Generic observability for port-Hamiltonian descriptor systems

Abstract

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021. https://doi.org/10.1007/s00498-021-00287-x), Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) on (relative) generic controllability of unstructured linear differential-algebraic systems and of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) on (relative) generic controllability of port-Hamiltonian descriptor systems. We extend their results to (relative) genericity of observability. For unstructured differential-algebraic systems, criteria for (relative) generic observability are derived from Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2023. https://doi.org/10.1007/s00498-021-00287-x) using duality. This is not possible for port-Hamiltonian systems. Hence, we tweak the results of Ilchmann et al. (Port-Hamiltonian descriptor systems are generically controllable and stabilizable. Submitted to Mathematics of Control, Signals and Systems, 2023. https://arxiv.org/abs/2302.05156) and derive similar criteria as for the unstructured case. Additionally, we consider certain rank constraints on the system matrices.

1 Introduction

Consider linear time-invariant input-state-output systems of the form

$$\begin{aligned}{} & {} \tfrac{\textrm{d}}{\textrm{d}t}x = Ax + Bu,\nonumber \\{} & {} y = Cx \end{aligned}$$

(1.1)

with $(A,B,C)\in \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}\times \mathbb {R}^{p\times n}$, $m,n,p\in \mathbb {N}_0$. A solution of (1.1) is a triple of functions $(x,y,u):\mathbb {R}\rightarrow \mathbb {R}^n\times \mathbb {R}^m\times \mathbb {R}^m$ in a suitable, but largely unspecified, function space with a fixed derivation $\tfrac{\textrm{d}}{\textrm{d}t}$, e.g. a local Sobolev space; the associated behaviour is the set of all solutions and denoted by $\mathfrak {B}_{(A,B,C)}$. Equation (1.1) is called observable if, and only if,

$$\begin{aligned} \forall (x,y,0),(x',y',0)\in \mathfrak {B}_{(A,B,C)}:\quad y = y'\implies x = x'. \end{aligned}$$

(1.2)

In case of classical or weak solutions, it is well known (see, for example, [16, Theorem 3.8]) that the purely analytic property (1.2) is equivalent to the algebraic property

$$\begin{aligned} \textrm{rk}\,[C^\top ,A^\top C^\top ,\ldots ,(A^{n-1})^\top C^\top ] = n. \end{aligned}$$

(1.3)

In the following, we specify our solution spaces insofar that we shall assume that (1.2) and (1.3) are equivalent, and that controllability of (1.1) is characterized by Kalman’s controllability criterion (see, for example, [16, Corollary 3.2]). Then, we see that the vector space isomorphism $(A,B,C)\mapsto (A^\top ,C^\top ,B^\top )$ establishes a bijection between

$$\begin{aligned} S_{\text {obs.}} := \left\{ (A,B,C)\in \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}\times \mathbb {R}^{p\times n}\,\big \vert \,~(1.1)~\text {observable} \right\} \end{aligned}$$

and

$$\begin{aligned} S_{\text {contr.}}&\!:=\! \big \{(A^\top \!,C^\top \!,B^\top )\!\in \!\mathbb {R}^{n\times n}\!\times \!\mathbb {R}^{n\times p}\!\times \!\mathbb {R}^{p\times n}\,\big \vert \,\tfrac{\textrm{d}}{\textrm{d}t}x \!=\! A^\top x \!+\! C^\top u~\text {controllable}\big \}. \end{aligned}$$

It is a classical result due to Wonham [17, Theorem 1.3] that the latter set is generic, which is defined as follows.

Definition 1.1

(see [17, p. 28], [14, pp. 50] and [8, Lemma 2.7]) A set $\mathbb {V}\subseteq \mathbb {R}^N$, $N\in \mathbb {N}$, is an algebraic set if, and only if,

$$\begin{aligned} \exists p_1,\ldots ,p_k\in \mathbb {R}[x_1,\ldots ,x_N]: \mathbb {V} = \bigcap _{i = 1}^k p_i^{-1}(\left\{ 0 \right\} ). \end{aligned}$$

The Zariski topology on $\mathbb {R}^N$ is then given as follows

$$\begin{aligned} \mathcal {Z}(\mathbb {R}^N) := \left\{ \mathbb {R}^N{\setminus }\mathbb {V}\,\big \vert \,\mathbb {V}~\text {is algebraic set} \right\} , \end{aligned}$$

i.e. a set is closed in the Zariski topology if, and only if, it is an algebraic set; for brevity, we call $O\in \mathcal {Z}(\mathbb {R}^N)$ Zariski-open.

A set $S\subseteq \mathbb {R}^N$ is generic if, and only if, the Zariski-interior of S is nonempty or, equivalently, S contains a nonempty Zariski-open set.

It is obvious that vector space isomorphisms are Zariski-homeomorphism (i.e. continuous with continuous inverse, where both spaces are equipped with the Zariski topology) and therefore preserve genericity. Hence, it can be concluded that $S_{\text {obs.}}$ is generic; see also [15] for a direct proof without using duality.

In particular, each proper algebraic subset of $\mathbb {R}^N$ is closed and nowhere dense with respect to the Euclidean topology. Hence, the observable linear systems of fixed number of inputs, states and outputs are closed under sufficiently small perturbations of the parameters. This means that for each observable system (1.1) with system matrices $(A,B,C)\in \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}\times \mathbb {R}^{p\times n}$, there exists a neighbourhood U of (A, B, C) with respect to the Euclidean topology on $\mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}\times \mathbb {R}^{p\times n}$ so that each system (1.1) with system matrices $(A',B',C')\in U$ is observable; and conversely, for each unobservable system (1.1), there exists arbitrarily small perturbations of the system matrices so that the perturbed system is observable. For controllability, this has also been shown earlier by Lee and Markus [10], who showed that the set of controllable systems is open and dense; see also [11] for a similar result for time-varying systems.

Moreover, proper algebraic sets are Lebesgue nullsets and hence the observable systems of fixed size are also closed under random, continuously distributed perturbations. This means, given a random, but continuously distributed perturbation $(\Delta _A,\Delta _B,\Delta _C)\in \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times m}\times \mathbb {R}^{p\times n}$, the system (1.1) with system matrices $(A+\Delta _A,B+\Delta _B,C+\Delta _C)$ is almost surely observable, regardless of whether (A, B, C) is observable.

Recently, Wonham’s result regarding controllability has been extended to unstructured linear time-invariant differential-algebraic equations (DAEs) of the form

$$\begin{aligned}{} & {} \tfrac{\textrm{d}}{\textrm{d}t} Ex = Ax + Bu,\nonumber \\{} & {} y = Cx \end{aligned}$$

(1.4)

with $(E,A,B,C)\in \Sigma _{\ell ,n,m,p}:= \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\times \mathbb {R}^{p\times m}$ and $\textrm{rk}\,E = r$ or $\textrm{rk}\,E\le r$ for some fixed $r\in \left\{ 1,\ldots ,\min \left\{ \ell ,n \right\} \right\}$, see [5, 7], and to port-Hamiltonian DAEs

$$\begin{aligned}{} & {} \tfrac{\textrm{d}}{\textrm{d}t} Ex = (J-R)Qx + Bu,\nonumber \\{} & {} y = B^\top Qx \end{aligned}$$

(1.5)

where $(E,J,R,Q,B)\in \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ with the structural properties $J^\top = -J$, $R = R^\top \ge 0$ and $E^\top Q = Q^\top E\ge 0$, see [8]. Due to the nonlinear character of the restriction $\textrm{rk}\,E\le r$ and the structural properties of port-Hamiltonian systems, the authors of the aforementioned contributions replaced Wonham’s original concept of genericity with the weaker concept of relative genericity, introduced in [9] and discussed in great detail in [7, 8].

Definition 1.2

(see [9, Definition I.2] and [7, Lemma 2.1]) A set $S\subseteq \mathbb {R}^N$ is relative generic in the reference set $V\subseteq \mathbb {R}^N$ if, and only if, the interior of $S\cap V$ with respect to the relative Zariski topology on V is dense with respect to the relative Euclidean topology on V.

The density with respect to the relative Euclidean topology guarantees in particular that relative generic sets are closed under sufficiently small structured perturbations and their complements can be left with arbitrarily small structured perturbations. While Definition 1.2 is valid for any reference set V, it would be advantageous to use the structure of the reference sets that are considered in this note to simplify the definition. In particular, the set of all $(E,A,B,C)\in \Sigma _{\ell ,n,m,p}$ with $\textrm{rk}\,E = r$ is an algebraic set which is irreducible, i.e. it can not be written as the union of two proper algebraic subsets. Therefore, the complement of each proper algebraic subset is dense with respect to the relative Zariski topology. This yields the question whether the explicit condition with respect to the Euclidean topology in Definition 1.2 is, in the particular case that the reference set is an irreducible algebraic set, automatically fulfilled. The Zariski topology is, in general, strictly coarser than the Euclidean topology, and therefore, density with respect to the former does not imply density with respect to the latter topology. When considering irreducible algebraic sets over the complex numbers, then this implication holds, see [13, Theorem 1, p. 58]; for real algebraic sets, however, this is not true. The book of Milnor [12] contains some examples of irreducible real algebraic sets containing a proper algebraic subset whose complement is not dense. A famous example is the Whitney umbrella $W:= \left\{ x\in \mathbb {R}^3\,\big \vert \, x_1^2-x_2^2x_3 = 0 \right\}$ which is an irreducible algebraic variety. W contains the proper subvariety $V_0:= \left\{ (0,0,x_3)\in \mathbb {R}^3 \right\}$, i.e. $V_0\subseteq W$ is an algebraic set and $V_0\ne W$ (e.g. $(1,1,1)\in W{\setminus } V_0$). If $z\in (-\infty ,0)$, then $x_1^2-z x_2^2 = 0$ has the unique solution $(0,0,z)\in V_0$ and hence $W{\setminus } V_0$ is not a dense subset of W with respect to the Euclidean topology. Thus, we can, in general, not readily use that our reference sets are irreducible algebraic sets, or composed of (closures of subsets with Zariski closed boundary of) such, to remove the explicit reference to the Euclidean topology in Definition 1.2.

While genericity of controllability has been analysed, genericity of observability for DAEs was neglected in the study of (relative) genericity of controllability for (1.4) and (1.5) and has, to the best of the knowledge of the author, not been studied elsewhere in the literature. The present note aims to resolve this shortcoming.

The paper is structured as follows. First, we recall the algebraic characterizations of different observability notions for DAEs and some properties of relative generic sets which are helpful in later proofs. Then, we discuss the reference sets which describe the parameter matrices for the unstructured DAE (1.4) and the port-Hamiltonian DAE (1.5), respectively, which allows to formulate the main results, that are characterizations of relative genericity of observability for unstructured and port-Hamiltonian DAEs in terms of the system dimensions $\ell ,n,m,p$. For unstructured DAEs, there exists a duality between observability and controllability, which allows to prove these characterizations using the results of [7] similarly to the introductory remarks on generic observability of ODEs. For port-Hamiltonian systems, however, the output has a different structure with respect to the (essentially unstructured) input, which prevents the straightforward usage of duality arguments. Therefore, we prove some auxiliary genericity results in Sect. 4.1. Lastly, we use this machinery to prove the characterizations of relative genericity for port-Hamiltonian systems.

2 Main results

As for controllability, there are various different notions of observability of DAEs. They are analytically defined as properties of the distributional behaviour of (1.4). Since they are quite involved, we omit them not to distract the reader from the subject of this note and refer for the definitions to the survey article [1]. The algebraic characterizations of the various observability notions may serve, only for the purpose of this note, as definitions.

Definition-Proposition 2.1

(see [1, Corollary 6.2]) We use the abbreviation

$$\begin{aligned} S_{\text {observable}} := \left\{ (E,A,B,C)\in \Sigma _{\ell ,n,m,p} \,\big \vert \,~(1.4)~\text {observable} \right\} , \end{aligned}$$

where observable is replaced by any of the following ten observability concepts, which are algebraically characterized (and, as far as we are concerned, defined) as follows

$$\begin{aligned} \begin{array}{lrclcl} (E,A,B,C)\in S_{\text {obs. at}~\infty } &{}:\iff &{}&{} \textrm{rk}\,[E^\top ,C^\top ] = n\ ;\\ (E,A,B,C)\in S_{\text {impulse obs.}} &{}:\iff &{}&{} \forall Z\in \mathbb {R}^{n\times (n-\textrm{rk}\,E)}~\text {with}~\textrm{im}\,_{\mathbb {R}}Z = \ker _{\mathbb {R}}E^\top \\ &{} &{} &{} : \textrm{rk}\,[E^\top , A^\top Z,C^\top ] = n\ ;\\ (E,A,B,C)\in S_{\text {behavioural obs.}} &{}:\iff &{}&{} \forall \lambda \in \mathbb {C}: \textrm{rk}\,[\lambda E^\top -A^\top ,C^\top ] = n\ ;\\ (E,A,B,C)\in S_{\text {completely obs.}} &{}:\iff &{}&{} \forall \lambda \in \mathbb {C}: \textrm{rk}\,[\lambda E^\top -A^\top ,C^\top ] =\textrm{rk}\,[E^\top ,C^\top ] = n\ ;\\ (E,A,B,C)\in S_{\text {strongly obs.}} &{}:\iff &{}&{} \forall Z\in \mathbb {R}^{n\times (n-\textrm{rk}\,E)}~\text {with}~\textrm{im}\,_{\mathbb {R}}Z = \ker _{\mathbb {R}}E^\top ~\forall \lambda \in \mathbb {C}\\ &{}&{}&{}: \textrm{rk}\,[\lambda E^\top -A^\top ,C^\top ] = \textrm{rk}\,[E^\top ,A^\top Z,C^\top ] = n \ ;\\ (E,A,B,C)\in S_{\text {RS obs. at}~\infty } &{}:\iff &{}&{} \textrm{rk}\,[E^\top ,C^\top ] = \textrm{rk}\,[E^\top ,A^\top ,B^\top ]\ ; \\ (E,A,B,C)\in S_{\text {RS impulse obs.}} &{}:\iff &{}&{} \forall Z\in \mathbb {R}^{n\times (n-\textrm{rk}\,E)}~\text {with}~\textrm{im}\,_{\mathbb {R}}Z = \ker _{\mathbb {R}}E^\top \\[2ex] &{} &{} &{} : \textrm{rk}\,[E^\top , A^\top Z,C^\top ] = \textrm{rk}\,[E^\top ,A^\top ,C^\top ]\ ;\\[4ex] (E,A,B,C)\in S_{\text {RS beh. obs.}} &{}:\iff &{}&{} \forall \lambda \in \mathbb {C}: \textrm{rk}\,[\lambda E^\top -A^\top ,C^\top ] = \textrm{rk}\,_{\mathbb {R}(x)}[xE^\top -A^\top ,C^\top ]\ ;\\[4ex] (E,A,B,C)\in S_{\text {RS compl. obs.}} &{}:\iff &{}&{} \forall \lambda \in \mathbb {C}: \textrm{rk}\,[\lambda E^\top -A^\top ,C^\top ]\\ &{}&{}&{} = \textrm{rk}\,[E^\top ,C^\top ] = \textrm{rk}\,[E^\top ,A^\top ,C^\top ]\ ;\\[4ex] (E,A,B,C)\in S_{\text {RS strongly obs.}} &{}:\iff &{}&{} \forall Z\in \mathbb {R}^{n\times (n-\textrm{rk}\,E)}~\text {with}~\textrm{im}\,_{\mathbb {R}}Z= \ker _{\mathbb {R}}E^\top ~\forall \lambda \in \mathbb {C}\\[2ex] &{}&{}&{}: \textrm{rk}\,[\lambda E^\top -A^\top ,C^\top ] = \textrm{rk}\,[E^\top ,A^\top Z,C^\top ] \\ &{}&{}&{}= \textrm{rk}\,[E^\top ,A^\top ,C^\top ] \ .\\[2ex] \end{array} \end{aligned}$$

Corollary 2.2

In view of the algebraic criteria for observability from Definition-Proposition 2.1, the identities

• (i) $S_{\text {completely obs.}} = S_{\text {behavioural obs.}}\cap S_{\text {obs. at}~\infty },$

• (ii) $S_{\text {strongly obs.}} = S_{\text {behavioural obs.}}\cap S_{\text {impulse obs.}},$

• (iii) $S_{\text {RS~compl. obs.}} = S_{\text {RS~beh. obs.}}\cap S_{\text {RS obs. at}~\infty },$

• (iv) $S_{\text {RS~strongly obs.}} = S_{\text {RS~beh. obs.}}\cap S_{\text {RS impulse obs.}}$

hold true.$\diamond$

One of our aims is the study of observability for the DAE (1.4) with $(E,A,B,C)\in \Sigma _{\ell ,n,m,p}$ and $\textrm{rk}\,E\le r$. To this end, we introduce, for $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$ and $\star \in \left\{ \le ,= \right\}$, the reference set

$$\begin{aligned} \Sigma _{\ell ,n,m,p}^{\star r} := \left\{ (E,A,B,C)\in \Sigma _{\ell ,n,m,p}\,\big \vert \,\textrm{rk}\,E \star r \right\} . \end{aligned}$$

This allows to formulate the first main result that is criteria for relative genericity of $S_{{\textit{observable}}}$ in $\Sigma _{\ell ,n,m,p}^{\star r}$, where observable is replaced by any of the observability notions in Definition-Proposition 2.1.

Theorem 2.3

Let $\ell ,n,p\in \mathbb {N}$, $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$ and $\star \in \left\{ \le ,= \right\}$. We have the following equivalences

$$\begin{aligned} \begin{array}{clccl} \text {(i)} &{}~ S_{\text {obs. at}~\infty } &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\le r+p,\\[2ex] \text {(ii)} &{}~ S_{\text {impulse obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\le \ell +p,\\[2ex] \text {(iii)} &{}~ S_{\text {behavioural obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\le \ell +p-\min \left\{ r,1 \right\} ,\\[2ex] \text {(iv)} &{}~ S_{\text {completely obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\le r+p~\wedge ~r\ne \ell ,\\[2ex] \text {(v)} &{}~ S_{\text {strongly obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n<\ell +p-\min \left\{ r,1 \right\} ,\\[2ex] \text {(vi)} &{}~ S_{\text {RS obs. at}~\infty } &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\le r+p,\\[2ex] \text {(vii)} &{}~ S_{\text {RS impulse obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\le \ell +p,\\[2ex] \text {(viii)} &{}~ S_{\text {RS beh. obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\ne \ell +p~\vee r = 0,\\[2ex] \text {(ix)} &{}~ S_{\text {RS completely obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n\le r+p~\wedge ~r\ne \ell ,\\[2ex] \text {(x)} &{}~ S_{\text {RS strongly obs.}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m,p}^{\star r} &{}~ \iff &{}~ n<\ell +p-\min \left\{ r,1 \right\} . \end{array} \end{aligned}$$

If the condition on the right-hand side of an equivalence is violated, then the complement of the corresponding set is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$.$\diamond$

The proof of Theorem 2.3 depends on a few auxiliary results and is postponed to Sect. 3.

