Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable

Abstract

The present work is a successor of Ilchmann and Kirchhoff (Math Control Signals Syst 33:359–377, 2021) on generic controllability and of Ilchmann and Kirchhoff (Math Control Signals Syst 35:45–76, 2022) on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

1 Introduction

In the present work we characterize (relative) genericity of controllability and stabilizability of linear finite-dimensional port-Hamiltonian descriptor systems described by linear differential-algebraic input-state-output systems of the following form [32]

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

(1)

where $B\in {\mathbb {K}}^{\ell \times m}$, $J = -J^*\in {\mathbb {K}}^{\ell \times \ell }$, ${\mathbb {K}}^{\ell \times \ell }\ni R = R^*\ge 0$ and $E,Q\in {\mathbb {K}}^{\ell \times n}$ with $E^* Q = Q^*E\ge 0$.

In this work, we study controllability as a fundamental property of control systems for the highly structured class of port-Hamiltonian systems which gained a lot of attention over the last decade. For unstructured systems, it has been shown that controllability is a prevalent property [18, 43], which is formalised in the concept of genericity. When studying genericity of controllability for port-Hamiltonian descriptor systems, one has to cope with the two aspects of the subtle structure of the underlying the port-Hamiltonian as well as the structure of the descriptor system. This requires a more sophisticated concept of genericity, namely relative, with respect to a reference set, genericity. The proofs of characterizations of genericity of controllability and stabilizability–in terms of the rank and dimension of certain matrices–are much more subtle than for DAEs, not to mention ordinary differential linear equation. In the end, our results may bring some “theoretical” insight into relative genericity of controllability or stabilizability for port-Hamiltonian DAEs. Moreover, we suggest that the structural ideas of the proofs, which are intimately related to perturbation theory, may yield insight for analysing numerically stable representations of port-Hamiltonian DAEs, cf. e.g. [30].

We give a brief literature review on both subjects: After Rudolf Kalman had introduced and characterized in 1960 [21, 22] controllability of linear systems of the form

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

(2)

where $(A,B)\in {\mathbb {K}}^{n\times n}\times {\mathbb {K}}^{n\times m}$ for $n,m\in {\mathbb {N}}^*$, Lee and Markus [26] proved in 1967 that the set of all controllable systems (2) is open and dense with respect to the Euclidean topology. In 1974, Wonham [43] showed that this set is not only open and dense but also conull w.r.t. the Lebesgue measure, i.e. its complement is a Lebesgue null set. In 2019, Hinrichsen and Oeljeklaus [15] improved Wonham’s result. They proved that in the case $m\ge 2$ the set of controllable systems (2) is generically convex, i.e. the set of pairs of tuples $((A_1,B_1),(A_2,B_2))$ so that all the systems (2) associated to the tuples $(A_1+\lambda (A_2-A_1),B_1+\lambda (B_2-B_1))$, $\lambda \in [0,1]$, are controllable is generic.

Recently, in 2021, Wonham’s results were generalized by Ilchmann and Kirchhoff [18, Thm. 2.3] to systems described by differential-algebraic equations of the form

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

(3)

where $(E,A,B)\in \varSigma _{\ell ,n,m}:= {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}$ for $\ell ,n,m\in {\mathbb {N}}^*$. It was shown that the DAE systems (3) are generically¹

$$\begin{aligned} \text {freely initializable} \quad{} & {} \text {if, and only if,} \quad ~\ell \le n+m;\nonumber \\ \text {impulse controllable}\quad{} & {} \text {if, and only if,} \quad ~\ell \le n+m;\nonumber \\ \text {behavioural controllable}\quad{} & {} \text {if, and only if}, \quad ~\ell \ne n+m;\nonumber \\ \text {completely controllable}\quad{} & {} \text {if, and only if,} \quad ~\ell< n+m;\nonumber \\ \text {strongly controllable} \quad{} & {} \text {if, and only if,} \quad ~\ell < n+m; \end{aligned}$$

(4)

and if one of the conditions on the right-hand side is violated, then the associated controllability notion is generically violated; see Proposition 2.1 for a definition of the five controllability concepts. In 2022, Hinrichsen and Oeljeklaus [16] proved that linear time-invariant ordinary delay-differential equations are generically controllable.

However, the concept of genericity of controllability for DAE systems has the drawback that the set $\varSigma _{\ell ,n,m}$ is too “large”: if $\ell =n$, then in each arbitrarily small neighbourhood of $(E,A,B)\in \varSigma _{\ell ,n,m}$ there is some invertible $E'\in {{\mathbb {K}}}^{n\times n}$ such that $(E',A,B)$ is an ordinary differential equation. To resolve this drawback, Kirchhoff [23] introduced the concept of relative genericity,² and Ilchmann and Kirchhoff [19, Thm. 3.2] showed in 2022 that a similar characterization to (4) holds for relative genericity of controllability for various reference sets such as $\left\{ (E,A,B)\in \varSigma _{\ell ,n,m}\,\big \vert \,\textrm{rk}\,_{{\mathbb {R}}} E \le r \right\}$ for $r\in {{\mathbb {N}}}$.

In 2021, Kirchhoff [23, Prop. III.1] has shown that the set of all controllable linear port-Hamiltonian systems described by

$$\begin{aligned} \begin{aligned} \tfrac{\textrm{d}}{\textrm{d}t}x&= JQx + Bu, \end{aligned} \end{aligned}$$

(5)

where $B\in {\mathbb {K}}^{\ell \times m}$, $J = -J^*\in {\mathbb {K}}^{\ell \times \ell }$, and $Q^*=Q\in {\mathbb {K}}^{\ell \times n}$, is relative generic in the set of all systems (5).

Similar results of the aforementioned hold for stabilizability.

The brief literature review about systems port-Hamiltonian systems is as follows. Port-Hamiltonian systems were introduced in the seminal work [28]. They provide a flexible and highly structured framework to model complex phenomena by means of input-state-output dynamical systems. Application areas encompass many physical domains, e.g. mechanics, electrical circuits, thermodynamics, and multi-energy systems, cf. [14, 27, 42] for recent applications. Due to their energy-based nature, a port-Hamiltonian (pH) structure inherently ensures passivity and stability and thus enables powerful analytic tools [20], numerical methods [24, 32] or passivity-based control techniques [33, 39], to analyse, simulate or control the corresponding systems. In view of optimal control, it was recently shown that pH structures can be used to overcome challenges in singular optimal control [8,9,10], allowing results that are not available for general input-state-output systems. Another advantage of the pH model class is its closedness under power-conserving interconnection: A power-conserving coupling of pH systems is again port-Hamiltonian. Last, the class of pH systems can be shown to be particularly robust w.r.t. structured perturbations [31].

Port-Hamiltonian models arise in various formulations, such as geometric (by means of, e.g. Dirac structures [5, 6, 25] and Lagrangian subspaces [29]) or explicit state space models, both in finite- [40] and infinite-dimensional [20, 35] spaces.

This is, so far, the state of the art in relative genericity of controllability/stabilizability of descriptor port-Hamiltonian systems. In the present paper we turn to linear port-Hamiltonian differential-algebraic systems described by (1). This class is a generalization of (5) and a restriction of (3). Due to the particular structure in (1), it is not readily possible to apply the results of [18, Theorem 2.3] and of [19, Thm. 3.2]. However, utilizing Kirchhoff’s [23] slight adaptation of Wonham’s original concept, we characterize–with striking resemblance to (4)–genericity and relative genericity of controllability/stabilizability of the system class (1). The latter is the main result of the paper.

Our article is organized as follows. In Sect. 2, we define subclasses of port-Hamiltonian systems; recall the definition and algebraic characterizations of the five controllability and three stabilizability concepts; recall the definitions of (relative) genericity and prove some properties that will be intensively exploited in the proofs of the upcoming results. Section 3 is an intermediate section; we derive various lemmata and a proposition to show that subsets of systems (1) which satisfy certain algebraic constraints (the latter induced by the characterizations of controllability/stabilizability) are relative generic in some reference sets of port-Hamiltonian systems. These results lay the ground for the proofs of the main results in Sect. 4. The latter are necessary and sufficient criteria on the dimensions $\ell ,n$ and m under which port-Hamiltonian systems are relative generically controllable. Similar results for relative generic stabilizability are shown in Sect. 5. Lastly, we will give a brief overview of possible future research in Sect. 6.

2 Controllability, stabilizability, and relative genericity

In this section, we define a class of port-Hamiltonian DAE systems, recall controllability and stabilizability concepts for differential-algebraic equations, and collect results on (relative) genericity needed for later purposes.

2.1 Port-Hamiltonian systems classes

A port-Hamiltonian descriptor system is a differential-algebraic equation of the form (1). If $R>0$, then the resulting DAE is called dissipative; otherwise it is called semi-dissipative, see [3, Definition 4.1.1(xxii) and (xxiii)] and [1]. We denote the sets of all matrix quintuples associated to (semi-)dissipative port-Hamiltonian systems by

$$\begin{aligned} \varSigma _{\ell ,n,m}^{sdH}:= \left\{ (E,J,R,Q,B)\,\big \vert \,J=-J^*,R = R^*\ge 0,E^*Q = Q^*E\ge 0 \right\} \end{aligned}$$

(6)

and

$$\begin{aligned} \varSigma _{\ell ,n,m}^{dH}:= \left\{ (E,J,R,Q,B)\,\big \vert \,J=-J^*,R = R^*> 0,E^*Q = Q^*E\ge 0 \right\} . \end{aligned}$$

(7)

An important subclass of port-Hamiltonian descriptor systems are conservative systems, i.e. $R = 0$, denoted by

$$\begin{aligned}{} & {} \varSigma _{\ell ,n,m}^{H}\nonumber \\ {}{} & {} :{=} \left\{ (E,J,Q,B)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell {\times } \ell }\times {\mathbb {K}}^{\ell {\times } n}\times {\mathbb {K}}^{\ell {\times } m}\,\big \vert \,J{=}-J^*,E^*Q {=} Q^*E{\ge } 0 \right\} .\nonumber \\ \end{aligned}$$

(8)

2.2 Controllability and stabilizability

To define controllability and stabilizability we need to define the behaviour of the differential-algebraic equation (3), where $\ell ,n,m\in {\mathbb {N}}^*$, that is the set

$$\begin{aligned} {\mathfrak {B}}_{[E,A,B]} := \left\{ (x,u)\in {\mathscr {L}}^1_{\textrm{loc}}({\mathbb {R}},{\mathbb {K}}^n)\times {\mathscr {L}}^1_{\textrm{loc}}({\mathbb {R}},{\mathbb {K}}^m)\,\left| \begin{array}{l} Ex\in AC({\mathbb {R}},{\mathbb {K}}^n)~\text {\ and}\\ \tfrac{\textrm{d}}{\textrm{d}t} (Ex) = Ax + Bu~\text {a.e.} \end{array} \right. \right\} \end{aligned}$$

wherein “a.e.” stands for “almost everywhere” with respect to the Lebesgue measure.

Next we recall the five definitions and characterizations of controllability for linear differential-algebraic systems (3).

Proposition 2.1

([2, Def. 2.1 and Cor. 4.3]) Consider the differential-algebraic equation (3) for some $\ell ,n,m\in {\mathbb {N}}^*$. Write

$$\begin{aligned} S_{{{controllable}}} := \left\{ (E,A,B)\in \varSigma _{\ell ,n,m}\,\big \vert \,~(3)~{{controllable}} \right\} \end{aligned}$$

for each of the following controllability concepts defined and algebraically characterized as follows:

$$\begin{aligned} \begin{array}{lrclcl} (E,A,B)\in S_{\text {freely initializable}} &{}:\iff &{}&{} \forall \, x^0\in {\mathbb {K}}^n~\exists \, (x,u)\in {\mathfrak {B}}_{[E,A,B]} : \ x(0) = x^0\\ &{}\iff &{}&{} \textrm{rk}\,[E,B] = \textrm{rk}\,[E,A,B]\ ;\\ (E,A,B){\in } S_{\text {impulse controllable}} &{}:\iff &{}&{} \forall \, x^0\in {\mathbb {K}}^n~\exists \, (x,u)\in {\mathfrak {B}}_{[E,A,B]} : \ Ex^0 {=} Ex(0)\\ &{}\iff &{} &{} \forall \, Z\in {\mathbb {K}}^{n\times n-\textrm{rk}\,E}~\text {with}~\textrm{im}_{{\mathbb {K}}} Z = \ker _{{\mathbb {K}}}E \\ &{}&{}&{} : \ \textrm{rk}\,[E,A,B] = \textrm{rk}\,[E,AZ,B]\ ;\\ \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{lrclcl} (E,A,B)\in S_{\text {behavioural controllable}} &{}:\iff &{}&{} \forall \, (x_1,u_1),(x_2,u_2)\in {\mathfrak {B}}_{[E,A,B]}\\ {} &{}&{}&{} \ \exists \, T>0~\exists \, (x,u)\in {\mathfrak {B}}_{[E,A,B]}\\ &{}&{}&{}: \ (x,u)(t) = {\left\{ \begin{array}{ll}(x_1,u_1)(t), &{} t<0\\ (x_2,u_2)(t), &{} t>T\end{array}\right. }\\ &{}\iff &{}&{} \forall \, \lambda \in {\mathbb {C}}\ : \textrm{rk}\,_{{\mathbb {K}}(x)}[xE-A,B]\\ &{}&{}&{}= \textrm{rk}\,_{{\mathbb {C}}}[\lambda E-A,B]\ ;\\ (E,A,B)\in S_{\text {completely controllable}} &{}:\iff &{}&{} \exists \, T>0~\forall \, x^0,x_T\in {\mathbb {R}}^n~\exists \, (x,u)\in {\mathfrak {B}}_{[E,A,B]}\\ &{}&{}&{} : \ x(0) = x^0\ \wedge \ x(T) = x_T \\ &{}\iff &{}&{} \forall \,\lambda \in {\mathbb {C}}: \textrm{rk}\,[E,A,B] = \textrm{rk}\,[E,B] \\ &{}&{}&{}= \textrm{rk}\,[\lambda E-A,B] \ ;\\ (E,A,B)\in S_{\text {strongly controllable}} &{}:\iff &{}&{} \exists \, T>0~\forall \, x^0,x_T\in {\mathbb {R}}^n ~\exists \, (x,u)\in {\mathfrak {B}}_{[E,A,B]}\\ &{}&{}&{} :\ Ex(0) = Ex^0 \ \wedge \ Ex(T) = Ex_T\\ &{}\iff &{}&{} \forall \, Z\in {\mathbb {K}}^{n\times (n-\textrm{rk}\,E)}\ \text {with}~\textrm{im}\,_{{\mathbb {K}}} Z \\ &{}&{}&{}= \ker _{{\mathbb {K}}}E~\forall \,\lambda \in {\mathbb {C}}\\ &{}&{}&{} : \ \textrm{rk}\,[E,A,B] = \textrm{rk}\,[E,AZ,B] \\ &{}&{}&{} = \textrm{rk}\,[\lambda E-A,B] \ .\\ \end{array} \end{aligned}$$

Furthermore,

$$\begin{aligned} S_{\text {completely~controllable}}&= S_{\text {freely initializable}}\cap S_{\text {behavioural controllable}}\quad \text {and}\\ S_{\text {strongly~controllable}}&= S_{\text {impulse~controllable}}\cap S_{\text {behavioural controllable}}. \end{aligned}$$

For port-Hamiltonian systems (1), we use, analogously to general differential-algebraic equations, the notation

$$\begin{aligned} S_{{\textit{controllable}}}^H:= \left\{ (E,J,Q,B)\in \varSigma _{\ell ,n,m}^H\,\big \vert \,(E,JQ,B)\in S_{{\textit{controllable}}} \right\} , \end{aligned}$$

(9)

and

$$\begin{aligned} S_{{\textit{controllable}}}^{dH}:= \left\{ (E,J,R,Q,B)\in \varSigma _{\ell ,n,m}^{dH}\,\big \vert \,(E,(J-R)Q,B)\in S_{{\textit{controllable}}} \right\} \nonumber \\ \end{aligned}$$

(10)

where controllable stands for either of the five aforementioned controllability concepts.

