is invertible)

$E_{22}\dot{x}_{2}=(A_{22}-A_{23}A_{33}^{-1}A_{32})x_{2},$ (112)

$x_{3}=-A_{33}^{-1}A_{32}x_{2}$, and $x_{4}$ is uniquely determined in terms
of $x_{2},\dot{x}_{2},x_{3}$. Initial conditions can be prescribed freely for
$x_{2}$ only.

###### Corollary 49

Consider a general regular pencil $\lambda E-A$ with $E,A\in\mathbb{R}^{n,n}$
that is of index at most two and for which all finite eigenvalues are in the
closed left half plane and those on the imaginary axis are semi-simple. Then
there exist invertible matrices $U\in\mathbb{R}^{n,n}$ and
$V\in\mathbb{R}^{n,n}$ such that

$U^{\top}EV=\left[\begin{array}[]{ccccc}I_{\hat{n}_{1}}&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&E_{33}&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right],\
U^{\top}AV=\left[\begin{array}[]{ccccc}0&0&0&I_{\hat{n}_{1}}&0\\\
0&0&0&0&I_{\hat{n}_{2}}\\\ 0&0&A_{33}&0&0\\\ 0&0&0&I_{\hat{n}_{4}}&0\\\
-I_{\hat{n}_{1}}&0&0&0&0\\\ 0&-I_{\hat{n}_{2}}&0&0&0\end{array}\right],$ (113)

where

$E_{33}=E_{33}^{\top}>0,\ A_{33}=J_{33}-R_{33},\ J_{33}=-J_{33}^{\top},\
R_{33}^{\top}=R_{33}\geq 0.$

Proof. The proof follows by considering the condensed form (111), and using
block elimination with the invertible matrices $A_{33}$, $A_{14}$, $A_{41}$,
$E_{22}$ to transform pencil in (111) to the form

$\lambda\left[\begin{array}[]{cccc}\tilde{E}_{11}&0&0&0\\\ 0&I_{n_{2}}&0&0\\\
0&0&0&0\\\
0&0&0&0\end{array}\right]-\left[\begin{array}[]{ccccc}0&0&0&I_{n_{1}}\\\
0&\tilde{A}_{22}&0&0\\\ 0&0&I_{n_{3}}&0\\\
-I_{n_{1}}&0&0&0\end{array}\right].$ (114)

Let
$U_{1}^{\top}\tilde{E}_{11}V_{1}=\left[\begin{array}[]{cc}I_{\hat{n}_{1}}&0\\\
0&0\end{array}\right]$ be the echelon form of $\tilde{E}_{11}$. We scale the
first block row of with $U_{1}^{\top}$, the fourth block row with
$V_{1}^{-1}$, the first block column by $V_{1}$, and the fourth block column
by $U_{1}^{-\top}$ and obtain a form

$U^{\top}EV=\left[\begin{array}[]{cccccc}I_{{\hat{n}}_{1}}&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&I_{n_{3}}&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\end{array}\right],\
U^{\top}AV=\left[\begin{array}[]{cccccc}0&0&0&0&I_{{\hat{n}}_{1}}&0\\\
0&0&0&0&0&I_{{\hat{n}}_{2}}\\\ 0&0&\hat{A}_{33}&0&0&0\\\
0&0&0&I_{{\hat{n}}_{4}}&0&0\\\ -I_{{\hat{n}}_{1}}&0&0&0&0&0\\\
0&-I_{{\hat{n}}_{2}}&0&0&0&0\end{array}\right],$

For any positive definite solution $X$ of the Lyapunov inequality

$-\tilde{A}_{33}^{\top}X-X\tilde{A}_{33}\geq 0$ (115)

one can multiply the second block row by $X$ and obtain that $E_{33}=X$ and
$A_{33}=X\hat{A}_{33}$ has the desired form, see e.g. [1, 3].

Note that the transformation to a system of the form (2) can also be achieved
in a similar way for singular pencils with zero minimal indices.

If there is no dissipation, i.e. if $R_{33}=0$, then $A_{33}$ is skew-
symmetric.