Opposed to unstructured DAEs, port-Hamiltonian DAEs warrant some more attention. In the introduction, we have given a quite general class of port-Hamiltonian systems (1.5). Associated with this equation is the Hamiltonian (“energy”) function

$$\begin{aligned} H:\mathbb {R}^n\rightarrow \mathbb {R},\quad x\mapsto \frac{1}{2}x^\top E^\top Qx \end{aligned}$$

which admits the power balance equation [3, Lemma 2.2]

$$\begin{aligned} \tfrac{\textrm{d}}{\textrm{d}t} H(x) = x^\top Q^\top JQx - x^\top Q^\top RQx + y^\top u \end{aligned}$$

(2.1)

along all solutions (x, u, y) of (1.5). By skew-symmetry of J, the first summand of (2.1) vanishes, and from positive semidefiniteness of R, the second summand is not positive. This allows for an interpretation of the system matrices J, R and B: J corresponds to the lossless power transport, R corresponds to the dissipation of energy within the system, and B conveys the exchange of energy at the “boundary” of the system. In particular, if the system is lossless in the sense that energy dissipation within the system cannot happen, then R should be zero. Therefore, and since generically R is nonzero, the author argues that lossless and (semi-)dissipative port-Hamiltonian systems should be treated separately. To this end, we consider the two reference sets

$$\begin{aligned} \Sigma _{\ell ,n,m}^{H} := \left\{ (E,J,Q,B)\in \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\,\left| \,\begin{array}{l} J = -J^\top , E^\top Q = Q^\top E\ge 0 \end{array}\right. \right\} \end{aligned}$$

corresponding to lossless port-Hamiltonian systems and

$$\begin{aligned} \Sigma _{\ell ,n,m}^{sdH} := \left\{ (E,J,R,Q,B)\in \mathbb {R}^{\ell \times n}\times \left( \mathbb {R}^{\ell \times \ell }\right) ^2\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\,\left| \,\begin{array}{l} J = -J^\top , R = R^\top \ge 0,\\ E^\top Q = Q^\top E\ge 0 \end{array}\right. \right\} \end{aligned}$$

corresponding to (semi-)dissipative port-Hamiltonian systems. Similar as to unstructured DAEs, we put for $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$ and $\star \in \left\{ =,\le \right\}$

$$\begin{aligned} \Sigma _{\ell ,n,m}^{H,\star r}&:= \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^H\,\big \vert \, \textrm{rk}\,E\star r \right\} \quad \text {and}\\ \Sigma _{\ell ,n,m}^{sdH,\star r}&:= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH}\,\big \vert \, \textrm{rk}\,E\star r \right\} . \end{aligned}$$

To talk about generic observability for port-Hamiltonian system, we define

$$\begin{aligned} S_{{\textit{observable}}}^{H}&:= \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^H\,\big \vert \, (E,JQ,B)\in S_{{\textit{observable}}} \right\} \quad \text {and}\\ S_{{\textit{observable}}}^{sdH}&:= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH}\,\big \vert \, (E,(J-R),B)\in S_{{\textit{observable}}} \right\} , \end{aligned}$$

where observable stands for any of the observability notions of Definition 2.1. Now, we may formulate our second main result that is criteria for relative genericity of $S_{{\textit{observable}}}^{H}$ in $\Sigma _{\ell ,n,m}^{H,\star r}$ and $S_{{\textit{observable}}}^{sdH}$ in $\Sigma _{\ell ,n,m}^{sdH,\star r}$.

Theorem 2.4

Let $\ell ,n,m\in \mathbb {N}$, $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$, $\star \in \left\{ \le ,= \right\}$ and $\mathcal {H}\in \left\{ H,sdH \right\}$. We have the following equivalences

$$\begin{aligned} \begin{array}{clccl} \text {(i)} &{}~ S_{\text {obs. at}~\infty }^{\mathcal {H}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m}^{\mathcal {H}, \star r} &{}~ \iff &{}~ n\le \min \left\{ \ell ,r+m \right\} ,\\[2ex] \text {(ii)} &{}~ S_{\text {impulse obs.}}^{\mathcal {H}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m}^{\mathcal {H}, \star r} &{}~ \iff &{}~ n\le \ell ,\\[2ex] \text {(iii)} &{}~ S_{\text {behavioural obs.}}^{\mathcal {H}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m}^{\mathcal {H}, \star r} &{}~ \iff &{}~ n\le \ell ,\\[2ex] \text {(iv)} &{}~ S_{\text {completely obs.}}^{\mathcal {H}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m}^{\mathcal {H}, \star r} &{}~ \iff &{}~ n\le \min \left\{ \ell ,r+m \right\} ,\\[2ex] \text {(v)} &{}~ S_{\text {strongly obs.}}^{\mathcal {H}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m}^{\mathcal {H}, \star r} &{}~ \iff &{}~ n\le \ell ,\\[2ex] \text {(vi)} &{}~ S_{\text {RS obs. at}~\infty }^{\mathcal {H}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m}^{\mathcal {H}, \star r} &{}~ \iff &{}~ r\ge \min \left\{ \ell ,n \right\} -m,\\[2ex] \text {(vii)} &{}~ S_{\text {RS completely observable}}^{\mathcal {H}} &{} is\, rel.\, gen.\, in~ \Sigma _{\ell ,n,m}^{\mathcal {H}, \star r} &{}~ \iff &{}~ r\ge \min \left\{ \ell ,n \right\} -m. \end{array} \end{aligned}$$

If the condition on the right-hand side of an equivalence is violated, then the complement of the corresponding set is relative generic in $\Sigma _{\ell ,n,m}^{\mathcal {H}, \star r}$. Furthermore, the following sets are relative generic in $\Sigma _{\ell ,n,m}^{\mathcal {H},\star r}$:

$$\begin{aligned} \begin{array}{ll} {\text {(viii)} } &{} S_{\text {RS impulse observable}}^{\mathcal {H}},\\ \\ {\text {(ix)} } &{} S_{\text {RS behavioural observable}}^{\mathcal {H}},\\ \\ {\text {(x)} } &{} S_{\text {RS strongly observable}}^{\mathcal {H}}.\\ \end{array} \end{aligned}$$

$\diamond$

The proof of Theorem 2.4 requires some auxiliary results and is postponed to Sect. 4.2. It is, however, noteworthy that the relative genericity of RS observability depends only on the maximal rank r of E and not on $\ell$, n and m, which separates port-Hamiltonian systems from unstructured systems. In particular, if we waive rank constraints, i.e. consider the case $r = \min \left\{ \ell ,n \right\}$, then port-Hamiltonian systems are generically RS observable for all choices of parameters $\ell ,n,m$.

3 Generic observability for unstructured DAEs

In this section, we use [7, Proposition 2.6] to derive the generic rank of the block matrices appearing in Definition-Proposition 2.1. This allows to prove Theorem 2.3 similarly to [7, Theorem 3.2] using the properties of relative generic sets which are discussed in detail in [7, Proposition 2.3].

Proposition 3.1

Let $\ell ,n,m,p\in \mathbb {N}$, $\star \in \left\{ \le ,= \right\}$ and $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$. Then, the sets

• $\widetilde{S}_{(i)}^r:= \left\{ (E,A,B,C)\in \Sigma _{\ell ,n,m,p}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,A^\top ,B^\top ] = \min \left\{ n,r+\ell +p \right\} \right\}$

• $\widetilde{S}_{(ii)}^r:= \left\{ (E,A,B,C)\in \Sigma _{\ell ,n,m,p}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,C^\top ] = \min \left\{ n,r+p \right\} \right\}$

• $\widetilde{S}_{(iii)}:= \left\{ (E,A,B,C)\in \Sigma _{\ell ,n,m,p}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}(x)}[xE^\top -A^\top ,C^\top ] = \min \left\{ n,\ell +p \right\} \right\}$

• $\widetilde{S}_{(iv)}:= \left\{ (E,A,B,C)\in \Sigma _{\ell ,n,m,p}\,\left| \,\begin{array}{l} \forall \, Z\in \mathbb {R}^{n\times (n-\textrm{rk}\,_{\mathbb {R}} E)}~\text {with}~\textrm{im}\,Z = \ker E^\top \\ : \textrm{rk}\,_{\mathbb {R}}[E^\top ,A^\top Z,B^\top ] = \min \left\{ n,\ell +p \right\} \end{array}\right. \right\}$

are relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$.

• $\widetilde{S}_{(v)}:= \left\{ (E,A,B,C)\in \Sigma _{\ell ,n,m,p}\,\big \vert \,\forall \,\lambda \in \mathbb {C}: \textrm{rk}\,_{\mathbb {C}}[\lambda E^\top -A^\top ,B^\top ] \right\}$ ${= \min \left\{ n,\ell +p \right\} }$

is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, $\big ( n\ne \ell +p \ \vee \ r = 0\big )$, otherwise $S_{(v)}^c$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$.

Proof

Recall that $\Sigma _{n,\ell ,p}:= \mathbb {R}^{n\times \ell }\times \mathbb {R}^{n\times \ell }\times \mathbb {R}^{n\times p}$. The function

$$\begin{aligned} \varphi : \Sigma _{n,\ell ,p}\times \mathbb {R}^{\ell \times m}\rightarrow \Sigma _{\ell ,n,m,p},\qquad ((E,A,C),B)\mapsto (E^\top ,A^\top ,B,C^\top ) \end{aligned}$$

is a homeomorphism with respect to the Zariski and Euclidean topology. Recall that [7, Proposition 2.6] studies relative genericity in the reference sets

$$\begin{aligned} \Sigma _{n,\ell ,p}^{\star r} := \left\{ (E,A,C)\in \Sigma _{\ell ,n,m}\,\big \vert \,\textrm{rk}\,E\le r \right\} ,\quad \star \in \left\{ \le , = \right\} , r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\} . \end{aligned}$$

We see that $\Sigma _{\ell ,n,m,p}^{\star r} = \varphi (\Sigma _{n,\ell ,p}^{\star r}\times \mathbb {R}^{\ell \times m})$. Furthermore, we have (using the notation of [7, Proposition 2.6]) the identities $\widetilde{S}_{(i)}^r = \varphi (S_{(i)}^r\times \mathbb {R}^{\ell \times m})$, $\widetilde{S}_{(ii)}^r = \varphi (S_{(ii)}^r\times \mathbb {R}^{\ell \times m})$, $\widetilde{S}_{(iii)} = \varphi (S_{(iii)}\times \mathbb {R}^{\ell \times m})$, $\widetilde{S}_{(iv)} = \varphi (S_{(iv)}\times \mathbb {R}^{\ell \times m})$ and $\widetilde{S}_{(v)} = \varphi (S_{(v)}\times \mathbb {R}^{\ell \times m})$. Since relative genericity is preserved under homeomorphisms with respect to the Zariski and Euclidean topology, the claim is a direct consequence of [7, Proposition 2.6]. $\square$

Using the notations and results of Proposition 3.1, we can straightforwardly prove Theorem 2.3, exploiting the known properties of relative generic sets with respect to a fixed reference set.

Proof of Theorem 2.3

The statements (i)–(iii) are direct consequences of Definition-Proposition 2.1 and Proposition 3.1.

(iv)   In view of Corollary 2.2 (i) and [7, Proposition 2.3 (c) and (d)], $S_{\text {completely obs.}}$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, $S_{\text {behavioural obs.}}$ and $S_{\text {obs. at}~\infty }$ are both relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. We have seen that the latter is the case if, and only if,

$$\begin{aligned} n\le r+p\quad \wedge \quad n\le \ell +p-\min \left\{ r,1 \right\} . \end{aligned}$$

The latter can be easily verified to be equivalent to the proposed condition $n\le r+p~\wedge ~r\ne \ell$.

(v)   From Corollary 2.2 (ii) and [7, Proposition 2.3 (c) and (d)], we conclude that $S_{\text {strongly obs.}}$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, $S_{\text {impulse obs.}}$ as well as $S_{\text {behavioural obs.}}$ are relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. In (ii) and (iii) of the present theorem, we have seen that the latter is the case if, and only if,

$$\begin{aligned} n\le \ell +p\quad \wedge \quad n\le \ell +p-\min \left\{ r,1 \right\} , \end{aligned}$$

which holds if, and only if, $n\le \ell +p-\min \left\{ r,1 \right\}$.

(vi)   By Proposition 3.1, $\widetilde{S}_{(i)}^r$ and $\widetilde{S}_{(ii)}^r$ are relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. By [7, Proposition 2.3 (d)], their intersection $\widetilde{S}_{(i)}^r\cap \widetilde{S}_{(ii)}^r$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ and fulfils, in view of Definition-Proposition 2.1, the inclusions

$$\begin{aligned} \widetilde{S}_{(i)}^r\cap \widetilde{S}_{(ii)}^r\subseteq {\left\{ \begin{array}{ll} S_{\text {RS obs. at}~\infty }, &{} n\le r+p,\\ \big (S_{\text {RS obs. at}~\infty }\big )^c, &{} n> r+p. \end{array}\right. } \end{aligned}$$

By [7, Proposition 2.3 (c)], the relative generic sets in a fixed reference set are closed under taking supersets. Thus, we conclude that $S_{\text {RS obs. at}~\infty }$ is relative generic in $\Sigma _{\ell ,n,m}^{\star r}$ if, and only if, $n\le r+p$; otherwise, its complement $\big (S_{\text {RS obs. at}~\infty }\big )^c$ is relative generic in $\Sigma _{\ell ,n,m}^{\star r}$.

(vii)   With Proposition 3.1, we have that $\widetilde{S}_{(i)}^r$ and $\widetilde{S}_{(iv)}$ are relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. Hence, [7, Proposition 2.3 (d)] yields that their intersection is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. By Definition-Proposition 2.1, we have the inclusions

$$\begin{aligned} \widetilde{S}_{(i)}^r\cap \widetilde{S}_{(iv)}^r \subseteq {\left\{ \begin{array}{ll} S_{\text {RS impulse obs}}, &{} n\le \ell +p,\\ \big (S_{\text {RS impulse obs}}\big )^c, &{} n>\ell +p. \end{array}\right. } \end{aligned}$$

Therefore, [7, Proposition 2.3 c] implies that $S_{\text {RS impulse obs}}$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, $n\le \ell +p$; and $\big (S_{\text {RS impulse obs}}\big )^c$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, $n>\ell +p$.

(viii)   We show that $S_{\text {RS beh. obs.}}$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, $n\ne \ell + p$ or $r = 0$.

Let $n\ne \ell +p$ or $r = 0$. In this case, Proposition 3.1 yields that $\widetilde{S}_{(iii)}^r$ and $\widetilde{S}_{(v)}$ are relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. Thus, since the inclusion

$$\begin{aligned} \widetilde{S}_{(iii)}^r\cap \widetilde{S}_{(v)}\subseteq S_{\text {RS beh. stab.}} \end{aligned}$$

holds true for $n\le \ell +p$, we conclude from [7, Proposition 2.3 (c) and (d)] that $S_{\text {RS beh. stab.}}$ is indeed relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$.

Let $\ell = n+p$ and $r\ne 0$. Then, Proposition 3.1 yields that $\widetilde{S}_{(v)}^c$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. Since $\widetilde{S}_{(iii)}$ is, by Proposition 3.1, relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$, and since the inclusion

$$\begin{aligned} \widetilde{S}_{(iii)}\cap \widetilde{S}_{(v)}^c\subseteq \big (S_{\text {RS beh. obs.}}\big )^c \end{aligned}$$

holds true, we conclude with [7, Proposition 2.3 (c) and (d)] that $\big (S_{\text {RS beh. obs.}}\big )^c$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$; in particular $S_{\text {RS beh. obs.}}$ is not relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$.

(ix)   In view of Corollary 2.2 (iii) and [7, Proposition 2.3 (c) and (d)], $S_{\text {RS completely obs.}}$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, both $S_{\text {RS obs. at}~\infty }$ and $S_{\text {RS beh. obs.}}$ are relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. By (vi) and (viii) of the present theorem, this is the case if, and only if,

$$\begin{aligned} n\le r+p\quad \wedge \quad n\le \ell +p-\min \left\{ r,1 \right\} . \end{aligned}$$

It is easy to verify that this is the case if, and only if, $n\le r + p$ and $r\ne \ell$.

(x)   Corollary 2.2 yields together with [7, Proposition 2.3 (c) and (d)] that $S_{\text {RS strongly obs.}}$ is relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$ if, and only if, $S_{\text {RS beh. obs.}}$ and $S_{\text {RS impulse obs.}}$ are both relative generic in $\Sigma _{\ell ,n,m,p}^{\star r}$. In view of (vii) and (viii) of the present theorem, the latter is equivalent to

$$\begin{aligned} n\le \ell +p\quad \wedge \quad n<\ell +p-\min \left\{ r,1 \right\} . \end{aligned}$$

The latter is equivalent to $n<\ell +p-\min \left\{ r,1 \right\}$. This completes the proof of the theorem. $\square$

4 Generic observability for port-Hamiltonian DAEs

The aim of this section is the proof of Theorem 2.4. The homeomorphism $(E,A,B,C)\mapsto (E^\top ,A^\top ,C^\top ,B)$, which maps, roughly speaking, controllable onto observable systems, does not preserve the port-Hamiltonian structure. This hinders us to directly apply the results of [8] in the spirit of Sect. 3. Therefore, we first prove some auxiliary results. Fix, throughout this section, $\ell ,n,m\in \mathbb {N}$ and $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$.

4.1 Auxiliary results: relative generic sets

The following lemma is a slight improvement of [8, Lemma 3.6] and characterizes the matrices E and Q that may appear in a port-Hamiltonian system (1.5).

Lemma 4.1

Let $E,Q\in \mathbb {R}^{\ell \times n}$ with $\textrm{rk}\,Q = r$. E and Q fulfil $E^\top Q = Q^\top E$ if, and only if, there are an orthogonal matrix $O\in \textbf{Gl}(\mathbb {R}^\ell )$, an invertible matrix $T\in \textbf{Gl}(\mathbb {R}^n)$, a symmetric matrix $\Sigma \in \mathbb {R}^{r\times r}$ and matrices $R_1\in \mathbb {R}^{(\ell -r)\times r}$ and $R_2\in \mathbb {R}^{(\ell -r)\times (n-r)}$ so that

$$\begin{aligned} OQT = \begin{bmatrix} I_r &{} 0_{r\times (n-r)}\\ 0_{(\ell -r)\times r} &{} 0_{(\ell -r)\times (n-r)} \end{bmatrix}\quad \text {and}\quad OET = \begin{bmatrix} \Sigma &{} 0_{r\times (n-r)}\\ R_1 &{} R_2 \end{bmatrix}; \end{aligned}$$

$E^\top Q$ is positive (negative) semidefinite if, and only if, $\Sigma$ is positive (negative) semidefinite.

Proof

“$\implies$” If $r = 0$, then nothing has to be proven; we consider the case $r\ge 1$. By the singular value theorem [4, Theorem 2.6.3], there are orthogonal matrices $O\in \textbf{Gl}(\mathbb {R}^\ell )$ and $T'\in \textbf{Gl}(\mathbb {R}^n)$ and $s_1,\ldots ,s_r\in \mathbb {R}{\setminus }\left\{ 0 \right\}$ so that

$$\begin{aligned} OQT' = \left[ \begin{array}{cc} \begin{array}{ccc} s_1\\ &{} \ddots \\ &{} &{} s_r \end{array} &{}\quad 0_{r\times (n-r)}\\ 0_{(\ell -r)\times r} &{}\quad 0_{(\ell -r)\times (n-r)} \end{array}\right] . \end{aligned}$$

Defining

$$\begin{aligned} T := T' \left[ \begin{array}{cc} \begin{array}{ccc} \frac{1}{s_1}\\ &{} \ddots \\ &{} &{} \frac{1}{s_r} \end{array}\\ &{} I_{n-r} \end{array}\right] , \end{aligned}$$

we have

$$\begin{aligned} OQT = \begin{bmatrix} I_r &{} 0_{r\times (n-r)}\\ 0_{(\ell -r)\times r} &{} 0_{(\ell -r)\times (n-r)} \end{bmatrix}. \end{aligned}$$

Since $E^\top Q = Q^\top E$ and since O is orthogonal, i.e. $O^\top = O^{-1}$, we have

$$\begin{aligned} (OQT)^\top (OET) = T^\top Q^\top ET = T^\top E^\top QT = (OET)^\top (OQT). \end{aligned}$$

(4.1)

Let $\Sigma \in \mathbb {R}^{r\times r}$, $P\in \mathbb {R}^{r\times (n-r)}$, $R_0\in \mathbb {R}^{(\ell -r)\times r}$ and $R_1\in \mathbb {R}^{(\ell -r)\times (n-r)}$ so that

$$\begin{aligned} OET = \begin{bmatrix} \Sigma &{} P\\ R_0 &{} R_1 \end{bmatrix}. \end{aligned}$$

Then, we have

$$\begin{aligned} (OQT)^\top (OET) = \begin{bmatrix} \Sigma &{} P\\ 0_{(\ell -r)\times r} &{} 0_{(\ell -r)\times (n-r)} \end{bmatrix} \end{aligned}$$

(4.2)

and hence (4.1) yields that $P = 0$ and $\Sigma$ is symmetric. Lastly, $E^\top Q$ is positive (negative) semidefinite if, and only if, $T^\top E^\top QT = (OET)^\top (OQT)$ is positive (negative) semidefinite, which is by (4.2) the case if, and only if, $\Sigma$ is semidefinite.

