situation, we also present the error for tangential interpolation via
structure-preserving IRKA with a modified Hamiltonian, denoted as IRKA (mod.
$\mathcal{H}$) in the figures, as described in (8.27), which yields a
significant improvement over the original formulation. This can also be seen
in the relative $\mathcal{H}_{2}$-errors displayed in Figure 6, where for
$r\geq 6$, the IRKA-reduced pHDAE with modified Hamiltonian yields an
approximation that is at least one order of magnitude better.

##  9\. Temporal discretization and linear system solvers

In this section, we discuss the time-discretization of dHDAE and pHDAE systems
and the associated linear system solves. The numerical solution of initial and
boundary value problems for general DAE systems of the form (2.1a) has been an
important research topic, see the monographs [41, 102, 104, 138]. Following
the approach discussed in [126, Cha. 6], we may assume that the DAE is
represented at every time-step in one of the strangeness-free forms (2.16) or
(2.18), and then it has been shown that many implicit one-step and multi-step
methods can be applied and have the same convergence order as for ODE systems.

###  9.1. Time discretization for pHDAEs

Most of the classical time-discretization techniques do not respect a given
dHDAE or pHDAE structure in such a way that the time-discretized system still
satisfies a power-balance equation or dissipation inequality. To analyze for
which approaches this is guaranteed is an active research area that is
proceeding in different directions. A very promising approach is the
discretization of pHDAE systems in such a way, that the time-discrete system
satisfies a discrete version of the power-balance equation, see [56, 120, 121]
for pHODE systems, and [162, 165] for pHDAE systems. Another class of methods,
particularly for non-dissipative ODE methods, is based on energy preserving
geometric integration, see _e.g._ , [57, 103, 180]. The analysis and
comparison of these techniques is a topic on its own, so we only briefly
discuss the approach based on collocation methods for pHDAEs of the form (4.1)
in [162].

Consider an autonomous pHDAE of the form (4.1) with a given input function
$u\colon\mathbb{T}\to\mathbb{R}^{m}$, a consistent initial value
$z(t_{0})=z_{0}$, and suppose that we want to approximate the solution in a
time interval $(t_{0},t_{\mathrm{f}}=t_{0}+\tau)$ by a polynomial
$\tilde{z}(t)$ of degree at most $s$. For a collocation method, the polynomial
$\tilde{z}(t)$ is chosen such that $\tilde{z}(t_{i})=z(t_{i})$ satisfies the
pHDAE (4.1) in the $s$ collocation points $t_{i}=t_{0}+\tau\gamma_{i}$ with
$\gamma_{i}\in[0,1]$ for $i=1,\ldots,s$.

Let $\ell_{i}$ denote the $i$-th Lagrange interpolation polynomial with
respect to the nodes $\gamma_{1},\ldots,\gamma_{s}$, _i.e._

$\ell_{i}(t)\vcentcolon=\prod_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{s}\frac{t-\gamma_{j}}{\gamma_{i}-\gamma_{j}}.$

Then for collocation methods one requires that

$\dot{\tilde{z}}(t_{0}+t\tau)=\sum_{i=1}^{s}\dot{z}_{i}\ell_{i}(t),\qquad\tilde{z}(t_{0}+t\tau)=z_{0}+\tau\sum_{j=1}^{s}\dot{z}_{j}\int_{0}^{t}\ell_{j}(\sigma)\,\text{d}\sigma,$

for the values $\dot{z}_{i}\vcentcolon=\dot{\tilde{z}}(t_{i})$, and also

$\displaystyle z_{i}$
$\displaystyle\vcentcolon=\tilde{z}(t_{i})=z_{0}+\tau\sum_{j=1}^{s}\alpha_{ij}\dot{z}_{j},$
$\displaystyle z_{f}$
$\displaystyle\vcentcolon=\tilde{z}(t_{\mathrm{f}})=z_{0}+\tau\sum_{j=1}^{s}\beta_{j}\dot{z}_{j},$

where the coefficients
$\alpha_{ij}\vcentcolon=\int_{0}^{\gamma_{i}}\ell_{j}(\sigma)\,\text{d}\sigma$
and $\beta_{j}\vcentcolon=\int_{0}^{1}\ell_{j}(\sigma)\,\text{d}\sigma$,
$i,j=1\ldots s$ are the coefficients of the Butcher diagram of the associated
Runge-Kutta method, see [103].

To preserve the pHDAE structure one uses the Dirac structure $\mathcal{D}_{z}$
as described in (6.4) associated to the pHDAE (4.1) and defines the Dirac
structure associated to the time-discretization

$\displaystyle\mathcal{D}_{z_{i}}$
$\displaystyle=\left\\{(f^{i},e^{i})\in\mathcal{V}_{z_{i}}\times\mathcal{V}_{z_{i}}^{*}\;\left|\;f^{i}+\begin{bmatrix}\Gamma(z_{i})&I_{\ell+m}\\\
-I_{\ell+m}&0\end{bmatrix}e^{i}=0\right.\right\\},\ i=1,\ldots,s$

with
$f^{i}=\begin{bmatrix}(f_{\mathrm{s}}^{i})^{T}&y_{i}^{T}&(f_{\mathrm{d}}^{i})^{T}\end{bmatrix}^{T}$
and
$e^{i}=\begin{bmatrix}(e_{\mathrm{s}}^{i})^{T},u_{i}^{T},(e_{\mathrm{d}}^{i})^{T}\end{bmatrix}^{T}$,
where

$\displaystyle f_{\mathrm{s}}^{i}$ $\displaystyle=-E(z_{i})\dot{z}_{i},$
$\displaystyle e_{\mathrm{s}}^{i}$ $\displaystyle=\eta(z_{i}),$ $\displaystyle
e_{\mathrm{d}}^{i}$ $\displaystyle=-W(z_{i})f_{\mathrm{d}}^{i},$
$\displaystyle u_{i}$ $\displaystyle=u(z_{i}).$