Proof

Berger and Reis [2] derive a feedback form and use this as a tool in conjunction with ‘canonical’ representatives of certain equivalence classes to prove all characterizations of controllability in their survey. Note that in their characterization of strongly controllability, the term ‘$+\textrm{im}\,_{{\mathbb {R}}}E$’ is missing in the first respective line in [2, Cor. 4.3]. $\square$

Additionally to controllability, we will characterize genericity of stabilizability for port-Hamiltonian descriptor systems. To this end, we recall the definitions and algebraic characterizations.

Proposition 2.2

([2, Def. 2.1 and Cor. 4.3]) Consider the differential-algebraic equation (3) for some $\ell ,n,m\in {\mathbb {N}}^*$. Writing, similar to the case of controllability,

$$\begin{aligned} S_{{stabilizable}} := \left\{ (E,A,B)\in \varSigma _{\ell ,n,m}\,\big \vert \,~(3)~{{{stabilizable}}} \right\} , \end{aligned}$$

the following definitions and characterizations hold true:

$$\begin{aligned} \begin{array}{lrlcl} (E,A,B)\in S_{\text {completely stabilizable}} &{}:\iff &{} \forall \, x^0\in {\mathbb {R}}^n~\exists \, (x,u)\in {\mathfrak {B}}_{(E,A,B)}\\ &{} &{} :\ x(0) {=} x^0 \ {\wedge }\ \lim _{t\rightarrow \infty }\mathrm {ess~sup}_{\tau \ge t}\left\| x(\tau ) \right\| {=} 0\\ &{} \iff &{} \forall \,\lambda \in \overline{{\mathbb {C}}}_{+}\ : \textrm{rk}\,_{{\mathbb {R}}}[E,A,B] = \textrm{rk}\,_{{\mathbb {R}}}[E,B] \\ &{}&{}=\textrm{rk}\,_{{\mathbb {C}}}[\lambda E-A,B];\\ (E,A,B)\in S_{\text {strongly stabilizable}} &{}:\iff &{} \forall \, x^0\in {\mathbb {R}}^n~\exists \, (x,u)\in {\mathfrak {B}}_{(E,A,B)}\\ &{} &{} :\ Ex(0) = Ex^0 \ \wedge \ \lim _{t\rightarrow \infty }Ex(t) = 0\\ &{}\iff &{} \forall \,\lambda \in \overline{{\mathbb {C}}}_+~\forall \,Z \in {\mathbb {K}}^{n\times (n-\textrm{rk}\,E)}\\ {} &{}&{}~\text {with}~\textrm{im}\,Z = \ker E:\\ &{}&{} \textrm{rk}\,_{{\mathbb {R}}}[E,A,B] = \textrm{rk}\,_{{\mathbb {R}}}[E,AZ,B] \\ {} &{}&{} = \textrm{rk}\,_{{\mathbb {C}}}[\lambda E-A,B] ; \\ (E,A,B)\in S_{\text {behavioural stabilizable}} &{}:\iff &{} \forall \, (x,u)\in {\mathfrak {B}}_{(E,A,B)}~\exists \, (x_1,u_1) \in {\mathfrak {B}}_{(E,A,B)}\\ &{}&{} :\ \left[ \forall \, t<0:(x(t),u(t)) {=} (x_1(t),u_1(t))\right] \\ &{} &{} \quad {\wedge }\quad \lim _{t\rightarrow \infty }\mathrm {ess~sup}_{\tau {\ge } t}\left\| (x_1(\tau ),u_1(\tau )) \right\| {=} 0 \\ &{}\iff &{} \forall \,\lambda \in \overline{{\mathbb {C}}}_+: \textrm{rk}\,_{{\mathbb {R}}(x)}[xE-A,B]\\ {} &{}&{} = \textrm{rk}\,_{{\mathbb {C}}}[\lambda E-A,B]. \end{array} \end{aligned}$$

For port-Hamiltonian systems (1), we use the notations

$$\begin{aligned} S_{{\textit{stabilizable}}}^H := \left\{ (E,J,Q,B)\in \varSigma _{\ell ,n,m}^H\,\big \vert \,(E,JQ,B)\in S_{{\textit{stabilizable}}} \right\} \end{aligned}$$

and

$$\begin{aligned} S_{{\textit{stabilizable}}}^{dH} := \left\{ (E,J,R,Q,B)\in \varSigma _{\ell ,n,m}^{dH}\,\big \vert \,(E,(J-R)Q,B)\in S_{{\textit{stabilizable}}} \right\} . \end{aligned}$$

Proof

For the definitions see [2, Definition 2.1] and for the algebraic criteria see [2, Corollary 4.3]. Please note that although we adapted the definition of behavioural stabilizability, in view of the normal form under feedback equivalence [2, Theorem 3.3], the algebraic characterization of Berger and Reis still holds, see also [19]. $\square$

2.3 Genericity and relative genericity

When it comes to genericity, there are various approaches available in the literature. Purely topological, genericity is commonly defined as follows.

Definition 2.3

[36, p. 1] Let $(X,{\mathscr {O}})$ be a topological space. A set $S\subseteq X$ is generic if, and only if, it contains a residual set, i.e. a family of open and dense sets $O_n\in {\mathscr {O}}$, $n\in {\mathbb {N}}$, such that $\bigcap \nolimits _{n\in {{\mathbb {N}}}} O_n \subset S$.

If the topological space $(X,{\mathscr {O}})$ is a Baire space [7, Definition 10.2, p. 249], then all generic sets are dense. Additionally to the purely topological approach to genericity, there is also the measure theoretic point of view. In this context, Vovk defines a typical point [41, p. 274]. This can be rephrased for genericity as follows.

Definition 2.4

Let $(X,{\mathscr {A}},\mu )$ be a complete (outer) measure space. A set $S\subseteq X$ is called generic if, and only if, S is $\mu$-conull, i.e. $S^c$ is measurable and $\mu (S^c) = 0$.

It would be nice to have a statement ‘If X admits a topology and $\mu$ is a continuous Borel measure so that all open sets have nonzero measure, then genericity in the measure theoretic sense implies genericity in the topological sense’. However, a classical result says that the real line admits a residual nullset, see [34, Theorem 1.6, p. 4]. Ergo, there exists some set that is generic in the Euclidean topological sense, but its complement is generic in the Lebesgue measure theoretical sense. Since the Lebesgue measure is non-trivial, we see that our set is not generic in the Lebesgue measure theoretical sense. Conversely, by the Baire category theorem [7, Theorem 10.1], there is no meagre set (i.e. a set, whose complement is residual) that is also residual. Thus, the complement of our set cannot contain any residual set and is therefore not generic in the Euclidean topological sense. Thus, the two aforementioned genericity concepts are, in general, unrelated. To this end, we will utilize a concept of genericity that unites the topological and measure theoretic point of view.

Consider the real finite-dimensional coordinate space ${\mathbb {R}}^n$, which is intrinsically a Baire space–equipped with the Euclidean topology–and intrinsically a complete measure space–equipped with the Lebesgue measure. In this case, Wonham introduced the following concept of genericity that unites the Euclidean topological and the Lebesgue measure theoretic concepts of genericity using the Zariski topology.

Definition 2.5

([43, p. 28] and [37]) Let $n\in {\mathbb {N}}^*$. A set ${\mathbb {V}}\subseteq {\mathbb {R}}^n$ is called an algebraic set if, and only if,

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

i.e. algebraic sets are the locus of the common zeros of finitely many polynomials in n indeterminants.

A set S is called generic in ${\mathbb {R}}^n$ if, and only if, there is some strict algebraic set ${\mathbb {V}}\subsetneq {\mathbb {R}}^n$ so that $S^c\subseteq {\mathbb {V}}$.

The polynomials $p_i$ in Definition 2.5 generate the ideal

$$\begin{aligned} I = \left\{ \left. \sum _{j = 1}^\kappa r_j p_{i_j}\,\right| \, \begin{array}{ll} \kappa \in {\mathbb {N}}_0, \ r_1,\ldots ,r_\kappa \in {\mathbb {R}}[x_1,\ldots ,x_n],\\ i_1,\ldots ,i_\kappa \in {\mathbb {N}}^*_{\le k} \end{array} \right\} , \end{aligned}$$

i.e. an additive subgroup that is closed under multiplication with any polynomial. It is easy to verify that all members of I vanish at the algebraic set ${\mathbb {V}}$. In later considerations, we make use of the correspondence of algebraic sets in ${\mathbb {R}}^n$ and ideals in the ring of n-variate polynomials. Since the latter is a commutative ring, there is no need to distinguish between left and right ideals.

Lemma 2.6

[37, pp. 50/1] Let $n\in {\mathbb {N}}^*$. A set ${\mathbb {V}}\subseteq {\mathbb {R}}^n$ is an algebraic set if, and only if, there is some ideal $I\subseteq {\mathbb {R}}[x_1,\ldots ,x_n]$ so that ${\mathbb {V}} = \left\{ x\in {\mathbb {R}}^n\,\big \vert \ \forall p\in I: p(x) = 0 \right\}$.

Genericity in the sense of Definition 2.5 can also be characterised by means of the Zariski topology. This topology is given by its closed sets, that are precisely the algebraic sets, see [37, p. 50].

Lemma 2.7

Let $n\in {\mathbb {N}}^*$. As set $S\subseteq {\mathbb {R}}^n$ is generic if, and only if, it contains a non-empty Zariski-open set $O\subseteq {\mathbb {R}}^n$.

Proof

A set ${\emptyset \ne } O\subseteq {\mathbb {R}}^n$ is Zariski-open if, and only if, $O^c{\ne {\mathbb {R}}^n}$ is Zariski-closed, i.e. a strict algebraic set. Thus, the assertion holds true since $O\subseteq S$ if, and only if, $S^c\subseteq O^c$. $\square$

The Zariski topology on ${\mathbb {R}}^n$ is strictly coarser than the Euclidean topology and has the property that each non-empty Zariski-open set is not only Euclidean dense but also Lebesgue conull, see [11, p. 240]. Further, there are residual conull sets that contain no non-empty Zariski open set (e.g. the natural numbers are an Euclidean closed Lebesgue nullset). Therefore Wonham’s concept from Definition 2.5 is strictly stronger than the Euclidean topological and Lebesgue measure theoretic approach to genericity. Additionally, from the tame nature of polynomials, the authors would like to argue that the concept of genericity in the sense of Definition 2.5 is rather handy to work with as opposed to the natural choice of defining genericity as residual (or even open and dense) conull sets. Unfortunately, a drawback of Wonham’s genericity as defined in Definition 2.5 is that its naive extension to genericity with respect to some reference set $V\subseteq {\mathbb {R}}^n$ fails in general. The relative Zariski topology on V has, in general, not the same favourable properties regarding the Euclidean topology (and the Lebesgue measure, if there is some reasonable way of defining such a measure) which allows to study genericity in the sense of Definition 2.5 as a reasonable concept; the simplest example is a discrete set with at least two points. To overcome this issue, Kirchhoff used in [23] an adapted concept for genericity in some reference se, which was refined in [19] and is defined as follows.

Definition 2.8

([23, Def. I.2] and [19, Def. 1.2]) Let $n\in {\mathbb {N}}^*$ and $S,V\subseteq {\mathbb {R}}^n$. S is relative generic in V if, and only if, there is some algebraic set ${\mathbb {V}}\subseteq {\mathbb {R}}^n$ so that

$$\begin{aligned} {{\mathbb {V}}^c\cap V\subseteq S\cap V}\quad \text {and}\quad {\mathbb {V}}^c\cap V~\text {is~Euclidean~dense~in}~V. \end{aligned}$$

Analogously to Wonham’s original concept as given in Definition 2.5, relative genericity can be characterised in terms of the relative Euclidean and relative Zariski topologies as follows.

Lemma 2.9

([19, Lemma 2.1]) Let $n\in {\mathbb {N}}^*$ and $S,V\subseteq {\mathbb {R}}^n$. S is relative generic in V if, and only if, $S\cap V$ contains some relative Zariski open, relative Euclidean dense set $O\subseteq V$.

In view of Lemma 2.7, relative genericity and genericity in the sense of Definition 2.5 coincide for $V = {\mathbb {R}}^n$. Moreover, when the reference set V admits enough structure (e.g. if V is an analytic submanifold with countable atlas), then the properties of the relative Zariski topology can be invoked to conclude that every relative generic set is conull w.r.t. a Lebesgue-type measure, see [19, Proposition 2.8]. Thus, the authors prefer this concept of relative genericity over an adaptation of the purely topological concept of Definition 2.3 w.r.t. the relative Euclidean topology on V.

Ilchmann and Kirchhoff [19] have collected the results needed in the following considerations. We recall the most important ones here.

Proposition 2.10

(see [19, Proposition 2.3]) Let $S_1,S_2,V,V'\subseteq {\mathbb {R}}^n$, $n\in {\mathbb {N}}^*$.

• (a) If $S_1$ is relative generic in V, then $S_1\cap V$ contains some open, dense set (in the relative Euclidean topology) and therefore $S_1^c\cap V$ is nowhere dense in V. The converse, however, is in general not true.

• (b) If $S_1$ is relative generic in V and $S_1\subseteq S_2$, then $S_2$ is also relative generic in V.

• (c) If $S_1$ and $S_2$ are relative generic in V, then $S_1\cap S_2$ and $S_1\cup S_2$ are relative generic in V.

• (d) If $S_1$ is relative generic in V and $S_3\subseteq {\mathbb {R}}^m$, $m\in {\mathbb {N}}^*$, is relative generic in $U\subseteq {\mathbb {R}}^m$, then $S_1\times S_3$ is relative generic in $V\times U$.

• (e) If $V'\subseteq V$ and $V'$ is relative generic in V, then $S_1$ is relative generic in V if, and only if, $S_1$ is relative generic in $V'$.

• (f) If $S_1$ is relative generic in V, then $S_1^c$ is not relative generic in V.

• (g) If V is open, then $S_1$ is relative generic in V if, and only if, $S_1$ is generic.

The statement (g) implies that relative genericity in open reference sets and genericity are equivalent. When considering relative open subsets of a given reference set–which is not necessarily ${\mathbb {R}}^n$, as it was in (g)–only one implication remains true.

Lemma 2.11

Let $V,{\widetilde{V}}\subseteq {\mathbb {R}}^n$ so that ${\widetilde{V}}\subseteq V$ is relative Euclidean open. If $S\subseteq {\mathbb {R}}^n$ is relative generic in V, then S is relative generic in ${\widetilde{V}}$. The converse fails in general.

Proof

Let S be relative generic in V. Then, there exists a Zariski-open set $O\subseteq {\mathbb {R}}^n$ so that $O\cap V$ is relatively Euclidean dense in V and $O\cap V\subseteq S\cap V$. In particular, we have $\textrm{int}_{{\widetilde{V}}}{\widetilde{V}}{\setminus } (O\cap {\widetilde{V}}) = {\widetilde{V}}\cap \textrm{int}_V V{\setminus } O = {\widetilde{V}}\cap \emptyset = \emptyset$, where ${\text {int}}_S$ denotes the interior with respect to the relative Euclidean topology of a set $S\subseteq {\mathbb {R}}^n$. This shows that $O\cap {\widetilde{V}}$ is a relative Zariski-open subset of ${\widetilde{V}}$ which is dense with respect to the relative Euclidean topology.