“” Let S, T, $\Sigma$, $R_1$ and $R_2$ as required. Since O is orthogonal, we have

$$\begin{aligned} (OET)^\top (OQT) = T^\top E^\top QT, \end{aligned}$$

and

$$\begin{aligned} T^\top Q^\top ET&= (OQT)^\top (OET)\\&= \begin{bmatrix} I_r &{} 0_{r\times (n-r)}\\ 0_{(\ell -r)\times r} &{} 0_{(\ell -r)\times (n-r)} \end{bmatrix}\begin{bmatrix} \Sigma &{} 0_{r\times (n-r)}\\ R_1 &{} R_2 \end{bmatrix} = \begin{bmatrix} \Sigma &{} 0_{r\times (n-r)}\\ 0_{(\ell -r)\times r} &{} 0_{(\ell -r)\times (n-r)} \end{bmatrix} \end{aligned}$$

is indeed symmetric and shares the definiteness properties of $\Sigma$. This completes the proof of the lemma. $\square$

Recall an elementary result from linear algebra.

Remark 4.2

Let $A\in \textbf{Gl}(\mathbb {R}^{n})$, $B\in \mathbb {R}^{\ell \times n}$ and $C\in \mathbb {R}^{\ell \times m}$. Then,

$$\begin{aligned} \textrm{rk}\,\begin{bmatrix} A &{} 0_{n\times m}\\ B &{} C \end{bmatrix} = n + \textrm{rk}\,C. \end{aligned}$$

Before we proceed, we shall briefly discuss the norm which we use in our following considerations.

Remark 4.3

The vector spaces $\mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ and $\mathbb {R}^{\ell \times n}\times \left( \mathbb {R}^{\ell \times \ell }\right) ^2\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ whose Euclidean topologies determine the relative Euclidean topologies on our reference sets $\Sigma _{\ell ,n,m}^{H,\star r}$ and $\Sigma _{\ell ,n,m}^{sdH,\star r}$, respectively, are finite-dimensional. Hence, all norms on them are equivalent and the Euclidean topology is induced by any norm. For convenience, we equip every matrix space $\mathbb {R}^{a\times b}$, $a,b\in \mathbb {N}$ with the supremum norm

$$\begin{aligned} \left\| \cdot \right\| : \mathbb {R}^{a\times b}\rightarrow \mathbb {R}_{\ge 0},\quad M\mapsto \max \left\{ \left| M_{i,j} \right| \,\big \vert \,i\in \underline{a},j\in \underline{b} \right\} . \end{aligned}$$

Then, we have, for all $a,b,c\in \mathbb {N}$, the inequality

$$\begin{aligned} \forall M_1\in \mathbb {R}^{a\times b}~\forall M_2\in \mathbb {R}^{b\times c}: \left\| M_1M_2 \right\| \le b\left\| M_1 \right\| \left\| M_2 \right\| . \end{aligned}$$

(4.3)

The spaces $\mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ and $\mathbb {R}^{\ell \times n}\times \left( \mathbb {R}^{\ell \times \ell }\right) ^2\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ will be equipped with the product norm, i.e. the maximum of the norms of their components.

Remark 4.4

Instead of viewing $\Sigma _{\ell ,n,m}^{H,\star r}$ and $\Sigma _{\ell ,n,m}^{sdH,\star r}$ as subsets of the vector spaces $\mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ and $\mathbb {R}^{\ell \times n}\times \left( \mathbb {R}^{\ell \times \ell }\right) ^2\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$, resp., we could equivalently (in terms of relative genericity, see [9, Lemma III.2]) view them as subsets of the vector spaces

$$\begin{aligned} \left\{ (E,J,Q,B)\in \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\,\Big \vert \, J = -J^\top \right\} \end{aligned}$$

and

$$\begin{aligned} \left\{ (E,J,R,Q,B)\in \mathbb {R}^{\ell \times n}\times \left( \mathbb {R}^{\ell \times \ell }\right) ^2\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}\,\Big \vert \, J = -J^\top , R = R^\top \right\} , \end{aligned}$$

respectively. The latter has been exploited, for example, in [8], where the authors aimed to find a “convenience reference set” (which is in view of relative genericity equivalent to $\Sigma _{\ell ,n,m}^{sdH}$) that allows to ignore some of the technical subtleties that are involved in the calculations. In the present note, however, we will not make use of such a construction and do not discuss the question of a “convenient” encompassing vector space further.

The following calculations rely heavily on the characterization of the rank by minors. For convenience, we recall this elementary result.

Lemma 4.5

(see [2, Section 3.3.6]) Let $R = \mathbb {R}$ or $R = \mathbb {R}[x]$. A minor of order $d\in \left\{ 1,\ldots ,\min \left\{ \ell ,n \right\} \right\}$ with respect to $R^{\ell \times n}$ is a function

$$\begin{aligned} M_{\sigma ,\pi }:R^{\ell \times n}\rightarrow \mathbb {R},\qquad A\mapsto \det [A_{\sigma (i),\pi (j)}]_{i,j\in \underline{d}}, \end{aligned}$$

where $\sigma :\underline{d}\rightarrow \underline{\ell }$ and $\pi :\underline{d}\rightarrow \underline{n}$ are strictly increasing functions. A minor of order zero is the constant function $A\mapsto 1$.

A matrix $A\in R^{\ell \times n}$ has $\textrm{rk}\,_R A\ge d$ if, and only if, there is a minor M of order d w.r.t. $R^{\ell \times n}$ which fulfils $M(A)\ne 0$.$\diamond$

The following lemma refines [8, Lemma 3.7].

Lemma 4.6

The following sets are relative generic in $\Sigma _{\ell ,n,m}^{H,\le r}$:

• (i) $\left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, \le r}\,\big \vert \,\textrm{rk}\,Q = \min \left\{ \ell ,n \right\} \right\}$,

• (ii) $\Sigma _{\ell ,n,m}^{H,= r} = \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, \le r}\,\big \vert \,\textrm{rk}\,E = r \right\}$.

Similarly, the following sets are relative generic in $\Sigma _{\ell ,n,m}^{sdH,\le r}$:

• (iii) $\left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, \le r}\,\big \vert \,\textrm{rk}\,Q = \min \left\{ \ell ,n \right\} \right\}$,

• (iv) $\Sigma _{\ell ,n,m}^{sdH,= r} = \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, \le r}\,\big \vert \,\textrm{rk}\,E = r \right\}$.

Proof

(i)   Put

$$\begin{aligned} S' := \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, \le r}\,\big \vert \,\textrm{rk}\,Q = \min \left\{ \ell ,n \right\} \right\} . \end{aligned}$$

Since the rank of elements of $\mathbb {R}^{\ell \times n}$ is bounded from above by $\min \left\{ \ell ,n \right\}$, we have

$$\begin{aligned} S' = \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, \le r}\,\big \vert \,\textrm{rk}\,Q \ge \min \left\{ \ell ,n \right\} \right\} . \end{aligned}$$

Since all minors are polynomials, Lemma 4.5 yields that $S'$ is a relative Zariski-open subset of $\Sigma _{\ell ,n,m}^{H, \le r}$. It remains to show that $S'$ is relative Euclidean dense in $\Sigma _{\ell ,n,m}^{H, \le r}$. Let $(E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, \le r}$ with $\textrm{rk}\,E = \rho \le r$ and $\varepsilon >0$. Applying Lemma 4.1 to the pair (E, Q), we find an orthogonal matrix $S\in \textbf{Gl}(\mathbb {R}^\ell )$, an invertible matrix $T\in \textbf{Gl}(\mathbb {R}^n)$, a symmetric and positive semidefinite matrix $\Sigma \in \mathbb {R}^{\rho \times \rho }$, and some matrices $R_1\in \mathbb {R}^{(\ell -\rho )\times \rho }$ and $R_2\in \mathbb {R}^{(\ell -\rho )\times (n-\rho )}$ so that

$$\begin{aligned} SET = \begin{bmatrix} I_\rho &{}\quad 0\\ 0 &{}\quad 0 \end{bmatrix}\quad \text {and}\quad SQT = \begin{bmatrix} \Sigma &{}\quad 0\\ R_1 &{}\quad R_2 \end{bmatrix}. \end{aligned}$$

Recall that the set of positive definite symmetric matrices is relative generic in the set of positive semidefinite symmetric matrices (see [8, Lemma 3.5] and that the set of matrices with full rank is due to Lemma 4.5 and Definition 1.1 generic. Hence, there are a positive definite symmetric $\Sigma '\in \mathbb {R}^{\rho \times \rho }$ and a matrix $R_2'\in \mathbb {R}^{(\ell - \rho )\times (n-\rho )}$ with $\textrm{rk}\,R_2' = \min \left\{ \ell -\rho ,n-\rho \right\}$ so that

$$\begin{aligned} \left\| \Sigma -\Sigma ' \right\|<\frac{\varepsilon }{\ell n\left\| S^{-1} \right\| \left\| T^{-1} \right\| }\quad \text {and}\quad \left\| R_2-R_2' \right\| <\frac{\varepsilon }{\ell n\left\| S^{-1} \right\| \left\| T^{-1} \right\| }. \end{aligned}$$

By Remark 4.2, we have

$$\begin{aligned} \textrm{rk}\,\underbrace{S^{-1}\begin{bmatrix} \Sigma ' &{} 0\\ R_1 &{} R_2' \end{bmatrix}T^{-1}}_{=:Q'} = \textrm{rk}\,\begin{bmatrix} \Sigma ' &{} 0\\ R_1 &{} R_2' \end{bmatrix} = \rho + \textrm{rk}\,R_2' = \min \left\{ \ell ,n \right\} \end{aligned}$$

and hence $(E,J,Q',B)\in S'$. Further, by (4.3), we have

$$\begin{aligned} \left\| Q'-Q \right\| \le \ell n\left\| S^{-1} \right\| \left\| T^{-1} \right\| \max \left\{ \left\| \Sigma '-\Sigma \right\| ,\left\| R_2'-R_2 \right\| \right\} <\varepsilon . \end{aligned}$$

This shows that $S'$ is indeed Euclidean dense and hence relative generic in $\Sigma _{\ell ,n,m}^{H, \le r}$.

(ii)   By definition of $\Sigma _{\ell ,n,m}^{H, \le r}$, we have

$$\begin{aligned} \Sigma _{\ell ,n,m}^{H, = r} = \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, \le r}\,\big \vert \,\textrm{rk}\,E \ge r \right\} \end{aligned}$$

and hence Lemma 4.5 yields that $\Sigma _{\ell ,n,m}^{H, = r}$ is a relative Zariski-open subset of $\Sigma _{\ell ,n,m}^{H, \le r}$. Thus, it remains to verify that $\Sigma _{\ell ,n,m}^{H, = r}$ is a relative Euclidean dense subset of $\Sigma _{\ell ,n,m}^{H, \le r}$. Let $(E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, \le r}$ and $\varepsilon >0$. By (i) of the present lemma, there is some $(\widehat{E},\widehat{J},\widehat{Q},\widehat{B})\in \Sigma _{\ell ,n,m}^{H, \le r}$ with $\textrm{rk}\,\widehat{Q} = \min \left\{ \ell ,n \right\}$ and

$$\begin{aligned} \left\| (E,J,Q,B)-(\widehat{E},\widehat{J},\widehat{Q},\widehat{B}) \right\| <\frac{\varepsilon }{2}. \end{aligned}$$

We distinguish the cases $\ell \le n$ and $n<\ell$.

Let $\ell \le n$. By Lemma 4.1, there is an orthogonal matrix $S\in \mathbb {R}^{\ell \times \ell }$, an invertible matrix $T\in \mathbb {R}^{n\times n}$ and a symmetric matrix $\Sigma \in \mathbb {R}^{\ell \times \ell }$ so that

$$\begin{aligned} S\widehat{Q}T = [I_\ell ,0_{\ell \times (n-\ell )}]\quad \text {and}\quad S\widehat{E}T = [\Sigma ,0_{\ell \times (n-\ell )}]. \end{aligned}$$

Due to Sylvester’s law of inertia [4, Theorem 4.5.8], there is some $S_\star \in \textbf{Gl}(\mathbb {R}^\ell )$ so that

$$\begin{aligned} S_\star \Sigma S_\star ^\top = \begin{bmatrix} I_\rho &{} 0{\rho \times (\ell -\rho )}\\ 0_{(\ell -\rho )\times \rho } &{} 0_{(\ell -\rho )\times (\ell -\rho )} \end{bmatrix} \end{aligned}$$

with $\rho = \textrm{rk}\,\widehat{E}$. Define

$$\begin{aligned} \delta := \frac{\varepsilon }{2\ell ^2 n^2\left\| S^{-1} \right\| \left\| S_\star ^{-1} \right\| ^2\left\| T^{-1} \right\| } \end{aligned}$$

and put

$$\begin{aligned} \widetilde{E} := \widehat{E} +\delta S^{-1}S_\star ^{-1}\begin{bmatrix} 0_{\rho \times \rho } &{} 0_{\rho \times (r-\rho )} &{} 0_{\rho \times (n-r)}\\ 0_{(r-\rho )\times \rho } &{} I_{r-\rho } &{} 0_{(r-\rho )\times (n-r)}\\ 0_{(\ell -r)\times \rho } &{} 0_{(\ell -r)\times (r-\rho )} &{} 0_{(\ell -r)\times (n-r)} \end{bmatrix}\begin{bmatrix}S_\star ^{-\top } &{} 0_{\ell \times (n-\ell }\\ 0_{(n-\ell )\times \ell } &{} I_{n-\ell } \end{bmatrix}T^{-1}. \end{aligned}$$

Then, we have $\left\| \widetilde{E}-E \right\| <\varepsilon$ and

$$\begin{aligned} S_\star S\widetilde{E}\begin{bmatrix}S_\star ^\top &{} 0_{\ell \times (n-\ell }\\ 0_{(n-\ell )\times \ell } &{} I_{n-\ell } \end{bmatrix}T = \begin{bmatrix} I_\rho &{} 0_{\rho \times (r-\rho )} &{} 0_{\rho \times (n-r)}\\ 0_{(r-\rho )\times \rho } &{} \frac{\varepsilon }{2\ell ^2 n^2\left\| S^{-1} \right\| \left\| S_\star ^{-1} \right\| ^2\left\| T^{-1} \right\| }I_{r-\rho } &{} 0_{(r-\rho )\times (n-r)}\\ 0_{(\ell -r)\times \rho } &{} 0_{(\ell -r)\times (r-\rho )} &{} 0_{(\ell -r)\times (n-r)} \end{bmatrix} \end{aligned}$$

yields $\textrm{rk}\,\widetilde{E} = r$. Further, Lemma 4.1 implies that $\widetilde{E}^\top \widehat{Q} = \widehat{Q}^\top \widetilde{E}>0$. Therefore,

$$\begin{aligned} (\widetilde{E},\widehat{J},\widehat{Q},\widehat{B})\in \Sigma _{\ell ,n,m}^{H,= r}\cap \mathbb {B}((E,J,Q,B),\varepsilon ). \end{aligned}$$

and hence $\Sigma _{\ell ,n,m}^{H, = r}$ is indeed Euclidean dense in $\Sigma _{\ell ,n,m}^{H, \le r}$.

It remains to consider the case $n<\ell$. In this case, Lemma 4.1 yields the existence of an orthogonal matrix $S\in \mathbb {R}^{\ell \times \ell }$, an invertible matrix $T\in \textbf{Gl}(\mathbb {R}^{n})$, a symmetric matrix $\Sigma \in \mathbb {R}^{n\times n}$ and some matrix $P\in \mathbb {R}^{(\ell -n)\times n}$ so that

$$\begin{aligned} S\widehat{Q}T = \begin{bmatrix} I_n\\ 0_{(\ell -n)\times n} \end{bmatrix} \quad \text {and}\quad S\widehat{E}T = \begin{bmatrix} \Sigma \\ P \end{bmatrix}. \end{aligned}$$

Applying Sylvester’s law of inertia, we find $\rho \in \mathbb {N}$ and a transformation $S_\star '\in \textbf{Gl}(\mathbb {R}^n)$ so that

$$\begin{aligned} \begin{bmatrix} S_\star ' &{} 0_{n\times (\ell -n)}\\ 0_{(\ell -n)\times n} &{} I_{\ell -n} \end{bmatrix}\begin{bmatrix} \Sigma \\ P \end{bmatrix}(S_\star ')^\top = \begin{bmatrix} I_\rho &{} 0_{\rho \times (n-\rho )}\\ 0_{(n-\rho )\times \rho } &{} 0_{(n-\rho )\times (n-\rho )}\\ R_1 &{} R_2 \end{bmatrix} \end{aligned}$$

for some $P_1\in \mathbb {R}^{(\ell -n)\times \rho }$ and $P_2\in \mathbb {R}^{(\ell -n)\times (n-\rho )}$. In this form, the rank of $\widehat{E}$ presents itself as $\rho +\textrm{rk}\,R_2$; in particular we have $\textrm{rk}\,R_2\le r-\rho$. Analogously to [7, Lemma 2.4] it can be shown that the set

$$\begin{aligned} \left\{ P\in \mathbb {R}^{(\ell -n)\times (n-\rho )}\,\big \vert \,\textrm{rk}\,P = r-\rho \right\} \end{aligned}$$

is relative generic in the set

$$\begin{aligned} \left\{ P\in \mathbb {R}^{(\ell -n)\times (n-\rho )}\,\big \vert \,\textrm{rk}\,P \le r-\rho \right\} . \end{aligned}$$

Therefore, we find some $\widetilde{R}_2\in \mathbb {R}^{(\ell -n)\times (n-\rho )}$ so that $\textrm{rk}\,\widetilde{R}_2 = r-\rho$ and

$$\begin{aligned} \left\| R_2-\widetilde{R}_2 \right\| <\frac{\varepsilon }{\ell ^2n^2\left\| S^{-1} \right\| \left\| (S_\star ')^{-1} \right\| ^2\left\| T^{-1} \right\| }. \end{aligned}$$

Put

$$\begin{aligned} \widetilde{E} := S^{-1}(S_\star ')^{-1}\begin{bmatrix} I_\rho &{} 0_{\rho \times (n-\rho )}\\ 0_{(n-\rho )\times \rho } &{} 0_{(n-\rho )\times (n-\rho )}\\ R_1 &{} \widetilde{R}_2 \end{bmatrix}(S_\star ')^{-\top }T^{-1}. \end{aligned}$$

Then, $\textrm{rk}\,\widetilde{E} = r$ and, by Lemma 4.1, $\widetilde{E}^\top \widehat{Q} = \widehat{Q}^\top \widetilde{E}\ge 0$. Further, the submultiplicativity property (4.3) yields

$$\begin{aligned} (\widetilde{E},\widehat{J},\widehat{Q},\widehat{B})\in \Sigma _{\ell ,n,m}^{H,= r}\cap \mathbb {B}((\widehat{E},\widehat{J},\widehat{Q},\widehat{B}),\varepsilon ). \end{aligned}$$

This proves that $\Sigma _{\ell ,n,m}^{H,=r}$ is indeed Euclidean dense in $\Sigma _{\ell ,n,m}^{H, \le r}$ and completes the proof of (ii).

(iii) and (iv)   Analogously to (i) and (ii), we see that the sets

$$\begin{aligned} S'':= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH,\le r}\,\big \vert \,\textrm{rk}\,Q = \min \left\{ \ell ,n \right\} \right\} \end{aligned}$$

and $\Sigma _{\ell ,n,m}^{sdH,= r}$ are relative Zariski-open subsets of $\Sigma _{\ell ,n,m}^{sdH,\le r}$. Thus, it remains to verify that they are relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH,\le r}$. Let $(E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH,\le r}$ and $\varepsilon >0$. By (i), there exist $(E',J',Q',B')\in S'$ so that $\left\| (E,J,Q,B)-(E',J',Q',B') \right\| <\varepsilon$. Then, $(E',J',R,Q',B')\in S''$ and

$$\begin{aligned} \left\| (E,J,R,Q,B)-(E',J',R,Q',B') \right\| = \left\| (E,J,Q,B)-(E',J',Q',B') \right\| <\varepsilon . \end{aligned}$$

Similarly, (ii) implies the existence of a quadruple $(E^\star ,J^\star ,Q^\star ,B^\star )\in \Sigma _{\ell ,n,m}^{H,=r}$ which fulfils $\left\| (E,J,Q,B)-(E^\star ,J^\star ,Q^\star ,B^\star ) \right\| <\varepsilon$. Thus, we conclude $(E^\star ,J^\star ,R,Q^\star ,B^\star )\in \Sigma _{\ell ,n,m}^{sdH,=r}$ and $\left\| (E,J,R,Q,B)-(E^\star ,J^\star ,R,Q^\star ,B^\star ) \right\| <\varepsilon$. $\square$

In particular, we have shown that $\Sigma _{\ell ,n,m}^{H,=r}$ and $\Sigma _{\ell ,n,m}^{sdH,=r}$ are relative generic subsets of $\Sigma _{\ell ,n,m}^{H,\le r}$ and $\Sigma _{\ell ,n,m}^{sdH,\le r}$, respectively. Therefore, [8, Proposition 2.10 (e)] implies that $\Sigma _{\ell ,n,m}^{H,= r}$ and $\Sigma _{\ell ,n,m}^{H,\le r}$ are equivalent reference sets, and so are $\Sigma _{\ell ,n,m}^{sdH,= r}$ and $\Sigma _{\ell ,n,m}^{sdH,\le r}$.