In Corollary 49 we have shown that general systems $E\dot{x}=Ax$ can be
transformed to a very special canonical form and the following remark shows
that for the case $R_{33}=0$ each of the blocks in the canonical form can be
expressed as a pencil of the form $\lambda KP-LS$.

###### Remark 50

Consider a regular pencil $\lambda E-A$ in the form (113) then after a
permutation one gets four blocks which all can be written in the form $\lambda
KP-LS$ as in (22).

  * 1)

We have

$\left[\begin{array}[]{cc}I_{{\hat{n}}_{1}}&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}\dot{z}_{1}\\\
\dot{z}_{5}\end{array}\right]-\left[\begin{array}[]{cc}0&I_{{\hat{n}}_{1}}\\\
-I_{{\hat{n}}_{1}}&0\end{array}\right]\left[\begin{array}[]{c}z_{1}\\\
z_{5}\end{array}\right]=K_{1}P_{1}\left[\begin{array}[]{c}\dot{z}_{1}\\\
\dot{z}_{5}\end{array}\right]-L_{1}S_{1}\left[\begin{array}[]{c}z_{1}\\\
z_{5}\end{array}\right]$

with

$K_{1}=\left[\begin{array}[]{cc}I_{{\hat{n}}_{1}}&0\\\
0&I_{{\hat{n}}_{1}}\end{array}\right],\
L_{1}=\left[\begin{array}[]{cc}0&I_{{\hat{n}}_{1}}\\\
-I_{{\hat{n}}_{1}}&0\end{array}\right],\
P_{1}=\left[\begin{array}[]{cc}I_{{\hat{n}}_{1}}&0\\\ 0&0\end{array}\right],\
S_{1}=\left[\begin{array}[]{cc}I_{{\hat{n}}_{1}}&0\\\
0&I_{{\hat{n}}_{1}}\end{array}\right]$

and Hamiltonian
$\mathcal{H}^{z}_{1}=\frac{1}{2}\left[\begin{array}[]{c}z_{1}\\\
z_{5}\end{array}\right]^{\top}\left[\begin{array}[]{cc}I_{{\hat{n}}_{1}}&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}z_{1}\\\
z_{5}\end{array}\right]={\frac{1}{2}z_{1}^{\top}z_{1}}$ which is actually $0$.

In this case we can insert the derivative of the second equation into the
first (index reduction) and obtain

$\left[\begin{array}[]{cc}0&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}\dot{z}_{1}\\\
\dot{z}_{5}\end{array}\right]=\left[\begin{array}[]{cc}0&I_{{\hat{n}}_{1}}\\\
-I_{{\hat{n}}_{1}}&0\end{array}\right]\left[\begin{array}[]{c}z_{1}\\\
z_{5}\end{array}\right]$

without changing the Hamiltonian.

  * 2)

We have

$\left[\begin{array}[]{cc}0&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}\dot{z}_{2}\\\
\dot{z}_{6}\end{array}\right]=\left[\begin{array}[]{cc}0&I_{{\hat{n}}_{2}}\\\
-I_{{\hat{n}}_{2}}&0\end{array}\right]\left[\begin{array}[]{c}z_{2}\\\
z_{6}\end{array}\right]=K_{2}P_{2}\left[\begin{array}[]{c}\dot{z}_{2}\\\
\dot{z}_{6}\end{array}\right]-L_{2}S_{2}\left[\begin{array}[]{c}z_{2}\\\
z_{6}\end{array}\right]$

with different possibilities of representation, e.g.
a)

$K_{2}=\left[\begin{array}[]{cc}I_{{\hat{n}}_{2}}&0\\\
0&I_{{\hat{n}}_{2}}\end{array}\right],\
L_{3}=\left[\begin{array}[]{cc}0&I_{{\hat{n}}_{2}}\\\
-I_{{\hat{n}}_{2}}&0\end{array}\right],\ P_{2}=\left[\begin{array}[]{cc}0&0\\\
0&0\end{array}\right],\ S_{3}=\left[\begin{array}[]{cc}I_{{\hat{n}}_{1}}&0\\\
0&I_{{\hat{n}}_{1}}\end{array}\right]$

and Hamiltonian
$\mathcal{H}^{z}_{2}=\frac{1}{2}\left[\begin{array}[]{c}z_{2}\\\
z_{6}\end{array}\right]^{\top}\left[\begin{array}[]{cc}0&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}z_{2}\\\
z_{6}\end{array}\right]=0$, or

b)

