1 Introduction

In a predecessor [10] of the present note, we characterized genericity of controllability and stabilizability of differential algebraic systems (DAEs) described by the equation

$$\begin{aligned} \tfrac{\textrm{d}}{\textrm{d}t}(Ex) = Ax + Bu, \end{aligned}$$
(1)

where

$$\begin{aligned} (E,A,B)\in \quad \Sigma _{\ell ,n,m}\ := \ \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m} \quad \text {and} \quad \ell ,n,m\in {\mathbb {N}}^*. \end{aligned}$$

The notion of genericity is defined as follows.

Definition 1.1

([18, p. 26] and [15, p. 50])  A set \( \mathbb {V} \subseteq \mathbb {R}^n\) is called an algebraic variety, if there exist finitely many polynomials

$$\begin{aligned} p_1(x_1,\ldots , x_n),\ldots ,p_k(x_1,\ldots , x_n) \in \mathbb {R}[x_1,\ldots , x_n] \end{aligned}$$

such that \(\mathbb {V}\) is the locus of their zeros, i. e.

$$\begin{aligned} \mathbb {V} = \left\{ x\in \mathbb {R}^n\ \big \vert \, \forall \, i\in \underline{k}: p_i(x)=0 \right\} = \bigcap _{i=1}^k p_i^{-1}(\left\{ 0 \right\} ). \end{aligned}$$
(2)

An algebraic variety \( \mathbb {V}\) is called proper if  \( \mathbb {V} \subsetneq \mathbb {R}^n\). The set of all algebraic varieties in \(\mathbb {R}^n\) is denoted as

$$\begin{aligned} \mathscr {V}(\mathbb {R}^n) := \left\{ \mathbb {V}\subseteq \mathbb {R}^n\,\left| \,\exists \, q_1(\cdot ),\ldots ,q_k(\cdot )\in \mathbb {R}[x_1,\ldots , x_n]: \bigcap _{i=1}^k q_i^{-1}(\left\{ 0 \right\} ) = \mathbb {V}\right. \right\} \end{aligned}$$
(3)

and the set of all proper algebraic varieties as

$$\begin{aligned} \mathscr {V}^{\text {prop}}(\mathbb {R}^n) := \mathscr {V}(\mathbb {R}^n)\setminus \left\{ \mathbb {R}^n \right\} . \end{aligned}$$
(4)

A set \(S\subseteq \mathbb {R}^n\) is called generic, if there exist a proper algebraic variety \(\mathbb {V}\in \mathscr {V}_n^{\text {prop}}(\mathbb {R})\) so that \(S^c\subseteq \mathbb {V}\). If the algebraic variety \(\mathbb {V}\) is known, then we call S generic with respect to (w.r.t.) \(\mathbb {V}\). \(\diamond \)

When the concept of genericity is applied to differential algebraic systems \((E,A,B)\in \Sigma _{\ell ,n,m}\), a drawback is that the set \(\Sigma _{\ell ,n,m}\) is too "large". By this we mean that if \(\ell = n\), then in each arbitrarily small neighbourhood of \((E,A,B)\in \Sigma _{\ell ,n,m}\) there is some \(E'\in \mathbb {R}^{\ell \times n}\) with full rank, and this yields that \((E',A,B)\) is an ordinary differential equations. To be more precise, the set

$$\begin{aligned} \Sigma ^{\mathrm{\tiny ODE}}_{n,m} = \left\{ (E,A,B)\in \Sigma _{n,n,m}\,\big \vert \,E\in {\textbf{G}}{\textbf{l}}(\mathbb {R}^n) \right\} \end{aligned}$$

is the nonempty complement of the preimage of zero under the polynomial

$$\begin{aligned} (E,A,B)\mapsto \det E\in \mathbb {R}[x_1,\ldots ,x_{n(2n+m)}]\setminus \left\{ 0 \right\} \end{aligned}$$

and therefore a generic set. Each system (1) that is described by a triple \((E,A,B)\in \Sigma ^{\mathrm{\tiny ODE}}_{n,m}\) corresponds to the ordinary differential equation

$$\begin{aligned} \dot{x} = E^{-1}Ax + E^{-1}Bu. \end{aligned}$$

For this case, Lee and Markus [12] proved that that the set of all controllable systems is open and dense w.r.t. the Euclidean topology, and Wonham [18, Thm. 1.3] showed in the first edition of his monograph that the set of all controllable systems is generic. We are aiming at reference sets where E does not have full rank, these are the systems (E, A, B) of (1) belonging to