In the following lemma, we explore the maximal rank of the block matrices appearing in the algebraic characterizations of observability in Definition-Proposition 2.1 for port-Hamiltonian systems.

Lemma 4.7

For all $(E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, \le r}$, we have

($\alpha$):

$\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top B] \le \min \left\{ \ell ,n,r+m \right\}$,

($\beta$):

$\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top (J+R),Q^\top B]\le \min \left\{ \ell ,n \right\}$,

($\gamma$):

$\textrm{rk}\,_{\mathbb {R}(x)}[xE^\top +Q^\top (J+R),Q^\top B]\le \min \left\{ \ell ,n \right\}$,

($\delta$):

$\forall Z\in \mathbb {R}^{\ell \times (\ell -\textrm{rk}\,E)}: \textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top (J+R)Z,Q^\top B]\le \min \left\{ \ell ,n \right\}$.

Further, we have for all $(E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H,\le r}$

($\varepsilon$):

$\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top B] \le \min \left\{ \ell ,n,r+m \right\}$,

($\zeta$):

$\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top J,Q^\top B]\le \min \left\{ \ell ,n \right\}$,

($\eta$):

$\textrm{rk}\,_{\mathbb {R}(x)}[xE^\top +Q^\top J,Q^\top B]\le \min \left\{ \ell ,n \right\}$,

($\vartheta$):

$\forall Z\in \mathbb {R}^{\ell \times (\ell -\textrm{rk}\,E)}: \textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top JZ,Q^\top B]\le \min \left\{ \ell ,n \right\}$.

Proof

Let $(E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, \le r}$ with $q:= \textrm{rk}\,Q$. Then, $E^\top Q = Q^\top E$ and by Lemma 4.1, there exists an orthogonal matrix $S\in \textbf{Gl}(\mathbb {R}^\ell )$, $T\in \textbf{Gl}(\mathbb {R}^n)$, $\Sigma = \Sigma ^\top \in \mathbb {R}^{q\times q}$, $P_1\in \mathbb {R}^{(\ell -q)\times q}$ and $P_2\in \mathbb {R}^{(\ell -q)\times (n-q)}$ so that

$$\begin{aligned} SQT = \begin{bmatrix} I_r &{} 0_{q\times (n-q)}\\ 0_{(\ell -q)\times q} &{} 0_{(\ell -q)\times (n-q)} \end{bmatrix}\quad \text {and}\quad SET = \begin{bmatrix} \Sigma &{} 0_{q\times (n-q)}\\ P_1 &{} P_2 \end{bmatrix}. \end{aligned}$$

Consider further $R_{1,1},J_{1,1}\in \mathbb {R}^{q\times q}$, $R_{1,2},J_{1,2}\in \mathbb {R}^{q\times (\ell -q)}$, $R_{2,2},J_{2,2}\in \mathbb {R}^{(\ell -q)\times (\ell -q)}$, $B_1\in \mathbb {R}^{q\times m}$ and $B_2\in \mathbb {R}^{(\ell -q)\times m}$ so that

$$\begin{aligned} SJS^\top = \begin{bmatrix} J_{1,1} &{} J_{1,2}\\ -J_{1,2}^\top &{} J_{2,2} \end{bmatrix},\quad SRS^\top = \begin{bmatrix} R_{1,1} &{} R_{1,2}\\ R_{1,2}^\top &{} R_{2,2} \end{bmatrix}\quad \text {and}\quad SB = \begin{bmatrix} B_1\\ B_2 \end{bmatrix}. \end{aligned}$$

Using these decompositions, the following inequalities hold in view of Remark 4.2:

$$\begin{aligned} \textrm{rk}\,_{\mathbb {R}} [E^\top ,Q^\top B]&= \textrm{rk}\,_{\mathbb {R}} \begin{bmatrix} \Sigma &{} P_1^\top &{} B_1\\ 0 &{} P_2^\top &{} 0 \end{bmatrix}\\&\le \textrm{rk}\,_{\mathbb {R}} [\Sigma ,P_1^\top ,B]+\textrm{rk}\,_{\mathbb {R}} P_2^\top \\&\le q + \min \left\{ \ell -q,n-q \right\} = \min \left\{ \ell ,n \right\} ,\\ \textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top B]&\le \textrm{rk}\,_{\mathbb {R}} E + \textrm{rk}\,_{\mathbb {R}} Q^\top B\le r + m,\\ \textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top (J+R),Q^\top B]&\le \textrm{rk}\,_{\mathbb {R}}[\Sigma ,P_1^\top ,J_{1,1}+R_{1,1},J_{1,2}+R_{1,2},B_1]\\&\quad +\textrm{rk}\,P_2^\top \\&\le q + \min \left\{ \ell -q,n-q \right\} = \min \left\{ \ell ,n \right\} ,\\ \textrm{rk}\,_{\mathbb {R}(x)}[xE^\top + Q^\top (J+R),Q^\top B]&\le \textrm{rk}\,_{\mathbb {R}(x)} [x\Sigma +J_{1,1}+R_{1,1},xP_1^\top +J_{1,2}\\&\quad +R_{1,2},B_1] + \textrm{rk}\,_{\mathbb {R}(x)} x P_2^\top \\&\le q + \min \left\{ \ell -q,n-q \right\} = \min \left\{ \ell ,q \right\} . \end{aligned}$$

Since $(E,J,0_{\ell \times \ell },Q,B)\in \Sigma _{\ell ,n,m}^{sdH,\le r}$ for all $(E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H,\le r}$, we conclude that the inequalities (v)–(viii) hold true. $\square$

We shall now show that the upper bounds given in Lemma 4.7 are generically attained. Since the method of proof in the cases ($\alpha$)–($\gamma$) and ($\varepsilon$)–($\eta$) differs from the cases ($\delta$) and ($\vartheta$), we shall first study the former.

Lemma 4.8

Let $\star \in \left\{ \le ,= \right\}$. The following sets are relative generic in $\Sigma _{\ell ,n,m}^{sdH, \star r}$

• $S_\alpha := \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top B] = \min \left\{ \ell ,n,r+m \right\} \right\}$,

• $S_{\beta }:= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top (J+R),Q^\top B] = \min \left\{ \ell ,n \right\} \right\}$,

• $S_{\gamma }{:=}\left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, {=} r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}(x)}[xE^\top {+}Q^\top (J+R),Q^\top B] \right\} {{=}\min \left\{ \ell ,n \right\} }$.

Analogously, the following sets are relative generic in $\Sigma _{\ell ,n,m}^{H, \star r}$

• $S_\varepsilon := \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top B] = \min \left\{ \ell ,n,r+m \right\} \right\}$,

• $S_{\zeta }:= \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top J,Q^\top B] = \min \left\{ \ell ,n \right\} \right\}$,

• $S_{\eta }:= \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}(x)}[xE^\top +Q^\top J,Q^\top B] = \min \left\{ \ell ,n \right\} \right\}$.

Proof

By Lemma 4.6 (ii) and (iv), $\Sigma _{\ell ,n,m}^{sdH,=r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$ are relative generic subsets of $\Sigma _{\ell ,n,m}^{sdH,\le r}$ and $\Sigma _{\ell ,n,m}^{H,\le r}$, respectively. By [8, Proposition 2.10 (e)], we conclude that it suffices to show relative genericity of $S_\alpha$, $S_\gamma$ and $S_\delta$ in the reference set $\Sigma _{\ell ,n,m}^{sdH,= r}$ and $S_\varepsilon$, $S_\zeta$ and $S_\eta$ in $\Sigma _{\ell ,n,m}^{H, = r}$, respectively.

From Lemma 4.7, we conclude that

$$\begin{aligned} S_\alpha&= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top B] \ge \min \left\{ \ell ,n,r+m \right\} \right\} ,\\ S_{\beta }&= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top (J+R),Q^\top B] \ge \min \left\{ \ell ,n \right\} \right\} ,\\ S_{\gamma }&= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}(x)}[xE^\top +Q^\top (J+R),Q^\top B] \ge \min \left\{ \ell ,n \right\} \right\} ,\\ S_\varepsilon&= \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top B] \ge \min \left\{ \ell ,n,r+m \right\} \right\} ,\\ S_{\zeta }&= \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top J,Q^\top B] \ge \min \left\{ \ell ,n \right\} \right\} ,\\ S_{\eta }&= \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H, = r}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}(x)}[xE^\top +Q^\top J,Q^\top B] \ge \min \left\{ \ell ,n \right\} \right\} . \end{aligned}$$

Since all minors are polynomials, we conclude from the characterization of the rank given in Lemma 4.5 that $S_\alpha$ and $S_\beta$, and $S_\varepsilon$ and $S_\zeta$ are relative Zariski-open subsets of $\Sigma _{\ell ,n,m}^{sdH,=r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$, respectively. Furthermore, since the coefficient functions of the minors of the polynomials $[xE^\top +Q^\top (J+R),Q^\top B]$ are polynomials in the entries of E, Q, J, R and B, Lemma 4.5 yields that $S_\gamma$ is relative Zariski-open in $\Sigma _{\ell ,n,m}^{sdH,=r}$; analogously we see that $S_\eta$ is a relative Zariski-open subset of $\Sigma _{\ell ,n,m}^{H,=r}$. In view of Definition 1.2, it remains to verify that these sets are dense with respect to the relative Euclidean topology on $\Sigma _{\ell ,n,m}^{sdH,\le r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$, respectively. We distinguish between the reference set $\Sigma _{\ell ,n,m}^{sdH,=r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$.

Reference set $\Sigma _{\ell ,n,m}^{sdH,=r}$: Let $(E^\star ,J^\star ,R^\star ,Q^\star ,B^\star )\in \Sigma _{\ell ,n,m}^{sdH, = r}$ and $\varepsilon >0$. By Lemma 4.6 (i), there are $(E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}$ so that $\textrm{rk}\,Q = \min \left\{ \ell ,n \right\}$ and

$$\begin{aligned} \left\| (E,J,R,Q,B)-(E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ) \right\| <\frac{\varepsilon }{2}. \end{aligned}$$

In the following, we distinguish further the cases $\ell \le n$ and $\ell >n$.

Case $\ell \le n$: Lemma 4.1 yields the existence of an orthogonal matrix $S\in \textbf{Gl}(\mathbb {R}^\ell )$, $T\in \textbf{Gl}(\mathbb {R}^n)$ and a symmetric and positive semidefinite $\Sigma \in \mathbb {R}^{\ell \times \ell }$ so that

$$\begin{aligned} SQT = [I_\ell ,0_{\ell \times (n-\ell )}]\quad \text {and}\quad SET = [\Sigma , 0_{\ell \times (n-\ell )}]. \end{aligned}$$

Due to Sylvester’s law of inertia [4, Theorem 4.5.8] and since $r = \textrm{rk}\,E = \textrm{rk}\,SET = \textrm{rk}\,\Sigma$, there is $\widetilde{S}\in \textbf{Gl}(\mathcal {R}^\ell )$ so that

$$\begin{aligned} \widetilde{S}\Sigma \widetilde{S}^\top = \begin{bmatrix} I_r &{}\quad 0\\ 0 &{}\quad 0 \end{bmatrix}. \end{aligned}$$

($\alpha$)   From the characterization of the rank with minors, we see that

$$\begin{aligned} S_\alpha ' := \left\{ \widetilde{B}\in \mathbb {R}^{\ell \times m}\,\left| \,\textrm{rk}\,_{\mathbb {R}} \begin{bmatrix} 0_{r\times r}\\ &{} I_{\ell -r} \end{bmatrix}\widetilde{S}S \widetilde{B} = \min \left\{ \ell -r,m \right\} \right. \right\} \end{aligned}$$

is Zariski-open, and since S and $\widetilde{S}$ are invertible, $S'_\alpha \ne \emptyset$. In view of Definition 1.1, we conclude that $S'_\alpha$ is a generic subset of $\mathbb {R}^{\ell \times m}$. Therefore, there exists $\widetilde{B}\in S_\alpha '$ so that $\left\| B-\widetilde{B} \right\| <\frac{\varepsilon }{2}$. Then, we have

$$\begin{aligned} \textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top \widetilde{B}]&= \textrm{rk}\,_{\mathbb {R}} \begin{bmatrix} \widetilde{S}\\ &{} I_{n-r} \end{bmatrix} T^\top [E^\top ,Q^\top \widetilde{B}]\begin{bmatrix} S^\top \\ &{} I_m \end{bmatrix}\begin{bmatrix} \widetilde{S}^\top \\ &{} I_{\ell -r}\\ &{} &{} I_m \end{bmatrix}\\&= \textrm{rk}\,_{\mathbb {R}} \begin{bmatrix} \widetilde{S}\\ &{} I_{n-r} \end{bmatrix}\begin{bmatrix} \Sigma ^\top &{} S\widetilde{B}\\ 0 &{} 0 \end{bmatrix}\begin{bmatrix} \widetilde{S}^\top \\ &{} I_{\ell -r}\\ &{} &{} I_m \end{bmatrix}\\&= r+\textrm{rk}\,_{\mathbb {R}} \begin{bmatrix} 0_{r\times r}\\ &{} I_{\ell -r} \end{bmatrix}\widetilde{S}S \widetilde{B}\\&= r+\min \left\{ \ell -r,m \right\} = \min \left\{ \ell ,m+r \right\} = \min \left\{ \ell ,n,m+r \right\} . \end{aligned}$$

Therefore, we conclude that $(E,J,R,Q,\widetilde{B})\in S_\alpha \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$. This shows that $S_\alpha$ is relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH, = r}$.

($\beta$)    From the characterization of the rank given in Lemma 4.5, we see that

$$\begin{aligned} S_\beta ' := \left\{ (J,R,B)\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}}[S(J+R)S^\top ,SB] = \ell \right\} \end{aligned}$$

is a relative Zariski-open subset of the nonempty convex set

$$\begin{aligned} V_\beta := \left\{ (J,R,B)\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\big \vert \, J = -J^\top , R = R^\top \ge 0 \right\} ; \end{aligned}$$

and since $S\in \textbf{Gl}(\mathbb {R}^\ell )$, $S_\alpha \ne \emptyset$. [8, Lemma 2.13] implies that $S_\beta '$ is relative generic in $V_\beta$. Thus, there exists $(\widehat{J},\widehat{R},\widehat{B})\in S_\beta '$ so that

$$\begin{aligned} \max \left\{ \left\| J-\widehat{J} \right\| ,\left\| R-\widehat{R} \right\| ,\left\| B-\widehat{B} \right\| \right\} <\frac{\varepsilon }{2}. \end{aligned}$$

The matrices $(\widehat{J},\widehat{R},\widehat{B})$ fulfil

$$\begin{aligned} \ell&= \textrm{rk}\,_{\mathbb {R}}[S(\widehat{J}+\widehat{R})S^\top ,S\widehat{B}]\\&= \textrm{rk}\,_{\mathbb {R}}[T^\top Q^\top S^\top S(\widehat{J}+\widehat{R})S^\top ,T^\top Q^\top S^\top S\widehat{B}]\\&\le \textrm{rk}\,_{\mathbb {R}} [T^\top E S^\top ,T^\top Q(\widehat{J}+\widehat{R})S^\top ,T^\top Q\widehat{B}]\\&= \textrm{rk}\,_{\mathbb {R}}[E^\top ,Q^\top (\widehat{J}+\widehat{R}),Q^\top \widehat{B}]\le \min \left\{ \ell ,n \right\} = \ell , \end{aligned}$$

and hence, the inequalities are equalities. This shows that

$$\begin{aligned} (E,\widehat{J},\widehat{R},Q,\widehat{B})\in S_\beta \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon ) \end{aligned}$$

and we have verified that $S_\beta$ is indeed relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH,= r}$.

($\gamma$)   Consider the quintuple $(E,\widehat{J},\widehat{R},Q,\widehat{B})\in \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$ that we constructed in ($\beta$). Since

$$\begin{aligned} \ell&= \textrm{rk}\,_{\mathbb {R}}[S(\widehat{J}+\widehat{R})S^\top ,S\widehat{B}]\\&= \textrm{rk}\,_{\mathbb {R}}[T^\top Q^\top S^\top S(\widehat{J}+\widehat{R})S^\top ,T^\top Q^\top S^\top S\widehat{B}]\\&= \textrm{rk}\,_{\mathbb {R}}[Q^\top (\widehat{J}+\widehat{R}),Q^\top \widehat{B}]\\&\le \textrm{rk}\,_{\mathbb {R}(x)}[xE^\top +Q^\top (J+R),Q^\top B]\le \min \left\{ \ell ,n \right\} = \ell , \end{aligned}$$

we conclude $(E,\widehat{J},\widehat{R},Q,\widehat{B})\in S_\gamma \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$. Therefore, $S_\gamma$ is indeed relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH,=r}$.

Case $\ell >n$: Lemma 4.1 and Sylvester’s law of inertia yield the existence of an orthogonal matrix $S\in \textbf{Gl}(\mathbb {R}^\ell )$, $T\in \textbf{Gl}(\mathbb {R}^n)$, a symmetric and positive semidefinite $\Sigma \in \mathbb {R}^{n\times n}$, $\widetilde{S}\in \textbf{Gl}(\mathbb {R}^n)$ and $P\in \mathbb {R}^{(\ell -n)\times n}$ so that

$$\begin{aligned} SQT = \begin{bmatrix} I_n\\ 0_{(\ell -n)\times n} \end{bmatrix},\quad SET = \begin{bmatrix} \Sigma \\ P \end{bmatrix}\quad \text {and}\quad \widetilde{S}\Sigma \widetilde{S}^\top = \begin{bmatrix} I_\rho &{} 0\\ 0 &{} 0 \end{bmatrix},\quad \rho = \textrm{rk}\,_{\mathbb {R}}\Sigma . \end{aligned}$$

Let $P_1\in \mathbb {R}^{(\ell -n)\times \rho }$, $P_2\in \mathbb {R}^{(\ell -n)\times (n-\rho )}$, $B_1\in \mathbb {R}^{\rho \times m}$, $B_2\in \mathbb {R}^{(n-\rho )\times m}$ and $B_3\in \mathbb {R}^{(\ell -n)\times m}$ so that

$$\begin{aligned} P\widetilde{S}^\top = [P_1,P_2]\quad \text {and}\quad \begin{bmatrix} \widetilde{S}\\ &{} I_{\ell -n} \end{bmatrix}SB = \begin{bmatrix} B_1\\ B_2\\ B_3 \end{bmatrix}; \end{aligned}$$

in particular, $r = \textrm{rk}\,E = \textrm{rk}\,\Sigma + \textrm{rk}\,P_2$ implies $\textrm{rk}\,_{\mathbb {R}} P_2 = r-\rho$.