To see that the converse implication fails in general, consider the reference set $V:= {\mathbb {B}}(0,1)\cup {\mathbb {R}}\times \left\{ 0 \right\} \subseteq {\mathbb {R}}^2$. The open ball ${\mathbb {B}}(0,1)$ is a relative Euclidean open subset of V. Further, ${\mathbb {B}}(0,1)$ is open and hence by Proposition 2.10 (g) the set ${\mathbb {R}}^2\setminus {\mathbb {R}}\times \left\{ 0 \right\}$ is relative generic in ${\mathbb {B}}(0,1)$. However, ${\mathbb {R}}^2\setminus {\mathbb {R}}\times \left\{ 0 \right\}$ is not Euclidean dense in V and by Proposition 2.10 (f) in particular not relative generic in V. $\square$

An even simpler counter-example would be a disconnected reference set: Each connected component of any set V is relative generic in itself, but not relative generic in V provided that V possesses at least two connected components.

So far, we have considered genericity in the sense of Wonham only in the real coordinate space, a restriction that seems odd. However, one should be careful when extending Wonham’s original definition (or the small adaptation towards arbitrary reference sets) to the complex case. Naively, the authors would try to use the complex Zariski topology (defined as in the real case but with complex polynomials) and get a reasonable concept. It should be kept in mind that the complex coordinate space ${\mathbb {C}}^n$ has naturally a real vector space structure and is isomorphic to ${\mathbb {R}}^{2n}$. When applying Wonham’s genericity to the ${\mathbb {R}}^{2n}$ representation of ${\mathbb {C}}^n$, we must be aware that there are 2n-dimensional real algebraic sets in ${\mathbb {C}}^n$ that can only contain complex algebraic sets up to dimension n. We must therefore distinguish between the complex genericity and the weaker concept of real genericity that is induced by any isomorphism between ${\mathbb {C}}^n$ and ${\mathbb {R}}^{2n}$. From the Euclidean topological and Lebesgue measure theoretic point of view, there seems to be no reason to prefer either of these concepts. For simplicity, however, we will use the real genericity for the complex coordinate space; a formal definition is the following.

Definition 2.12

Let $n\in {\mathbb {N}}^*$, $S,V\in {\mathbb {C}}^n$ and $\varphi :{\mathbb {C}}^n\rightarrow {\mathbb {R}}^{2n}$ a real vector space isomorphism. We call S relative generic in V if, and only if, $\varphi (S)$ is relative generic in $\varphi (V)$.

We stress that we will be very informal when identifying ${\mathbb {C}}^n$ with ${\mathbb {R}}^{2n}$ and equipping the former with its real Zariski topology. Since isomorphisms are bijective linear maps and especially polynomial vectors, Definition 2.12 is independent of the explicit choice of $\varphi$. Therefore we do not state the used isomorphism explicitly. Since we have defined relative genericity in the complex coordinate space via its representation as real coordinate space, the results of Proposition 2.10 hold true for ${\mathbb {C}}$ instead of ${\mathbb {R}}$.

Before we proceed with our main results, we need an additional lemma on relative genericity.

Lemma 2.13

Let $n\in {\mathbb {N}}^*$ and $V\subseteq {\mathbb {R}}^n$ be convex and non-empty. Then $S\subseteq {\mathbb {R}}^n$ is relative generic in V if, and only if, there is some Zariski open set O so that $O\cap V\ne \emptyset$ and $O\cap V\subseteq S\cap V$.

Proof

Necessity is a consequence of Definition 2.8; we show sufficiency. Let O be Zariski open with $O\cap V\ne \emptyset$ and $O\cap V\subseteq S\cap V$. It remains to show that $O\cap V$ is dense in the relative Euclidean topology of V. Evidently, if V is a singleton set, then $O\cap V = V$ is in particular dense. In the following, we consider the case that V contains at least two (and hence, due to convexity, uncountably many) points. Since relative genericity is by Definition 2.8 invariant under simultaneous translations of S and V, we may assume, without loss of generality, that $0\in V$; and since S is relative generic in V if, and only if, $S\cap V$ is relative generic in V, we may assume, without loss of generality, that $S\subseteq V$. In [23, Lemma III.2] it is shown that, when considering relative genericity w.r.t. a reference set that is contained in a linear subspace $W\subseteq {\mathbb {R}}^n$, it can, without loss of generality, be assumed that $W = {\mathbb {R}}^n$. Hence, we may assume without loss of generality that ${\mathbb {R}}^n$ and $\textrm{span}\,V$, the linear span of V, coincide. Then [38, Theorem 6.3] implies that the interior of V is dense with respect to the relative Euclidean topology of V. Seeking a contradiction, assume that $\textrm{int}_V (V{\setminus } O)\ne \emptyset$ and hence $\textrm{int}\,V\cap \textrm{int}_V (V\setminus O)$ is a non-empty open subset of the algebraic set ${\mathbb {R}}^n\setminus O$. This yields ${\mathbb {R}}^n\setminus O = {\mathbb {R}}^n$ and hence $O = \emptyset$, which contradicts the assumption $O\cap V\ne \emptyset$.

$\square$

3 Relative generic sets

The present section is an intermediate section. We prove those results on relative genericity which are needed for proving the main result in Sect. 4. The main result under these lemmata is Proposition 3.11; it is shown that various subsets of differential-algebraic systems (1) satisfying an algebraic constraint associated to controllability/stabilizability are (relative) generic sets with respect to (semi-)dissipative or conservative port-Hamiltonian descriptor systems.

The most important tool in our analysis is the well-known concept of a minor.

Definition 3.1

Let $d,\ell ,n\in {\mathbb {N}}^*$ so that $d\le \min \left\{ \ell ,n \right\}$. Let

$$\begin{aligned} \sigma :{\mathbb {N}}^*_{\le d}\rightarrow {\mathbb {N}}^*_{\le \ell } \end{aligned}$$

and

$$\begin{aligned} \pi :{\mathbb {N}}^*_{\le d}\rightarrow {\mathbb {N}}^*_{\le n} \end{aligned}$$

be strictly increasing, where we use the abbreviation ${\mathbb {N}}^*_{\le k}:= \left\{ 1,\ldots ,k \right\}$ for all $k\in {\mathbb {N}}^*$. We consider either $R = {\mathbb {K}}$ or $R = {\mathbb {K}}[x]$. The function

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

is called minor of order d w.r.t. $R^{\ell \times n}$. The minor of order 0 w.r.t. $R^{\ell \times n}$ is the constant function $A\mapsto 1$.

Recall the correspondence between lower bounds of the rank of matrices and minors.

Lemma 3.2

(see [12, Sect. 3.3.6]) Let $d\in {\mathbb {N}}_0$ and $\ell ,n\in {\mathbb {N}}^*$ with $d\le \min \left\{ \ell ,n \right\}$ and either $R = {\mathbb {K}}$ or $R = {\mathbb {K}}[x]$. A matrix $A\in R^{\ell \times n}$ has rank at least d if, and only if, there is some minor of order d w.r.t. $R^{\ell \times n}$ that does not vanish at A.

Motivated by the algebraic characterizations of controllability and stabilizability in Propositions 2.1 and 2.2, we make use of Lemma 3.2 to construct algebraic sets defined by the rank of certain polynomial matrices. This is possible since determinants and thus minors are in particular polynomials.

Lemma 3.3

Let $d,g\in {\mathbb {N}}_0$ and $\ell ,n\in {\mathbb {N}}^*$ with $d\le \min \left\{ \ell ,n \right\}$. Let ${\widetilde{M}}$ be a minor of order d w.r.t. ${\mathbb {K}}[x]^{\ell \times n}$ and consider the induced function

$$\begin{aligned} M:\left( {\mathbb {K}}^{\ell \times n}\right) ^{g+1}\rightarrow {\mathbb {K}}[x],\qquad (P_0,\ldots ,P_g)\mapsto {\widetilde{M}}\left( \sum _{i = 0}^gP_i x^i\right) . \end{aligned}$$

Then there are multivariate polynomials $M_0,\ldots ,M_{dg}\in {\mathbb {K}}[x_1,\ldots ,x_{\ell n(g+1)}]$ so that

$$\begin{aligned} \forall (P_0,\ldots ,P_g)\in \left( {\mathbb {K}}^{\ell \times n}\right) ^{g+1}: M(P_0,\ldots ,P_g) = \sum _{i = 0}^{dg} M_i(P_0,\ldots ,P_g)x^i. \end{aligned}$$

(11)

In particular,

$$\begin{aligned} M_0(P_0,\ldots ,P_g) = {\widetilde{M}}(P_0)\quad \text {and}\quad M_{gd}(P_0,\ldots ,P_g) = {\widetilde{M}}(P_g). \end{aligned}$$

(12)

Proof

By definition, minors are determinants of certain submatrices. Therefore, it suffices to consider the case $\ell = n = d$ and ${\widetilde{M}} = \det (\cdot )$. In that case, the Leibniz formula for the determinant yields

$$\begin{aligned} M(P_0,\ldots ,P_g) = {\widetilde{M}}\left( \sum _{i = 0}^gP_i x^i\right) = \sum _{\sigma \in S_d}\textrm{sgn}(\sigma )\prod _{j = 1}^d\sum _{i = 0}^g(P_i)_{j,\sigma (j)} x^i, \end{aligned}$$

where $S_d$ denotes the set of all permutations of all d-tuples. From this, (11) and (12) follow by expanding the products and sorting by the degree of the resulting monomials. The fact that the functions $M_i$ are polynomials in the entries of $P_0,\ldots ,P_g$, follows likewise from the Leibniz formula. $\square$

The properties of minors in Lemmas 3.2 and 3.3 will help to understand Zariski-open sets within the sets $S^H_{\textit{controllable}}$ and $S^{sdH}_{\textit{controllable}}$ that are given by the rank of particular block matrices. As a last algebraic ingredient, we recall the important concept of the Sylvester resultant.

Lemma 3.4

([13, Thm. 3.3.1, p. 61]) The resultant of two polynomials $p,q\in {\mathbb {K}}[x]\setminus \left\{ 0_{{\mathbb {K}}[s]} \right\}$ with $\deg p = n\ge 0$ and $\deg q = m\ge 0$ and coefficients $p_1,\ldots ,p_n,q_1,\ldots ,q_m\in {\mathbb {K}}$ is defined as

$$\begin{aligned} \textrm{Res}(p,q) = \det \underbrace{\left[ \begin{array}{ccccc|ccccc} p_0 &{} &{} &{} &{} &{} q_0 &{} &{} &{}\\ p_1 &{} p_0 &{} &{} &{} &{} q_1 &{} \cdot &{}\\ \cdot &{} \cdot &{} &{} &{} &{} \cdot &{} \cdot &{} \cdot &{} \\ \cdot &{} \cdot &{} \cdot &{} &{} &{} \cdot &{} \cdot &{} \cdot &{} q_0 \\ p_n &{} p_{n-1} &{} \cdot &{} \cdot &{} &{} q_n &{} \cdot &{} \cdot &{} q_1 \\ &{} p_n &{} \cdot &{} \cdot &{} &{} \cdot &{} \cdot &{} \cdot &{} \cdot \\ &{} &{} \cdot &{} \cdot &{} p_0 &{} q_{m-1} &{} \cdot &{} \cdot &{} \cdot \\ &{} &{} \cdot &{} \cdot &{} \cdot &{} q_{m} &{} \cdot &{} \cdot &{} \cdot &{} \\ &{} &{} \cdot &{} \cdot &{} \cdot &{} &{} \cdot &{} \cdot &{} \cdot \\ &{} &{} &{} \cdot &{} \cdot &{} &{} &{} \cdot &{} \cdot \\ &{} &{} &{} &{} p_n &{} &{} &{} &{} q_m \end{array}\right] }_{\in {\mathbb {K}}^{(n+m)\times (m+n)}}. \end{aligned}$$

The matrix above contains m columns with the coefficients of p and n columns with the coefficients of q, so that it is in ${\mathbb {K}}^{(n+m)\times (m+n)}$. All other entries are zero. Please note that the diagram illustrates the case $n<m$.

The resultant of p and q vanishes if, and only if, p and q are not coprime $\big ($i.e. there is some common zero $z\in {\mathbb {C}}$ such that $p(z) = q(z) = 0\big )$.

Since relative genericity strongly depends on the reference set, we seek to simplify (in the sense of Proposition 2.10 (e)) the reference sets $\varSigma _{\ell ,n,m}^H$ and $\varSigma _{\ell ,n,m}^{sdH}$ as much as possible. To this end, we observe that $\varSigma _{\ell ,n,m}^H$ and $\varSigma _{\ell ,n,m}^{sdH}$ are isomorphic to the products

$$\begin{aligned} \varSigma _{\ell ,n,m}^H&\cong \left\{ J\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,J = -J^* \right\} \times \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E\ge 0 \right\} \\ {}&\quad \,\times {\mathbb {K}}^{\ell \times m} \end{aligned}$$

and

$$\begin{aligned} \varSigma _{\ell ,n,m}^{sdH}&\cong \left\{ J\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,J = -J^* \right\} \times \left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,R = R^*\ge 0 \right\} \\&\quad \,\times \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E\ge 0 \right\} \times {\mathbb {K}}^{\ell \times m}. \end{aligned}$$

Then, relative genericity is equivalent to relative genericity (of the image under the isomorphisms) in each factor, and the products of   “convenient” relative generic subsets of each factor is a “convenient” relative generic subset of our reference sets. First, we show that $\left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,R = R^*> 0 \right\}$ is relative generic in $\left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,R = R^*\ge 0 \right\}$.

Lemma 3.5

Let $\ell \in {\mathbb {N}}^*$. The set $S = \left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,R^* = R > 0 \right\}$ of Hermitian (symmetric) positive definite matrices is relative generic in $V = \left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,R^* = R \ge 0 \right\}$, the set of Hermitian (symmetric) positive semidefinite matrices. The same holds true if we consider negative instead of positive (semi-)definite matrices.

Proof

V is a convex set and $S = V{\setminus }\left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,\det R = 0 \right\}$ is a non-empty relative Zariski-open set. The assertion is thus a consequence of Lemma 2.13. $\square$

Now, we turn our attention to the set $\left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E\ge 0 \right\}$. Our aim is to show that this set contains the relative generic subset

$$\begin{aligned} \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E\ge 0,\textrm{rk}\,E = \textrm{rk}\,Q = \textrm{rk}\,E^*Q = \min \left\{ \ell ,n \right\} \right\} . \end{aligned}$$

To this end, we study perturbations of $E,Q\in {\mathbb {K}}^{\ell \times n}$ with $E^*Q = Q^*E$ which preserve this property. The following consequence of the singular value theorem reveals a transformation of E and Q which aids the finding of such perturbations.

Lemma 3.6

Let $\ell ,n\in {\mathbb {N}}^*$ and $E,Q\in {\mathbb {K}}^{\ell \times n}$. There exist unitary (orthogonal) matrices $P\in {\mathbb {K}}^{\ell \times \ell }$, $T\in {\mathbb {K}}^{n\times n}$ and

$$\begin{aligned} \varSigma _k = \begin{bmatrix} \sigma _1\\ &{} \ddots \\ &{} &{} \sigma _k \end{bmatrix} \in \textbf{Gl}({\mathbb {R}}^k),\quad \text {where } k = \textrm{rk}\,E \end{aligned}$$

(13)

such that, for some ${\widetilde{Q}}\in {\mathbb {K}}^{k\times k}$ with $\varSigma _k {\widetilde{Q}} = {\widetilde{Q}}^* \varSigma _k$ and  $R_1\in {\mathbb {K}}^{(\ell -k)\times k}$, $R_2\in {\mathbb {K}}^{(\ell -k)\times (n-k)}$,

$$\begin{aligned} PET = \begin{bmatrix} \varSigma _k&{}0_{k\times (n-k)}\\ 0_{(\ell -k)\times k}&{}0_{(\ell -k)\times (n-k)} \end{bmatrix},\qquad PQT= \begin{bmatrix} {\widetilde{Q}}&{}0_{k\times (n-k)}\\ R_1&{}R_2 \end{bmatrix} \end{aligned}$$

(14)

if, and only if, $E^*Q = Q^*E$. In that case, $\textrm{rk}\,{\widetilde{Q}} = \textrm{rk}\,E^* Q$ and

$$\begin{aligned} E^*Q=Q^*E \ge 0 \quad \text {if, and only if, } \quad \varSigma _K{\widetilde{Q}} = {\widetilde{Q}}^*\varSigma _k \ge 0. \end{aligned}$$

(15)

Furthermore,

$$\begin{aligned} V_{\varSigma _k} := \left\{ \left. \begin{bmatrix} M &{}\quad 0_{k\times (n-k)}\\ H_1 &{}\quad H_2 \end{bmatrix} \in {\mathbb {K}}^{\ell \times n}\,\right| \, \varSigma _k M = M^*\varSigma _k\right\} \end{aligned}$$

is a real vector space.