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

or

$$\begin{aligned} \Sigma _{\ell ,n,m}^{= r} \ := \ \left\{ (E,A,B)\in \Sigma _{\ell ,n,m}\,\big \vert \,\textrm{rk}\,_{\mathbb {R}} E = r \right\} , \qquad \text {where} \ r\in {\mathbb {N}}. \end{aligned}$$

However, the question as to whether controllability and stabilizability holds for one of these system classes is not well posed. If \(\Sigma _{\ell ,n,m}^{\le r}\) and \(\Sigma _{\ell ,n,m}^{= r}\) were (affine) subspaces, then we could identify them with a real coordinate space and study whether the sets of matrix triples \((E,A,B)\in \Sigma _{\ell ,n,m}^{\le r}\) and \((E,A,B)\in \Sigma _{\ell ,n,m}^{= r}\) whose corresponding DAE is controllable are generic subsets in the sense of Definition 1.1. Unfortunately, these sets are (in general) not a subspace and hence Definition 1.1 is not readily applicable. To resolve this, we propose to study relative genericity of controllability and stabilizability – a concept recently introduced by Kirchhoff [11].

Definition 1.2

([11, Definition I.2]) Let \(S,V\subseteq \mathbb {R}^n\). We call S relative generic (rel. gen.) in V if, and only if, there is some proper algebraic variety \(\mathbb {V}\in \mathscr {V}^{\text {prop}}(\mathbb {R}^n)\) so that

$$\begin{aligned} S^c\cap V = V\setminus S \subseteq \mathbb {V}\cap V \end{aligned}$$
(RG1)

and

$$\begin{aligned} \mathbb {V}^c\cap V = V\setminus \mathbb {V} \ \text { \ is~dense~in}~V. \end{aligned}$$
(RG2)

 \(\diamond \)

Relative genericity of controllability has already been shown for linear port Hamiltonian systems without dissipation, see [11]. In the following, we relate relative genericity to other concepts of genericity known in the literature.

First of all, recall what “generic” means in a topological context. In a Baire space \((X,\mathscr {O})\), a set \(S\subseteq X\) is called generic if, and only if, there are open, dense sets \(D_i\in \mathscr {O}\), \(i\in \mathbb {N}\), so that \(\bigcap _{i\in \mathbb {N}}D_i\subseteq S\), see [9, p. 45]. A stronger formulation of this, which can be extended to arbitrary topological spaces, is that a set is generic if, and only if, it contains some open and dense set. This definition was used e. g. by Banaszuk and Przyłuski [3] to show that the systems (1) with \((E,A,B)\in \Sigma _{\ell ,n,m}^{\le r}\) are generically controllable and stabilizable. Unfortunately, their notions of controllability and stabilizability are different to the standard concepts treated by the control theory community, see for example the survey by Berger and Reis [4, Cor. 4.3].