($\alpha$)   Analogously to [7, Proposition 2.6 (i)], we see that

$$\begin{aligned} S_\alpha ''&:= \left\{ ( \right\} A_1,A_2)\in \mathbb {R}^{(\ell -n)\times (n-\rho )}\times \mathbb {R}^{(n-\rho )\times m}\,\big \vert \,\textrm{rk}\,[A_1^\top ,A_2] \\&= \min \left\{ n-\rho ,r-\rho +m \right\} \end{aligned}$$

is relative generic in the set

$$\begin{aligned} \left\{ (A_1,A_2)\in \mathbb {R}^{(\ell -n)\times (n-\rho )}\times \mathbb {R}^{(n-\rho )\times m}\,\big \vert \,\textrm{rk}\,A_1 = r-\rho \right\} ; \end{aligned}$$

replace E by $A_1$ and the block matrix [A, B] by $A_2$ in the proof of [7, Proposition 2.6 (i)]. Hence, there are $(\widetilde{P}_2,\widetilde{B}_2)$ so that

$$\begin{aligned} \max \left\{ \left\| P_2-\widetilde{P}_2 \right\| ,\left\| B_2-\widetilde{B}_2 \right\| \right\} <\frac{\varepsilon }{2\ell n^2\max \left\{ \left\| S \right\| \left\| T \right\| ^{-1}\left\| \widetilde{S}^{-1} \right\| ,\left\| S \right\| \left\| \widetilde{S}^{-1} \right\| \right\} }. \end{aligned}$$

Define

$$\begin{aligned} \widehat{E} := S^\top \begin{bmatrix} \Sigma \\ [P_1, \widetilde{P}_2]\widetilde{S}^{-\top } \end{bmatrix} T^{-1}\quad \text {and}\quad \widehat{B} := S^{-1}\begin{bmatrix} \widetilde{S}^{-1}\\ &{} I_{\ell -n} \end{bmatrix}\begin{bmatrix} B_1\\ \widetilde{B}_2\\ B_3 \end{bmatrix}. \end{aligned}$$

Then, (4.3) implies

$$\begin{aligned} \left\| E-\widehat{E} \right\|&= \left\| S^\top S(E-\widehat{E})TT^{-1} \right\| \\&\le \ell n \left\| S \right\| \left\| T^{-1} \right\| \left\| P-[P_1,P_2]\widetilde{S}^{-\top } \right\| \\&\le \ell n^2\left\| S \right\| \left\| T^{-1} \right\| \left\| \widetilde{S}^{-1} \right\| \left\| [P_1,P_2]-[P_1,\widetilde{P}_2] \right\| \\&\le \ell n^2\left\| S \right\| \left\| T^{-1} \right\| \left\| \widetilde{S}^{-1} \right\| \left\| P_2-\widetilde{P}_2 \right\| <\frac{\varepsilon }{2} \end{aligned}$$

and analogously $\left\| B-\widehat{B} \right\| <\frac{\varepsilon }{2}$. Furthermore, we have

$$\begin{aligned} \textrm{rk}\,_{\mathbb {R}}[\widehat{E}^\top ,Q^\top \widehat{B}]&= \textrm{rk}\,_{\mathbb {R}} \begin{bmatrix} \widetilde{S}\\ &{} I_{\ell -n} \end{bmatrix}T^\top [\widehat{E}^\top ,Q^\top \widehat{B}]\begin{bmatrix} S^\top \widetilde{S}^\top \\ &{} I_m \end{bmatrix}\\&= \textrm{rk}\,_{\mathbb {R}}\begin{bmatrix} I_\rho &{} 0 &{} \widetilde{P}_1^\top &{} B_1\\ 0 &{} 0 &{} \widetilde{P}_2^\top &{} \widetilde{B}_2 \end{bmatrix}\\&= \rho + \textrm{rk}\,_{\mathbb {R}}[\widetilde{P}_2^\top ,\widetilde{B}_2]\\&= \min \left\{ n,r+m \right\} = \min \left\{ \ell ,n,r+m \right\} . \end{aligned}$$

Thus, $(\widehat{E},J,R,Q,\widehat{B})\in S_\alpha \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$. This shows that $S_\alpha$ is relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH, = r}$.

($\beta$)   Let $J_{1,1},R_{1,1}\in \mathbb {R}^{n\times n}$, $J_{1,2},R_{1,2}\in \mathbb {R}^{n\times (\ell -n)}$ and $J_{2,2},R_{2,2}\in \mathbb {R}^{(\ell -n)\times (\ell -n)}$ so that

$$\begin{aligned} SJS^\top = \begin{bmatrix} J_{1,1} &{} J_{1,2}\\ -J_{1,2}^\top &{} J_{2,2} \end{bmatrix}\quad \text {and}\quad SRS^\top = \begin{bmatrix} R_{1,1} &{} R_{1,2}\\ R_{1,2}^\top &{} R_{2,2} \end{bmatrix}. \end{aligned}$$

Put further $B^1:= \widetilde{S}^{-1} \begin{bmatrix} B_1\\ B_2 \end{bmatrix}$ and $B^2:= B_3$. Similar to ($\beta$) in case $\ell \le n$, [8, Lemma 2.13] implies that the set

$$\begin{aligned} S_\beta '' := \left\{ (\widehat{J}_{1,1},\widehat{R}_{1,1},\widehat{B}^1)\in \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times n}\times \mathbb {R}\,\big \vert \,\textrm{rk}\,[\widehat{J}_{1,1}+\widehat{R}_{1,1},\widehat{B}^1] = n \right\} \end{aligned}$$

is relative generic in the nonempty convex reference set

$$\begin{aligned} V_\beta ' := \left\{ (\widehat{J}_{1,1},\widehat{R}_{1,1},\widehat{B}^1)\in \mathbb {R}^{n\times n}\times \mathbb {R}^{n\times n}\times \mathbb {R}\,\left| \, \widehat{J}_{1,1}^\top = -\widehat{J}_{1,1}, \widehat{R}_{1,1}^\top = \widehat{R}_{1,1}, \begin{bmatrix} \widehat{R}_{1,1} &{} R_{1,2}\\ R_{1,2}^\top &{} R_{2,2} \end{bmatrix}\ge 0\right. \right\} . \end{aligned}$$

Therefore, there are $(\widehat{J}_{1,1},\widehat{R}_{1,1},\widehat{B}^{1})\in S_\beta ''$ with

$$\begin{aligned} \max \left\{ \left\| J_{1,1}-\widehat{J}_{1,1} \right\| ,\left\| R_{1,1}-\widehat{R}_{1,1} \right\| ,\left\| B^1-\widehat{B}^1 \right\| \right\} <\frac{\varepsilon }{2\ell ^2\max \left\{ \left\| S \right\| ,\left\| S \right\| ^2 \right\} }. \end{aligned}$$

Define

$$\begin{aligned} \widehat{J} := S^\top \begin{bmatrix} \widehat{J}_{1,1} &{} J_{1,2}\\ -J_{1,2}^\top &{} J_{2,2} \end{bmatrix}S,\quad \widehat{R} := S^\top \begin{bmatrix} \widehat{R}_{1,1} &{} R_{1,2}\\ R_{1,2}^\top &{} R_{2,2} \end{bmatrix}S,\quad \text {and}\quad \widehat{B} := S^\top \begin{bmatrix} \widehat{B}^1\\ B^2 \end{bmatrix}. \end{aligned}$$

Then, we conclude from (4.3)

$$\begin{aligned} \left\| J-\widehat{J} \right\| \le \ell ^2\left\| S \right\| ^2\left\| SJS^\top -S\widehat{J}S^\top \right\| = \ell ^2\left\| S \right\| ^2\left\| J_{1,1}-\widehat{J}_{1,1} \right\| <\frac{\varepsilon }{2} \end{aligned}$$

and analogously $\left\| R-\widehat{R} \right\| <\frac{\varepsilon }{2}$ and $\left\| B-\widehat{B} \right\| <\frac{\varepsilon }{2}$. Furthermore, we have

$$\begin{aligned} n&= \textrm{rk}\,[\widehat{J}_{1,1}^\top +\widehat{R}_{1,1}^\top ,\widehat{B}^1]\\&\le \textrm{rk}\,[\Sigma ^\top ,P^\top ,\widehat{J}_{1,1}^\top +\widehat{R}_{1,1}^\top ,J_{1,2}+R_{1,2},\widehat{B}^1]\\&\le \textrm{rk}\,[(SET)^\top (SQT)^\top S(\widehat{J}-\widehat{R})S^\top ,(SQT)^\top S\widehat{B}]\\&= \textrm{rk}\,[E^\top ,Q^\top (\widehat{J}+\widehat{R}),Q^\top \widehat{B}] \le n. \end{aligned}$$

Hence, all inequalities are equalities and $(E,\widehat{J},\widehat{R},Q,\widehat{B})\in S_\beta \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$. This verifies that $S_\beta$ is relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH,=r}$.

($\gamma$) Analogously to ($\gamma$) in case $n\le \ell$, the quintuple $(E,\widehat{J},\widehat{R},Q,\widehat{B})\in \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$ that we constructed in ($\beta$) fulfils

$$\begin{aligned} n&= \textrm{rk}\,_{\mathbb {R}}[\widehat{J}_{1,1}^\top +\widehat{R}_{1,1}^\top ,\widehat{B}^1]\\&\le \textrm{rk}\,_{\mathbb {R}(x)}\Sigma ^\top +\widehat{J}_{1,1}^\top +\widehat{R}_{1,1}^\top ,xP^\top + \widehat{J}_{1,2}+\widehat{R}_{1,2},\widehat{B}^1]\\&\le \textrm{rk}\,[x(SET)^\top +(SQT)^\top S(\widehat{J}+\widehat{R})S^\top ,(SQT)^\top SB]\\&= \textrm{rk}\,[xE^\top +Q^\top (\widehat{J}+\widehat{R}),Q^\top \widehat{B}]\le n. \end{aligned}$$

Therefore, $(E,\widehat{J},\widehat{R},Q,\widehat{B})\in S_\gamma \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$ and we conclude that $S_\gamma$ is dense in $\Sigma _{\ell ,n,m}^{sdH,=r}$.

Reference set $\Sigma _{\ell ,n,m}^{H,=r}$: The proof in this case is analogous to the proof for the reference set $\Sigma _{\ell ,n,m}^{sdH,=r}$; simply remove all Cartesian factors containing the dissipation matrix R and observe that all claims still remain true. $\square$

Recall a helpful result on matrices with bounded rank.

Lemma 4.9

Let $n,m\in \mathbb {N}$ and $r\in \left\{ 0,\ldots ,\min \left\{ n,m \right\} \right\}$. The set

$$\begin{aligned} S = \left\{ A\in \mathbb {R}^{n\times m}\,\left| \,\exists H\in \textbf{Gl}(\mathbb {R}^n)~\exists M\in \mathbb {R}^{r\times (m-r)}: HA = \begin{bmatrix} I_r &{} M\\ 0_{(n-r)\times r} &{} 0_{(n-r)\times (m-r)} \end{bmatrix}\right. \right\} \end{aligned}$$

is relative generic in

$$\begin{aligned} V = \left\{ A\in \mathbb {R}^{n\times m}\,\big \vert \,\textrm{rk}\,A = r \right\} . \end{aligned}$$

Proof

In [6], the authors construct a set $D_{=r}^{r}\subseteq V$ and functions

$$\begin{aligned} T_{=r}^{r}: D_{=r}^{r}\rightarrow \mathbb {R}^{n\times m},\qquad \tau _{=r}^r:D_{=r}^r\rightarrow \textbf{Gl}(\mathbb {R}^{n}) \end{aligned}$$

so that for each $A\in D_{=r}^{r}$ the matrix $T_{=r}^r(A)$ has the structure

$$\begin{aligned} T_{=r}^{r}(A) = \tau _{=r}^r(A) A \begin{bmatrix} \varepsilon _1(A) &{} * &{} \cdots &{} * &{} * &{} \cdots &{} *\\ 0 &{} \varepsilon _2(A) &{} \cdots &{} * &{} * &{} \cdots &{} * \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} \varepsilon _r(A) &{} * &{} \cdots &{} *\\ 0 &{} 0 &{} \cdots &{} 0 &{} * &{} \cdots &{} *\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} 0 &{} * &{} \cdots &{} * \end{bmatrix}, \qquad \varepsilon _1(A),\ldots ,\varepsilon _r(A)\in \mathbb {R}{\setminus }\left\{ 0 \right\} . \end{aligned}$$

Since $\textrm{rk}\,A = r$ for all $A\in V$, we see that the last $n-r$ rows of $T_{=r}^{r}(A)$ are zero. Since the $\varepsilon _1(A),\ldots ,\varepsilon _r(A)$ are nonzero, we conclude that there is a matrix $H'\in \textbf{Gl}(\mathbb {R}^n)$ so that

$$\begin{aligned} H'\tau _{=r}^r(A)A = \begin{bmatrix} I_r &{} M\\ 0_{(n-r)\times r} &{} 0_{(n-r)\times (m-r)} \end{bmatrix} \end{aligned}$$

for some matrix $M\in \mathbb {R}^{r\times (m-r)}$. Thus $D_{=r}^r \subseteq S$. By [6, Proposition 2], $D_{=r}^r$ is relative generic in S. Hence, [7, Proposition 2.3 (c)] yields that S is indeed relative generic in V. $\square$

The following lemma is closely related to [7, Proposition 2.6 (v)] and [8, Proposition 3.11 (j)]. These results, however, cannot be straightforwardly applied, so that a refined conclusion is made.

Lemma 4.10

Let $\star \in \left\{ \le ,= \right\}$. The set

$$\begin{aligned} S_{\iota } := \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,\forall \lambda \in \mathbb {C}: \textrm{rk}\,_{\mathbb {C}}\lambda E^\top +Q^\top (J+R),Q^\top B] = \min \left\{ \ell ,n \right\} \right\} \end{aligned}$$

is relative generic in $\Sigma _{\ell ,n,m}^{sdH, \star r}$, and the set

$$\begin{aligned} S_{\kappa } := \left\{ (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,\forall \lambda \in \mathbb {C}: \textrm{rk}\,_{\mathbb {C}}\lambda E^\top +Q^\top J,Q^\top B] = \min \left\{ \ell ,n \right\} \right\} \end{aligned}$$

is relative generic in $\Sigma _{\ell ,n,m}^{H, \star r}$

Proof

By Lemma 4.6, $\Sigma _{\ell ,n,m}^{sdH,=r}$ is a relative generic subset of $\Sigma _{\ell ,n,m}^{sdH,\le r}$ and $\Sigma _{\ell ,n,m}^{H, = r}$ is a relative generic subset of $\Sigma _{\ell ,n,m}^{H,\le r}$. Therefore, [8, Proposition 2.10 (e)] yields that it suffices to show the assertion for the reference sets $\Sigma _{\ell ,n,m}^{sdH, = r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$. The proof is divided into steps.

Step 1:   We construct relative Zariski-open sets $\widetilde{S}_\iota \subseteq \Sigma _{\ell ,n,m}^{sdH,=r}$, and $\widetilde{S}_\kappa \subseteq \Sigma _{\ell ,n,m}^{H,=r}$ so that $\widetilde{S}_\iota \subseteq S_\iota$ and $\widetilde{S}_\kappa \subseteq S_\kappa$. Let $\widetilde{M}_1,\ldots ,\widetilde{M}_q$ be all minors of order $d:= \min \left\{ \ell ,n \right\}$ with respect to $\mathbb {R}[x]^{n\times (\ell +m)}$. These minors induce for each $i\in \underline{q}$, $O\in \textbf{Gl}(\mathbb {R}^\ell )$ and $T\in \textbf{Gl}(\mathbb {R}^n)$ the functions

$$\begin{aligned}&M_i^{\iota ,O,T}:\overbrace{\left\{ (E,J,R,Q,B)\,\big \vert \, J^\top = -J, R^\top = R \right\} }^{=:V^\iota } \rightarrow \mathbb {R}[x],\\&\quad (E,J,R,Q,B) \mapsto \widetilde{M}_i\left( T^\top [xE^\top +Q^\top (J+R),Q^\top B]\begin{bmatrix} S^\top &{} 0\\ 0 &{} I_m \end{bmatrix}\right) \end{aligned}$$

and

$$\begin{aligned}&M_i^{\kappa ,O,T}:\overbrace{\left\{ (E,J,Q,B)\,\big \vert \, J^\top = -J \right\} }^{=:V^\kappa } \rightarrow \mathbb {R}[x],\\&\quad (E,J,R,Q,B) \mapsto \widetilde{M}_i\left( T^\top [xE^\top +Q^\top J,Q^\top B]\begin{bmatrix} S^\top &{} 0\\ 0 &{} I_m \end{bmatrix}\right) . \end{aligned}$$

Using the Leibniz formula for the determinant, it is easy to see that the $M_i^{\iota ,O,T}$ have pointwise degree at most d (as they are the determinant of a $d\times d$ polynomial matrix with degree at most one) and that the coefficients of $M_i^{\iota ,O,T}$ are polynomials in the coefficients of E, J, R, Q and B, i.e. for each $i\in \underline{q}$ and $O\in \textbf{Gl}(\mathbb {R}^\ell )$, $T\in \textbf{Gl}(\mathbb {R}^n)$, there are polynomials $M_{i,j}^{\iota ,O,T}\in \mathbb {R}[x_1,\ldots ,x_{\ell (2n+2\ell +m)}]$, $j\in \left\{ 0,\ldots ,d \right\}$, so that

$$\begin{aligned} \forall (E,J,R,Q,B)\in V: M_{i}^{\iota ,O,T}(E,J,R,Q,B) = \sum _{j = 0}^d M_{i,j}^{\iota ,O,T}(E,J,R,Q,B)x^j. \end{aligned}$$

Analogously, we see that for each $i\in \underline{q}$, $O\in \textbf{Gl}(\mathbb {R}^\ell )$ and $T\in \textbf{Gl}(\mathbb {R}^n)$ there are polynomials $M_{i,j}^{\kappa ,O,T}\in \mathbb {R}[x_1,\ldots ,x_{\ell (2n+\ell +m)}]$, $j\in \left\{ 0,\ldots ,d \right\}$, so that

$$\begin{aligned} \forall (E,J,R,Q,B)\in V: M_{i}^{\kappa ,O,T}(E,J,Q,B) = \sum _{j = 0}^d M_{i,j}^{\kappa ,O,T}(E,J,Q,B)x^j. \end{aligned}$$

Define further, for each $i\in \underline{q}$, $O\in \textbf{Gl}(\mathbb {R}^\ell )$ and $T\in \textbf{Gl}(\mathbb {R}^n)$,

$$\begin{aligned} \gamma _i^{\iota ,O,T}&:= \max \left\{ j\in \left\{ 0,\ldots ,d \right\} \,\vert \,\exists (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH,=r}: M_{i,j}^{S,T}(E,J,R,Q,B)\ne 0 \right\} ,\\ \gamma _i^{\kappa ,O,T}&:= \max \left\{ j\in \left\{ 0,\ldots ,d \right\} \,\vert \,\exists (E,J,Q,B)\in \Sigma _{\ell ,n,m}^{H,=r}: M_{i,j}^{S,T}(E,J,Q,B)\ne 0 \right\} , \end{aligned}$$

as the maximal degree of the restriction of $M_{i}^{\iota ,O,T}$ to $\Sigma _{\ell ,n,m}^{sdH,=r}$ and $M_{i}^{\kappa ,O,T}$ to $\Sigma _{\ell ,n,m}^{H,=r}$, respectively. In particular, we have $\gamma _i^{\iota ,O,T}\le r$ and $\gamma _i^{\kappa ,O,T}\le r$ for all $i\in \underline{q}$, $O\in \textbf{Gl}(\mathbb {R}^{\ell })$ and $T\in \textbf{Gl}(\mathbb {R}^n)$. Define, analogously to [7, (31)] and [8, (21)], for all $\xi \in \left\{ \iota ,\kappa \right\}$, $i,j\in \underline{q}$, $O\in \textbf{Gl}(\mathbb {R}^\ell )$ and $T\in \textbf{Gl}(\mathbb {R}^n)$, the polynomials

$$\begin{aligned} p_{i,j}^{\xi ,O,T}(\cdot )=\det \underbrace{\left[ \begin{array}{ccccc|ccccc} M_{i,0}^{\xi ,O,T}(\cdot ) &{} &{} &{} &{} &{} M_{j,0}^{\xi ,O,T}(\cdot ) &{} &{} &{}\\ M_{i,1}^{\xi ,O,T}(\cdot ) &{} M_{i,0}^{\xi ,O,T}(\cdot ) &{} &{} &{} &{} M_{j,1}^{\xi ,O,T}(\cdot ) &{} \cdot &{}\\ \cdot &{} \cdot &{} &{} &{} &{} \cdot &{} \cdot &{} \cdot &{} \\ \cdot &{} \cdot &{} \cdot &{} &{} &{} \cdot &{} \cdot &{} \cdot &{} M_{j,0}^{\xi ,O,T}(\cdot ) \\ M_{i,{\gamma _i^{\xi ,O,T}}}^{\xi ,O,T}(\cdot ) &{} M_{i,{{\gamma _i}-1}}^{\xi ,O,T}(\cdot ) &{} \cdot &{} \cdot &{} &{} \cdot &{} \cdot &{} \cdot &{} \cdot \\ &{} M_{i,{\gamma _i^{\xi ,O,T}}}^{\xi ,O,T}(\cdot ) &{} \cdot &{} \cdot &{} &{} \cdot &{} \cdot &{} \cdot &{} \cdot \\ &{} &{} \cdot &{} \cdot &{} M_{i,0}^{\xi ,O,T}(\cdot ) &{} M_{j,{{\gamma _j^{\xi ,O,T}}-1}}^{\xi ,O,T}(\cdot ) &{} \cdot &{} \cdot &{} \cdot \\ &{} &{} \cdot &{} \cdot &{} \cdot &{} M_{j,{{\gamma _j^{\xi ,O,T}}}}^{\xi ,O,T}(\cdot ) &{} \cdot &{} \cdot &{} \cdot &{} \\ &{} &{} \cdot &{} \cdot &{} \cdot &{} &{} \cdot &{} \cdot &{} \cdot \\ &{} &{} &{} \cdot &{} \cdot &{} &{} &{} \cdot &{} \cdot \\ &{} &{} &{} &{} M_{i,{\gamma _i^{\xi ,O,T}}}^{\xi ,O,T}(\cdot ) &{} &{} &{} &{} M_{j,{\gamma _j^{\xi ,O,T}}}^{\xi ,O,T}(\cdot ) \end{array}\right] }_{\in \mathbb {R}[x_1,\ldots ,x_{\ell (2n+\ell _\xi +m)}]^{\left( \gamma _i^{\xi ,O,T}+\gamma _j^{\xi ,O,T}\right) \times \left( \gamma _i^{\xi ,O,T}+\gamma _j^{\xi ,O,T}\right) }}, \end{aligned}$$