In this way one obtains a system that is equivalent to the approach of
applying the collocation method and then computing discrete inputs and outputs
$u_{i},y_{i}$, for $i=1,\ldots,s$, and then introduces the associated
collocation flows, efforts, input and output as

$\displaystyle\tilde{f}_{\mathrm{s}}(t_{0}+\tau t)$
$\displaystyle=\sum_{i=1}^{s}f_{\mathrm{s}}^{i}\ell_{i}(t),$
$\displaystyle\tilde{e}_{\mathrm{s}}(t_{0}+\tau t)$
$\displaystyle=\sum_{i=1}^{s}e_{\mathrm{s}}^{i}\ell_{i}(t),$
$\displaystyle\tilde{f}_{\mathrm{d}}(t_{0}+\tau t)$
$\displaystyle=\sum_{i=1}^{s}f_{\mathrm{d}}^{i}\ell_{i}(t),$
$\displaystyle\tilde{e}_{\mathrm{d}}(t_{0}+\tau t)$
$\displaystyle=\sum_{i=1}^{s}e_{\mathrm{d}}^{i}\ell_{i}(t),$
$\displaystyle\tilde{y}(t_{0}+\tau t)$
$\displaystyle=\sum_{i=1}^{s}y_{i}\ell_{i}(t),\ $
$\displaystyle\tilde{u}(t_{0}+\tau t)$
$\displaystyle=\sum_{i=1}^{s}u_{i}\ell_{i}(t).$

In this way the discrete values are in $\mathcal{D}_{\tilde{z}}$ in all
collocation points $t_{i}$ and the discretization preserves the structure. To
see this, let us consider the evolution of the Hamiltonian $\mathcal{H}$ along
the collocation polynomial $\tilde{z}(t)$. For this, let
$\tilde{\mathcal{H}}(t)\vcentcolon=\mathcal{H}(\tilde{z}(t))$. Then we have

$\tilde{\mathcal{H}}(t)-\tilde{\mathcal{H}}(t_{0})=\int_{t_{0}}^{t}\dot{\tilde{\mathcal{H}}}(\sigma)\,\text{d}\sigma,$

and in the collocation points the power balance equation is satisfied, _i.e._

$\dot{\tilde{\mathcal{H}}}(t_{i})=\tfrac{\partial}{\partial{z_{i}}}\tilde{\mathcal{H}}^{T}\dot{z}_{i}=\eta(z_{i})^{T}E(z_{i})\dot{z}_{i}=-\left\langle\left.e_{\mathrm{s}}^{i}\,\right|\,f_{\mathrm{s}}^{i}\right\rangle=\left\langle\left.e_{\mathrm{d}}^{i}\,\right|\,f_{\mathrm{d}}^{i}\right\rangle+\left\langle\left.y_{i}\,\right|\,u_{i}\right\rangle,$

for $i=1,\ldots,s$. Applying the quadrature rule associated to the collocation
method to evaluate the integral, we get

$\displaystyle\tilde{\mathcal{H}}(t_{\mathrm{f}})-\tilde{\mathcal{H}}(t_{0})$
$\displaystyle=\tau\sum_{j=1}^{s}\beta_{j}\dot{\tilde{\mathcal{H}}}(t_{j})+\mathcal{O}(\tau^{p+1})=-\tau\sum_{j=1}^{s}\beta_{j}\left\langle\left.e_{\mathrm{s}}^{j}\,\right|\,f_{\mathrm{s}}^{j}\right\rangle+\mathcal{O}(\tau^{p+1})$
$\displaystyle=\tau\sum_{j=1}^{s}\beta_{j}\left\langle\left.e_{\mathrm{d}}^{j}\,\right|\,f_{\mathrm{d}}^{j}\right\rangle+h\sum_{j=1}^{s}\beta_{j}\left\langle\left.y_{j}\,\right|\,u_{j}\right\rangle+\mathcal{O}(\tau^{p+1}),$

where $p\in\mathbb{N}$ is the approximation order of the quadrature rule. With
the same argument we get

$\displaystyle\tau\sum_{j=1}^{s}\beta_{j}\left\langle\left.e_{\mathrm{d}}^{j}\,\right|\,f_{\mathrm{d}}^{j}\right\rangle$
$\displaystyle=\int_{t_{0}}^{t_{\mathrm{f}}}\left\langle\left.\tilde{e}_{\mathrm{d}}(\sigma)\,\right|\,\tilde{f}_{\mathrm{d}}(\sigma)\right\rangle\,\text{d}\sigma+\mathcal{O}(\tau^{p+1}),$
(9.2a)
$\displaystyle\tau\sum_{j=1}^{s}\beta_{j}\left\langle\left.y_{j}\,\right|\,u_{j}\right\rangle$
$\displaystyle=\int_{t_{0}}^{t_{\mathrm{f}}}\left\langle\left.\tilde{y}(\sigma)\,\right|\,\tilde{u}(\sigma)\right\rangle\,\text{d}\sigma+\mathcal{O}(\tau^{p+1}),$
(9.2b)

and hence

$\tilde{\mathcal{H}}(t_{\mathrm{f}})-\tilde{\mathcal{H}}(t_{0})=\int_{t_{0}}^{t_{\mathrm{f}}}\Big{(}\left\langle\left.\tilde{e}_{\mathrm{d}}(\sigma)\,\right|\,\tilde{f}_{\mathrm{d}}(\sigma)\right\rangle+\left\langle\left.\tilde{y}(\sigma)\,\right|\,\tilde{u}(\sigma)\right\rangle\Big{)}\,\text{d}\sigma+\mathcal{O}(\tau^{p+1}).$

If $p\geq 2s-2$, then (9.2a) and (9.2b) are even exact, and, if $\beta_{j}\geq
0$ for $j=1,\ldots,s$, as is the case for many collocation methods, we have
that
$\tau\sum_{j=1}^{s}\left\langle\left.e_{\mathrm{d}}^{j}\,\right|\,f_{\mathrm{d}}^{j}\right\rangle\leq
0$, thus the discrete system satisfies the same qualitative behaviour as the
continuous problem.