The two definitions of genericity given in the latter paragraph are purely topological. However, the \(\mathbb {R}^n\) is not only intrinsically a Baire space (with the Euclidean topology) but also a complete measure space (with the Lebesgue measure). Therefore, we should also consider genericity from a measure theoretic point of view. If \((X,\mathscr {A},\mu )\) is a complete measure space (or \(\mu \) just an outer measure), then we could call some set \(S\subseteq X\) generic if, and only if, there is some \(\mu \)-nullset \(N\in \mathscr {A}\) so that \(X\setminus S\subseteq N\). This concept is e. g. used by Vovk [17]. A reference from mathematical systems theory might be the survey article by Hunt and Kaloshin [9], where prevalence is considered: a subset S of the complete metric Abelian group X is prevalent if, and only if, its complement is a shy set, i. e. a nullset with respect to a nontrivial measure on X so that all its translations are nullsets, too. These properties are essentially the properties of the Lebesgue-measue on \(\mathbb {R}^n\), i. e. Lebesgue-almost-everywhere implies prevalence. It can be shown that the converse holds true. [9, Proposition 2.5] The concept of prevalence has been extended to relative prevalence on completely metrizable convex subsets of X, see [2]. Unfortunately, neither of our reference sets \(\Sigma _{\ell ,n,m}^{= r}\) and \(\Sigma _{\ell ,n,m}^{\le r}\) is convex. Furthermore, these measure-theoretic concepts do largely not consider the topological properties of generic sets.

Therefore, we would like to see a concept that includes both the topological and the measure theoretic point of view. Intuitively, such a definition is of the form

$$\begin{aligned} S\subseteq \mathbb {R}^n~\text {is~generic} :\iff \begin{array}{l} O\subseteq S~\text {for~some~Euclidean~open,~dense}~O\\ \text {and}~\lambda ^n(S^c) = 0, \end{array} \end{aligned}$$
(5)

where \(\lambda ^n\) denotes the n-dimensional Lebesgue measure. Genericity as in Definition 1.1 does fulfil this requirement: each proper algebraic variety is a nowhere dense Lebesgue nullset, see [6, p. 240]. Therefore, Wonham’s definition is stronger than the intuitive definition (5). Considering any co-countable set, we see that it is strictly stronger. The drawback that some sets such as \(\mathbb {R}\setminus \mathbb {N}\) should intuitively be generic can be resolved when considering analytic everywhere defined functions instead of polynomials; the resulting concept remains strictly stronger than the intuitive definition (5) and is strictly weaker than Wonham’s definition. A problem that occurs in generalizing this concept to arbitrary subsets of \(\mathbb {R}^n\) is that not each subset is intrinsically a measure space with a nontrivial measure. We choose to ignore this fact and try to define a genericity that is topological with respect to the Euclidean relative topology and measure theoretic for special submanifolds. When sticking to the algebraic varieties used in Wonham’s definition, this leads to relative genericity as in Definition 1.2.

The paper is organized as follows. Section 2 is devoted to the properties of relative generic sets and some illustrating examples. We give a topological characterization of relative generic sets and briefly discuss some measure theoretic properties. The main results, that are the characterizations of relative genericity of controllability and stabilizability for sets with respect to the reference set \(\Sigma _{\ell ,n,m}^{\le r}\) in terms of the dimensions \(\ell ,n,m\) and the maximal rank r, are given in Sections 3 and 4 , resp. The proofs in these sections rely heavily on the well-known algebraic characterizations of controllability and stabilizability (see Propositions 3.1 and 4.1 ) and on subtle and involved technicalities, which we have relegated to “Appendix A”.

2 Relative generic sets

In the present section, we present elementary properties of relative generic sets. In the introduction, we claimed that relative genericity as in Definition 1.2 can be viewed from a topological and, provided that the reference set admits some “structure”, from a measure theoretic point of view. Since we consider subsets of the n-dimensional Euclidean space, we equip each reference set with its Euclidean relative topology. Additionally to the Euclidean relative topology we consider the coarser Zariski relative topology. Recall [15, pp. 50] that the Zariski topology is defined by the property that all closed sets are the algebraic varieties. In terms of these topologies, we can characterize relative generic sets as follows.

Lemma 2.1

Let \(S,V\subseteq \mathbb {R}^n\), \(n\in \mathbb {N}^*\). The set S is relative generic in the reference set V if, and only if, S contains a Zariski open, Euclidean dense set from the respective relative topology on V.