$K_{2}=\left[\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right],\
L_{2}=\left[\begin{array}[]{cc}0&I_{{\hat{n}}_{2}}\\\
-I_{{\hat{n}}_{2}}&0\end{array}\right],\ P_{2}=\left[\begin{array}[]{cc}0&0\\\
0&0\end{array}\right],\ S_{2}=\left[\begin{array}[]{cc}I_{{\hat{n}}_{2}}&0\\\
0&I_{{\hat{n}}_{2}}\end{array}\right]$

and Hamiltonian
$\mathcal{H}^{z}_{2}=\frac{1}{2}\left[\begin{array}[]{c}z_{2}\\\
z_{6}\end{array}\right]^{\top}\left[\begin{array}[]{cc}0&0\\\
0&0\end{array}\right]\left[\begin{array}[]{c}z_{2}\\\
z_{6}\end{array}\right]=0$.

  * 3)

We have $\lambda E_{33}-A_{33}=\lambda K_{3}P_{3}-L_{3}S_{3}$ with
$K_{3}=I_{\hat{n_{3}}}$, $P_{3}=E_{33}$, $L_{3}=A_{33}$,
$S_{3}=I_{{\hat{n}}_{3}}$. Here the Hamiltonian is
$\mathcal{H}^{z}_{3}=\frac{1}{2}z_{3}^{\top}E_{33}z_{3}$.

  * 4)

We have $\lambda 0-I_{{\hat{n}}_{4}}=\lambda K_{4}P_{4}-L_{4}S_{4}$ with
$K_{4}=0$, $P_{4}=I_{{\hat{n}}_{4}}$, $L_{4}=I_{{\hat{n}}_{4}}$,
$S_{4}=I_{{\hat{n}}_{4}}$. Here the Hamiltonian is
$\mathcal{H}^{z}_{4}=\frac{1}{2}z_{4}^{\top}z_{4}$.

Note that the presented representations are in no way unique, but if the
condensed form is available or computable, and the properties of Theorem 48
hold then we can express the general DAE in the representation (22) or (2).

This discussion yields the following useful corollary.

###### Corollary 51

Consider a regular pencil of the form $\lambda KP-LS$ associated with the
dHDAE (22). Then it has index at most two, and index two can only occur if the
system has a singular Lagrange structure.

Proof. Consider the representations in Remark 50. Then the index two structure
occurs only in the first case where $K_{1},L_{1}$ are invertible, but the
product $P_{1}^{T}S_{1}$ is singular. Thus index two arises only from a
singular Lagrange structure.

## Conclusion and Outlook

Different definitions of (extended, dissipative) Hamiltonian or port
Hamiltonian differential-algebraic systems lead to different representations.
We have collected all the known representations as well as a few new ones and
analyzed them from a geometric as well as an algebraic point of view. The
latter leads to condensed forms that can be directly implemented in numerical
algorithms to compute the structural properties of the systems. We have also
studied the effect that the different representations have on the index of the
differential-algebraic system as well as on the associated Hamilton function.
In general it can be seen that certain algebraic constraints do not contribute
to the Hamiltonian and therefore can be separated from the system in an
appropriate coordinate system. We have also characterized when a general
differential-algebraic system can be transformed to the different
representations. Several important tasks remain open. These include extensions
to the case of non-regular systems. These can be based on the results and
methods in Appendices A and B that are already proved for the non-square case.
For systems with inputs and outputs, linear time-varying and nonlinear systems
the extensions are currently under consideration.

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## Appendix

### Appendix A: Proof of Lemma 37

Proof. We present a proof in form of an algorithmic procedure that can be
implemented as a numerical algorithm.