Proof

If E and Q can be simultaneously transformed to (14), then

$$\begin{aligned} T^*E^*QT = (PET)^*(PQT) {\mathop {=}\limits ^{(14)}} \begin{bmatrix} \varSigma _k {\widetilde{Q}}&{}\quad 0_{k\times (n-k)}\\ 0_{(\ell -k)\times k}&{}\quad 0_{(\ell -k)\times (n-k)} \end{bmatrix} \end{aligned}$$

(16)

shows that $E^*Q = Q^*E$. Conversely, let $E^*Q = Q^*E$. By the singular value theorem [17, Theorem 2.6.3], there are orthogonal (unitary) matrices $P\in {\mathbb {K}}^{\ell \times \ell }$ and $T\in {\mathbb {K}}^{n\times n}$ such that (14) holds with $\varSigma _k$ as in (13). Since P is unitary (orthogonal) and T invertible, $E^*Q = Q^*E$ is equivalent to

$$\begin{aligned} (PET)^*(PQT) = (PQT)^*(PET). \end{aligned}$$

(17)

Inserting the first equation of (14) into (17) yields the second equation in (14). A repeated application of $E^*Q=Q^*E$ yields

which shows that $\varSigma _k{\widetilde{Q}} = {\widetilde{Q}}^*\varSigma _k$. Using (16), $\textrm{rk}\,E^*Q = \textrm{rk}\,T^*E^*QT = \textrm{rk}\,\varSigma _k{\widetilde{Q}}$ and, since $\varSigma _k \in \textbf{Gl}({\mathbb {R}}^k)$, $\textrm{rk}\,E^*Q = \textrm{rk}\,{\widetilde{Q}}$. The equivalence in (15) is a consequence of (16). Finally, it is easy to see that $V_{\varSigma _k}$ is a real vector space. $\square$

In the next lemma, we show that the rank of matrices $E,Q\in {\mathbb {K}}^{\ell \times n}$ with the property $E^*Q = Q^*E$ is generically full.

Lemma 3.7

Let $\ell ,n\in {\mathbb {N}}^*$. Then each of the sets

• (i)  $S_{(i)}\, = \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,\textrm{rk}\,Q = \min \left\{ \ell ,n \right\} \right\}$

• (ii)  $S_{(ii)} = \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,\textrm{rk}\,E = \min \left\{ \ell ,n \right\} \right\}$

is relative generic in the reference set

$$\begin{aligned} V \ = \ \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E{\ge 0} \right\} . \end{aligned}$$

Proof

Since the statements of (i) and (ii) are symmetric, it suffices in view of Proposition 2.10 (b) to show that $S_{(i)}$ is relative generic in V. Due to Lemmata 3.2 and 3.3, the set

$$\begin{aligned} \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,\textrm{rk}\,Q \ge \min \left\{ \ell ,n \right\} \right\} \end{aligned}$$

is Zariski-open. As $\textrm{rk}\,Q \le \min \left\{ \ell ,n \right\}$ for all $Q \in {\mathbb {K}}^{\ell \times n}$, $S_{(i)}$ is Zariski-open, as well. By Lemma 2.9, it suffices to show that $S_{(i)}\cap V$ is Euclidean dense in V. Let $(E,Q)\in V$. Lemma 3.6 yields that there are unitary (orthogonal) matrices $P\in {\mathbb {K}}^{\ell \times \ell }$ and $T\in {\mathbb {K}}^{n\times n}$, a diagonal matrix $\varSigma \in \textbf{Gl}({\mathbb {R}}^k)$ with $k = \textrm{rk}\,E$, ${\widetilde{Q}}\in {\mathbb {K}}^{k\times k}$ with $\varSigma ^*{\widetilde{Q}} = {\widetilde{Q}}^*\varSigma \ge 0$, $R_1\in {\mathbb {R}}^{(\ell -k)\times k}$ and $R_2\in {\mathbb {K}}^{(\ell -k)\times (n-k)}$ so that

$$\begin{aligned} PET = \begin{bmatrix} \varSigma &{}\quad 0_{k\times (n-k)}\\ 0_{(\ell -k)\times k} &{}\quad 0_{(\ell -k)\times (n-k)} \end{bmatrix}\quad \text {and}\quad PQT = \begin{bmatrix} {\widetilde{Q}} &{}\quad 0_{k\times (n-k)}\\ R_1 &{}\quad R_2 \end{bmatrix}. \end{aligned}$$

Let $\varepsilon >0$ and consider ${\widehat{Q}}:= {\widetilde{Q}}+\frac{\varepsilon }{2\left\| P \right\| \left\| T \right\| \left\| \varSigma \right\| }\varSigma$ so that ${\widehat{Q}}^*\varSigma = {\widetilde{Q}}^*\varSigma + \varSigma ^2 = \varSigma {\widehat{Q}}$ is positive definite. Since $\left\{ {\widehat{R}}_2\in {\mathbb {K}}^{(\ell -k)\times (n-k)}\,\big \vert \,\textrm{rk}\,{\widehat{R}}_2 = \min \left\{ \ell ,n \right\} -k \right\}$ is generic, there exists ${\widehat{R}}_2\in {\mathbb {K}}^{(\ell -k)\times (n-k)}$ with $\textrm{rk}\,{\widehat{R}}_2 = \min \left\{ \ell ,n \right\} -k$ and $\left\| {\widehat{R}}_2-R_2 \right\| <\frac{\varepsilon }{\left\| P \right\| \left\| T \right\| }$. Then

$$\begin{aligned} Q' := P^{-1}\begin{bmatrix} {\widehat{Q}} &{}\quad 0_{k\times (n-k)}\\ R_1 &{}\quad {\widehat{R}}_2 \end{bmatrix}T^{-1} \end{aligned}$$

(18)

has rank $\textrm{rk}\,Q' = \textrm{rk}\,{\widehat{Q}} + \textrm{rk}\,{\widehat{R}}_2 = k+\min \left\{ \ell ,n \right\} -k = \min \left\{ \ell ,n \right\}$, so that $(E,Q')\in S_{(i)}\cap V$, and fulfils, due to unitarity of P and T,

$$\begin{aligned} \left\| Q'-Q \right\| \le \left\| P \right\| \left\| T \right\| \left\| \begin{bmatrix} \frac{\varepsilon }{2\left\| P \right\| \left\| T \right\| \left\| \varSigma \right\| }\varSigma &{}\quad 0_{k\times (n-k)}\\ 0_{(\ell -k)\times k} &{}\quad {\widehat{R}}_2-R_2 \end{bmatrix} \right\| <\varepsilon . \end{aligned}$$

This shows that $S_{(i)}\cap V$ is Euclidean dense in V.

$\square$

It is noteworthy that we only perturb the matrix Q in the proof of Lemma 3.7. This shows that the following stronger result holds. We omit the proof; it would be a repetition of the proof of Lemma 3.7.

Lemma 3.8

Let $\ell ,n\in {\mathbb {N}}^*$ and $Q\in {\mathbb {K}}^{\ell \times n}$. The set

$$\begin{aligned} S = \left\{ E\in {\mathbb {K}}^{\ell \times n}\,\big \vert \,\textrm{rk}\,E = \min \left\{ \ell ,n \right\} \right\} \end{aligned}$$

is relative generic in the set

$$\begin{aligned} V_Q = \left\{ E\in {\mathbb {K}}^{\ell \times n}\,\big \vert \, E^*Q = Q^*E\ge 0 \right\} . \end{aligned}$$

Using the results of Lemma 3.7, we show that the rank of $E^*Q$ is generically full.

Proposition 3.9

Let $\ell ,n\in {\mathbb {N}}^*$. The set

$$\begin{aligned} S = \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,\textrm{rk}\,E^*Q = \min \left\{ \ell ,n \right\} \right\} \end{aligned}$$

is relative generic in

$$\begin{aligned} V = \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E{\ge 0} \right\} . \end{aligned}$$

Proof

For brevity put $k = \min \left\{ \ell ,n \right\}$. In view of Lemma 3.7 and Proposition 2.10 (c), the set

$$\begin{aligned} V' {:=} \left\{ (E,Q){\in }{\mathbb {K}}^{\ell \times n}{\times }{\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q {=} Q^*E\ge 0,\textrm{rk}\,E = \textrm{rk}\,Q {=} k \right\} = V\cap S_{(i)}\cap S_{(ii)} \end{aligned}$$

is relative generic in V, so that it remains to show that S is relative generic in $V'$. Since S is Zariski-open, it remains to show that $S\cap V'$ is Euclidean dense in $V'$. Let $\varepsilon >0$ and $(E,Q)\in V'$. By Lemma 3.6, there exist unitary (orthogonal) matrices $P\in {\mathbb {K}}^{\ell \times \ell }$ and $T\in {\mathbb {K}}^{n\times n}$, a diagonal matrix $\varSigma \in \textbf{Gl}({\mathbb {R}}^k)$, ${\widetilde{Q}}\in {\mathbb {K}}^{k\times k}$ with $\varSigma ^*{\widetilde{Q}} = {\widetilde{Q}}^*\varSigma \ge 0$, $R_1\in {\mathbb {R}}^{(\ell -k)\times k}$ and $R_2\in {\mathbb {K}}^{(\ell -k)\times (n-k)}$ so that

$$\begin{aligned} PET = \begin{bmatrix} \varSigma &{}\quad 0_{k\times (n-k)}\\ 0_{(\ell -k)\times k} &{}\quad 0_{(\ell -k)\times (n-k)} \end{bmatrix}\quad \text {and}\quad PQT = \begin{bmatrix} {\widetilde{Q}} &{}\quad 0_{k\times (n-k)}\\ R_1 &{}\quad R_2 \end{bmatrix}, \end{aligned}$$

where we view matrices with zero rows or columns as empty matrices which do not change the block matrices. Consider ${\widehat{Q}}:= {\widetilde{Q}}+\frac{\varepsilon }{2\left\| P \right\| \left\| T \right\| \left\| \varSigma \right\| }\varSigma$ so that ${\widehat{Q}}^*\varSigma = {\widetilde{Q}}^*\varSigma + \varSigma ^*\varSigma = \varSigma ^*{\widehat{Q}}$ is positive definite, and put $Q'$ as in (18). Then, $\textrm{rk}\,Q'\ge \textrm{rk}\,{\widehat{Q}} = k$ and hence Lemma 3.6 implies that $(E,Q')\in S\cap V'$ with

$$\begin{aligned} \left\| (E,Q)-(E,Q') \right\| = \left\| Q-Q' \right\| = \left\| P \right\| \left\| T \right\| \left\| \frac{\varepsilon }{2\left\| P \right\| \left\| T \right\| \left\| \varSigma \right\| }\varSigma \right\| <\varepsilon . \end{aligned}$$

This shows that S is indeed Euclidean dense in $V'$. $\square$

In the case $\ell \ge n$, we readily conclude that $E^*Q$ is generically invertible and hence positive definite.

Corollary 3.10

Let $\ell \ge n\in {\mathbb {N}}^*$. The set

$$\begin{aligned} S = \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E>0 \right\} \end{aligned}$$

is relative generic in

$$\begin{aligned} V = \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E\ge 0 \right\} . \end{aligned}$$

Proof

We have

$$\begin{aligned} S\cap V = \left\{ (E,Q)\in V\,\big \vert \,\textrm{rk}\,E^*Q = \min \left\{ \ell ,n \right\} \right\} \end{aligned}$$

and the latter is, by Proposition 3.9, relative generic in V. $\square$

We have now sufficiently simplified our reference sets. Before we may prove that port-Hamiltonian descriptor systems are generically controllable, however, we need an auxiliary result, where we show that the rank of the block matrices that appear in the algebraic criteria in Proposition 2.1 is generically full. Recall that our reference sets of port-Hamiltonian matrix tuples are defined as

$$\begin{aligned} \varSigma _{\ell ,n,m}^{H}&{=} \left\{ (E,J,Q,B){\in }{\mathbb {K}}^{\ell \times n}{\times }{\mathbb {K}}^{\ell {\times } \ell }{\times }{\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}\,\big \vert \,J{=}-J^*,E^*Q = Q^*E{\ge } 0 \right\} ,\\ \varSigma _{\ell ,n,m}^{sdH}&= \left\{ (E,J,R,Q,B)\,\big \vert \,J=-J^*,R = R^*\ge 0,E^*Q = Q^*E\ge 0 \right\} . \end{aligned}$$

Proposition 3.11

Let $\ell ,n,m\in {\mathbb {N}}^*$. The sets

• (a)  $S_{(a)}:= \left\{ (E,J,Q,B)\,\big \vert \,\textrm{rk}\,[E,B] = \min \left\{ \ell ,n+m \right\} \right\}$

• (b)  $S_{(b)}:= \left\{ (E,J,Q,B)\,\big \vert \,\textrm{rk}\,[E,JQ,B] = \min \left\{ \ell ,2n+m \right\} \right\}$

• (c)  $S_{(c)}:= \left\{ (E,J,Q,B)\,\big \vert \,\textrm{rk}\,_{{\mathbb {K}}(x)}[xE-JQ,B] = \min \left\{ \ell ,n+m \right\} \right\}$

• (d)  $S_{(d)}:= \left\{ (E,J,Q,B)\,\left| \,\begin{array}{l}\forall Z\in {\mathbb {K}}^{n-\textrm{rk}\,E}~\text {with}~\textrm{im}\,Z = \ker E\\ : \textrm{rk}\,[E,JQZ,B] = \min \left\{ \ell ,n+m \right\} \end{array}\right. \right\}$ are relative generic in $\varSigma _{\ell ,n,m}^H$. The sets

• (e)  $S_{(e)}:= \left\{ (E,J,R,Q,B)\,\big \vert \,\textrm{rk}\,[E,B] = \min \left\{ \ell ,n+m \right\} \right\}$

• (f)  $S_{(f)}:= \left\{ (E,J,R,Q,B)\,\big \vert \,\textrm{rk}\,[E,(J-R)Q,B] = \min \left\{ \ell ,2n+m \right\} \right\}$

• (g)  $S_{(g)}:= \left\{ (E,J,R,Q,B)\,\big \vert \,\textrm{rk}\,_{{\mathbb {R}}(x)}[xE-(J-R)Q,B] = \min \left\{ \ell ,2n+m \right\} \right\}$

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

are relative generic in $\varSigma _{\ell ,n,m}^{sdH}$ and $\varSigma _{\ell ,n,m}^{dH}$. The sets

• (i)  $S_{(i)}:= \left\{ (E,J,Q,B)\,\big \vert \,\forall \lambda \in {\mathbb {C}}:\textrm{rk}\,_{{\mathbb {C}}}[\lambda E-JQ,B] = \min \left\{ \ell ,n+m \right\} \right\}$

• (ii)  $S_{(j)}{:=} \left\{ (E,J,R,Q,B)\,\big \vert \,\forall \lambda {\in }{\mathbb {C}}:\textrm{rk}\,_{{\mathbb {C}}}[\lambda E-(J-R)Q,B] {=} \min \left\{ \ell ,n+m \right\} \right\}$

are relative generic in $\varSigma _{\ell ,n,m}^H$ and $\varSigma _{\ell ,n,m}^{sdH}$, resp., if, and only if, $\ell \ne n+m$; otherwise their complement is relative generic in the respective reference set.