Proof

Let S be relative generic in V. By Definition 1.2, there is some algebraic variety \(\mathbb {V}\in \mathscr {V}^{\text {prop}}(\mathbb {R}^n)\) so that \(S^c\cap V \subseteq \mathbb {V}\cap S\) and \(\mathbb {V}^c\cap V\) is Euclidean dense in V. The inclusion \(S^c\cap V \subseteq \mathbb {V}\cap S\) is equivalent to the inclusion \(\mathbb {V}^c\cap V\subseteq S\cap V\subseteq S\). By definition of the Zariski topology, \(\mathbb {V}\) is Zariski closed and hence \(\mathbb {V}^c\) is Zariski open. The relative topology on V induced by some topology \(\mathscr {O}\) on \(\mathbb {R}^n\) is by definition the set of all \(V\cap O\), where \(O\in \mathscr {O}\). Therefore, \(\mathbb {V}^c\cap V\) is a subset of S that is open w.r.t. the relative Zariski topology on V and dense w.r.t. the relative Euclidean topology on V.

Conversely, if there is some set \(\widetilde{O}\subseteq S\) that is open w.r.t. the relative Zariski topology on V and dense w.r.t. the relative Euclidean topology on V, then \(\widetilde{O}\) is necessarily of the form \(\widetilde{\mathbb {V}}^c\cap V\) for some algebraic variety \(\widetilde{\mathbb {V}}\). Unless \(V = \emptyset \), Euclidean density of \(\widetilde{O}\) yields that \(\widetilde{O}\) is nonempty or, equivalently, \(\widetilde{\mathbb {V}}\) a proper algebraic variety. If \(V = \emptyset \), however, then \(\widetilde{O} = \widehat{\mathbb {V}}\cap V\) for all \(\widehat{\mathbb {V}}\in \mathscr {V}^{\text {prop}}(\mathbb {R}^n)\) and hence we can w.l.o.g. assume that \(\widetilde{\mathbb {V}}\in \mathscr {V}^{\text {prop}}(\mathbb {R}^n)\). Therefore, we conclude that there is indeed some proper algebraic variety \(\widetilde{\mathbb {V}}\) so that the properties (RG1) and (RG2) are fullfilled. Thus, S is relative generic in V. \(\square \)

This characterization yields that genericity and relative genericity coincide for the reference set \(\mathbb {R}^n\). In this case Zariski open and Zariski dense sets are Euclidean dense. In [11] we have already mentioned that we need to impose some additional condition on relative generic sets beside containing a nonempty Zariski open set; otherwise we can show that each co-finite subset of a discrete reference set is relative generic, which is not consistent with the intuitive meaning of genericity. Since the Euclidean topology is the intrinsic topology on \(\mathbb {R}^n\), we would like to see density with respect to the Euclidean relative topology on the reference set. Here we ask as to whether it suffices to require density in (RG2) in the relative Zariski topology. This, however, is generally not the case.

Example 2.2

Consider the reference set \(V = \mathbb {R}\times \left\{ 0 \right\} \cup [0,1]\times [0,1]\). Then \(\mathbb {V} = \mathbb {R}\times \left\{ 0 \right\} \) is a proper algebraic variety and hence \(V\setminus \mathbb {V}\) is nonempty and open in the relative Zariski topology on V. Let O be Zariski-open such that \(O\cap V\setminus \mathbb {V} = \emptyset \). Since \(V\setminus \mathbb {V}\) contains some inner point, each polynomial that vanishes on \(V\setminus \mathbb {V}\) vanishes everywhere and therefore, \(O = \emptyset \). This shows that \(V\setminus \mathbb {V}\) is a Zariski-open, Zariski-dense subset of V which is not Euclidean dense.

The following proposition contains a small collection of properties of relative generic sets.

Proposition 2.3

Let \(S_1,S_2,V,V'\subseteq \mathbb {R}^n\), \(n\in \mathbb {N}^*\) be arbitrary.

  1. (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.