Step 1. Let $V_{1}$ and $W_{1}$ be real orthogonal matrices such that

$V_{1}^{\top}PW_{1}=\left[\begin{array}[]{cc}\tilde{P}_{11}&0\\\
0&0\end{array}\right],\
V_{1}^{\top}SW=\left[\begin{array}[]{cc}\tilde{S}_{11}&\tilde{S}_{12}\\\
\tilde{S}_{21}&\tilde{S}_{22}\end{array}\right],$

where $\tilde{P}_{11}\in\mathbb{R}^{\tilde{m}_{1},\tilde{m}_{1}}$ is diagonal
with positive diagonal elements. This transformation can be constructed via a
singular value decomposition of $P$ and a numerical rank decision.

Then, using the structure (28), it follows that
$\tilde{P}_{11}^{\top}\tilde{S}_{11}=\tilde{S}_{11}^{\top}\tilde{P}_{11}$ and
$\tilde{S}_{12}=0$.

Step 2. Let $\tilde{V}_{2}$, $\tilde{W}_{2}$ be real orthogonal matrices such
that

$\tilde{V}_{2}^{\top}\tilde{S}_{22}W_{2}=\left[\begin{array}[]{cc}\hat{S}_{22}&0\\\
0&0\end{array}\right]$

where $\hat{S}_{22}\in\mathbb{R}^{m_{4},m_{4}}$ is diagonal with positive
diagonal elements. This transformation can be constructed again via a singular
value decomposition and a numerical rank decision. Set

$V_{2}=\left[\begin{array}[]{cc}I_{\tilde{m}_{1}}&0\\\
0&\tilde{V}_{2}^{\top}\end{array}\right],\
W_{2}=\left[\begin{array}[]{cc}I_{\tilde{m}_{1}}&0\\\
0&\tilde{W}_{2}\end{array}\right]$

and form

$V_{2}^{\top}V_{1}^{\top}PW_{1}W_{2}=\left[\begin{array}[]{ccc}\hat{P}_{11}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right],\
V_{2}^{\top}V_{1}^{\top}SW_{1}W_{2}=\left[\begin{array}[]{ccc}\hat{S}_{11}&0&0\\\
\hat{S}_{21}&\hat{S}_{22}&0\\\ \hat{S}_{31}&0&0\end{array}\right],$

with $\hat{P}_{11}^{\top}=\tilde{P}_{11}$.

Step 3. Let

$V_{3}=\left[\begin{array}[]{ccc}\hat{P}_{11}&0&0\\\
0&\hat{S}_{22}^{-\top}&0\\\ 0&0&I_{n-\tilde{m}_{1}-m_{4}}\end{array}\right],\
W_{3}=\left[\begin{array}[]{ccc}I_{\tilde{m}_{1}}&0&0\\\
-\hat{S}_{22}^{-1}\hat{S}_{21}&I_{m_{4}}&0\\\
0&0&I_{m-\tilde{m}_{1}-m_{4}}\end{array}\right],$

and form

$V_{3}^{-1}V_{2}^{\top}V_{1}^{\top}PW_{1}W_{2}W_{3}=\left[\begin{array}[]{ccc}I_{\tilde{m}_{1}}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right],\
V_{3}^{\top}V_{2}^{\top}V_{1}^{\top}SW_{1}W_{2}W_{3}=\left[\begin{array}[]{ccc}\check{S}_{11}&0&0\\\
0&I_{m_{4}}&0\\\ \check{S}_{31}&0&0\end{array}\right],$

where by the structure (28) now $\check{S}_{11}$ is symmetric. Note that
although we are working with nonorthogonal transformation matrices in this
step the numerical errors can be controlled, since we are inverting diagonal
matrices.

Step 4. Let

$\check{V}_{11}^{\top}\check{S}_{11}\check{V}_{11}=\left[\begin{array}[]{ccc}I_{m_{1}}&0&0\\\
0&-I_{m_{2}}&0\\\ 0&0&0\end{array}\right]$

be the canonical form of the skew-symmetric matrix $\check{S}_{11}$ under
congruence which can be obtained by first computing the spectral decomposition
and then scaling the nonsingular diagonal parts by congruence to be $\pm I$,
see e.g. [27].