If the Hamiltonian $\mathcal{H}$ is quadratic, _i.e._

$\mathcal{H}(z)=\tfrac{1}{2}z^{T}Ez+v^{T}z+c,$

for some $E=E^{T}\in\mathbb{R}^{n,n}$, $v\in\mathbb{R}^{n}$ and
$c\in\mathbb{R}$, then we have that
$\tilde{\mathcal{H}}=\mathcal{H}(\tilde{z})$ and $\dot{\tilde{\mathcal{H}}}$
are polynomials of degree $2s$ and degree $2s-1$, respectively. Using the
well-known fact, see _e.g._ [104], that the maximum degree of exactness for
quadrature rules with $s$ nodes is $2s-1$, and that it is attained only with
Gauß-Legendre collocation methods, it follows that for these methods the
integration of $\dot{\tilde{\mathcal{H}}}$ is exact, _i.e._

$\begin{split}\tilde{\mathcal{H}}(t_{\mathrm{f}})-\tilde{\mathcal{H}}(t_{0})&=\tau\sum_{j=1}^{s}\beta_{j}\left\langle\left.e_{\mathrm{d}}^{j}\,\right|\,f_{\mathrm{d}}^{j}\right\rangle+\tau\sum_{j=1}^{s}\beta_{j}\left\langle\left.y_{j}\,\right|\,u_{j}\right\rangle=\\\
&=\int_{t_{0}}^{t_{\mathrm{f}}}\left(\left\langle\left.\tilde{e}_{\mathrm{d}}(s)\,\right|\,\tilde{f}_{\mathrm{d}}(s)\right\rangle+\left\langle\left.\tilde{y}(\sigma)\,\right|\,\tilde{u}(\sigma)\right\rangle\right)\,\text{d}\sigma.\end{split}$

Since furthermore, for these methods we have $\beta_{j}\geq 0$, it follows
that the dissipation term is always non-positive, and we obtain the discrete
version of the dissipation inequality

$\tilde{\mathcal{H}}(t_{\mathrm{f}})-\tilde{\mathcal{H}}(t_{0})\leq\tau\sum_{j=1}^{s}\beta_{j}\left\langle\left.y_{j}\,\right|\,u_{j}\right\rangle=\int_{t_{0}}^{t_{\mathrm{f}}}\left\langle\left.\tilde{y}(\sigma)\,\right|\,\tilde{u}(\sigma)\right\rangle\,\text{d}\sigma,$

hence for quadratic Hamiltonians the pHDAE structure is fully preserved.

######  Example 9.1.

Consider the numerical solution of the strangeness-free dHDAE system, given by
the power network presented in Section 5.2. We use the artificial constants
$E_{G}=0$, $L=2$, $C_{1}=0.01$, $C_{2}=0.02$, $R_{\mathrm{L}}=0.1$,
$R_{\mathrm{G}}=6$ and $R_{\mathrm{R}}=3$, see [162]. For the time integration
we choose $\tau=0.01$ and $t_{\mathrm{f}}=1$ and use the implicit midpoint
rule, _i.e._ , the Gauß-Legendre collocation method with $s=1$ stages and
order $p=2$. The numerical result for the consistent initial value

$z_{0}=\sqrt{\tfrac{10}{R_{\mathrm{R}}}}\begin{bmatrix}1&-R_{\mathrm{R}}-R_{\mathrm{L}}&-R_{\mathrm{R}}&-\tfrac{R_{\mathrm{R}}-R_{\mathrm{L}}}{R_{\mathrm{R}}}&-1\end{bmatrix}^{T}$

is presented in Figure 7. One observes that after an initial phase the state
converges to zero and the Hamiltonian decreases monotonically.

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$-5$$0$$5$$I$$V_{1}$$V_{2}$$I_{\mathrm{G}}$$I_{\mathrm{R}}$$\mathcal{H}$
Figure 7. Evolution of state components (solid lines) and Hamiltonian (dashed
line).

###  9.2. Linear System Solvers

In every step of an implicit time-discretization method for a finite
dimensional pHDAE system, it is necessary to solve linear algebraic systems of
the form

$Ax=\Big{(}E+\tau(R-J)\Big{)}x=b,$ (9.3)

where $\tau$ is the time-stepsize. The matrix $A$ can be split in its
symmetric and skew-symmetric part $A=H+S$, where $H=\frac{1}{2}((E+\tau
R)^{T}+(E+\tau R))\geq 0$ and $S=\frac{1}{2}((E+\tau R)^{T}-(E+\tau R))$. An
analogous linear system structure occurs in discretized linear time-varying
and nonlinear pHDAE systems, in the construction of reduced models [71], and
by multiplying some equations with $-1$ in optimization methods,see also [92,
153].

The fact that the symmetric part $H$ is positive semi-definite can be
exploited in direct or iterative solution methods. In the small and medium
scale case we can make use of the following staircase form, see [2], which we
present here in the real case.

######  Lemma 9.2.

Consider $A=H+S\in\mathbb{R}^{n,n}$, where $H=H^{T}\geq 0$ and $S=-S^{T}$.
Then there exist a real orthogonal matrix $U\in\mathbb{R}^{n,n}$, and integers
$n_{1}\geq n_{2}\geq\cdots\geq n_{r-1}\geq 0$ and $n_{r}\geq 0$, such that

$U^{T}HU=\begin{bmatrix}H_{11}&0\\\ 0&0\end{bmatrix},\
U^{T}SU=\begin{bmatrix}S_{11}&S_{12}&&&0\\\ S_{21}&S_{22}&\ddots&&0\\\
&\ddots&\ddots&S_{r-2,r-1}&\vdots\\\ &&S_{r-1,r-2}&S_{r-1,r-1}&0\\\
0&\cdots&\cdots&0&S_{r,r}\end{bmatrix},$ (9.4)

where $H_{11}=H_{11}^{T}\in\mathbb{R}^{n_{1},n_{1}}$ is positive definite,
$S_{ii}=-S_{ii}^{T}\in\mathbb{R}^{n_{i},n_{i}}$ for $i=1,\dots,r$, and
$S_{i,i-1}=-S_{i-1,i}^{T}=[\Sigma_{i,i-1}\;0]\in\mathbb{R}^{n_{i},n_{i-1}}$
with $\Sigma_{i,i-1}$ being nonsingular for $i=2,\dots,r-1$.