  2. (b)

    The empty set is relative generic only in the empty set itself.

  3. (c)

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

  4. (d)

    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.

  5. (e)

    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 \(V\times U\).

  6. (f)

    If \(S_2\subseteq V\) 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 \({V'}\).

  7. (g)

    If \(S_1\) is relative generic in V, then \(S_1^c\) is not relative generic in V.

Proof

(a) is shown in [11, Proposition I.3], and (b) and (c) follow from Lemma 2.1. We show (d): Since the Zariski topology is coarser than the Euclidean topology, the intersection and union of two Zariski open Euclidean dense sets is Zariski open and Euclidean dense. Thus, Lemma 2.1 yields that if \(S_1\) and \(S_2\) are relative generic in V, then their intersection and union is likewise relative generic in V. Conversely, if \(S_1\cap S_2\) is relative generic in V, then (c) yields with the trivial inclusions \(S_1\cap S_2\subseteq S_1\) and \(S_1\cap S_2\subseteq S_2\) that both \(S_1\) and \(S_2\) are relative generic in V.

(e) is consequence of the properties of the product topology and Lemma 2.1.

We show (f): With Lemma 2.1, we may assume w.l.o.g. that \(S_1\) and \(S_2\) are Zariski-open (and hence Euclidean open); therefore it remains to show that \(S_1\) is Euclidean dense in \(S_2\) if, and only if, \(S_1\) is Euclidean dense in V. This equivalence, however, is a general property of open sets.

It remains to prove (g): Assume that both \(S_1\) and \(S_1^c\) are relative generic in V. By Lemma 2.1, we conclude that there are two disjoint Euclidean open and dense sets in the relative topology on V. This yields that either one of these sets is empty and therefore \(V = \emptyset \), a contradiction to the assumption. Therefore there is no non-trivial partition of a given reference set in relative generic sets. \(\square \)

A first result on the reference sets \((E,A,B)\in \Sigma _{\ell ,n,m}^{\le r} \) and \((E,A,B)\in \Sigma _{\ell ,n,m}^{= r} \) is the following.

Lemma 2.4

\(\Sigma _{\ell ,n,m}^{= r} \) is relative generic in \(\Sigma _{\ell ,n,m}^{\le r}\) for each \(\ell ,n,m\in \mathbb {N}^*, r\in \mathbb {N}\).

Proof

Let \(\widetilde{M}_1,\ldots ,\widetilde{M}_q\) be all minors of order r with respect to \(\mathbb {R}^{\ell \times n}\) and consider the mappings

$$\begin{aligned} M_i:\Sigma _{\ell ,n,m}\rightarrow \mathbb {R},\qquad (E,A,B)\mapsto \widetilde{M}_i(E),\qquad i\in \underline{q}. \end{aligned}$$

Then the identity

$$\begin{aligned} \Sigma _{\ell ,n,m}^{\le r}\setminus \Sigma _{\ell ,n,m}^{= r} \ = \ \underbrace{\left( \bigcap _{i = 1}^q M_i^{-1}(\left\{ 0 \right\} )\right) }_{=:\mathbb {V}\in \mathscr {V}^{\text {prop}}(\mathbb {R}^n)} \ \cap \quad \Sigma _{\ell ,n,m}^{\le r} \end{aligned}$$

holds true. In view of Definition 1.2, it remains to prove that

$$\begin{aligned} \mathbb {V}^c \cap \Sigma _{\ell ,n,m}^{\le r} = \Sigma _{\ell ,n,m}^{=r} \quad \text {is Euclidean dense in} \quad \Sigma _{\ell ,n,m}^{\le r}. \end{aligned}$$