Furthermore let

$\check{S}_{31}=\check{V}_{4}\left[\begin{array}[]{ccc}S_{51}&S_{5,2}&S_{5,3}\\\
0&0&0\end{array}\right]$

be a full rank decomposition partitioned accordingly, with $\check{V}_{3}$
real orthogonal. Then set

$V_{4}=\left[\begin{array}[]{ccc}\check{V}_{11}^{-\top}&0&0\\\
0&I_{m_{4}}&0\\\ 0&0&\check{V}_{4}\end{array}\right],\
W_{4}=\left[\begin{array}[]{ccc}\check{V}_{11}&0&0\\\ &I_{m_{4}}&0\\\
0&0&I_{m-\tilde{m}_{1}-m_{4}}\end{array}\right],$

and form

$\displaystyle V_{4}^{-1}V_{3}^{-1}V_{2}^{-1}V_{1}^{-1}PW_{1}W_{2}W_{3}W_{4}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}I_{m_{1}}&0&0&0&0\\\
0&I_{m_{2}}&0&0&0\\\ 0&0&I_{m_{3}}&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right],$ $\displaystyle
V_{4}^{-1}V_{3}^{-1}V_{2}^{-1}V_{1}^{-1}SW_{1}W_{2}W_{3}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccc}I_{m_{1}}&0&0&0&0\\\
0&-I_{m_{2}}&0&0&0\\\ 0&0&0&0&0\\\ 0&0&0&I_{m_{4}}&0\\\
S_{51}&S_{5,2}&S_{5,3}&0&0\\\ 0&0&0&0&0\end{array}\right],$

which is as claimed.

### Appendix B: Proof of Lemma 39

Proof. The proof is similar to that of Lemma 37 just adapting to the different
symmetries and transformation structure. The following algorithmic procedure
can be directly implemented as a numerical algorithm.

Step 1. Let $U_{1}$ and $V_{1}$ be real orthogonal matrices such that

$U_{1}^{\top}KV_{1}=\left[\begin{array}[]{cc}\tilde{K}_{11}&0\\\
0&0\end{array}\right],\
U_{1}^{\top}LV_{1}^{-\top}=\left[\begin{array}[]{cc}\tilde{L}_{11}&\tilde{L}_{12}\\\
\tilde{L}_{21}&\tilde{L}_{22}\end{array}\right],$

where $\tilde{K}_{11}\in\mathbb{R}^{\tilde{\ell}_{1},\tilde{\ell}_{1}}$ is
diagonal with positive diagonal elements. This transformation can be
constructed via a singular value decomposition of $K$ and a numerical rank
decision.

Then, using the structure (26), it follows that
$\tilde{K}_{11}^{\top}\tilde{L}_{11}=\tilde{L}_{11}^{\top}\tilde{K}_{11}$ and
$\tilde{L}_{21}=0$.

Step 2. Let $\tilde{U}_{2}$, $\tilde{V}_{2}$ be real orthogonal matrices such
that

$\tilde{U}_{2}^{\top}\tilde{L}_{22}V_{2}^{-\top}=\left[\begin{array}[]{cc}\hat{L}_{22}&0\\\
0&0\end{array}\right],$

where $\hat{L}_{22}\in\mathbb{R}^{\ell_{4},\ell_{4}}$ is diagonal with
positive diagonal elements. This transformation can be constructed again via a
singular value decomposition and a numerical rank decision. Set

$U_{2}=\left[\begin{array}[]{cc}I_{\tilde{\ell}_{1}}&0\\\
0&\tilde{U}_{2}\end{array}\right],\
V_{2}=\left[\begin{array}[]{cc}I_{\tilde{\ell}_{1}}&0\\\
0&\tilde{V}_{2}\end{array}\right]$

and form

$U_{2}^{\top}U_{1}^{\top}KV_{1}V_{2}=\left[\begin{array}[]{ccc}\hat{K}_{11}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right],\
U_{2}^{\top}U_{1}^{\top}LV_{1}^{-\top}V_{2}^{-\top}=\left[\begin{array}[]{ccc}\hat{L}_{11}&\hat{L}_{12}&\hat{L}_{13}\\\
0&\hat{L}_{22}&0\\\ 0&0&0\end{array}\right],$

with $\hat{K}_{11}=\tilde{K}_{11}$.