Proof

This proof is split into the two cases $\ell \ge n$ and $\ell <n$.

Case $\ell \ge n$: We find convenient reference sets to simplify our calculations. Note that

$$\begin{aligned} \varSigma _{\ell ,n,m}^{H}&\cong \left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E\ge 0 \right\} \times \left\{ J\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,J = -J^* \right\} \\ {}&\quad \,\,\times {\mathbb {K}}^{\ell \times m}. \end{aligned}$$

In Lemma 3.10 we have shown that $\left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E > 0 \right\}$ is relative generic in $\left\{ (E,Q)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times n}\,\big \vert \,E^*Q = Q^*E\ge 0 \right\}$. In view of Proposition 2.10 (d), we conclude that

$$\begin{aligned} V_1 := \left\{ (E,J,Q,B)\in \varSigma _{\ell ,n,m}^{H}\,\big \vert \, E^*Q = Q^*E>0 \right\} \subseteq \varSigma _{\ell ,n,m}^{H} \end{aligned}$$

is a relative generic subset of $\varSigma _{\ell ,n,m}^H$. Thus, in view of Proposition 2.10 (e), relative genericity in $\varSigma _{\ell ,n,m}^H$ and $V_1$ are equivalent. By definition of $\varSigma _{\ell ,n,m}^H$ and continuity of the spectrum, $V_1$ is a relatively open subset of

$$\begin{aligned} V_1' := \left\{ (E,J,Q,B)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}\,\big \vert \,E^*Q = Q^*E, J^* = -J \right\} . \end{aligned}$$

By Lemma 2.11 it suffices to prove the assertion for the reference set $V_1'$ instead of $\varSigma _{\ell ,n,m}^H$. For $\varSigma _{\ell ,n,m}^{sdH}$ we have

$$\begin{aligned} \varSigma _{\ell ,n,m}^{sdH} \cong&\varSigma _{\ell ,n,m}^{H}\times \left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert \,R^* = R\ge 0 \right\} . \end{aligned}$$

Since $V_1$ is a relative generic subset of $\varSigma _{\ell ,n,m}^H$, we conclude from Lemma 3.5 and Proposition 2.10 (d) that

$$\begin{aligned} \left\{ (E,J,R,Q,B)\,\big \vert \,E^*Q {=} Q^*E> 0, J^* {=} {-}J, R^* {=} R> 0 \right\} \cong V_1{\times }\left\{ R\,\big \vert \,R^* {=} R {>} 0 \right\} \end{aligned}$$

is a relative generic subset of $\varSigma _{\ell ,n,m}^{sdH}$. By continuity of the spectrum, we conclude with Proposition 2.10 (e) and Lemma 2.11 that it suffices to consider the reference set

$$\begin{aligned} V_2' {:=}&\left\{ (E,J,R,Q,B)\,\big \vert \,E^*Q {=} Q^*E, J^* {=} {-}J, R^* {=} R \right\} \cong V_1'{\times }\left\{ R{\in }{\mathbb {K}}^{\ell {\times } \ell }\,\big \vert \,R^* {=} R \right\} \end{aligned}$$

instead of $\varSigma _{\ell ,n,m}^{sdH}$. We consider therefore, without loss of generality, the reference set $V_1'$ instead of $\varSigma _{\ell ,n,m}^{H}$, and the reference set $V_2'$ instead of $\varSigma _{\ell ,n,m}^{sdH}$.

(a)   Since all minors are polynomials, $S_{(a)}$ is in view of Lemma 3.3 Zariski-open. In view of Lemma 2.9, it remains to prove that $S_{(a)}\cap V_1'$ is Euclidean dense in $V_1'$. Let $(E,J,Q,B)\in V_1'$ and $\varepsilon >0$. Since relative generic sets are especially Euclidean dense, see Lemma 2.10 (a), Lemma 3.7 (ii) yields that there are some ${\widetilde{E}},{\widetilde{Q}}\in {\mathbb {K}}^{\ell \times n}$ so that ${\widetilde{E}}^*{\widetilde{Q}} = {\widetilde{Q}}^*{\widetilde{E}}$, $\textrm{rk}\,{\widetilde{E}} = \min \left\{ \ell ,n \right\} = n$, and

$$\begin{aligned} \max \left\{ \left\| E-{\widetilde{E}} \right\| _{\ell ,n},\left\| Q-{\widetilde{Q}} \right\| _{\ell ,n} \right\} <\varepsilon . \end{aligned}$$

Consider the set

$$\begin{aligned} S_{(a)}({\widetilde{E}}) := \left\{ B\in {\mathbb {K}}^{\ell \times m}\,\big \vert \,\textrm{rk}\,[{\widetilde{E}},B] = \min \left\{ \ell ,n+m \right\} \right\} . \end{aligned}$$

By Lemma 3.3, $S_{(a)}({\widetilde{E}})$ is Zariski-open. Furthermore, in view of the basis completion lemma and since $\textrm{rk}\,{\widetilde{E}} = n$, it can be readily seen that $S_{(a)}({\widetilde{E}})\ne \emptyset$. Thus Lemma 2.7 yields that $S_{(a)}({\widetilde{E}})$ is generic. Especially, we conclude that there is some ${\widetilde{B}}\in {\mathbb {K}}^{\ell \times m}$ so that $\textrm{rk}\,[{\widetilde{E}},{\widetilde{B}}] = \min \left\{ \ell ,n+m \right\}$ and $\left\| B-{\widetilde{B}} \right\| _{\ell ,m}<\varepsilon$. Thus, we have found some $({\widetilde{E}},J,{\widetilde{Q}},{\widetilde{B}})\in S_{(a)}\cap V_1'$ with

$$\begin{aligned} \max \left\{ \left\| E-{\widetilde{E}} \right\| _{\ell ,n},\left\| Q-{\widetilde{Q}} \right\| _{\ell ,n},\left\| B-{\widetilde{B}} \right\| _{\ell ,m} \right\} <\varepsilon . \end{aligned}$$

This shows that $S_{(a)}\cap V_1'$ is indeed Euclidean dense and therefore relative generic in $V_1'$.

(b)   We distinguish further between $\ell \le n+m$ and $\ell >n+m$.

If $\ell \le n+m$, then we have $\ell = \min \left\{ \ell ,n+m \right\} = \min \left\{ \ell ,2n+m \right\}$ and thus the inclusion $S_{(a)}\subseteq S_{(b)}$ holds true. Since we have already shown that $S_{(a)}$ is relative generic in $V_1'$, Proposition 2.10 (b) yields that $S_{(b)}$ is relative generic in $V_1'$.

Let $\ell >n+m$. In view of Lemma 3.3, $S_{(b)}$ is Zariski open. By Lemma 2.9, it remains to prove that $S_{(b)}\cap V_1'$ is Euclidean dense in $V_1'$. Let $(E,J,Q,B)\in V_1'$ and $\varepsilon >0$. Analogously to (a), Lemma 3.7 (ii) implies the existence of some ${\widetilde{E}},{\widetilde{Q}}\in {\mathbb {K}}^{\ell \times n}$ so that $\textrm{rk}\,{{\widetilde{E}}} = n$, ${\widetilde{E}}^*{\widetilde{Q}} = {\widetilde{Q}}^*{\widetilde{E}}$ and

$$\begin{aligned} \max \left\{ \Vert {E-{\widetilde{E}}}\Vert _{\ell ,n}, \Vert {Q-{\widetilde{Q}}}\Vert _{\ell ,n} \right\} < \varepsilon /{2}. \end{aligned}$$

Consider the convex reference set

$$\begin{aligned} V_{(b)}({\widetilde{E}}) = \left\{ (J,Q,B)\in {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}\,\big \vert \,J = -J^*, {\widetilde{E}}^*Q = Q^*{\widetilde{E}} \right\} \end{aligned}$$

and the set

$$\begin{aligned} S_{(b)}({\widetilde{E}}) = \left\{ (J,Q,B)\in V_{(b)}({\widetilde{E}})\,\big \vert \,\textrm{rk}\,[{\widetilde{E}},JQ,B] = \min \left\{ \ell ,2n+m \right\} \right\} . \end{aligned}$$

We show that $S_{(b)}({\widetilde{E}})$ is relative generic in $V_{(b)}({\widetilde{E}})$. In view of Lemma 3.3, $S_{(b)}({\widetilde{E}})\subseteq V_{(b)}({\widetilde{E}})$ is relative Zariski-open in $V_{(b)}({\widetilde{E}})$. Since $V_{(b)}({\widetilde{E}})$ is convex, Lemma 2.13 yields that it suffices to prove that $S_{(b)}({\widetilde{E}})\ne \emptyset$. In view of Lemma 3.6, there are unitary (orthogonal) matrices $P\in \textbf{Gl}({\mathbb {K}}^\ell )$, $T\in \textbf{Gl}({\mathbb {K}}^n)$ and some diagonal matrix $\varSigma \in {\mathbb {R}}^{n\times n}$ so that

$$\begin{aligned} P{\widetilde{E}}T = \begin{bmatrix} \varSigma \\ 0_{(\ell -n)\times n} \end{bmatrix}. \end{aligned}$$

Choose the skew-symmetric matrix $J'\in {\mathbb {K}}^{\ell \times \ell }$ with

$$\begin{aligned} \forall i,j\in {\mathbb {N}}^*_{\le \ell }: J'_{i,j} = {\left\{ \begin{array}{ll} 1, &{} i = n+m+j,\\ -1, &{} j = n+m+i,\\ 0, &{} \text {else}, \end{array}\right. } \end{aligned}$$

and put

$$\begin{aligned} {\widehat{J}}' = P^*J'P,\quad {\widehat{Q}}' = {\widetilde{E}},\quad \text {and}\quad {\widehat{B}}' := P^*\begin{bmatrix} 0_{n\times m}\\ I_m\\ 0_{(\ell -n-m)\times m} \end{bmatrix}. \end{aligned}$$

Let $\kappa :=\min \left\{ \ell -n-m,n \right\}$. Since P and T are unitary and $\varSigma \in {\mathbb {K}}^{n\times n}$ is an invertible diagonal matrix, we have

$$\begin{aligned}&\textrm{rk}\,[{\widetilde{E}},{\widehat{J}}'{\widehat{Q}}',{\widehat{B}}']= \textrm{rk}\,P [{\widetilde{E}},{\widehat{J}}'{\widehat{Q}}',{\widehat{B}}']\begin{bmatrix} T\\ {} &{} T\\ {} &{} &{} I_m \end{bmatrix}\\&= \textrm{rk}\,[P{\widetilde{E}}T,P{\widehat{J}}'P^*P{\widehat{Q}}'T,P{\widehat{B}}'] \\&= \textrm{rk}\,\left[ \begin{array}{c|c|c} \varSigma &{} 0_{n\times n} &{} 0_{n\times m}\\ 0_{m\times n} &{} 0_{m\times n} &{} I_m\\ \hline 0_{\kappa \times n} &{} \begin{array}{cccccccc} \varSigma _{1,1} &{} \star &{} \cdots &{} \star &{} \star &{} \cdots &{} \star \\ &{} \ddots &{} \ddots &{} \vdots &{} \ddots &{} \ddots &{} \vdots \\ &{} &{} \varSigma _{\kappa -1,\kappa -1} &{} \star &{} \star &{} \cdots &{} \star \\ &{} &{} &{} \varSigma _{\kappa ,\kappa } &{} \star &{} \cdots &{} \star \end{array} &{} 0_{\kappa \times m}\\ 0_{(\ell -n-m-\kappa )\times n} &{} 0_{(\ell -n-m-\kappa )\times n} &{} 0_{(\ell -n-m-\kappa )\times n} \end{array}\right] \\&= n+m+\kappa \\&= \min \left\{ \ell ,2n+m \right\} . \end{aligned}$$

This shows that $({\widehat{J}}',{\widehat{Q}}',{\widehat{B}}')\in S_{(b)}({\widetilde{E}})$ and thus the latter set is indeed non-empty. We conclude that $S_{(b)}({\widetilde{E}})$ is relative generic in $V_{(b)}({\widetilde{E}})$. Hence, there are $({\widehat{J}},{\widehat{Q}},{\widehat{B}})\in S_{(b)}({\widetilde{E}})$ so that

$$\begin{aligned} \max \left\{ \left\| {\widehat{J}}-J \right\| _{\ell ,\ell },\left\| {\widehat{Q}}-{\widetilde{Q}} \right\| _{\ell ,n},\left\| {\widehat{B}}-B \right\| _{\ell ,m} \right\} <\frac{\varepsilon }{2}. \end{aligned}$$

From the definition of $S_{(b)}({\widetilde{E}})$, we conclude that $({\widetilde{E}},{\widehat{J}},{\widehat{Q}},{\widehat{B}})\in S_{(b)}\cap V_1'$, and the triangle inequality yields

$$\begin{aligned} \max \left\{ \left\| {\widetilde{E}}-E \right\| _{\ell ,n},\left\| {\widehat{J}}-J \right\| _{\ell ,\ell },\left\| {\widehat{Q}}-Q \right\| _{\ell ,n},\left\| {\widehat{B}}-B \right\| _{\ell ,m} \right\} <\varepsilon . \end{aligned}$$

This shows that $S_{(b)}\cap V_1'$ is indeed Euclidean dense in $V_1'$, and therefore relative generic in $V_1'$.

(c)   Analogously to the proof of Lemma 3.3, a simple application of the Leibniz formula for the determinant yields with Lemma 3.2 that the implication

$$\begin{aligned} \textrm{rk}\,_{{\mathbb {K}}}[E,B] \!=\! \min \left\{ \ell ,n\!+\!m \right\} \implies \forall A\!\in \!{\mathbb {K}}^{\ell \!\times \! n}: \textrm{rk}\,_{{\mathbb {K}}(x)}[xE\!-\!A,B] = \min \left\{ \ell ,n\!+\!m \right\} \end{aligned}$$

holds true for all $E\in {\mathbb {K}}^{\ell \times n}$ and $B\in {\mathbb {K}}^{\ell \times m}$. Thus, we have $S_{(a)}\subseteq S_{(c)}$. Since we have proven that $S_{(a)}$ is relative generic in $V_1'$, we conclude from Proposition 2.10 (b) that $S_{(c)}$ is relative generic in $V_1'$.

(d)   Since $\textrm{rk}\,E = n$ yields $\ker E = \left\{ 0 \right\}$, the inclusion

$$\begin{aligned} S_{(a)}\cap \left\{ (E,J,Q,B)\,\big \vert \,\textrm{rk}\,E = n \right\} \subseteq S_{(d)} \end{aligned}$$

holds true. Since $S_{(a)}$ is relative generic in $V_1'$ by (a) of the present proposition, we conclude from Lemma 3.7 (ii) and Proposition 2.10 (c) that the intersection $S_{(a)}\cap \left\{ (E,J,Q,B)\,\big \vert \,\textrm{rk}\,E = n \right\}$ is relative generic in $V_1'$. Thus, Proposition 2.10 (b) yields that $S_{(d)}$ is relative generic in $V_1'$.

The proof of (e)–(h) is analogous to the proof of (a)–(d): While the proof of (a), (c), and (d) does not depend on J, their counterparts (e), (g), and (h) follow analogously. To prove (f), we can simply add a summand $\left\{ R\in {\mathbb {K}}^{\ell \times \ell }\,\big \vert R = R^* \right\}$ to the real vector space $V_{(b)}({\widetilde{E}})$ in the proof of (b) and proceed analogously.