###### Proof.

We present the proof for completeness, see also [92]. The result is trivial
when $H$ is nonsingular (and thus positive definite), since in this case it
holds with $U=I$, $r=2$, $n_{1}=n$, and $n_{2}=0$.

Let $H=H^{T}\geq 0$ be singular. We consider a full rank decomposition of $H$
with a real orthogonal $U$ such that

$U_{1}^{T}HU_{1}=\begin{bmatrix}\widehat{H}_{11}&0\\\ 0&0\end{bmatrix},$

where we assume that
$\widehat{H}_{11}=\widehat{H}_{11}^{T}\in\mathbb{R}^{n_{1},n_{1}}$, with
$0\leq n_{1}\leq n$, is void or positive definite. Applying the same
orthogonal congruence transformation to $S$ gives the matrix

$\widehat{S}=U_{1}^{T}SU_{1}=\begin{bmatrix}\widehat{S}_{11}&\widehat{S}_{12}\\\
\widehat{S}_{21}&\widehat{S}_{22}\end{bmatrix},$ (9.5)

where $\widehat{S}_{11}\in\mathbb{R}^{n_{1},n_{1}}$, and
$\widehat{S}_{21}=-\widehat{S}_{12}^{T}$, since $S$ is skew-symmetric. If
$\widehat{H}_{11}$ is void or $\widehat{S}_{21}=0$, then the proof is
complete. Otherwise, let

$\widehat{S}_{21}=W_{2}\begin{bmatrix}\Sigma_{21}&0\\\
0&0\end{bmatrix}V_{2}^{T}$

be a singular value decomposition, where $\Sigma_{21}$ is nonsingular (and
diagonal), and $W_{2}\in\mathbb{R}^{n_{1},n_{1}}$,
$V_{2}\in\mathbb{R}^{n-n_{1},n-n_{1}}$ are real orthogonal. We define
$U_{2}\vcentcolon=\operatorname{diag}(V_{2},W_{2})\in\mathbb{R}^{n,n}$ and
form

$U_{2}^{T}U_{1}^{T}HU_{1}U_{2}=\begin{bmatrix}V_{2}^{T}\widehat{H}_{11}V_{2}&0\\\
0&0\end{bmatrix},$

where $V_{2}^{T}\widehat{H}_{11}V_{2}\in\mathbb{R}^{n_{1},n_{1}}$ is void or
symmetric positive definite, and

$U_{2}^{T}U_{1}^{T}SU_{1}U_{2}=\begin{bmatrix}V_{2}^{T}\widehat{S}_{11}V_{2}&V_{2}^{T}\widehat{S}_{12}W_{2}\\\
W_{2}^{T}\widehat{S}_{21}V_{2}&W_{2}^{T}\widehat{S}_{22}W_{2}\end{bmatrix}=\begin{bmatrix}\widetilde{S}_{11}&\widetilde{S}_{12}&0\\\
\widetilde{S}_{21}&\widetilde{S}_{22}&\widetilde{S}_{23}\\\
0&\widetilde{S}_{32}&\widetilde{S}_{33}\end{bmatrix}$

with $\widetilde{S}_{21}=[\Sigma_{21}\;0]$. If $\widetilde{S}_{32}=0$ or is
void then the proof is again complete. Otherwise we continue inductively and
after finitely many steps we obtain a decomposition of the required form. ∎

When the staircase form (9.4) has been computed, then the transformed linear
system $(U^{T}AU)(U^{T}x)=U^{T}b$ can be solved using block Gaussian
elimination.

######  Lemma 9.3.

Consider the matrix $U^{T}AU$ in (9.4). Then there exist invertible lower and
upper block bi-diagonal matrices $L_{s},R_{s}$, respectively, such that

$L_{s}U^{T}AUR_{s}=\begin{bmatrix}{H}_{11}+S_{11}&&&&\\\
&\mathcal{S}_{1}&&&\\\ &&\ddots&&\\\ &&&\mathcal{S}_{r-2}&\\\
&&&&\mathcal{S}_{r,r}\end{bmatrix},$

with Schur complements $\mathcal{S}_{1},\dots,\mathcal{S}_{r-2}$ that have
positive definite symmetric parts.

###### Proof.

A constructive proof via a sequence of block Gaussian elimination steps is
given in [92]. It relies on the fact that in every step (except the last one)
the Schur complement has a symmetric part which is positive definite. ∎

The properties of the linear system resulting from the dHDAE structure also
have an immediate advantage in the context of iterative methods. If the
symmetric part is positive definite then in [229] it is suggested to solve,
instead of $Ax=b$, the equivalent system

$(I+K)x=\hat{b},\quad\mbox{where}\quad K=H^{-1}S,\quad\hat{b}=H^{-1}b.$ (9.6)

This transformation is a left preconditioning of the original system with its
positive definite symmetric part, which defines the $H$-inner product

$\langle x,y\rangle_{H}=y^{T}Hx.$

This implies that one can construct optimal Krylov subspace methods based on
three-term recurrences for the system (9.6), see [187, 229]. If the symmetric
part is semi-definite but singular, then one has to identify the nullspace,
which is actually easy in many applications. The advantages of this approach
and the fact that one obtains a rigorous convergence analysis and optimality
conditions is discussed in detail in [92] and illustrated with several
numerical examples including those discussed in Section 5.