Step 3. Let

$U_{3}=\left[\begin{array}[]{ccc}I_{\tilde{\ell}_{1}}&-\hat{L}_{22}^{-\top}\hat{L}_{12}^{\top}&0\\\
0&\hat{L}_{22}^{-\top}&0\\\
0&0&I_{\ell-\tilde{\ell}_{1}-\ell_{4}}\end{array}\right],\
V_{3}=\left[\begin{array}[]{ccc}\hat{K}_{11}^{-1}&0&0\\\ 0&I_{\ell_{4}}&0\\\
0&0&I_{n-\tilde{\ell}_{1}-\ell_{4}}\end{array}\right],$

and form

$U_{3}^{\top}U_{2}^{\top}U_{1}^{\top}KV_{1}V_{2}V_{3}=\left[\begin{array}[]{ccc}I_{\tilde{\ell}_{1}}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right],\
U_{3}^{\top}U_{2}^{\top}U_{1}^{\top}LV_{1}^{-\top}V_{2}^{-\top}V_{3}^{-\top}=\left[\begin{array}[]{ccc}\check{L}_{11}&0&\check{L}_{13}\\\
0&I_{\ell_{4}}&0\\\ \check{0}&0&0\end{array}\right],$

where by the structure (26) now $\check{L}_{11}$ is skew-symmetric. Note that
although we are working with nonorthogonal transformation matrices in this
step, the numerical errors can be controlled since we are inverting diagonal
matrices.

Step 4. Let

$\check{U}_{11}^{\top}\check{L}_{11}\check{U}_{11}=\left[\begin{array}[]{ccc}0&I_{\ell_{1}}&0\\\
-I_{\ell_{1}}&0&0\\\ 0&0&0\end{array}\right]$

be the canonical form of the skew-symmetric matrix $\check{L}_{11}$ under
congruence which can be obtained by first computing the spectral decomposition
and then scaling the nonsingular diagonal parts by congruence to be $\pm I$.
This procedure is implemented in a numerically robust way in [5]. Furthermore
let

$\check{L}_{13}=\left[\begin{array}[]{cc}L_{15}&0\\\ L_{25}&0\\\
L_{35}&0\end{array}\right]\check{V}_{3}$

be a full rank decomposition partitioned accordingly, with $\check{V}_{3}$
real orthogonal. Then set

$U_{4}=\left[\begin{array}[]{ccc}\check{U}_{11}&0&0\\\ 0&I_{\ell_{4}}&0\\\
0&0&I_{\ell-\tilde{\ell}_{1}-\ell_{4}}\end{array}\right],\
V_{4}=\left[\begin{array}[]{ccc}\check{U}_{11}^{-\top}&0&0\\\
&I_{\ell_{4}}&0\\\ 0&0&\check{V}_{3}\end{array}\right],$

and form

$\displaystyle
U_{4}^{T}U_{3}^{T}U_{2}^{\top}U_{1}^{\top}KV_{1}V_{2}V_{3}V_{4}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{ccccc}I_{\ell_{1}}&0&0&0&0\\\
0&I_{\ell_{1}}&0&0&0\\\ 0&0&I_{\ell_{3}}&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right],$ $\displaystyle
U_{4}^{T}U_{3}^{T}U_{2}^{\top}U_{1}^{\top}LV_{1}^{-\top}V_{2}^{-\top}V_{3}^{-\top}V_{4}^{-\top}$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccccc}0&I_{\ell_{1}}&0&0&L_{15}&0\\\
-I_{\ell_{1}}&0&0&0&L_{25}&0\\\ 0&0&0&0&L_{35}&0\\\ 0&0&0&I_{\ell_{4}}&0&0\\\
0&0&0&0&0&0\end{array}\right],$

which is as claimed.

### Appendix C: Proof of Theorem 48

Proof. We again present a proof that can be implemented as a numerical
algorithm.