(i)   We proceed with the proof of (i) in steps (i$_1$)–(i$_5$).

Step (i$_1$):    We show that if $\ell = n+m$, then $S_{(i)}^c$ is relative generic in $V_1'$.

In view of Laplace’s expansion formula, for all $(E,J,Q,B)\in S_{(a)}$,

$$\begin{aligned} \det [xE-JQ,B] = x^n\underbrace{\det [E,B]}_{\ne 0} + \text {lower-order~terms}. \end{aligned}$$

Therefore, the fundamental theorem of algebra yields that there is some $\lambda \in {\mathbb {C}}$ so that $\det [\lambda E-JQ,B] = 0$. This shows that $S_{(a)}\subseteq S_{(i)}^c$. Since $S_{(a)}$ is relative generic in $V_1'$, Proposition 2.10 (b) yields that $S_{(i)}^c$ is relative generic in $V_{1}'$. Especially $S_{(i)}$ is not relative generic in $V_1'$.

Step (i$_2$):    Let $\ell \ne n+m$ and $\ell \ge n$. We construct an algebraic set ${\mathbb {V}}$ so that $V_1'\setminus S_{(i)}\subseteq {\mathbb {V}}$.

This construction is analogous to the construction in the proof of [18, Proposition B.8], where a similar result for matrices without structural constraints is shown. Let ${\widetilde{M}}_1,\ldots ,{\widetilde{M}}_q$, $q\in {\mathbb {N}}$ be all minors of order $d:= \min \left\{ \ell ,n+m \right\}$ w.r.t. ${\mathbb {K}}[x]^{\ell \times (n+m)}$ and put, for all $i\in {\mathbb {N}}^*_{\le q}$ and $P\in \textbf{Gl}({\mathbb {K}}^\ell )$,

$$\begin{aligned} M_i^{P}:{\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}&\rightarrow {\mathbb {K}}[x],\\ (E,J,Q,B)&\mapsto {\widetilde{M}}_i\left( P[xE-JQ,B]\right) . \end{aligned}$$

Since the rank of a matrix is invariant under regular transformations from the left and from the right, we have by Lemma 3.2 $(E,J,Q,B)\in S_{(i)}$ if, and only if,

$$\begin{aligned} \forall \lambda \in {\mathbb {C}}~\exists i\in {\mathbb {N}}^*_{\le q}~\exists P\in \textbf{Gl}({\mathbb {K}}^\ell ): M_i^{P}(E,J,Q,B)(\lambda )\ne 0. \end{aligned}$$

Define the maximal degree of $M_i^{P}$ as

$$\begin{aligned}&\gamma _i^{P}{:=} \max \left\{ \deg M_i^{P}(E,J,Q,R)\,\big \vert \,(E,J,Q,B)\in {\mathbb {K}}^{\ell {\times } n}\times {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m} \right\} {\ge } 0. \end{aligned}$$

In passing, note that the maximal degree $\gamma _i^{P}$ is attained at (E, J, Q, B) if, and only if, $\deg M_i^{P}(E,0,0,B) = \deg {\widetilde{M}}_i([xPET,B]) = \gamma _i$. Since $(E,0,0,B)\in V_1'$ for all $E\in {\mathbb {K}}^{\ell \times n}$ and $B\in {\mathbb {K}}^{\ell \times m}$,

$$\begin{aligned} \gamma _i^{P} = \max \left\{ \deg M_i^{P}(E,J,Q,R)\,\big \vert \,(E,J,Q,B)\in V_1' \right\} \end{aligned}$$

and, for all ${\mathscr {S}}\subseteq {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}$,

$$\begin{aligned}&\left\{ (E,J,Q,B){\in }{\mathscr {S}}\,\big \vert \,\deg M_i^{P}(E,J,Q,B) {=} \gamma _i \right\} \\ {}&=\left\{ (E,J,Q,B){\in }{\mathscr {S}}\,\big \vert \, M_i^{P}(E,0,0,B){\ne } 0 \right\} . \end{aligned}$$

By Lemma 3.3, there are polynomials $M_{i,k}^{P}\in {\mathbb {K}}[x_1,\ldots ,x_{\ell (2n+m+\ell )}]$, $k \in \left\{ 0,\ldots ,\gamma _i^{P} \right\}$, $i\in {\mathbb {N}}^*_{\le q}$, so that each $M_i^{P}$ has the representation

$$\begin{aligned} \forall (E,J,Q,B)\in \mathbb {K}^{\ell \times n}\times \mathbb {K}^{\ell \times \ell }\times \mathbb {K}^{\ell \times n}\times \mathbb {K}^{\ell \times m} \end{aligned}$$

The $M_{i,k}^{P}$, $k \in \left\{ 0,\ldots ,\gamma _{i}^{P} \right\}$, which are multivariate polynomials in the entries of E, J, Q, B, are the coefficients of $M_i^{P}$ in the monomial basis of the univariate polynomials. Using these representations, define, for all $i,j\in {\mathbb {N}}^*_{\le q}$ and $P\in \textbf{Gl}({\mathbb {K}}^\ell )$, the polynomials

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

(19)

If  $M_{i,\gamma _i}^{P}(E,J,Q,B)\ne 0$ and $M_{j,\gamma _j}^{P}(E,J,Q,B)\ne 0$, then $p_{i,j}^{P}(E,J,Q,B)$ is the Sylvester resultant (see Lemma 3.4) of the polynomials $M_i(E,J,Q,B)$ and $M_j(E,J,Q,B)$. Invoking Lemma 3.4 and Laplace’s expansion formula [12, p. 203], we have, for all ${\mathfrak {E}} = (E,J,Q,B)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}$, the equivalence

$$\begin{aligned} p_{i,j}^{P}({\mathfrak {E}}) = 0 \iff \big [ M_{i,\gamma _i}^{P}({\mathfrak {E}}) = M_{j,\gamma _j}^{P}({\mathfrak {E}}) = 0\ \vee \ M_i({\mathfrak {E}}),M_j({\mathfrak {E}})~\text {not~coprime}\big ]. \end{aligned}$$

(20)

Thus, Lemma 3.2 yields that the following chain of implications holds true for all $(E,J,Q,B)\in {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}$:

$$\begin{aligned}&\exists P\in \textbf{Gl}({\mathbb {K}}^\ell )~\exists i,j\in {\mathbb {N}}^*_{\le q}: p_{i,j}^{P}(E,J,Q,B)\ne 0\\&\implies \exists i,j\in {\mathbb {N}}^*_{\le q}: M_i^{P}(E,J,Q,B), M_j^{P}(E,J,Q,B)~\text {are~coprime}\\&\implies \forall \lambda {\in }{\mathbb {C}}~\exists \iota {\in }{\mathbb {N}}^*_{\le q}: {\widetilde{M}}_\iota [\lambda PET-PJP^*P^{-*}QT,B] {=} M_\iota ^{P}(E,J,Q,B)(\lambda ) \ne 0\\&\implies \forall \lambda \in {\mathbb {C}}: \textrm{rk}\,_{{\mathbb {C}}}[\lambda E-JQ,B]) = d. \end{aligned}$$

Let $I\subseteq {\mathbb {R}}[x_1,\ldots ,x_{\ell (2n+m+\ell )}]$ be the ideal generated by the set

$$\begin{aligned} \left\{ p_{i,j}^{P}\,\big \vert \,i,j\in {\mathbb {N}}^*_{\le q}, P\in \textbf{Gl}({\mathbb {K}}^\ell ) \right\} . \end{aligned}$$

Since multivariable polynomials are a commutative ring, the ideal I has the form

$$\begin{aligned} I = \left\{ \left. \sum _{\alpha = 1}^k r_\alpha p_{i_\alpha ,j_\alpha }^{P_\alpha }\,\right| \,\begin{array}{l} k\in {\mathbb {N}}_0, i_\alpha ,j_\alpha \in {\mathbb {N}}^*_{\le q},r_\alpha \in {\mathbb {R}}[x_1,\ldots ,x_{\ell (2n+m+\ell )}],\\ P_\alpha \in \textbf{Gl}({\mathbb {K}}^{\ell }),\alpha \in {\mathbb {N}}^*_{\le k} \end{array} \right\} , \end{aligned}$$

see [4, p. 538]. Define

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

as the algebraic set generated by the ideal I, see Lemma 2.6. Due to our chain of implications after (20), we have

$$\begin{aligned} (E,J,Q,B)\in {\mathbb {V}}^c&\implies \exists P\in \textbf{Gl}({\mathbb {K}}^\ell )~\exists i,j\in {\mathbb {N}}^*_{\le q}: p_{i,j}^{P}(E,J,Q,B)\ne 0\\&\implies (E,J,Q,B)\in S_{(i)}; \end{aligned}$$

hence,  ${\mathbb {V}}^c\cap V_1'\subseteq S_{(i)}\cap V_1'$ or, equivalently, $V_1'\cap S_{(i)}^c\subseteq {\mathbb {V}}\cap V_1'$.

Step (i$_3$):    We show that the set ${\mathbb {V}}^c\cap V_1'$ is Euclidean dense in $V_1'$.

The arguments used are similar to those in the proof of [18, Proposition B.8]. However [18, Proposition B.8] deals with the unrestrained case; a more involved proof has to obey the reference set $V_1'$. Let $(E,J,Q,B)\in V_1'$ and $\varepsilon >0$. By Lemma 3.7, there are ${\widetilde{E}},{\widetilde{Q}}\in {\mathbb {K}}^{\ell \times n}$ with $\textrm{rk}\,{\widetilde{E}} = n$, ${\widetilde{E}}^*{\widetilde{Q}} = {\widetilde{Q}}^*{\widetilde{E}}$ and $\max \left\{ \left\| E-{\widetilde{E}} \right\| _{\ell ,n},\left\| Q-{\widetilde{Q}} \right\| _{\ell ,n} \right\} <\frac{\varepsilon }{2}$. Consider the set

$$\begin{aligned} V_{(i)}({\widetilde{E}}) := \left\{ (J,Q,B)\in {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}\,\big \vert \,J^* = -J, {\widetilde{E}}^*Q = Q^*{\widetilde{E}} \right\} . \end{aligned}$$

Assume, for a moment, that we had already proven that the set

$$\begin{aligned} S_{(i)}({\widetilde{E}}) := \left\{ (J,Q,B)\in V_{(i)}({\widetilde{E}})\,\big \vert \,({\widetilde{E}},J,Q,B)\in {\mathbb {V}}^c \right\} \end{aligned}$$

is relative generic in $V_{(i)}({\widetilde{E}})$. Since $(J,{\widetilde{Q}},B)\in V_{(i)}({\widetilde{E}})$, we find some $({\widehat{J}},{\widehat{Q}},{\widehat{B}})\in S_{(i)}({\widetilde{E}})$ so that

$$\begin{aligned} \max \left\{ \left\| {\widehat{J}}-J \right\| _{\ell ,\ell },\left\| {\widehat{Q}}-{\widetilde{Q}} \right\| _{\ell ,n},\left\| {\widehat{B}}-B \right\| _{\ell ,m} \right\} <\frac{\varepsilon }{2}. \end{aligned}$$

By definition of $S_{(i)}({\widetilde{E}})$ and $V_{(i)}({\widetilde{E}})$, we have $({\widetilde{E}},{\widehat{J}},{\widehat{Q}},{\widehat{B}})\in {\mathbb {V}}^c\cap V_1'$ so that

$$\begin{aligned} \max \left\{ \left\| E-{\widetilde{E}} \right\| _{\ell ,n},\left\| {\widehat{J}}-J \right\| _{\ell ,\ell },\left\| {\widehat{Q}}-Q \right\| _{\ell ,n},\left\| {\widehat{B}}-B \right\| _{\ell ,m} \right\} <\varepsilon . \end{aligned}$$

This shows that ${\mathbb {V}}^c\cap V_1'$ is indeed Euclidean dense in $V_1'$. It remains to prove that $S_{(i)}({\widetilde{E}})$ is relative generic in $V_{(i)}({\widetilde{E}})$. The latter is convex and hence it remains, in view of Lemma 2.13, to show that $S_{(i)}({\widetilde{E}})$ is non-empty. As in the proof of [18, Proposition B.8], we distinguish between the two cases $\ell <n+m$ and $\ell >n+m$; the case $\ell = n+m$ was dealt with in Step (i$_1$).

Step (i$_4$):   Let $\ell <n+m$.

Since $\ell \ge n$, there is some $T\in \textbf{Gl}({\mathbb {K}}^\ell )$ so that

$$\begin{aligned} T{\widetilde{E}} = \begin{bmatrix} I_n\\ 0_{(\ell -n)\times n} \end{bmatrix}. \end{aligned}$$

(21)

Since T is regular we have, for all $\lambda \in {\mathbb {C}}$ and $(J,Q,B)\in V_{(i)}({\widetilde{E}})$,

$$\begin{aligned} \textrm{rk}\,[\lambda {\widetilde{E}}-JQ,B] = \textrm{rk}\,T[\lambda {\widetilde{E}}-JQ,B] = \textrm{rk}\,[\lambda T{\widetilde{E}}-TJT^*T^{-*}Q,TB]. \end{aligned}$$

Moreover, $Q\in {\mathbb {K}}^{\ell \times n}$ fulfills ${\widetilde{E}}^*Q$ if, and only if, $(T{\widetilde{E}})^*(T^{-*}Q) = (T^{-*}Q)^*(T{\widetilde{E}})$. In view of (21), the latter is the case if, and only if, $T^{-*}Q$ allows a block-representation

$$\begin{aligned} T^{-*}Q = \begin{bmatrix} Q^1\\ Q^2 \end{bmatrix} \end{aligned}$$

for some $Q^2\in {\mathbb {K}}^{(\ell -n)\times n}$ and some Hermitian $Q^1\in {\mathbb {K}}^{n\times n}$. Consider the matrices

$$\begin{aligned} J_0 = T^{-1}\begin{bmatrix} 0 &{} -1\\ 1 &{} 0 &{} -1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} 1 &{} 0 &{} -1\\ &{} &{} &{} 1 &{} 0 \end{bmatrix}T^{-*}\in {\mathbb {K}}^{\ell \times \ell }\quad \text {and}\quad Q(\beta ) := T^*\begin{bmatrix} \beta _1\\ &{} \ddots \\ &{} &{} \beta _n\\ 0 &{} \cdots &{} 0\\ \vdots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} 0 \end{bmatrix}\in {\mathbb {K}}^{\ell \times n} \end{aligned}$$

for $\beta \in ({\mathbb {R}}\setminus \left\{ 0 \right\} )^{n}$. $J_0$ is skew-Hermitian and $Q(\beta )$ fulfils, for all $\beta \in {\mathbb {R}}^n$, ${\widetilde{E}}^*Q(\beta ) = Q(\beta )^*{\widetilde{E}}$. Our choice of $Q(\beta )$ and $J_0$ guarantees that $(J_0,Q(\beta ))\in V_{(i)}({\widetilde{E}})$ for all $\beta \in {\mathbb {R}}^n$ and $B\in {\mathbb {K}}^{\ell \times m}$. Furthermore,

$$\begin{aligned} J_0Q(\beta ) = {\left\{ \begin{array}{ll} \begin{bmatrix} 0 &{} -\beta _2\\ \beta _1 &{} 0 &{} -\beta _3\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} \beta _{n-2} &{} 0 &{} -\beta _n\\ &{} &{} &{} \beta _{n-1} &{} 0 \end{bmatrix}, &{} \ell = n,\\ \left[ \begin{array}{c} \begin{array}{ccccc} 0 &{} -\beta _2\\ \beta _1 &{} 0 &{} -\beta _3\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} \beta _{n-2} &{} 0 &{} -\beta _n\\ &{} &{} &{} \beta _{n-1} &{} 0\\ &{} &{} &{} &{} \beta _n \end{array}\\ \hline 0_{(\ell -n-1)\times n} \end{array}\right] ,&\ell >n. \end{array}\right. } \end{aligned}$$