######  Example 9.4.

The finite element model of the disk brake discussed in Section 5.7 leads to a
second order DAE of the form

$M\ddot{p}+D\dot{p}+Kp=f,$

with $p$ the coefficient vector of displacements of the structure, with
frequency dependent mass matrix $M=M^{T}>0$, damping matrix $D=D^{T}\geq 0$,
and stiffness matrix $K=K^{T}>0$, see [89] evaluated for
$\omega_{\mathrm{ref}}=500$. For $f=0$, after a first-order reformulation and
discretization with the implicit mid-point rule one obtains a linear system
with $n=9338$ and a positive definite symmetric part.

As shown in Table 2, a preconditioned GMRES methods (preconditioned with the
inverse of the symmetric part), even though using fewer iterations, takes a
significantly longer time than the method in [229], due to the full
recurrences in the algorithm compared to three-term recurrences in Widlund’s
method. This effect becomes even more pronounced for smaller stepsizes.

| $\tau=0.001$ | $\tau=0.0001$
---|---|---
method | time [s] | # of iter. | time [s] | # of iter
Widlund | 3.54 | 85 | 0.72 | 16
GMRES | 31.10 | 65 | 10.51 | 13
Table 2. Brake squeal problem. Run times and iteration numbers.

##  10\. Control methods for pHDAE systems

One of the main advantages to introduce pH descriptor systems is its direct
base in control theory. In this section we therefore discuss classical control
applications for pHDAE systems.

Consider a linear pHDAE system as in Definition 4.8 and a linear output
feedback $u=F(t)y$. Then, we can write the system in behavior form by
introducing $\xi=\begin{bmatrix}z^{T}&u^{T}&y^{T}\end{bmatrix}^{T}$ and new
block matrices

$\displaystyle\mathcal{E}$
$\displaystyle\vcentcolon=\operatorname{diag}(E,0,0),$
$\displaystyle\mathcal{Q}$
$\displaystyle\vcentcolon=\operatorname{diag}(Q,I,I),$
$\displaystyle\mathcal{K}$
$\displaystyle\vcentcolon=\operatorname{diag}(K,0,0),$

as well as

$\mathcal{J}=\begin{bmatrix}J&G&0\\\ -G^{T}&N&I\\\
0&-I&\tfrac{1}{2}(F-F^{T})\end{bmatrix},\quad\mathcal{R}=\begin{bmatrix}R&P&0\\\
P^{T}&S&0\\\ 0&0&-\tfrac{1}{2}(F+F^{T})\end{bmatrix},$ (10.1)

which gives the closed-loop descriptor system

$\mathcal{E}\dot{\xi}=((\mathcal{J}-\mathcal{R})\mathcal{Q}-\mathcal{E}\mathcal{K})\xi.$
(10.2)

This is a dHDAE if and only if $-\tfrac{1}{2}(F+F^{T})\geq 0$ pointwise.

Analogously, for the general nonlinear pHDAE structure in Definition 4.1, we
introduce

$\displaystyle\xi$
$\displaystyle\vcentcolon=\begin{bmatrix}z^{T}&u^{T}&y^{T}\end{bmatrix}^{T},$
$\displaystyle\tilde{\eta}$
$\displaystyle\vcentcolon=\begin{bmatrix}\eta^{T}&u^{T}&y^{T}\end{bmatrix}^{T},$
$\displaystyle\tilde{r}$
$\displaystyle\vcentcolon=\begin{bmatrix}r^{T}&0&0\end{bmatrix}^{T},$

the Hamiltonian $\tilde{\mathcal{H}}(t,\xi)\vcentcolon=\mathcal{H}(t,z)$ and
matrix functions $\mathcal{E}\vcentcolon=\operatorname{diag}(E,0,0)$, and
$\mathcal{J}$ and $\mathcal{R}$ as defined in (10.1). This gives the system in
behavior form

$\mathcal{E}\dot{\xi}+\tilde{r}=(\mathcal{J}-\mathcal{R})\tilde{\eta}(\xi)$
(10.3)

satisfying
$\tfrac{\partial}{\partial\xi}\tilde{\mathcal{H}}=\mathcal{E}^{T}\tilde{\eta}$
and $\tfrac{\partial}{\partial
t}\tilde{\mathcal{H}}=\tilde{\eta}^{T}\tilde{r}$ pointwise, which is a pHDAE
if and only if $\tfrac{1}{2}(F+F^{T})\leq 0$ pointwise.

In this way, using a parameterization via the output feedback matrix
(function) $F$, we can introduce control methods via the dHDAE systems (10.2)
or (10.3), respectively.

###  10.1. Robust stabilization/passivation

We have seen in Section 7.6 that for dHDAE systems, stability and passivity
can be easily characterized, while for asymptotic stability or strict
passivity, in general, we have only sufficient conditions.

Considering a linear pHDAE system of the form (4.5) with $Q=I$ that has no
feedthrough term (cf. Section 4.4), and using a linear output feedback
$u=Fy+w$ we obtain the closed-loop system

$\displaystyle E\dot{z}+EKz$ $\displaystyle=(J-R+GFG^{T})z+Gw,$ $\displaystyle
y$ $\displaystyle=G^{T}z.$

If $F=F_{\mathrm{H}}+F_{\mathrm{S}}$ is chosen to have a negative semi-
definite symmetric part $F_{H}$, then the new dissipation coefficient becomes
$\hat{R}=R-GF_{\mathrm{H}}G^{T}$ and the skew-symmetric part becomes
$J-GF_{\mathrm{S}}G^{T}$. If $\hat{R}$ is positive definite then the system is
asymptotically stable. The same approach can also be applied in the general
nonlinear case, if an output feedback leads to a positive definite $R(t,z)$ in
(4.1). We only have this sufficient condition to guarantee asymptotic
stability, see Corollary 7.29. At current, the problem to find a necessary and
sufficient condition that guarantees that a pHDAE system can be made
asymptotically stable by output feedback is under investigation.