Step 1. Let $U_{1}$ and $V_{1}$ be real orthogonal matrices such that

$U_{1}^{\top}EV_{1}=\left[\begin{array}[]{cc}\tilde{E}_{11}&0\\\
0&0\end{array}\right],\
U_{1}^{\top}AV_{1}=\left[\begin{array}[]{cc}\tilde{A}_{11}&\tilde{A}_{12}\\\
\tilde{A}_{21}&\tilde{A}_{22}\end{array}\right],$

where $\tilde{E}_{11}\in\mathbb{R}^{\tilde{n}_{1},\tilde{n}_{1}}$ is diagonal
with positive diagonal elements. This transformation can be constructed via a
singular value decomposition of $E$ and a numerical rank decision.

Step 2. Let $\tilde{U}_{2}$, $\tilde{V}_{2}$ be real orthogonal matrices such
that

$\tilde{U}_{2}^{\top}\tilde{A}_{22}\tilde{V}_{2}=\left[\begin{array}[]{cc}\hat{A}_{22}&0\\\
0&0\end{array}\right],$

where $\hat{A}_{22}\in\mathbb{R}^{n_{3},n_{3}}$ is diagonal with positive
diagonal elements. This transformation can be constructed again via a singular
value decomposition and a numerical rank decision. Set

$U_{2}=\left[\begin{array}[]{cc}I_{\tilde{n}_{1}}&0\\\
0&\tilde{U}_{2}\end{array}\right],\
V_{2}=\left[\begin{array}[]{cc}I_{\tilde{n}_{1}}&0\\\
0&\tilde{V}_{2}\end{array}\right]$

and form

$U_{2}^{\top}U_{1}^{\top}EV_{1}V_{2}=\left[\begin{array}[]{ccc}\hat{E}_{11}&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right],\
U_{2}^{\top}U_{1}^{\top}AV_{1}^{-\top}V_{2}=\left[\begin{array}[]{ccc}\hat{A}_{11}&\hat{A}_{12}&\hat{A}_{13}\\\
A_{21}&\hat{A}_{22}&0\\\ A_{31}&0&0\end{array}\right],$

with $\hat{E}_{11}=\tilde{E}_{11}$.

The regularity of the pencil implies that $A_{13}$ has full column rank
$n_{1}=\tilde{n}_{1}-n_{3}$ and that $A_{31}$ has full row rank
$n_{1}=\tilde{n}_{1}-n_{3}$, because otherwise there would be common right or
left nullspace, respectively.

Step 3. Let $\tilde{U}_{31}$, $\tilde{V}_{31}$, $\tilde{U}_{13}$,
$\tilde{V}_{13}$ be real orthogonal matrices of appropriate dimensions such
that

$\tilde{U}_{13}^{\top}\tilde{A}_{13}\tilde{V}_{13}=\left[\begin{array}[]{c}\hat{A}_{14}\\\
0\end{array}\right],\
\tilde{U}_{31}^{\top}\tilde{A}_{31}\tilde{V}_{31}=\left[\begin{array}[]{cc}\hat{A}_{41}&0\end{array}\right]$

where $\hat{A}_{41}\in\mathbb{R}^{n_{1},n_{1}}$ and
$\hat{A}_{14}\in\mathbb{R}^{n_{1},n_{1}}$ are diagonal with positive diagonal
elements. This transformation can be constructed again via a singular value
decomposition and numerical rank decisions. Set

$U_{3}=\left[\begin{array}[]{ccc}\tilde{U}_{13}&0&0\\\ 0&I_{\ell_{3}}&0\\\
0&0&\tilde{U}_{31}\end{array}\right],\
V_{3}=\left[\begin{array}[]{ccc}\tilde{V}_{31}&0&0\\\ 0&I_{\ell_{3}}&0\\\
0&0&\tilde{V}_{13}\end{array}\right],$

then $U_{3}^{\top}U_{2}^{\top}U_{1}^{\top}EV_{1}V_{2}V_{3}$ and
$U_{3}^{\top}U_{2}^{\top}U_{1}^{\top}AV_{1}V_{2}V_{3}$ are as claimed in
(111). The invertibility of $E_{22}$ then follows from the assumption that the
pencil has index at most two, see [25].