(4.4)

where we used $\ell _\xi := 2\ell$ for $\xi = \iota$ and $\ell _\xi := \ell$ for $\xi = \kappa$. Using Laplace’s expansion formula and the properties of the Sylvester resultant, we have for all $\mathfrak {E} = (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}$ the following chain of implications

$$\begin{aligned}&\exists i,j\in \underline{q}~\exists O\in \textbf{Gl}(\mathbb {R}^\ell )~\exists T\in \textbf{Gl}(\mathbb {R}^n): p_{i,j}^{\iota ,O,T}(\mathfrak {E})\ne 0 \end{aligned}$$

(4.5)

$$\begin{aligned}&\quad \iff \exists i,j\in \underline{q}~\exists O\in \textbf{Gl}(\mathbb {R}^\ell )~\exists T\in \textbf{Gl}(\mathbb {R}^n): \nonumber \\&\qquad \left\{ \begin{array}{l} \big (M_{i,\gamma _i^{\iota ,O,T}}^{\iota ,O,T}(\mathfrak {E})\ne 0 \vee M_{j,\gamma _j^{\iota ,O,T}}^{\iota ,O,T}(\mathfrak {E})\ne 0\big )\\ \wedge ~M_i^{\iota ,O,T}(\mathfrak {E}),M_j^{\iota ,O,T}(\mathfrak {E})~\text {coprime} \end{array}\right. \nonumber \\&\quad \implies \exists O\in \textbf{Gl}(\mathbb {R}^\ell )~\exists T\in \textbf{Gl}(\mathbb {R}^n)~\forall \lambda \in \mathbb {C}~\exists i\in \underline{q}: M_i^{\iota ,O,T}(\mathfrak {E})(\lambda )\ne 0\nonumber \\&\quad \iff \exists O\in \textbf{Gl}(\mathbb {R}^\ell )~\exists T\in \textbf{Gl}(\mathbb {R}^n)~\forall \lambda \in \mathbb {C}: \textrm{rk}\,T^\top [\lambda E^\top \nonumber \\&\qquad + Q^\top (J+R),Q^\top B]\begin{bmatrix} S^\top \\ {} &{} I_m \end{bmatrix}\ge \min \left\{ \ell ,n \right\} \nonumber \\&\quad \iff \forall \lambda \in \mathbb {C}: \textrm{rk}\,[\lambda E^\top +Q^\top (J+R),Q^\top B] = \min \left\{ \ell ,n \right\} , \end{aligned}$$

(4.6)

where we used

$$\begin{aligned} \textrm{rk}\,_{\mathbb {R}(x)} [xE^\top +Q^\top (J{+}R),Q^\top B] {=} \max \left\{ \left. \textrm{rk}\,[\lambda E^\top +Q^\top (J+R),Q^\top B]\,\right| \,\lambda \in \mathbb {C} \right\} \end{aligned}$$

and the upper bound on $\textrm{rk}\,_{\mathbb {R}(x)} [xE^\top +Q^\top (J+R),Q^\top B]$ derived in Lemma 4.7. Define

$$\begin{aligned} \mathcal {I}^\iota := \left\langle \left\{ p_{i,j}^{\iota ,O,T}\,\big \vert \, i,j\in \underline{q}, O\in \textbf{Gl}(\mathbb {R}^\ell ), T\in \textbf{Gl}(\mathbb {R}^n) \right\} \right\rangle _{\mathbb {R}[x_1,\ldots ,x_{\ell (2n+2\ell +m)}]} \end{aligned}$$

as the ideal of polynomials spanned by the $p_{i,j}^{\iota ,O,T}$. Let

$$\begin{aligned} \mathbb {V}^\iota := \left\{ (E,J,R,Q,B)\in V^\iota \,\big \vert \,\forall p\in \mathcal {I}^\iota : p(E,J,R,Q,B) = 0 \right\} \end{aligned}$$

be the algebraic set induced by $\mathcal {I}$ and put

$$\begin{aligned} \widetilde{S}_\iota := \Sigma _{\ell ,n,m}^{sdH, = r}{\setminus }\mathbb {V}^\iota . \end{aligned}$$

Then, $\widetilde{S}_\iota$ is a relative Zariski-open subset of $\Sigma _{\ell ,n,m}^{sdH, = r}$. Due to the chain of implications from above, the inclusion $\widetilde{S}_\iota \subseteq S_\iota$ holds true. Analogously, we define the ideal

$$\begin{aligned} \mathcal {I}^\kappa := \left\langle \left\{ p_{i,j}^{\kappa ,O,T}\,\big \vert \, i,j\in \underline{q}, O\in \textbf{Gl}(\mathbb {R}^\ell ), T\in \textbf{Gl}(\mathbb {R}^n) \right\} \right\rangle _{\mathbb {R}[x_1,\ldots ,x_{\ell (2n+\ell +m)}]} \end{aligned}$$

and the algebraic set

$$\begin{aligned} \mathbb {V}^\kappa := \left\{ (E,J,Q,B)\in V^\kappa \,\big \vert \,\forall p\in \mathcal {I}^\kappa : p(E,J,Q,B) = 0 \right\} \end{aligned}$$

and see that $\widetilde{S}_\kappa := \Sigma _{\ell ,n,m}^{H,=r}{\setminus }\mathbb {V}^\kappa$ is a relative Zariski-open subset of $S_\kappa$.

Step 2:    We show that $\widetilde{S}_\iota$ is dense in $\Sigma _{\ell ,n,m}^{sdH,=r}$ with respect to the relative Euclidean topology. Let $(E^\star ,J^\star ,R^\star ,Q^\star ,B^\star )\in \Sigma _{\ell ,n,m}^{sdH,=r}$ and $\varepsilon >0$. By Lemma 4.6 (i), there exists $(E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH,=r}$ so that $\textrm{rk}\,Q = \min \left\{ \ell ,n \right\}$ and $\left\| (E,J,R,Q,B)-(E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ) \right\| <\frac{\varepsilon }{2}$. In the following, we distinguish further the cases $\ell \le n$ and $n<\ell$.

Step 2.1:    Let $\ell \le n$. By Lemma 4.1, there are an orthogonal $O\in \textbf{Gl}(\mathbb {R}^\ell )$, $T\in \textbf{Gl}(\mathbb {R}^n)$ and a $\Sigma \in \mathbb {R}^{\ell \times \ell }$ so that

$$\begin{aligned} OET = [\Sigma ,0_{n\times (n-\ell )}]\quad \text {and}\quad OQT = [I_\ell ,0_{\ell \times (n-\ell )}]. \end{aligned}$$

Since $\Sigma$ is symmetric and positive semidefinite, Sylvester’s law of inertia implies that there exists $\widetilde{O}\in \textbf{Gl}(\mathbb {R}^\ell )$ so that

$$\begin{aligned} \widetilde{O}\Sigma \widetilde{O}^\top = \begin{bmatrix} I_r &{} \quad 0_{r\times (\ell -r)}\\ 0_{(\ell -r)\times r} &{}\quad 0_{(\ell -r)\times (\ell -r)} \end{bmatrix}. \end{aligned}$$

We show that the set

$$\begin{aligned} \widetilde{S}_\iota (E,Q) := \left\{ (\widehat{J},\widehat{R},\widehat{B})\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\Big \vert \, (E,\widehat{J},\widehat{R},Q,\widehat{B})\in \widetilde{S}_\iota \right\} \end{aligned}$$

is relative generic in the reference set

$$\begin{aligned} V_\iota (E,Q) := \left\{ (\widehat{J},\widehat{R},\widehat{B})\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\Big \vert \, (E,\widehat{J},\widehat{R},Q,\widehat{B})\in \Sigma _{\ell ,n,m}^{sdH, = r} \right\} . \end{aligned}$$

Since $V_\iota (E,Q)$ is nonempty and convex, and since $\widetilde{S}_{\delta }(E,Q)$ is by construction of $\widetilde{S}_\iota$ a relative Zariski-open subset of $V_\iota (E,Q)$, [8, Lemma 2.13] yields that $\widetilde{S}_\iota (E,Q)$ is relative generic in $V_\iota (E,Q)$ if, and only if, $\widetilde{S}_\iota (E,Q)$ is nonempty. Consider the matrices

$$\begin{aligned}&\widehat{R} := 0_{\ell \times \ell },\quad \widehat{J} := O^{-1}\widetilde{O}^{-1}\begin{bmatrix} 0 &{} 1\\ -1 &{} 0 &{} 1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} -1 &{} 0 &{} 1\\ &{} &{} &{} -1 &{} 0 \end{bmatrix}\widetilde{O}^{-\top }O^{-\top },\\&\widehat{B} := O^{-1}[e_\ell ,0_{\ell \times (m-1)}]. \end{aligned}$$

This choice guarantees that $(\widehat{J},\widehat{R},\widehat{B})\in V_{\delta }(E,Q)$. Define further

$$\begin{aligned} \widehat{O} := \widetilde{O}O\quad \text {and}\quad \widehat{T} := T\widetilde{O}^\top . \end{aligned}$$

Then, using the orthogonality of O, $[xE^\top +Q^\top (\widehat{J}+\widehat{R}),Q^\top \widehat{B}]$ is transformed into

$$\begin{aligned}&\widehat{T}^\top [xE^\top +Q^\top (\widehat{J}+\widehat{R}),Q^\top \widehat{B}]\begin{bmatrix} \widehat{S}^\top \\ &{} I_m \end{bmatrix}\\&\quad = \left[ \begin{array}{c|cc} \begin{array}{cccccccc} x &{} 1\\ -1 &{} x &{} 1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} -1 &{} x &{} 1\\ &{} &{} &{} -1 &{} 0 &{} 1\\ &{} &{} &{} &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} &{} &{} &{} -1 &{} 0 &{} 1\\ &{} &{} &{} &{} &{} &{} -1 &{} 0 \end{array} &{} \begin{array}{c} 0\\ 0\\ \vdots \\ 0\\ 0\\ \vdots \\ 0\\ 1 \end{array} &{} 0_{\ell \times (m-1)}\\ \hline 0_{(n-\ell )\times \ell } &{} 0_{(n-\ell )\times 1} &{} 0_{(n-\ell )\times (m-1)} \end{array}\right] . \end{aligned}$$

Without losing generality, we may assume that the $\widetilde{M}_i$ are ordered in such a manner that

$$\begin{aligned} \widetilde{M}_1 =&P(x)\mapsto \det [P_{i,j}(x)]_{i\in \underline{\ell },j\in \left\{ 1,\ldots ,r,r+2,\ldots ,\ell +1 \right\} },\quad \text {and}\quad \\ \widetilde{M}_2 =&P(x)\mapsto \det [P_{i,j}(x)]_{i,j-1\in \underline{\ell }}. \end{aligned}$$

Then, we have

$$\begin{aligned} M_1^{\iota ,\widehat{O},\widehat{T}}(E,\widehat{J},\widehat{R},Q,\widehat{B})&= x^r + \text {lower order terms},\quad \text {and}\quad \\ M_2^{\iota ,\widehat{O},\widehat{T}}(E,\widehat{J},\widehat{R},Q,\widehat{B})&= 1. \end{aligned}$$

Since $\gamma _1^{\iota ,\widehat{O},\widehat{T}}\le r$, we find that $M_{1,\gamma _1^{\iota ,\widehat{O},\widehat{T}}}^{\widehat{O},\widehat{T}}(E,\widehat{J},\widehat{R},Q,\widehat{B})\ne 0$. Further, $M_1^{\iota ,\widehat{O},\widehat{T}}(E,\widehat{J},\widehat{R},Q,\widehat{B})$ and $M_2^{\iota ,\widehat{O},\widehat{T}}(E,\widehat{J},\widehat{R},Q,\widehat{B})$ are coprime and hence $p_{1,2}^{\iota ,\widehat{O},\widehat{T}}(E,\widehat{J},\widehat{R},Q,\widehat{B})\ne 0$. From our definition of $\widetilde{S}_\iota (E,Q)$, we conclude that $(\widehat{J},\widehat{R},\widehat{B})\in \widetilde{S}_\iota (E,Q)$ and thus $\widetilde{S}_\iota$ is indeed nonempty. [8, Lemma 2.13] yields that $\widetilde{S}_\iota (E,Q)$ is relative generic in $V_\iota (E,Q)$. Therefore, there are $(\widetilde{J},\widetilde{R},\widetilde{B})\in \widetilde{S}_\iota (E,Q)$ so that

$$\begin{aligned} \left\| (\widetilde{J},\widetilde{R},\widetilde{B})-(J,R,B) \right\| <\frac{\varepsilon }{2}. \end{aligned}$$

By definition of $\widetilde{S}_\iota (E,Q)$, we conclude that $(E,\widetilde{J},\widetilde{R},Q,\widetilde{B})\in \widetilde{S}_\iota \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$. This verifies that $\widetilde{S}_\iota$ is relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH,=r}$.

Step 2.2:   Let $\ell >n$. Lemma 4.1 yields the existence of an orthogonal $O\in \textbf{Gl}(\mathbb {R}^\ell )$, $T\in \textbf{Gl}(\mathbb {R}^n)$, a symmetric and positive semidefinite $\Sigma \in \mathbb {R}^{n\times n}$ and $P\in \mathbb {R}^{(\ell -n)\times n}$ so that

$$\begin{aligned} OQT = \begin{bmatrix} I_n\\ 0_{(\ell -n)\times n} \end{bmatrix}\quad \text {and}\quad OET = \begin{bmatrix} \Sigma \\ P \end{bmatrix}. \end{aligned}$$

Since $\Sigma$ is symmetric and positive semidefinite, there is a matrix $\widetilde{O}\in \textbf{Gl}(\mathbb {R}^{n})$ so that

$$\begin{aligned} \widetilde{O}\Sigma \widetilde{O}^\top = \begin{bmatrix} I_\rho &{} 0_{\rho \times (n-\rho )}\\ 0_{(n-\rho )\times \rho } &{} 0_{(n-\rho )\times (n-\rho )} \end{bmatrix},\quad \rho = \textrm{rk}\,\Sigma \end{aligned}$$

Let $P_1\in \mathbb {R}^{(\ell -n)\times \rho }$ and $P_2\in \mathbb {R}^{(\ell -n)\times (n-\rho )}$ with $P\widetilde{O}^\top = [P_1,P_2]$. In particular, we have $\textrm{rk}\,P_2 = r-\rho$. By Lemma 4.9, there are some $\widetilde{P}_2\in \mathbb {R}^{(\ell -n)\times (n-\rho )}$, $H\in \textbf{Gl}(\mathbb {R}^{\ell -n})$ and $M\in \mathbb {R}^{(r-\rho )\times (n-r)}$ with

$$\begin{aligned} H\widetilde{P}_2 = \begin{bmatrix} I_{r-\rho } &{} M\\ 0 &{} 0 \end{bmatrix}\quad \text {and}\quad \left\| \widetilde{P}_2-P_2 \right\| <\frac{\varepsilon }{2\ell ^2 n^2\left\| T^{-1} \right\| \left\| O^{-1} \right\| \left\| \widetilde{O}^{-1} \right\| ^2}. \end{aligned}$$

Consider the matrix

$$\begin{aligned} \widetilde{E} := O^{-1}\begin{bmatrix} \widetilde{O}^{-1}\\ &{} I_{\ell -n} \end{bmatrix}\begin{bmatrix} I_\rho &{} 0_{\rho \times (n-\rho )}\\ 0_{(n-\rho )\times \rho } &{} 0_{(n-\rho )\times (n-\rho )}\\ P_1 &{} \widetilde{P}_2 \end{bmatrix}\widetilde{O}^{-\top } T^{-1}. \end{aligned}$$

Then, (4.3) implies

$$\begin{aligned} \left\| E-\widetilde{E} \right\|&= \left\| O^{-1}\begin{bmatrix} \widetilde{O}^{-1}\\ &{} I_{\ell -n} \end{bmatrix}\begin{bmatrix} \widetilde{O}\\ &{} I_{\ell -n} \end{bmatrix}(E-\widetilde{E})T\widetilde{O}^\top \widetilde{O}^{-\top } T^{-1} \right\| \\&\le \ell ^2 n^2\left\| O^{-1} \right\| \left\| T^{-1} \right\| \left\| \widetilde{O}^{-1} \right\| ^2\left\| R_2-\widetilde{R}_2 \right\| <\frac{\varepsilon }{2} \end{aligned}$$

and, in view of Lemma 4.1, $(\widetilde{E},J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}$. Furthermore, by putting $HP_1 = \begin{bmatrix} M_1\\ M_2 \end{bmatrix}$ for matrices $M_1\in \mathbb {R}^{(r-\rho )\times \rho }$ and $M_2\in \mathbb {R}^{(\ell -n-r+\rho )\times \rho }$, we find

$$\begin{aligned} \begin{bmatrix} I_n\\ {} &{} H \end{bmatrix}\begin{bmatrix} \widetilde{O}\\ {} &{} I_{\ell -n} \end{bmatrix}O\widetilde{E}T\widetilde{O}^\top = \begin{bmatrix} I_\rho &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ M_1 &{} I_{r-\rho } &{} M\\ M_2 &{} 0 &{} 0 \end{bmatrix}. \end{aligned}$$

We show, similar to the case $\ell \le n$, that the set

$$\begin{aligned} \widetilde{S}_\iota (\widetilde{E},Q) := \left\{ (\widehat{J},\widehat{R},\widehat{B})\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\Big \vert \, (\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})\in \widetilde{S}_\iota \right\} \end{aligned}$$

is relative generic in the reference set

$$\begin{aligned} V_\iota (\widetilde{E},Q) := \left\{ (\widehat{J},\widehat{R},\widehat{B})\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\Big \vert \, (\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})\in \Sigma _{\ell ,n,m}^{sdH, = r} \right\} . \end{aligned}$$

Define $\widehat{R}:= 0_{\ell \times \ell }$ and

$$\begin{aligned} \widehat{J}_{1,1} := \widetilde{O}^{-1} \begin{bmatrix} 0 &{} 1\\ -1 &{} 0 &{} 1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} -1 &{} 0 &{} 1\\ &{} &{} &{} -1 &{} 0 \end{bmatrix}\widetilde{O}^{-\top }\in \mathbb {R}^{n\times n}, \end{aligned}$$

$\widehat{J}_{1,2}:= 0_{n\times (\ell -n)}$ and $\widehat{J}_{2,2}:= 0_{(\ell -n)\times (\ell -n)}$, from which we obtain the matrix

$$\begin{aligned} \widehat{J} := O^{-1}\begin{bmatrix} \widehat{J}_{1,1} &{} \widehat{J}_{1,2}\\ -\widehat{J}_{1,2}^\top &{} \widehat{J}_{2,2} \end{bmatrix}O^{-\top }\in \mathbb {R}^{\ell \times \ell }. \end{aligned}$$