The situation is much better understood in the case of LTI pHDAE systems,
where we have a characterization via the hypocoercivity index being finite in
Corollary 7.22. Then in view of Theorem 7.20 (ii), we have different options
to obtain asymptotic stability.

Again, if we can achieve $\hat{R}=R-GF_{\mathrm{H}}G^{T}>0$, then we have
asymptotic stability. But we can also use the skew part $F_{\mathrm{S}}$ to
change the eigenvectors of the pair $(E,J+GF_{\mathrm{S}}G^{T})$ in such a way
that no eigenvector is in the kernel of $R-GF_{\mathrm{H}}G^{T}$, or we can
use a combination of both. It is clear from the classical theory of
unstructured DAE systems, see Section 3.3, that if the system is strongly
stabilizable and strongly detectable, then such an output feedback exists and
can be computed by ignoring the structure and solving an optimal control
problem.

In constructing an output feedback via optimal control methods, we have the
freedom to choose the cost functional (3.20) and, furthermore, we also have
some freedom in choosing the pHDAE representation, which is not unique (see,
for instance, the discussion at the end of Section 8.4). This flexibility can
be used to make the resulting closed-loop system maximally robust against
perturbations, which for general LTI control systems recently has been an
important research topic, see _e.g._ , [164] and the references therein. For
pHODE systems, this topic has recently been of great importance by introducing
measures like the distance to instability for the robustness of pHODE
representations, see [6], as well as their optimization [87, 88]. For pHDAE
systems, this is currently an active research topic.

The analogous question arises in the context of passivity. We have that a
regular strangeness-free pHDAE system is passive, but in general not
necessarily strictly passive, since $W$ in (4.8) is only assumed positive
semi-definite. To obtain strict passivity, it is necessary to consider the
system in the formulation with feedthrough term and similarly we can analyze
how to obtain robust representations as is done for pHODE systems in [20, 24,
161]. For pHDAE systems, this is again an active research topic.

###  10.2. Optimal control

Due to the many interesting properties of pHDAE systems, one may investigate
whether some extra advantages can be obtained also in the context of optimal
control problems. Clearly one can just use the general results in Section 3.5
and obtain the same optimality conditions. However, it has been observed in
two recent papers, see [173, 76], that some surprising results arise for
optimal control problems with LTI pHODE and pHDAE, when as very special cost
functional the supplied energy is minimized, _i.e._

$\tfrac{1}{2}z(t_{\mathrm{f}})^{T}Mz(t_{\mathrm{f}})+\tfrac{1}{2}\int^{t_{\mathrm{f}}}_{t_{0}}2y^{T}u\,\text{d}t=\tfrac{1}{2}z(t_{\mathrm{f}})^{T}Mz(t_{\mathrm{f}})+\tfrac{1}{2}\int^{t_{\mathrm{f}}}_{t_{0}}2z^{T}G^{T}u\,\text{d}t,$
(10.4)

subject to the constraint

$E\dot{z}=(J-R)z+Gu,\qquad E^{\dagger}Ez(t_{0})=z_{0}$ (10.5)

and the output equation is $y=G^{T}z$. Note that in the cost functional (3.32)
we then have $W_{\mathrm{z}}=0$, $W_{\mathrm{u}}=0$, and $S=G^{T}$. Since we
are in the LTI case, we can insert the data into the optimality system (3.34)
and obtain the following result.

######  Corollary 10.1.

Consider the optimal control problem to minimize (10.4) subject to the
constraint (10.5). Assume that the pair $(E,J-R)$ is regular and of
(Kronecker) index at most one (as a free system with $u\equiv 0$) and that $M$
is in $\operatorname{cokernel}E$. If $(z,u)\in\mathbb{Z}\times\mathbb{U}$ is a
solution to this optimal control problem, then there exists a Lagrange
multiplier
$\lambda\in\mathcal{C}^{1}_{E^{\dagger}E}(\mathbb{T},\mathbb{R}^{n})$, such
that $(z,\lambda,u)$ satisfy the boundary value problem

$\begin{bmatrix}0&E&0\\\ -E^{T}&0&0\\\
0&0&0\end{bmatrix}\begin{bmatrix}\dot{\lambda}\\\ \dot{z}\\\
\dot{u}\end{bmatrix}=\begin{bmatrix}0&J-R&G\\\ (J-R)^{T}&0&G\\\
G^{T}&G^{T}&0\end{bmatrix}\begin{bmatrix}\lambda\\\ z\\\ u\end{bmatrix},\ $
(10.6)

with boundary conditions

$E^{\dagger}Ez(t_{0})=z_{0},\qquad
EE^{\dagger}\lambda(t_{\mathrm{f}})=-(E^{\dagger})^{T}Mz(t_{\mathrm{f}}).$

Use a full rank decomposition $E=U_{E}\left[\begin{smallmatrix}E_{11}&0\\\
0&0\end{smallmatrix}\right]U_{E}^{T}$ with $E_{11}=E_{11}^{T}>0$ and transform
the other coefficients accordingly as

$\displaystyle U^{T}_{E}(J-R)U_{E}=\begin{bmatrix}L_{11}&L_{12}\\\
L_{21}&L_{22}\end{bmatrix},\quad
U^{T}_{E}MU_{E}=\begin{bmatrix}M_{11}&M_{12}\\\ M_{21}&M_{22}\end{bmatrix},$
$\displaystyle U^{T}_{E}G=\begin{bmatrix}G_{1}\\\
G_{2}\end{bmatrix},\qquad\begin{bmatrix}\hat{z}_{1}\\\
\hat{z}_{2}\end{bmatrix}=U_{E}^{T}\hat{z},\qquad\begin{bmatrix}\hat{\lambda}_{1}\\\
\hat{\lambda}_{2}\end{bmatrix}=U_{E}^{T}\hat{\lambda},\qquad\begin{bmatrix}\hat{z}_{1,0}\\\
\hat{z}_{2,0}\end{bmatrix}=U^{T}_{E}z_{0}.$