Let $\delta \in ({\mathbb {R}}\setminus \left\{ 0 \right\} )^{\ell -n}$ and $\xi \in ({\mathbb {R}}\setminus \left\{ 0 \right\} )^{\ell -n+1}$, and put

$$\begin{aligned} B(\delta ,\xi ) := T^{-1}\left[ \begin{array}{c|c} 0_{(n-1)\times (\ell -n+1)} &{} 0_{(n-1)\times (m-\ell +n-1)}\\ \hline \begin{array}{cccc} \xi _1\\ \delta _1 &{} \xi _2\\ &{} \ddots &{} \ddots \\ &{} &{} \delta _{\ell -n} &{} \xi _{\ell -n+1} \end{array}&0_{(\ell -n)\times (m-\ell +n-1)} \end{array}\right] . \end{aligned}$$

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

$$\begin{aligned} {\widetilde{M}}_1 = P(x)\mapsto \det [P(x)_{i,j}]_{i,j\in {\mathbb {N}}^*_{\le \ell }}, \end{aligned}$$

i.e. $\sigma$ and $\pi$ in Definition 3.1 are the respective identity functions, and

$$\begin{aligned} {\widetilde{M}}_2 = P(x)\mapsto \det [P(x)_{i,j}]_{i,j-1\in {\mathbb {N}}^*_{\le \ell }}, \end{aligned}$$

i.e. $\pi$ in Definition 3.1 is the increment by one and $\sigma$ the identity function. Then, our choice of $J_0, Q(\beta )$, and $B(\delta ,\xi )$ yields

$$\begin{aligned} M_1^{T}({\widetilde{E}},J_0,Q(\beta ),B(\delta ,\xi ))&= {\widetilde{M}}_1([xT{\widetilde{E}}-TJ_0T^{-*}TQ(\beta ),B(\delta ,\xi )])\\&= x^n\prod _{j = 1}^{\ell -n}\delta _j + \text {lower-order terms}. \end{aligned}$$

Since $\delta _j\ne 0$, $j\in {\mathbb {N}}^*_{\le \ell -n}$, we have $\deg M_1^{T}(E,J_0,Q(\beta ),B(\delta ,\xi )) = n = \gamma _1^{T}$. Further, we have

$$\begin{aligned} M_2^{T}({\widetilde{E}},J_0,Q(\beta ),B(\delta ,\xi )) = \prod _{i = 2}^n \beta _i\prod _{j = 1}^{\ell -n+1}\xi _j\ne 0. \end{aligned}$$

Thus, the equivalence (20) yields that $p_{1,2}^{T}({\widetilde{E}},J_0,Q(\beta ),B(\delta ,\xi ))\ne 0$. By definition of $S_{(i)}({\widetilde{E}})$ and ${\mathbb {V}}$, we conclude $(J_0,Q(\beta ),B(\delta ,\xi ))\in S_{(i)}({\widetilde{E}})$ and hence the latter set is non-empty. This proves the assertion for the case $\ell <n+m$.

Step (i$_5$):   Let $\ell >n+m$. We show that $S_{(i)}({\widetilde{E}})$ is non-empty.

Analogously to Step (i$_4$), we use the decomposition

$$\begin{aligned} T{\widetilde{E}} = \begin{bmatrix} I_n\\ 0_{(\ell -n)\times n} \end{bmatrix} \end{aligned}$$

for some $T\in \textbf{Gl}({\mathbb {R}}^\ell )$. Define, additionally to the matrices $J_0$ and $Q(\beta )$ defined in Step (i$_4$), the matrix

$$\begin{aligned} B_0' := T^{-1}\left[ \begin{array}{c} 0_{n\times m}\\ \hline \begin{array}{cccc} 1\\ 1 &{} 1\\ &{} \ddots &{} \ddots \\ &{} &{} 1 &{} 1\\ &{} &{} &{} 1 \end{array}\\ \hline 0_{(\ell -n-m-1)\times m} \end{array}\right] . \end{aligned}$$

Analogously to the case $\ell < n+m$ in Step (i$_4$), we can, without loss of generality, assume that the minors ${\widetilde{M}}_i$ are ordered in such a manner that

$$\begin{aligned} {\widetilde{M}}_1 = P(x)\mapsto \det [P(x)_{i,j}]_{i,j\in {\mathbb {N}}^*_{\le n+m}}\quad \text {and}\quad {\widetilde{M}}_2 {=} P(x)\mapsto \det [P(x)_{i,j}]_{i-1,j\in {\mathbb {N}}^*_{\le n{+}m}}. \end{aligned}$$

Then, it is straightforward to verify

$$\begin{aligned} M_1^{T}({\widetilde{E}},J_0',Q'({\textbf{1}}),B_0') = \det \underbrace{\begin{bmatrix} x &{} 1 &{} &{} &{} &{} &{}\\ -1 &{} x &{} 1 &{} &{} &{}\\ &{} \ddots &{} \ddots &{} \ddots &{} &{} &{} \\ &{} &{} -1 &{} x &{} 1\\ &{} &{} &{} -1 &{} x \end{bmatrix}}_{\in {\mathbb {K}}[x]^{n\times n}}. \end{aligned}$$

Since E has n columns, it is easy to see that the maximal degree of any minor of $[xPE-PJQ,B]$ of order d does not exceed n. Hence we have

$$\begin{aligned} \deg M_1^{P}(E,J_0,Q({\textbf{1}}),B_0') = n = \gamma _1^{T,S}\quad \text {and}\quad M_2^{P}({\widetilde{E}},J_0,Q({\textbf{1}}),B_0') = (-1)^n\ne 0. \end{aligned}$$

This shows that $p_{1,2}^{P}({\widetilde{E}},J_0,Q({\textbf{1}}),B_0')\ne 0$ and therefore $(J_0,Q({\textbf{1}}),B_0')\in S_{(i)}({\widetilde{E}})$. This shows that the latter set is indeed non-empty. As discussed earlier, this shows that ${\mathbb {V}}^c$ is indeed Euclidean dense in $V_1'$.

(j)   The proof of (j) is analogous to the proof of (i): We can construct an algebraic set $\widehat{{\mathbb {V}}}$ of the same type as in (i). It is just necessary to redefine the $M_i^{P}$ as

$$\begin{aligned} M_i^{P} := (E,J,R,Q,B)\mapsto {\widetilde{M}}_i([xPET-P(J-R)P^*P^{-*}QT,B]) \end{aligned}$$

to incorporate the R. Then we show that for given $E\in {\mathbb {K}}^{\ell \times n}$ with full rank the set of all matrices (J, R, Q, B) with $(E,J,R,Q,B)\in \widehat{{\mathbb {V}}}^c$ has non-empty intersection with the convex set $\left\{ (J,R,Q,B)\,\big \vert \,(E,J,R,Q,B)\in \varSigma _{\ell ,n,m}^{sdH} \right\}$. This can be directly concluded from the proof in Step (i$_3$) since each matrix triple $(J,Q,B)\in S(E)$ yields a feasible matrix quadruple (J, 0, Q, B) for the proof of this step. This completes the proof of (a)–(j) for the case $\ell \ge n$.

Case $\ell < n$: We prove (a)–(h).

In view of Lemma 3.8, the set

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

is relative generic in $\varSigma _{\ell ,n,m}^H$ and

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

is relative generic in $\varSigma _{\ell ,n,m}^{sdH}$. Since $\ell = \min \left\{ \ell ,n+m \right\} = \min \left\{ \ell ,2n+m \right\}$, the inclusions $S^H\subseteq S_{(a)}\cap S_{(b)}\cap S_{(c)}\cap S_{(d)}$ and $S^{sdH}\subseteq S_{(e)}\cap S_{(f)}\cap S_{(g)}\cap S_{(h)}$ hold, where the inclusion $S_{(a)}\subseteq S_{(c)}$ can be verified analogously to (c) in the case $\ell \ge n$. Hence, Proposition 2.10 (b) yields that $S_{(a)}$, $S_{(b)}$, $S_{(c)}$, and $S_{(d)}$ are relative generic in $\varSigma _{\ell ,n,m}^H$, and $S_{(e)}$, $S_{(f)}$, $S_{(g)}$, and $S_{(h)}$ are relative generic in $\varSigma _{\ell ,n,m}^{sdH}$.

(i)   Analogously to the case $\ell \ge n$, we construct the polynomials $p_{i,j}$ associated to all minors ${\widetilde{M}}_1,\ldots ,{\widetilde{M}}_q$, $q\in {\mathbb {N}}^*$, of order $\ell$ w.r.t. ${\mathbb {R}}[x]^{\ell \times (n+m)}$ and their induced functions

$$\begin{aligned} M_i^{P,T}&:{\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}\rightarrow {\mathbb {K}}[x],\\ {}&(E,J,Q,B) \mapsto {\widetilde{M}}_i([xPET-PJP^*P^{-*}QT,B]). \end{aligned}$$

We omit the details of this construction. The ideal I generated by the polynomials $p_{i,j}^{P,T}$ defined as in (19) generates the algebraic set

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

which fulfills ${\mathbb {V}}^c\cap \varSigma _{\ell ,n,m}^H\subseteq S_{(i)}\cap \varSigma _{\ell ,n,m}^H$. By Definition 2.8, it remains to prove that ${\mathbb {V}}^c\cap \varSigma _{\ell ,n,m}^H$ is Euclidean dense in $\varSigma _{\ell ,n,m}^H$. Let $(E,J,Q,B)\in \varSigma _{\ell ,n,m}^H$. As discussed in detail in the proof of (i) in the case $\ell \ge n$, it suffices to show that

$$\begin{aligned} S(E) := \left\{ (J,Q,B)\in {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}\,\big \vert \,(E,J,Q,B)\in {\mathbb {V}}^c \right\} \end{aligned}$$

is relative generic in

$$\begin{aligned} V(E) := \left\{ (J,Q,B)\in {\mathbb {K}}^{\ell \times \ell }\times {\mathbb {K}}^{\ell \times n}\times {\mathbb {K}}^{\ell \times m}\,\big \vert \,J^* = -J, Q^*E = E^*Q\ge 0 \right\} . \end{aligned}$$

Furthermore, an application of Lemma 3.8 analogously to Lemma 3.7 (ii) in the case $\ell \ge n$ yields that it suffices to consider the case $\textrm{rk}\,E = \ell$. Since ${\mathbb {V}}$ is an algebraic set, S(E) is Zariski-open. Since V(E) is a non-empty convex set, it suffices therefore in view of Lemma 2.13 to prove that $S(E)\cap V(E) \ne \emptyset$. There are $P\in \textbf{Gl}({\mathbb {K}}^\ell )$ and $T\in \textbf{Gl}({\mathbb {K}}^n)$ so that $PET = [I_\ell , 0_{\ell \times (n-\ell )}]$. Choose $Q_1:= P^*[I_\ell , 0_{\ell \times (n-\ell )}]T^{-1}$. Since T is regular, we conclude from

$$\begin{aligned} T^*E^*QT = T^*E^*P^*P^{-*}QT = \begin{bmatrix} I_\ell &{}\quad 0_{\ell \times (n-\ell )}\\ 0_{(n-\ell )\times \ell } &{}\quad 0_{(n-\ell )\times (n-\ell )} \end{bmatrix} \end{aligned}$$

that $E^*Q_1$ is Hermitian and positive semi-definite. Choose further the skew-symmetric matrix

$$\begin{aligned} J_1 = P^{-1}\begin{bmatrix} 0 &{} -1\\ 1 &{} 0 &{} -1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} 1 &{} 0 &{} -1\\ &{} &{} &{} 1 &{} 0 \end{bmatrix}P^{-*}\in {\mathbb {K}}^{\ell \times \ell }\quad \text {and}\quad {\widehat{B}} := P^{-1}[e_\ell ,0_{\ell \times (m-1)}]\in {\mathbb {K}}^{\ell \times m} \end{aligned}$$

where $e_\ell = [0,\ldots ,0,1]^\top$. With these particular matrices, we get

$$\begin{aligned} P\big [xE-J_1Q_1,{\widehat{B}}]\begin{bmatrix} T\\ {} &{} I_m \end{bmatrix} = \left[ \begin{array}{c|c|c|c} \begin{array}{ccccc} x &{} 1\\ -1 &{} x &{} 1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} -1 &{} x &{} 1\\ &{} &{} &{} -1 &{} x \end{array} &{} \, 0_{\ell \times (n-\ell )}\, &{} \begin{array}{c} 0\\ 0\\ \vdots \\ 0\\ 1 \end{array}&\, 0_{\ell \times (m-1)} \end{array}\right] . \end{aligned}$$

Without loss of generality we can assume that the ${\widetilde{M}}_i$ are ordered in such a manner that

$$\begin{aligned}&{\widetilde{M}}_1 = P(x)\mapsto \det [P(x)_{i,j}]_{i,j\in {\mathbb {N}}^*_{\le \ell }}\quad \text {and}\\ {}&{\widetilde{M}}_2 = P(x)\mapsto \det [P(x)_{i,j}]_{i\in {\mathbb {N}}^*_{\le \ell },j\in \left\{ 2,\ldots ,\ell ,n+1 \right\} }. \end{aligned}$$

Then, it is easy to verify that $M_1^{P,T}(E,J_1,Q_1,{\widehat{B}})$ and $M_2^{P,T}(E,J_1,Q_1,{\widehat{B}})$ are coprime and $\deg M_1^{P,T}(E,J_1,Q_1,{\widehat{B}}) = \gamma _1^{P,T}$. We can, analogously to the case $\ell \ge n$, conclude from (20) that $p_{1,2}^{P,T}(E,J_1,Q_1,{\widehat{B}})\ne 0$. Thus, we have $(J_1,Q_1,{\widehat{B}})\in S(E)\cap V(E)$ and therefore the latter is non-empty. We conclude that S(E) is relative generic in V(E). Therefore, we conclude that ${\mathbb {V}}^c$ is indeed Euclidean dense in $\varSigma _{\ell ,n,m}^H$.

(j)   As in the case $\ell \ge n$, we can modify the proof of (i) to incorporate the additional matrix R and readily conclude that $S_{(j)}$ is relative generic in $\varSigma _{\ell ,n,m}^{sdH}$. This completes the proof of the proposition. $\square$

4 Relative genericity of controllability

We are now in the position to derive necessary and sufficient conditions on the dimensions $\ell ,n,m$ of linear finite-dimensional port-Hamiltonian descriptor systems described by (1) so that they are relative generically controllable (with respect to the five concepts defined in Proposition 2.1) in the reference sets $\varSigma _{\ell ,n,m}^{H}$,  $\varSigma _{\ell ,n,m}^{sdH}$, and $\varSigma _{\ell ,n,m}^{dH}$.

Theorem 4.1

Consider, for $\ell ,n,m\in {\mathbb {N}}^*$, the port-Hamiltonian descriptor systems described by (1), subsets of controllable systems $S_{\text {controllable}}^H$ defined in (9), and reference sets $\varSigma _{\ell ,n,m}^{H}$,  $\varSigma _{\ell ,n,m}^{sdH}$ and $\varSigma _{\ell ,n,m}^{dH}$ defined in (6)–(8). Then the following equivalences hold:

$$\begin{aligned} \begin{array}{lllll} \mathrm{(i)} &{}~S_{\text {freely initializable}}^H &{} is\, rel.\, gen.\, in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell \le n+m,\\ \mathrm{(ii)} &{}~S_{\text {impulse controllable}}^H &{} is\, rel.\, gen.\, in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell \le n+m,\\ \mathrm{(iii)} &{}~S_{\text {behavioural controllable}}^H &{} is\, rel.\, gen. \,in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell \ne n+m,\\ \mathrm{(iv)} &{}~S_{\text {completely controllable}}^H &{} is\, rel.\, gen.\, in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell<n+m,\\ \mathrm{(v)} &{}~S_{\text {strongly controllable}}^H &{} is\, rel.\, gen. \,in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell <n+m.\\ \end{array} \end{aligned}$$

Moreover, if $S_{{\textit{controllable}}}^H$ is not relative generic in $\varSigma _{\ell ,n,m}^H$, then $(S_{{\textit{controllable}}}^H)^c$ is relative generic in $\varSigma _{\ell ,n,m}^H$, where controllable stands for either one of the studied controllability notions.