Choose

$$\begin{aligned} \widehat{B}&:= O^{-1}\widetilde{O}^{-1}[e_n,0_{\ell \times (m-1)}],\quad \text {and}\quad \widehat{O} := \begin{bmatrix} I_n\\ &{} H \end{bmatrix}\begin{bmatrix}\widetilde{O}\\ {} &{}I_{\ell -n}\end{bmatrix}O,\\&\quad \text {and}\quad \widehat{T} := T\widetilde{O}^\top , \end{aligned}$$

and consider the set of indices

$$\begin{aligned} \mathcal {M} := {\left\{ \begin{array}{ll} \underline{n}, &{} \rho = n,\\ \left\{ 1,\ldots ,\rho ,n-r+1,\ldots ,n+r-\rho ,\ell +1 \right\} , &{} \rho <n. \end{array}\right. } \end{aligned}$$

Without loss of generality, we may assume that the $\widetilde{M}_i$ are arranged in such a way that

$$\begin{aligned} \widetilde{M}_1 = P(x)\mapsto \det [P_{i,j}(x)]_{i\in \underline{n},j\in \mathcal {M}},\quad \text {and}\quad \widetilde{M}_2 = P(x)\mapsto \det [P_{i,j}(x)]_{i\in \underline{n},j-1\in \underline{n}}. \end{aligned}$$

Then,

$$\begin{aligned} M_1^{\iota ,\widehat{O},\widehat{T}}(\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})&= -\det \begin{bmatrix} x I_\rho &{} 0 &{} xM_1^\top \\ 0 &{} 0 &{} xI_{r-\rho }\\ 0 &{} I_{n-r} &{} xM^\top \end{bmatrix} + \text {lower order terms}\\&= \pm x^r + \text {lower order terms} \end{aligned}$$

and hence $\deg M_1^{\iota ,\widehat{O},\widehat{T}}(\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B}) = \gamma _1^{\widehat{O},\widehat{T}}$. Furthermore, our choice of $\widehat{R},\widehat{J}$ and $\widehat{B}$ guarantees that

$$\begin{aligned} M_2^{\iota ,\widehat{O},\widehat{T}}(\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B}) = 1. \end{aligned}$$

Therefore, $M_1^{\iota ,\widehat{O},\widehat{T}}(\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})$ and $M_2^{\iota ,\widehat{O},\widehat{T}}(\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})$ are coprime. Using the equivalence of (4.5) and (4.6), we conclude that $p_1^{\iota ,\widehat{O},\widehat{T}}(\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})\ne 0$ or, equivalently, $(\widehat{J},\widehat{R},\widehat{B})\in \widetilde{S}_\iota (\widetilde{E},Q)$. Hence, [8, Lemma 2.13] yields that $\widetilde{S}_\iota (\widetilde{E},Q)$ is relative generic in $V_\delta (\widetilde{E},Q)$. In particular, we find $(\widetilde{J},\widetilde{R},\widetilde{B})\in \widetilde{S}_\iota (\widetilde{E},Q)$ so that

$$\begin{aligned} \left\| (\widetilde{J},\widetilde{R},\widetilde{B})-(J,R,B) \right\| <\frac{\varepsilon }{2}. \end{aligned}$$

By construction of $\widetilde{S}_\iota (\widetilde{E},Q)$, we conclude that $(\widetilde{E},\widetilde{J},\widetilde{R},Q,\widetilde{B})\in \widetilde{S}_\iota \cap \mathbb {B}((E^\star ,J^\star ,R^\star ,Q^\star ,B^\star ),\varepsilon )$. This verifies that $\widetilde{S}_\iota$ is relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH,=r}$.

Step 3: Analogously to Step 2, we may show that $\widetilde{S}_\kappa$ is dense in $\Sigma _{\ell ,n,m}^{H, = r}$ with respect to the relative Euclidean topology. We omit the details and give only a brief sketch of the proof. In both cases $\ell \le n$ and $n<\ell$, we may remove in all considered sets the Cartesian factor corresponding to the dissipation matrix R, i.e. we consider the sets

$$\begin{aligned} \widetilde{S}_{\kappa }(\widetilde{E},Q)&:= \left\{ (\widehat{J},\widehat{B})\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\Big \vert \, (\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})\in \widetilde{S}_\iota \right\} ,\\ V_\kappa (\widetilde{E},Q)&:= \left\{ (\widehat{J},\widehat{B})\in \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times m}\,\Big \vert \, (\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})\in \Sigma _{\ell ,n,m}^{sdH, = r} \right\} \end{aligned}$$

and show that $\widetilde{S}_{\kappa }(\widetilde{E},Q)$ is relative generic in $V_\kappa (\widetilde{E},Q)$ or, in view of [8, Lemma 2.13] equivalently, $\widetilde{S}_{\kappa }(\widetilde{E},Q)\ne \emptyset$. We observe that the quintuples $(\widetilde{E},\widehat{J},\widehat{R},Q,\widehat{B})\in \widetilde{S}_{\iota }(\widetilde{E},Q)$ constructed in Step 2 are such that $(\widetilde{E},\widehat{J},Q,\widehat{B})\in \widetilde{S}_{\kappa }(\widetilde{E},Q)$. Therefore, we may proceed as in Step 2 to see that $\widetilde{S}_\kappa$ is indeed relative Euclidean dense in $\Sigma _{\ell ,n,m}^{H, = r}$.

Step 4: In Step 1, we constructed relative Zariski-open sets $\widetilde{S}_\iota \subseteq \Sigma _{\ell ,n,m}^{sdH, = r}$ and $\widetilde{S}_\kappa \subseteq \Sigma _{\ell ,n,m}^{H, = r}$ so that $\widetilde{S}_\iota \subseteq S_\iota$ and $\widetilde{S}_\kappa \subseteq S_\kappa$. In Step 2, we have verified that $\widetilde{S}_\iota$ and $\widetilde{S}_\kappa$ are relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH, = r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$, respectively. Therefore, we conclude that $\widetilde{S}_\iota$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH,=r}$ and $\widetilde{S}_\kappa$ is relative generic in $\Sigma _{\ell ,n,m}^{H,=r}$. Hence, [7, Proposition 2.3 (c)] implies that $S_\iota$ and $S_\kappa$ are relative generic in $\Sigma _{\ell ,n,m}^{sdH,=r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$, respectively. $\square$

For convenience, recall the definition of elementary matrices.

Definition 4.11

Let $n\in \mathbb {N}$, $i,j\in \underline{n}$ and $\lambda \in \mathbb {R}$. We denote with $E^{i,j,n}\in \mathbb {R}^{n\times n}$ the matrix defined as

$$\begin{aligned} \forall a,b\in \underline{n}: \left( E^{i,j,n}\right) _{a,b} := \delta _{i,a}\delta _{j,b}, \end{aligned}$$

where $\delta _{\alpha ,\beta }$ denotes the Kronecker delta symbol. We consider the following elementary matrices

$$\begin{aligned} T_{i\leftrightarrow j}&:= I_n-E^{i,i,n}-E^{j,j,n}+E_{i,j,n}+E_{j,i,n},\\ T_{i\rightarrow i+\lambda j}&:= I_n + \lambda E^{i,j,n},\\ T_{i\rightarrow \lambda i}&:= I_n+(\lambda -1) E^{i,i,n}. \end{aligned}$$

Lemma 4.12

Let $\star \in \left\{ \le ,= \right\}$. The set

$$\begin{aligned} S_{\delta } := \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\left| \,\begin{array}{l}\forall Z\in \mathbb {R}^{\ell \times (\ell -r)}~\text {with}~\textrm{im}\,Z = \ker E^\top \\ \ : \textrm{rk}\,[E^\top ,Q^\top (J+R)Z,Q^\top B] = \min \left\{ \ell ,n \right\} \end{array}\right. \right\} \end{aligned}$$

is relative generic in $\Sigma _{\ell ,n,m}^{sdH, \star r}$. Similarly,

$$\begin{aligned} S_{\vartheta } := \left\{ (E,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\left| \,\begin{array}{l}\forall Z\in \mathbb {R}^{\ell \times (\ell -r)}~\text {with}~\textrm{im}\,Z = \ker E^\top \\ \ : \textrm{rk}\,[E^\top ,Q^\top JZ,Q^\top B] = \min \left\{ \ell ,n \right\} \end{array}\right. \right\} \end{aligned}$$

is relative generic in $\Sigma _{\ell ,n,m}^{H, \star r}$.

Proof

By Lemma 4.6 and [8, Proposition 2.10 (e)], it suffices to show the assertion for the reference sets $\Sigma _{\ell ,n,m}^{sdH,=r}$ and $\Sigma _{\ell ,n,m}^{H,=r}$, respectively. This proof is an adapted version of the proof of [7, Proposition 2.6 (iv)]. In each step of the original proof, we have to keep in mind that, in addition to the condition $\textrm{rk}\,E\le r$, the condition $E^\top Q = Q^\top E\ge 0$ has to be met. We distinguish between the cases $r = \ell$ and $r<\ell$. If $r = \ell$, then we have the inclusions

$$\begin{aligned} S_{\alpha }\subseteq S_{\delta }\quad \text {and}\quad S_{\varepsilon }\subseteq S_{\vartheta }. \end{aligned}$$

In view of [7, Proposition 2.3 (c)]), we conclude from Lemma 4.8 that $S_\delta$ and $S_\vartheta$ are relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$ and $\Sigma _{\ell ,n,m}^{H, = r}$, respectively. In the remainder of the proof, let $r<\ell$. Equivalently, for all quintuples $(E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}$, $\ker E^\top \ne \left\{ 0 \right\}$. As in the proof of Lemma 4.8, we distinguish further between the reference sets $\Sigma _{\ell ,n,m}^{sdH, = r}$ and $\Sigma _{\ell ,n,m}^{H, = r}$.

Reference set $\Sigma _{\ell ,n,m}^{sdH, = r}$: We proceed in steps.

Step 1: The following is almost an exact copy of Step (iv).1 in the proof of [7, Proposition 2.6] with minor changes due to the fact that the reference set does (unless $r = 0$) not allow a representation of the form $\Sigma _{\ell ,n,m}^{sdH, = r} = \left\{ E\in S_1 \right\} \times \left\{ (J,R,Q,B)\in S_2 \right\}$ for some sets $S_1\subseteq \mathbb {R}^{\ell \times n}$ and $S_2\subseteq \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times \ell }\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$. We construct, for all $\rho \in \left\{ 0,\ldots ,r \right\}$, a function $T^\rho$ that applies the first $\rho$ steps of the Gaussian elimination without interchange of rows and without normalization to $E^\top$. Let $D_0:= \Sigma _{\ell ,n,m}^{sdH, = r}$ and define the functions

$$\begin{aligned} T_0:\Sigma _{\ell ,n,m}^{sdH, = r}\rightarrow \mathbb {R}^{n\times \ell },\quad (E,J,R,Q,B)\mapsto E^\top \end{aligned}$$

and

$$\begin{aligned} \tau _0: \Sigma _{\ell ,n,m}^{sdH, = r}\rightarrow \textbf{Gl}(\mathbb {R}^n),\quad (E,J,R,Q,B)\mapsto I_n. \end{aligned}$$

For $i\in \underline{r}$, we consider the sets $D_i:= \left\{ (E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}\,\big \vert \,E_{i,i}\ne 0 \right\}$ and the functions

$$\begin{aligned} \tau _i: D_i\rightarrow \textbf{Gl}(\mathbb {R}^n),\quad (E,J,R,Q,B)\mapsto \begin{bmatrix} 1\\ &{} \ddots \\ &{} &{} 1\\ &{} &{} -\frac{E_{i,i+1}}{E_{i,i}} &{} 1\\ &{} &{} \vdots &{} &{} \ddots \\ &{} &{} -\frac{E_{i,n}}{E_{i,i}} &{} &{} &{} 1 \end{bmatrix}. \end{aligned}$$

Using $\tau _i$, define the function

$$\begin{aligned} T_i: D_i\rightarrow \mathbb {R}^{n\times \ell },\quad (E,J,R,Q,B)\mapsto \tau _i(E,J,R,Q,B) E^\top . \end{aligned}$$

The functions $\tau _i$ (and hence $T_i$) are rational matrices, i.e. the concatenation of each canonical projection with $\tau _i$ (and $T_i$) is the restriction of a rational function (in the entries of (E, J, R, Q, B)) to $D_i$. Define, recursively, for $\rho \in \left\{ 0,\ldots ,r \right\}$ the functions $\tau ^\rho$, $T^\rho$ and sets $D^\rho$ as

$$\begin{aligned} D^0&:= D_0,\\ T^0&:= T_0,\\ \tau ^0&:= \tau _0,\\ D^{i+1}&:= \left\{ (E,J,R,Q,B)\in D^i\,\big \vert \, T^i(E,J,R,Q,B)\in D_{i+1} \right\} ,\\ T^{i+1}&:= T_{i+1}\circ T^i,\\ \tau ^{i+1}&: D^{i+1}\rightarrow \textbf{Gl}(\mathbb {R}^n), \quad (E,J,R,Q,B)\mapsto \tau _{i+1}\\&\quad (T^i(E,J,R,Q,B),J,R,Q,B)\tau ^i(E,J,R,Q,B). \end{aligned}$$

Analogously to Step (iv).1 in the proof of [7, Proposition 2.6], it can be easily verified that $T^\rho$ formalizes the first $\rho$ steps of the Gaussian algorithm applied to $E^\top$ for $(E,J,R,Q,B)\in D^\rho$; we omit the details.

Step 2: We show, by induction on $\rho \in \left\{ 0,\ldots ,r \right\}$, that $D^\rho$ is dense with respect to the Euclidean relative topology on $\Sigma _{\ell ,n,m}^{sdH, = r}$. For $\rho = 0$, this is trivial as $D^\rho = \Sigma _{\ell ,n,m}^{sdH, = r}$. If $r = 0$, the step is complete. Let, in the following, $r\ge 1$ and $\rho \in \underline{r}$. By induction assumption, $D^{\rho -1}$ is relative Euclidean dense in $\Sigma _{\ell ,n,m}^{sdH, = r}$. Therefore, it remains to verify that $D^{\rho }$ is relative Euclidean dense in $D^{\rho -1}$. Let $\varepsilon >0$ and $(E,J,R,Q,B)\in \Sigma _{\ell ,n,m}^{sdH, = r}$. If

$$\begin{aligned} \forall i\in \left\{ \rho ,\ldots ,n \right\} ~\forall j\in \left\{ \rho ,\ldots ,\ell \right\} : T^{\rho -1}(E,J,R,Q,B)_{i,j} = 0, \end{aligned}$$

then we have from our definition of $T^{\rho -1}$ (and the definition of the $\tau _0,\ldots ,\tau _{\rho -1}$)

$$\begin{aligned} r = \textrm{rk}\,E&= \textrm{rk}\,E^\top = \textrm{rk}\,\tau ^{\rho -1}(E,J,R,Q,B)E^\top = \textrm{rk}\,T^{\rho -1}(E,J,R,Q,B) \\&= \rho -1, \end{aligned}$$

which is impossible due to $\rho \in \underline{r}$. Hence, we conclude

$$\begin{aligned} \exists (\iota ,\kappa )\in \left\{ \rho ,\ldots ,n \right\} \times \left\{ \rho ,\ldots ,\ell \right\} : T^{\rho -1}(E,J,R,Q,B)_{\iota ,\kappa }\ne 0. \end{aligned}$$

For the sake of brevity, put $\vartheta _{\rho -1}:= \tau ^{\rho -1}(E,J,R,Q,B)$. Let $\delta >0$ with

$$\begin{aligned} \delta < \min \left\{ 1,\frac{\varepsilon }{2n\left\| T^{\rho -1}(E,J,R,Q,B) \right\| \left\| \vartheta _{\rho -1}^{-1} \right\| },\frac{\varepsilon }{2n\left\| Q\vartheta _{\rho -1}^\top \right\| \left\| \vartheta _{\rho -1}^{-1} \right\| } \right\} \end{aligned}$$

so that

$$\begin{aligned} T^{\rho -1}(E,J,R,Q,B)_{\rho ,\kappa }+\delta T^{\rho -1}(E,J,R,Q,B)_{\iota ,\kappa }\ne 0. \end{aligned}$$

Defining the matrices

$$\begin{aligned} \widetilde{E} := T_{\rho \rightarrow \rho +\delta \kappa }^\top E \vartheta _{\rho -1}^\top T_{\rho \rightarrow \rho +\delta \iota }^\top \vartheta _{\rho -1}^{-\top }\quad \text {and}\quad \widetilde{Q} := T_{\rho \rightarrow \rho -\delta \kappa } Q \vartheta _{\rho -1}^\top T_{\rho \rightarrow \rho +\delta \iota }^\top \vartheta _{\rho -1}^{,-\top } \end{aligned}$$

we have

$$\begin{aligned} \left\| E-\widetilde{E} \right\|&= \left\| E^\top -\widetilde{E}^\top \right\| \\&= \left\| \vartheta _{\rho -1}^{-1}\left( T^{\rho -1}(E,J,R,Q,B)-\vartheta _{\rho -1}\widetilde{E}^\top \right) \right\| \\&\le n\left\| \vartheta _{\rho -1} \right\| \left\| T^{\rho -1}(E,J,R,Q,B) \right\| \\&\quad {- T_{\rho \rightarrow \rho +\delta \iota }T^{\rho -1}(E,J,R,Q,B)T_{\rho \rightarrow \rho +\delta \kappa }}\\&\le n\left\| \vartheta _{\rho -1} \right\| \delta 2\left\| T^{\rho -1}(E,J,R,Q,B) \right\| < \varepsilon . \end{aligned}$$

Analogously, it is verified that $\left\| Q-\widetilde{Q} \right\| <\varepsilon$. Furthermore, since

$$\begin{aligned} T_{\rho \rightarrow \rho -\delta \kappa }T_{\rho \rightarrow \rho +\delta \kappa } = I_n, \end{aligned}$$

we have

$$\begin{aligned} \widetilde{E}^\top \widetilde{Q} = \vartheta _{\rho -1}^{-1} T_{\rho \rightarrow \rho +\delta \iota } \vartheta _{\rho -1}E^\top Q \vartheta _{\rho -1}^\top T_{\rho \rightarrow \rho +\delta \iota }^\top \vartheta _{\rho -1}^{-\top }, \end{aligned}$$

which is symmetric and positive semidefinite as $E^\top Q$ is symmetric and positive semidefinite. Furthermore, the nature of our multiplicative perturbation of E ensures that

$$\begin{aligned} \forall i,j\in \underline{\rho -1}: \widetilde{E}_{i,j} = E_{i,j}. \end{aligned}$$

Hence, we conclude

$$\begin{aligned} T^{\rho -1}(\widetilde{E},J,R,\widetilde{Q},B)_{\rho ,\rho }&= \left( \tau ^{\rho -1}(\widetilde{E},J,R,\widetilde{Q},B)\widetilde{E}^\top \right) _{\rho ,\rho }\\&= \left( \tau ^{\rho -1}(E,J,R,Q,B)\widetilde{E}^\top \right) _{\rho ,\rho }\\&{=} \delta \left( T^{\rho {-}1}(E,J,R,Q,B)_{\rho ,\kappa }{+}\delta T^{\rho -1}(E,J,R,Q,B)_{\iota ,\kappa }\right) \\&\ne 0. \end{aligned}$$

This verifies that

$$\begin{aligned} (\widetilde{E},J,R,\widetilde{Q},B)\in D^\rho \cap \mathbb {B}((E,J,R,Q,B),\varepsilon ) \end{aligned}$$

and thus we conclude that $D^{\rho }$ is indeed dense with respect to the Euclidean relative topology on $\Sigma _{\ell ,n,m}^{sdH,=r}$.

Step 3: Analogously to steps (iv).2 and (iv).3 in the proof of [7, Proposition 2.6], it can be verified that $D^\rho$ is, for all $\rho \in \left\{ 0,\ldots ,r \right\}$, a relative Zariski-open subset of $\Sigma _{\ell ,n,m}^{sdH, = r}$; we omit the details. Hence, we conclude with Step 2 of the present proof that each $D^\rho$—and especially $D^r$—is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$.