After some permutations we can express (10.6) in the form

$\begin{bmatrix}0&E_{11}&0&0&0\\\ -E_{11}^{T}&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\end{bmatrix}\begin{bmatrix}\dot{\lambda}_{1}\\\
\dot{\hat{z}}_{1}\\\ \dot{\hat{\lambda}}_{2}\\\ \dot{\hat{z}}_{2}\\\
\dot{u}\end{bmatrix}=\begin{bmatrix}0&L_{11}&0&L_{12}&G_{1}\\\
L_{11}^{T}&0&L_{21}^{T}&0&G_{1}\\\ 0&L_{21}&0&L_{22}&G_{2}\\\
L_{12}^{T}&0&L_{22}^{T}&0&G_{2}\\\
G_{1}^{T}&G_{1}^{T}&G_{2}^{T}&G_{2}^{T}&0\end{bmatrix}\begin{bmatrix}\hat{\lambda}_{1}\\\
\hat{z}_{1}\\\ \hat{\lambda}_{2}\\\ \hat{z}_{2}\\\ u\end{bmatrix},$ (10.7)

with boundary conditions $\hat{z}_{1}(t_{0})=\hat{z}_{1,0}$ and
$\hat{\lambda}_{1}(t_{\mathrm{f}})=-E_{11}^{-1}M_{11}\hat{z}_{1}(t_{\mathrm{f}})$,
and consistency condition $\hat{z}_{2,0}=0$. Here we have used the condition
that $M=M^{T}$ is in $\operatorname{cokernel}E$, which implies that
$M_{12}=0$, $M_{22}=0$.

For the structured matrix pencil associated with (10.7) there exists a
condensed form under real orthogonal congruence transformations, which has
been introduced in [50], and from which the spectral properties, the
(Kronecker) index and the regularity can be read off. If this pencil is
regular, then we directly obtain that the system has (Kronecker) index one if
and only if

$\hat{W}_{u}=\begin{bmatrix}0&J_{22}-R_{22}&G_{2}\\\ -J_{22}-R_{22}&0&G_{2}\\\
G_{2}^{T}&G_{2}^{T}&0\end{bmatrix}$

is invertible, see Section 2.3. To simplify the algebraic equations, we can
perform a congruence transformation with the orthogonal matrix

$U_{S}=\begin{bmatrix}\tfrac{1}{\sqrt{2}}I&-\tfrac{1}{\sqrt{2}}I&0\\\
\tfrac{1}{\sqrt{2}}I&\tfrac{1}{\sqrt{2}}I&0\\\ 0&0&I\end{bmatrix},$

_i.e._ , we multiply the system with $U_{S}^{T}$ from the left and set

$\begin{bmatrix}\tilde{\lambda}_{2}\\\ \tilde{z}_{2}\\\
u\end{bmatrix}=\begin{bmatrix}\frac{1}{\sqrt{2}}(\hat{z}_{2}+\hat{\lambda}_{2})\\\
\frac{1}{\sqrt{2}}(\hat{z}_{2}-\hat{\lambda}_{2})\\\
u\end{bmatrix}=U_{S}^{T}\begin{bmatrix}\hat{\lambda}_{2}\\\ \hat{z}_{2}\\\
u\end{bmatrix}.$

Then we get that

$U_{S}^{T}\hat{W}_{\mathrm{u}}U_{S}=\begin{bmatrix}-R_{22}&J_{22}&G_{2}\\\
-J_{22}&R_{22}&0\\\ G_{2}^{T}&0&0\end{bmatrix}$

Clearly, for this to be invertible, we need that $G_{2}$ has full column rank,
which implies that $u$ is fixed as a linear combination of $\hat{\lambda}_{2}$
and $\hat{z}_{2}$. Considering the application examples in Section 5, where we
typically have $G_{2}=0$, we cannot expect the DAE associated with the
optimality system to be of (Kronecker) index one. Thus we are in the case of a
singular control problem, see _e.g._ , [44].

The case that $\hat{W}_{\mathrm{u}}$ is not invertible has been analyzed in
[200] for the pHODE case and in [76] for the pHDAE case, where $\mu E-(J-R)$
is regular and of (Kronecker) index at most one. In these papers, it is
assumed that the image of the matrix $G$ does not intersect with the kernel of
$R$ and that even though this is a singular control problem, the optimal
solution is still a feedback control that can be obtained via the solution of
a Riccati equation. We refer to [15] for the analysis of such problems. The
extension of this analysis to the LTV pHDAE case and the case that this
assumption is not valid is currently under investigation.

##  11\. Summary and open problems

This survey paper discusses the model class of port-Hamiltonian descriptor
systems (differential-algebraic systems) for numerical simulation and control.
We have demonstrated that this model class has many nice properties:

  * •

It allows for automated modeling in a modularized and network-based fashion.

  * •

It allows the coupling of mathematical models across different scales and
physical domains.

  * •

It incorporates the properties of the real (open or closed) physical system.

  * •

It has nice algebraic, geometric, and analytical properties and allows
analysis concerning existence, uniqueness, robustness, stability, and
passivity.

  * •

It is invariant under local variable (coordinate) transformations which leads
to local canonical and condensed forms.

  * •

Furthermore, it allows for structure-preserving (space-time) discretization
and model reduction methods as well as fast solvers for the resulting linear
and nonlinear systems of equations.

Despite the many promising results already available for pHDAEs, there are
still many open problems that are either under investigation or pose
challenging problems to be tackled in the future. In the following, we present
an incomplete list to stimulate further research.

Many of the control theoretical concepts presented within this paper rely on
instantaneous feedback. While this is a convenient theoretical approach, it is
not always possible to realize in applications, where the states or outputs
first have to be measured, the control action computed, and then fed back to
the system, thus resulting in a necessary intrinsic time delay; see also
Remark 3.5. Although some first results for pHODEs with delays are available
in the literature, see for instance [202], a general model class for time-
delayed pHDAEs is not yet available and the results presented in this paper
have to be extended to the time-delay case.