The above remains true, if we consider $S_{{\textit{controllable}}}^{sdH}$ with reference set $\varSigma _{\ell ,n,m}^{sdH}$ or $S_{{\textit{controllable}}}^{dH}$ with reference set $\varSigma _{\ell ,n,m}^{dH}$.

Proof

In this proof, we will use the notions introduced in Proposition 3.11.

(i)    By Propositions 3.11 and 2.10 (c), the set $S_{(a)}\cap S_{(b)}$ is relative generic in $\varSigma _{\ell ,n,m}^H$. In view of the algebraic characterization of impulse controllability in Proposition 2.1, we have the inclusions

$$\begin{aligned} S_{(a)}\cap S_{(b)}\subseteq {\left\{ \begin{array}{ll} S_{\textit{freely~initializable}}^H, &{} \ell \le n+m,\\ (S_{\textit{freely~initializable}}^H)^c, &{} \ell >n+m. \end{array}\right. } \end{aligned}$$

By Proposition 2.10 (b), we conclude that $S_{\textit{freely~initializable}}^H$ is relative generic in $\varSigma _{\ell ,n,m}^H$ if, and only if, $\ell \le n+m$ and that $(S_{\textit{freely~initializable}}^H)^c$ is relative generic in $\varSigma _{\ell ,n,m}^H$ if, and only if, $\ell >n+m$.

(ii)   By Proposition 3.11 and Proposition 2.10 (c), the set $S_{(b)}\cap S_{(d)}$ is relative generic in $\varSigma _{\ell ,n,m}^H$. Proposition 2.1 yields the inclusions

$$\begin{aligned} S_{(b)}\cap S_{(d)}\subseteq {\left\{ \begin{array}{ll} S_{\textit{impulse~controllable}}^H, &{} \ell \le n+m,\\ (S_{\textit{impulse~controllable}}^H)^c, &{} \ell >n+m. \end{array}\right. } \end{aligned}$$

Thus, Proposition 2.10 (b) yields the assertion for $S_{\textit{impulse~controllable}}^H$.

(iii)   In view of Propositions 3.11 and 2.10 (c), the set $S_{(c)}\cap S_{(i)}$ is relative generic in $\varSigma _{\ell ,n,m}^H$ if, and only if, $\ell \ne n+m$; otherwise the set $S_{(c)}\cap S_{(i)}^c$ is relative generic in $\varSigma _{\ell ,n,m}^H$. By Proposition 2.1, the inclusion

$$\begin{aligned} S_{(c)}\cap S_{(i)}\subseteq S_{\textit{behavioural~controllable}}^H \end{aligned}$$

holds true. Thus, Proposition 2.10 (b) yields that $S_{\textit{behavioural~controllable}}^H$ is relative generic in $\varSigma _{\ell ,n,m}^H$ if, and only if, $\ell \ne n+m$; otherwise its complement is relative generic in $\varSigma _{\ell ,n,m}^H$.

(iv)   In view of Proposition 2.1, the identity

$$\begin{aligned} S_{\textit{completely~controllable}}^H = S_{\textit{freely~initializable}}^H\cap S_{\textit{behavioural~controllable}}^H \end{aligned}$$

holds true. By Proposition 2.10 (c), the assertion follows from (i) and (iii) of the present proposition.

(v)   Proposition 2.1 yields that

$$\begin{aligned} S_{\textit{strongly~controllable}}^H = S_{\textit{impulse~controllable}}^H\cap S_{\textit{behavioural~controllable}}^H. \end{aligned}$$

Hence, Proposition 2.10 (c), and (ii) and (iii) of the present proposition imply the proposed equivalence for relative genericity of $S_{\textit{strongly~controllable}}$.

The assertion for the reference set $\varSigma _{\ell ,n,m}^{sdH}$ can be analogously proven by replacing $S_{(a)}, S_{(b)}, S_{(c)}, S_{(d)}$ and $S_{(i)}$ by $S_{(e)}, S_{(f)}, S_{(g)}, S_{(h)}$ and $S_{(j)}$, respectively.

Finally, in view of Lemma 3.5 and Proposition 2.10 (d), $\varSigma _{\ell ,n,m}^{dH}$ is a relative generic subset of $\varSigma _{\ell ,n,m}^{sdH}$. Therefore, Proposition 2.10 (e) yields the assertion for the reference set $\varSigma _{\ell ,n,m}^{dH}$. This shows the proposition. $\square$

We finally observe that generic controllability holds for port-Hamiltonian descriptor systems if, and only if, relative generic controllability holds for unconstrained differential-algebraic systems. This is made precise in the following corollary.

Corollary 4.2

Let $r,\ell ,n,m\in {\mathbb {N}}^*$. Then the port-Hamiltonian descriptor systems (1) are generically controllable if, and only if, the unconstrained descriptor systems (3) are relative generically controllable in $\varSigma _{\ell ,n,m}^{H}$, where controllable stands for either of the five controllability concepts defined in Proposition 2.1.

Proof

This is a consequence of the conjunction of [18, Thm. 2.3] and Theorem 4.1. $\square$

5 Relative genericity of stabilizability

In this section we study–similar to relative generic controllability in Sect. 4–relative generic stabilizability. Recall that there are three different concepts of stabilizability–see Proposition 2.2. We derive necessary and sufficient conditions on the dimensions $\ell ,n,m$ of linear finite-dimensional port-Hamiltonian descriptor systems described by (1) so that they are relative generically stabilizable in the reference sets $\varSigma _{\ell ,n,m}^{H}$,  $\varSigma _{\ell ,n,m}^{sdH}$, and $\varSigma _{\ell ,n,m}^{dH}$.

Theorem 5.1

Let $\ell ,n,m\in {\mathbb {N}}^*$. Then the following equivalences hold:

$$\begin{aligned} \begin{array}{lllll} (i) &{}~S_{\text {behavioural stabilizable}}^H &{} is\, rel.\, gen.\, in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell \ne n+m,\\ (ii) &{}~S_{\text {completely stabilizable}}^H &{} is\, rel.\, gen. in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell<n+m,\\ (iii) &{}~S_{\text {strongly stabilizable}}^H &{} is\, rel.\, gen. in~\varSigma _{\ell ,n,m}^{H} &{}~\iff &{}~\ell <n+m.\\ \end{array} \end{aligned}$$

The above remains true, if we consider $S_{{\textit{stabilizable}}}^{sdH}$ with reference set $\varSigma _{\ell ,n,m}^{sdH}$. Moreover, if $\ell >n+m$, then $(S_{{\textit{completely stabilizable}}}^*)^c$ and $(S_{{\textit{strongly stabilizable}}}^*)^c$ are relative generic in $\varSigma _{\ell ,n,m}^*$, where $*$ stands for either H or dH.

Proof

We distinguish the two cases $\ell \ne n+m$ and $\ell = n+m$.

Case $\ell \ne n+m$: In view of Proposition 2.2, the inclusions

$$\begin{aligned} S_{(a)}\cap S_{(b)}\cap S_{(i)}\subseteq {\left\{ \begin{array}{ll} S_{\text {completely stabilizable}}^H, &{} \ell <n+m,\\ (S_{\text {completely stabilizable}}^H)^c, &{} \ell >n+m, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} S_{(b)}\cap S_{(d)}\cap S_{(i)}\subseteq {\left\{ \begin{array}{ll} S_{\text {strongly stabilizable}}^H, &{} \ell <n+m,\\ (S_{\text {strongly stabilizable}}^H)^c, &{} \ell >n+m, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} S_{(c)}\cap S_{(i)}\subseteq S_{\text {behavioural stabilizable}}^H \end{aligned}$$

hold true. Therefore, Propositions 3.11 and 2.10(b) and (c) yield the assertion for $\ell \ne n+m$.

Case $\ell = n+m$: We show that the set

$$\begin{aligned} S^H {:=} \left\{ (E,J,Q,B){\in }\varSigma _{\ell ,n,m}^H\,\big \vert \,\exists \lambda {\in }\textrm{int}\,\overline{{\mathbb {C}}}_{+}: \textrm{rk}\,[\lambda E{-}JQ,B] < \ell {=} \min \left\{ \ell ,n{+}m \right\} \right\} \end{aligned}$$

contains a inner point w.r.t. the Euclidean relative topology on $\varSigma _{\ell ,n,m}^H$. Then, we can show that the complement of $S_{{\textit{controllable}}}^H$ has a relative inner point and is therefore especially not nowhere dense.

In passing we note that $\ell = n+m\ge 2$. Since the spectrum of a matrix depends continuously from its entries, $S^H$ is open and thus it suffices to show that $S^H\ne \emptyset$. Put

$$\begin{aligned}&E: {=} \begin{bmatrix} I_n\\ 0_{m\times n} \end{bmatrix}, \ J :{=} \begin{bmatrix} 0 &{} -1\\ 1 &{} 0 &{} -1\\ &{} \ddots &{} \ddots &{} \ddots \\ &{} &{} 1 &{} 0 &{} -1\\ &{} &{} &{} 1 &{} 0 \end{bmatrix}\in {\mathbb {K}}^{\ell \times \ell }, \ Q :{=} \begin{bmatrix} 0_{n\times n}\\ -e_n^\top \\ 0_{(m-1)\times n} \end{bmatrix},\\ {}&\ B :{=} \begin{bmatrix} 0_{n\times m}\\ I_m \end{bmatrix}. \end{aligned}$$

Then $E^*Q = 0_{n\times n}\in {{\mathbb {R}}}^{n\times n}$ is symmetric and positive semi-definite. Furthermore,

$$\begin{aligned} JQ = {\left\{ \begin{array}{ll}\begin{bmatrix} 0_{(n-1)\times n}\\ e_n^\top \\ 0_{1\times n}\\ -e_n^\top \\ 0_{(m-2)\times n} \end{bmatrix}, &{} m\ge 2 \\ \begin{bmatrix} 0_{(n-1)\times n}\\ e_n^\top \\ 0_{1\times n} \end{bmatrix},&m = 1. \end{array}\right. } \end{aligned}$$

This yields

$$\begin{aligned}{}[xE-JQ,B] = {\left\{ \begin{array}{ll} \begin{bmatrix} x\\ &{} &{}\ddots \\ &{} &{} &{}&{}x &{} \\ &{} &{} &{}&{} &{} x-1\\ &{} &{} &{}&{} &{} 0 &{}1\\ &{} &{} &{}&{} &{} 1 &{} 0 &{} 1\\ &{} &{} &{} &{}&{} &{} &{}&{} \ddots \\ &{} &{} &{} &{} &{}&{}&{}&{}&{} 1 \end{bmatrix}, &{} m\ge 2\\ \begin{bmatrix} x\\ &{} &{}\ddots \\ &{} &{} &{}&{}x &{} \\ &{} &{} &{}&{} &{} x-1\\ &{} &{} &{}&{} &{} 0 &{}1 \end{bmatrix},&m = 1. \end{array}\right. } \end{aligned}$$

Hence, we conclude

$$\begin{aligned} \det [xE-JQ,B] = x^{n-1}(x-1), \end{aligned}$$

which vanishes at $x=1$; equivalently, since $[xE-JQ,B]\in {\mathbb {K}}[x]^{\ell \times \ell }$, $\textrm{rk}\,[1\cdot E-JQ,B]<\ell$. Thus $(E,J,Q,B)\in S^H$ and the latter set is non-empty. In view of Propositions 3.11 and 2.10(c), the sets $S_{(a)}\cap S_{(b)}$, $S_{(b)}\cap S_{(d)}$, and $S_{(c)}$ are relative generic in $\varSigma _{\ell ,n,m}^H$. Therefore, we conclude that the sets $S_{(a)}\cap S_{(b)}\cap S^H$, $S_{(b)}\cap S_{(d)}\cap S^H$, and $S_{(c)}\cap S^H$ have non-empty relative interior. By Proposition 2.1, the inclusions

$$\begin{aligned} S_{(a)}\cap S_{(b)}\cap S^H&\subseteq (S_{\text {completely stabilizable}}^H)^c\\ S_{(b)}\cap S_{(d)}\cap S^H&\subseteq (S_{\text {strongly stabilizable}}^H)^c\\ S_{(c)}\cap S^H&\subseteq (S_{\text {behavioural stabilizable}}^H)^c \end{aligned}$$

hold true. Hence, the sets on the right-hand side contain an inner point and are especially not nowhere dense. By Proposition 2.10 (a), we conclude that neither of the sets $S_{\text {completely stabilizable}}^H, S_{\text {strongly stabilizable}}^H$ and $S_{\text {behavioural stabilizable}}^H$ is relative generic in $\varSigma _{\ell ,n,m}^H$.

The respective statement for $\varSigma _{\ell ,n,m}^{sdH}$ can be proven similarly by considering

$$\begin{aligned} S^{sdH} := \left\{ (E,J,Q,B)\in \varSigma _{\ell ,n,m}^{sdH}\,\big \vert \,\exists \lambda \in \overline{{\mathbb {C}}}_+: \textrm{rk}\,[\lambda E-JQ,B] < \min \left\{ \ell ,n+m \right\} \right\} . \end{aligned}$$

Since $\varSigma _{\ell ,n,m}^{dH}$ is a relative generic subset of $\varSigma _{\ell ,n,m}^{sdH}$, Proposition 2.10 (e) yields the assertion for the reference set $\varSigma _{\ell ,n,m}^{dH}$. This completes the proof of the theorem. $\square$

Finally we observe–as for controllability in Corollary 4.2–that port-Hamiltonian descriptor systems (1) are generically stabilizable if, and only if, the unconstrained systems (3) are relative generically stabilizable in $\varSigma _{\ell ,n,m}^{H}$.

Corollary 5.2

Let $\ell ,n,m\in {\mathbb {N}}^*$. Then the port-Hamiltonian descriptor systems (1) are generically stabilizable if, and only if, the unconstrained descriptor systems (3) are relative generically stabilizable in $\varSigma _{\ell ,n,m}^{H}$, where stabilizable stands for either of the three concepts in Proposition 2.2.

Proof

This is a consequence of the conjunction of [18, Thm. 3.3] and Theorem 5.1. $\square$

6 Conclusion and outlook

We have studied relative genericity of controllability and stabilizability for port-Hamiltonian systems and derived necessary and sufficient conditions for relative genericity of five controllability and three stabilizability concepts. This extends the result of [23, Theorem II.1] to descriptor systems with arbitrary dimensions and non-trivial dissipation matrix.

For square port-Hamiltonian systems, that is, $\ell = n$, the matrix E in (1) is generically invertible. In this case, Theorem 4.1 is very closely related to [23, Theorem II.1], although with the addition of a possibly (generically) nonzero R. In future work, we will put particular care on the rank of E and consider the reference sets

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

and

$$\begin{aligned} \varSigma _{\ell ,n,m}^{dH,\le r} := \left\{ (E,J,R,Q,B)\in \varSigma _{\ell ,n,m}^{dH}\,\big \vert \,\textrm{rk}\,E\le r \right\} . \end{aligned}$$

When $\ell = n$ and $r<n$, this excludes especially the case that the considered systems can be (generically) reduced to an ODE system. We are confident that we can extend [19, Theorems 3.2 and 4.2], where unstructured DAEs with the same restriction on the rank of E are considered, to these reference sets for port-Hamiltonian descriptor systems.