Step 4: We construct, analogously to step (iv).6 in the proof of [7, Proposition 2.6], a polynomial matrix

$$\begin{aligned} Z: D^r\rightarrow \mathbb {R}^{\ell \times (\ell -r)} \end{aligned}$$

so that for all $(E,J,R,Q,B)\in D^r$, $\textrm{im}\,Z(E,J,R,Q,B) = \ker E^\top$. Here, for Z being a polynomial function means that there is a polynomial matrix $P\in \mathbb {R}[x_1,\ldots ,x_{\ell (2\ell +2n+m}]^{\ell \times (\ell -r)}$ so that its restriction to $D^r$ is Z, i.e. $P\vert _{D^r} = Z$. For all $i\in \left\{ r+1,\ldots ,\ell \right\}$ and $j\in \underline{n-r}$, define the (constant) functions

$$\begin{aligned} z_{i,j}: D^r\rightarrow \mathbb {R},\qquad (E,J,R,Q,B)\mapsto \delta _{i,j+r}, \end{aligned}$$

where $\delta _{i,j+r}$ denotes the Kronecker delta symbol of i and $j+r$. Define further, recursively, for $i\in \underline{r}$ and $j\in \underline{n-r}$ the functions

$$\begin{aligned} z_{i,j}:D^r\rightarrow \mathbb {R},\quad \mathcal {E} = (E,J,R,Q,B)\mapsto -\frac{1}{T^r(\mathcal {E})_{i,i}}\sum _{k = i+1}^\ell T^r(\mathcal {E})_{i,k}z_{k,j}(\mathcal {E}). \end{aligned}$$

From our definition of $D^r$, we do, in fact, not divide by zero and the functions $z_{i,j}$ are well defined. Furthermore, since $T^r$ is a rational matrix, we conclude that the $z_{i,j}$ are rational functions (or rather the restrictions of rational functions to $D^r$). Hence, there are polynomials $p_{i,j},q_{i,j}$ with $q_{i,j}\ne 0$, $i\in \underline{\ell }$, $j\in \underline{n-r}$ so that

$$\begin{aligned} z_{i,j} = \frac{p_{i,j}\vert _{D^r}}{q_{i,j}\vert _{D^r}}. \end{aligned}$$

Define the polynomial matrix Z by

$$\begin{aligned} \forall i\in \underline{\ell }~\forall j\in \underline{\ell -r}: \big (Z(\cdot )\big )_{i,j} := z_{i,j}(\cdot )\prod _{s = 1}^{\ell }\prod _{t = 1}^{\ell -r} q_{i,j}(\cdot ). \end{aligned}$$

Then, for all $(E,J,R,Q,B)\in D^r$, we have

$$\begin{aligned} \tau ^{r}(E,J,R,Q,B)E^\top Z(E,J,R,Q,B) = 0 \end{aligned}$$

or, equivalently (since $\tau ^{r}(E,J,R,Q,B)\in \textbf{Gl}(\mathbb {R}^n)$), $\textrm{im}\,Z(E,J,R,Q,B)\subseteq \ker R$. By construction, the columns of Z(E, J, R, Q, B) are linearly independent and thus the rank-nullity theorem yields that $\textrm{im}\,Z(E,J,R,Q,B) = \ker R$.

Step 5: We show that the set

$$\begin{aligned} \widetilde{S}_\delta&:= \left\{ ( \right\} E,J,R,Q,B)\in D^r\,\big \vert \,\textrm{rk}\,[E^\top ,Q^\top (J+R)Z(E,J,R,Q,B),Q^\top B] \\&= \min \left\{ \ell ,n \right\} \end{aligned}$$

is relative generic in $D^r$. Due to Lemma 4.7, $\widetilde{S}_\delta$ is a relative Zariski-open subset of $D^r$. Therefore, it remains to verify that $\widetilde{S}_\delta$ is dense with respect to the relative Euclidean topology on $D^r$. Let $\varepsilon >0$ and $\mathcal {E}:= (E,J,R,Q,B)\in D^r$. By Lemma 4.1, there are an orthogonal matrix $O\in \textbf{Gl}(\mathbb {R}^\ell )$, an invertible matrix $T\in \textbf{Gl}(\mathbb {R}^n)$, a symmetric and positive semidefinite matrix $\Sigma \in \mathbb {R}^{r\times r}$ and matrices $P_1\in \mathbb {R}^{(\ell -r)\times r}$ and $P_2\in \mathbb {R}^{(\ell -r)\times (n-r)}$ so that

$$\begin{aligned} OET = \begin{bmatrix} I_r &{} 0\\ 0 &{} 0 \end{bmatrix}\quad \text {and}\quad OQT = \begin{bmatrix} \Sigma &{} 0\\ P_1 &{} P_2 \end{bmatrix}. \end{aligned}$$

With this transformation in mind, we conclude from the orthogonality of O

$$\begin{aligned} 0 = T^\top E^\top Z(\mathcal {E}) = T^\top E^\top O^\top OZ(\mathcal {E}) = \begin{bmatrix} I_r &{} 0\\ 0 &{} 0 \end{bmatrix}SZ(\mathcal {E}) \end{aligned}$$

and therefore we conclude from invertibility of T that there is a matrix $\widetilde{Z}(\mathcal {E})\in \mathbb {R}^{(\ell -r)\times (\ell -r)}$ so that

$$\begin{aligned} SZ(\mathcal {E}) = \begin{bmatrix} 0_{r\times (\ell -r)}\\ \widetilde{Z}(\mathcal {E}) \end{bmatrix}. \end{aligned}$$

Since $Z(\mathcal {E})$ has full (column-)rank, we conclude that $\widetilde{Z}(\mathcal {E})$ is invertible. Furthermore, $Z(\mathcal {E})$ (and hence $\widetilde{Z}(\mathcal {E})$ depends by construction only on E, which we highlight by writing $Z_E:= \widetilde{Z}(\mathcal {E})$. We decompose J, R and B into the form

$$\begin{aligned} OJO^\top = \begin{bmatrix} J_{1,1} &{} J_{1,2}\\ -J_{1,2}^\top &{} J_{2,2} \end{bmatrix},\quad ORO^\top = \begin{bmatrix} R_{1,1} &{} R_{1,2}\\ R_{1,2}^\top &{} R_{2,2} \end{bmatrix},\quad \text {and}\quad OB = \begin{bmatrix} B_1\\ B_2 \end{bmatrix} \end{aligned}$$

for suitable matrices $J_{1,1},R_{1,1}\in \mathbb {R}^{r\times r}$, $J_{1,2},R_{1,2}\in \mathbb {R}^{r\times (\ell -r)}$, $J_{2,2},R_{2,2}\in \mathbb {R}^{(\ell -r)\times (\ell -r)}$, $B_1\in \mathbb {R}^{r\times m}$ and $B_2\in \mathbb {R}^{(\ell -r)\times n}$. Then, we have

$$\begin{aligned}&\textrm{rk}\,[E^\top ,Q^\top (J+R)Z,Q^\top B] \\&\quad = \textrm{rk}\,T^\top [E^\top ,Q^\top (J+R)Z,Q^\top B]\begin{bmatrix} O^\top \\ {} &{}I_{(\ell -r)}\\ {} &{}&{}I_m \end{bmatrix}\\&\quad = \textrm{rk}\,[(OET)^\top ,(OQT)^\top O(J+R)O^\top OZ(\mathcal {E}),(OQT)^\top OB]\\&\quad = \textrm{rk}\,\left[ \begin{array}{c|c|c} \begin{array}{cc} I_r &{} 0\\ 0 &{} 0 \end{array} &{} \begin{bmatrix} \Sigma ^\top &{} P_1^\top \\ 0 &{} P_2^\top \end{bmatrix}\begin{bmatrix} (J_{1,2}+R_{1,2})Z_E\\ (J_{2,2}+R_{2,2})Z_E \end{bmatrix} &{} \begin{bmatrix} \Sigma ^\top &{} P_1^\top \\ 0 &{} P_2^\top \end{bmatrix}\begin{bmatrix} B_1\\ B_2 \end{bmatrix} \end{array}\right] . \end{aligned}$$

We conclude that if $r = n$, then $(E,J,R,Q,B)\in \widetilde{S}_\varepsilon$ and therefore $\widetilde{S}_\varepsilon = D^r$ so that the former is especially relative generic in the latter. In the following, we assume that $r<n$. In this case, we have $(E,J,R,Q,B)\in \widetilde{S}_\varepsilon$ if, and only if,

$$\begin{aligned} \textrm{rk}\,[P_2^\top (J_{2,2}+R_{2,2})Z_E,P_2^\top B_2] = \min \left\{ \ell ,n \right\} -r \end{aligned}$$

(4.7)

Since $Z_E$ is invertible, (4.7) is fulfilled if, and only if,

$$\begin{aligned} \textrm{rk}\,P_2^\top [(J_{2,2}+R_{2,2}),B_2] = \min \left\{ \ell ,n \right\} -r \end{aligned}$$

Consider the convex nonempty set

$$\begin{aligned} C := \left\{ (\widetilde{P}_2,\widetilde{J},\widetilde{R},\widetilde{B}_2)\in \mathbb {R}^{(\ell -r)\times (n-r)}\times \left( \mathbb {R}^{\ell \times \ell }\right) ^2\times \mathbb {R}^{(\ell -r)\times m}\,\left| \,\begin{array}{l} \widetilde{J}^\top = -\widetilde{J},\\ \widetilde{R}^\top = \widetilde{R}\ge 0 \end{array}\right. \right\} . \end{aligned}$$

Let

$$\begin{aligned} \pi : \mathbb {R}^{\ell \times \ell }\rightarrow \mathbb {R}^{(\ell -r)\times (\ell -r)},\quad A\mapsto [A_{i,j}]_{i,j\in \left\{ r+1,\ldots ,\ell \right\} } \end{aligned}$$

and define the set

$$\begin{aligned} C_\delta := \left\{ (\widetilde{P}_2,\widetilde{J},\widetilde{R},\widetilde{B}_2)\in C\,\big \vert \,\textrm{rk}\,\widetilde{P}_2^\top \left[ \left( \pi (\widetilde{J})+\pi (\widetilde{R})\right) Z_E, \widetilde{B}_2\right] = \min \left\{ \ell ,n \right\} -r \right\} , \end{aligned}$$

which is a relative Zariski-open subset of C. Considering the matrices

$$\begin{aligned} \widetilde{R} =&0,\quad \widetilde{J} = \begin{bmatrix} 0 &{} 1\\ -1 &{} 0 &{} 1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} -1 &{} 0 &{} 1\\ &{} &{} &{} -1 &{} 0 \end{bmatrix},\quad \widetilde{B}_2 = [e_{\ell -r},0_{(\ell -r)\times (m-1)}],\quad \\ \widetilde{P}_2 =&\begin{bmatrix} I_{\min \left\{ \ell -r,n-r \right\} } &{} 0\\ 0 &{} 0 \end{bmatrix}, \end{aligned}$$

we see that $C_\varepsilon$ is nonempty. Therefore, [8, Lemma 2.13] yields that $C_\varepsilon$ is relative generic in C. Particularly, we conclude that there are $(\widetilde{P}_2,\widetilde{J},\widetilde{R},\widetilde{B}_2)\in C_\varepsilon$ so that

$$\begin{aligned}&\max \left\{ \left\| \widetilde{P}_2-P_2 \right\| ,\left\| OJO^\top -\widetilde{J} \right\| ,\left\| ORO^\top -\widetilde{R} \right\| ,\left\| B_2-\widetilde{B_2} \right\| \right\} \\&\quad <\frac{\varepsilon }{\ell ^2 n\left\| O \right\| \max \left\{ 1,\left\| O \right\| ,\left\| T^{-1} \right\| \right\} }. \end{aligned}$$

Define

$$\begin{aligned} \widehat{Q} := O^\top \begin{bmatrix} \Sigma &{} 0\\ P_1 &{} \widetilde{P}_2 \end{bmatrix}T^{-1},\quad \widehat{J} := O^\top \widetilde{J}O,\quad \widehat{R} := O^\top \widetilde{R}O,\quad \widehat{B} := O^\top \begin{bmatrix} B_1\\ \widetilde{B}_2 \end{bmatrix}. \end{aligned}$$

From our construction of $C_\delta$, we conclude that $(E,\widehat{J},\widehat{R},\widehat{Q},\widehat{B})\in \widetilde{S}_\delta$. Furthermore, we have

$$\begin{aligned} \left\| J-\widehat{J} \right\| = \left\| O^\top O(J-\widehat{J})O^\top O \right\| \le \ell ^2\left\| O \right\| ^2\left\| OJO-\widetilde{J} \right\| <\varepsilon ; \end{aligned}$$

analogously we can prove that $(E,\widehat{J},\widehat{R},\widehat{Q},\widehat{B})\in \mathbb {B}((E,J,R,Q,B),\varepsilon )$. This proves that $\widetilde{S}_\delta$ is indeed dense with respect to the relative Euclidean topology on $D^r$.

Step 6: In Step 3, we have verified that $D^r$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$ and in Step 5, we have verified that $\widetilde{S}_{\delta }$ is relative generic in $D^r$. Therefore, [7, Proposition 2.3 (f)] yields that $\widetilde{S}_\delta$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$. Furthermore, the obvious inclusion $\widetilde{S}_{\delta }\subseteq S_\delta$ yields in view of [7, Proposition 2.3 (c)] that $S_\delta$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$. This completes the proof of the lemma for the reference set $\Sigma _{\ell ,n,m}^{sdH,=r}$

Reference set $\Sigma _{\ell ,n,m}^{H,=r}$: This may be treated analogously to the reference set $\Sigma _{\ell ,n,m}^{sdH, = r}$; simply remove the Cartesian factor associated with the dissipation matrix R and observe that all statements remain true. We omit the details. $\square$

4.2 Proof of Theorem 2.4

In view of [8, Proposition 2.10 (e)] and Lemma 4.6, it suffices to consider the reference sets $\Sigma _{\ell ,n,m}^{sdH, = r}$ and $\Sigma _{\ell ,n,m}^{H,= r}$.

Reference set $\Sigma _{\ell ,n,m}^{sdH,=r}$:   (i)   By Definition-Proposition 2.1, we have

$$\begin{aligned} S_{\text {obs. at}~\infty }^{sdH} {\left\{ \begin{array}{ll} = S_\alpha , &{} n = \min \left\{ \ell ,n,r+m \right\} ,\\ \subseteq S_\alpha ^c, &{} \text {else}. \end{array}\right. } \end{aligned}$$

Therefore, if $n = \min \left\{ \ell ,n,r+m \right\}$ or, equivalently, $n \le \min \left\{ \ell ,r+m \right\}$, Lemma 4.8 ($\alpha$) yields that $S_{\text {obs. at}~\infty }^{sdH}$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$. Otherwise, Lemma 4.8 ($\alpha$) and the inclusion $S_\alpha \subseteq (S_{\text {obs. at}~\infty }^{sdH})^c$ yield with [7, Proposition 2.3 (c)] that the latter is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$.

(ii)   For $S_{\text {impulse obs.}}^{sdH}$, Definition-Proposition 2.1 yields

$$\begin{aligned} S_{\text {impulse obs.}}{\left\{ \begin{array}{ll} = S_\delta , &{} n\le \ell \\ \subseteq S_\delta ^c, &{} \text {else}. \end{array}\right. } \end{aligned}$$

The assertion can now be derived from Lemma 4.12 analogously as to (i).

(iii)   In this case, we have

$$\begin{aligned} S_{\text {behavioural obs.}}^{sdH} {\left\{ \begin{array}{ll} = S_\iota , &{} n\le \ell ,\\ \subseteq S_\iota ^c, &{} \text {else}, \end{array}\right. } \end{aligned}$$

by virtue of Definition-Proposition 2.1. Analogously to (i) and (ii), the assertion is a direct consequence of Lemma 4.10.

(iv) Corollary 2.2 (ii) noted the identity

$$\begin{aligned} S_{\text {completely obs.}}^{sdH} = S_{\text {obs. at}~\infty }^{sdH}\cap S_{\text {behavioural obs.}}^{sdH}. \end{aligned}$$

Thus, [7, Proposition 2.3 (d)] yields that $S_{\text {completely obs.}}^{sdH}$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH,=r}$ if, and only if, both $S_{\text {obs. at}~\infty }^{sdH}$ and $S_{\text {behavioural obs.}}^{sdH}$ are relative generic in $\Sigma _{\ell ,n,m}^{sdH,=r}$. In view of (i) and (iii), this is the case if, and only if, $n\le \min \left\{ \ell ,r+m \right\}$.

(v)   Analogously to (iv), the identity

$$\begin{aligned} S_{\text {strongly obs.}}^{sdH} = S_{\text {impulse obs.}}^{sdH}\cap S_{\text {behavioural obs.}}^{sdH}, \end{aligned}$$

which was observed in Corollary 2.2 (i), yields with (ii) and (iii) of the present theorem in convolution with [7, Proposition 2.3 (d)] the assertion.

(vi)   The sets $S_\alpha$ and $S_\beta$ are, due to Lemma 4.8 ($\alpha$) and ($\beta$), relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$. Furthermore, Definition-Proposition 2.1 yields the inclusions

$$\begin{aligned} S_\alpha \cap S_\beta \subseteq {\left\{ \begin{array}{ll} S_{\text {RS obs. at}~\infty }^{sdH}, &{} r\ge \min \left\{ \ell ,n \right\} -m,\\[2ex] \left( S_{\text {RS obs. at}~\infty }^{sdH}\right) ^c, &{} \text {else}. \end{array}\right. } \end{aligned}$$

Hence, [7, Proposition 2.3 (c) and (d)] yields that $S_{\text {RS obs. at}~\infty }^{sdH}$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH,=r}$ if, and only if, $r\ge \min \left\{ \ell ,n \right\} -m$, and otherwise, its complement is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$.

(ix) Definition-Proposition 2.1 yields that the inclusion

$$\begin{aligned} S_{\gamma }\cap S_\iota \subseteq S_{\text {RS behavioural observable}}^{sdH} \end{aligned}$$

holds true. Therefore, [7, Proposition 2.3 (c) and (d)] yields with Lemma 4.8 ($\gamma$) and Lemma 4.10 that $S_{\text {RS behavioural observable}}^{sdH}$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$.

(vii) With [7, Proposition 2.3 (c) and (d)], this is an immediate consequence of Corollary 2.2 (iii) and (vi) and (ix) of the present theorem.

(viii) Lemma 4.8 ($\beta$) and ($\gamma$) imply that $S_\beta$ and $S_\gamma$ are relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$. Since the inclusion

$$\begin{aligned} S_\beta \cap S_\gamma \subseteq S_{\text {RS impulse observable}}^{sdH} \end{aligned}$$

holds true in view of Definition-Proposition 2.1, [7, Proposition 2.3 (c) and (d)] implies that $S_{\text {RS impulse observable}}^{sdH}$ is relative generic in $\Sigma _{\ell ,n,m}^{sdH, = r}$.

(x) Using [7, Proposition 2.3 (c) and (d)], we see that this is an immediate consequence of (viii) and (ix) of the present theorem.

Reference set $\Sigma _{\ell ,n,m}^{H,=r}$: This may be analogously treated as the reference set $\Sigma _{\ell ,n,m}^{sdH,=r}$. One must only replace “sdH” by “H” and the indices $\alpha$, $\beta$, $\gamma$, $\delta$ and $\iota$ by $\varepsilon$, $\eta$, $\vartheta$ and $\kappa$, respectively. This completes the proof of the theorem. $\square$

5 Conclusion and outlook

We have derived criteria for (relative) genericity of observability for linear time-invariant differential-algebraic systems and in particular for port-Hamiltonian descriptor systems. This extends on the one hand the well-known genericity of observability of linear time-invariant ordinary differential equations [15] to DAEs, and on the other hand the (relative) genericity of controllability of unstructured [5,6,7] and port-Hamiltonian DAEs [8] to observability.

The methods developed in Sect. 4.1 may be applied to extend [8, Theorem 4.1] to the reference sets $\Sigma _{\ell ,n,m}^{sdH,\le r}$ and $\Sigma _{\ell ,n,m}^{H,\le r}$, i.e. characterize the genericity of controllability for port-Hamiltonian DAEs with rank constraint on the system matrix E. Furthermore, this note restricts itself to systems with real coefficients. Using the notion of genericity on complex coordinate spaces suggested in [8, Definition 2.12], we may replace the may replace the transposition with the Hermitian conjugation in the proofs of this note and see that Theorem 2.3 and Theorem 2.4 should still hold true.