In terms of MOR, the impact of the Hamiltonian on the approximation quality,
see [38, 40], needs to be further investigated, in particular with an emphasis
on nonlinear MOR methods. Structure-preserving balancing methods are still
under investigation; see [39] for some first results for LTI systems. Another
open problem in structure-preserving MOR is the construction of optimal
projection spaces, in the sense that they minimize the Kolmogorov $n$-widths,
respectively the Hankel singular values, cf. [222]. First gradient-based
optimization procedures are discussed for pHODEs in [166, 199, 208]. If the
$n$-widths do not decay rapidly, then one cannot expect accurate ROMs with a
small dimension and current efforts in the reduction of transport-dominated
phenomena, see for instance [36] and the references therein, need to be
adapted to the pH framework.

A closely related topic to MOR is identifying a pHDAE realization from
measurements. In view of large data sets, modern artificial intelligence
approaches, and automatized machine learning methods used within the digital
twin paradigm, this is an important topic requiring further research, with
only a few available results for LTI systems, see Remark 8.15.

It is an open problem to derive necessary and sufficient conditions under
which output feedback can make a general nonlinear pHDAE system asymptotically
stable. A natural research question in this context would be to extend the
concept of hypocoercivity to the LTV and nonlinear case.

The characterization of distance measures for general dHDAE systems, like the
distance from an asymptotically stable dHDAE system to a system which is only
stable or the distance of a strictly passive pHDAE system to the nearest
system that is only passive, is an important research topic because it is a
requirement for the design of real-world systems, as _e.g._ , in Section 5.7.
Even if such a characterization is available, then one needs computational
methods to compute these distances. In the large-scale setting, this is a
challenging issue that can only be achieved with a clever combination with
model reduction techniques; see [6] for a first attempt.

Another important research problem is the exact characterization of the
relationship between passivity, positive realness, and the port-Hamiltonian
structure for pHDAEs, see [24, 25] for the pHODE case, as well as the
characterization via Kalman-Yakobovich-Popov inequalities as it has been done
for general LTI DAE systems in [190, 193].

A research field that has so far not received much attention are discrete-time
pH descriptor systems. The primary research in this direction arises from
discretizations of pHDAE systems, see [120, 162] for the most recent
approaches. Since discrete-time systems not only arise from discretization but
also from sampling or realization, it is an open question how to properly
define discrete-time pH systems in a general way. This also concerns the
stability and passivity analysis. An approach to modifying the concept of
hypocoercivity for use in discrete-time systems is currently under
investigation.

Since, in many cases, mathematical models are obtained from data via
measurements, and the resulting parameters, as well as model coefficients
(including the inputs and outputs), are only available with some uncertainty
or randomness, it is an open question to include such uncertainties adequately
into the pHDAE framework and also to study robust control methods that deal
with such uncertainties, see [38, 39] for the modification of classical
balanced truncation methods and [117] for the modification of robust control
methods in the pHODE case. Another research area is the perturbation theory
for the problems and the error analysis of the relevant numerical methods.

As we have seen, also the area of optimal control for pHDAE systems requires a
lot of further research. This concerns, in particular, the optimal use of the
structure for nonlinear systems, the choice of an appropriate cost functional,
and also structure-preserving numerical methods in particular for large-scale
problems.

Finally, we would like to discuss a topic that has only briefly been touched
on in this survey: the extension of pH modeling to partial differential
equations that has been pursued in many different directions in recent years.
One can take the approach to replace the coefficient matrices with linear
differential operators, one can follow a differential geometric approach via
the extension of Lagrange or Dirac structures, or one can formulate all
classical partial differential equations from physics in the areas of
elasticity, electromagnetism, fluid dynamics, structural mechanics,
geomechanics, poroelasticity, gas or water transport, to name a few
directions, in a structure that resembles the pHDAE structure via the given
symmetries or differential forms. These efforts have been the topic of many
recent research papers on modeling, numerical methods, optimization, and
control. Discussing these developments would be a considerable survey on its
own, mainly since the field is growing immensely fast. An essential question
that is widely open is to incorporate the boundary appropriately into the
structure, to obtain well-posed partial differential equations and so that
they, on the one hand, can be treated as controls and, on the other hand, also
be used for the interconnection of subsystems. Another important topic that is
actively pursued is appropriate time space-time discretization methods like
finite element or finite volume approaches that preserve the structure.

Instead of discussing this further, we present the following incomplete list
of references that describes many different research directions pursued: [7,
8, 14, 16, 21, 55, 68, 69, 70, 71, 75, 115, 83, 84, 91, 119, 121, 135, 141,
148, 149, 150, 151, 154, 165, 168, 167, 170, 171, 184, 226, 203, 210, 228,
234, 235].

### Acknowledgments

The work of V. Mehrmann has been supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) CRC 910 _Control of
self-organizing nonlinear systems: Theoretical methods and concepts of
application_ : Project No. 163436311, CRC TRR 154 _Mathematical modeling,
simulation and optimization at the example of gas networks_ : Project No.
239904186, priority programs SPP 1984 _Hybrid and multimodal energy systems:
System theoretical methods for the transformation and operation of complex
networks_ : Project No. 361092219, and SPP 1897, _Calm, Smooth and Smart -
Novel Approaches for Influencing Vibrations by Means of Deliberately
Introduced Dissipation_ : Project No. 273845692, DFG Excellence Cluster 2046
Math+: Project No. 390685689, as well as the BMBF (German Ministry of
Education and Research) via the Project EiFer.

B. Unger acknowledges funding from the DFG under Germany’s Excellence Strategy
– EXC 2075 – 390740016 and is thankful for support by the Stuttgart Center for
Simulation Science (SimTech).

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