Let \((E,A,B)\in \Sigma _{\ell ,n,m}^{\le r}\) and \(\varepsilon >0\). Then

$$\begin{aligned} \exists S\in {\textbf{G}}{\textbf{l}}(\mathbb {R}^\ell ) \ \exists T\in {\textbf{G}}{\textbf{l}}(\mathbb {R}^n) \ : SET = \begin{bmatrix} I_{d}\\ &{} 0 \end{bmatrix},\qquad \text {where}\ d = \textrm{rk}\,E, \end{aligned}$$

and the perturbation matrix

$$\begin{aligned} \varDelta := \delta \, S^{-1} \begin{bmatrix} 0_{d\times d}\\ &{} \quad I_{r-d}\\ &{} &{} \quad 0 \end{bmatrix}T^{-1}\in \mathbb {R}^{\ell \times n} \quad \text {for} \quad \delta >0 \quad \text {so that} \left\| \varDelta \right\| _{\ell \times n} = 1 \end{aligned}$$

satisfies

$$\begin{aligned} \textrm{rk}\,(E+\tfrac{\varepsilon }{2}\varDelta )=r \qquad \text {and}\qquad \left\| \left( E+\tfrac{\varepsilon }{2}\varDelta ,A,B\right) -(E,A,B) \right\| = \frac{\varepsilon }{2} <\varepsilon . \end{aligned}$$

Therefore, \(\Sigma _{\ell ,n,m}^{= r}\) is a dense subset of \(\Sigma _{\ell ,n,m}^{\le r}\). \(\square \)

As a consequence of Proposition 2.3(v) and Lemma 2.4 we derive the following corollary that \(\Sigma _{\ell ,n,m}^{\le r}\) and \(\Sigma _{\ell ,n,m}^{=r}\) are interchangeable reference sets.

Corollary 2.5

A set \(S\subseteq \Sigma _{\ell ,n,m}\) is relative generic in \(\Sigma _{\ell ,n,m}^{\le r}\) if, and only if, \(S\subseteq \Sigma _{\ell ,n,m}\) is relative generic in \(\Sigma _{\ell ,n,m}^{=r}\).

The proofs of our main Theorems 3.2 and 4.2 depend crucially on the relative genericity of some rank properties. This is the content of the following proposition. Since the proofs are rather lengthy, we only state the result and relegate the proofs to “Appendix A”.

Proposition 2.6

Let \(r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\} \) be arbitrary. Then the sets

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

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

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

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

are relative generic in \(\Sigma _{\ell ,n,m}^{= r}\).

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

is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(\big ( \ell \ne n+m \ \vee \ r = 0\big )\),

otherwise \(S_{(v)}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\).

  • \(S_{(vi)} := \left\{ (E,A,B)\in \Sigma _{\ell ,n,m}\,\big \vert \,\forall \,\lambda \in \overline{\mathbb {C}}_{+}: \textrm{rk}\,_{\mathbb {C}}[\lambda E-A,B] = \min \left\{ \ell ,n+m \right\} \right\} \)

is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(\big ( \ell \ne n+m \ \vee \ r = 0\big )\).

An interesting observation about Proposition 2.6 is that only the sets \(S_{(i)}^r\) and \(S_{(ii)}^r\) depend on r and that relative genericity of \(S_{(v)}\) and \(S_{(vi)}\) depends (unless \(E = 0\)) not on r at all. Intuitively, this is clear for \(S_{(iii)}\), \(S_{(v)}\) and \(S_{(vi)}\), since we do not impose any restriction onto A and B so that we have enough degrees of freedom left to fill the rank of the matrix \([xE-A,B]\) in the polynomial or pointwise sense. For \(S_{(iv)}\), the intuitive reasoning is that the rank defect of E can be compensated by AZ. Another noteworthy fact is that we recover for \(r = \min \left\{ \ell ,n \right\} \) the results of [10, Proposition B.3, B.5 and B.8].

In the remainder of this section we interprete relative generic sets from a measure theoretic point of view for special submanifolds. In the next proposition it is shown that these special submanifolds are at least those, who admit a countable induced atlas. We recall the necessary definitions.

Definition 2.7

(Submanifold, see [13, p. 125] and [1, p. 252]) A set \(M\subseteq \mathbb {R}^n\) is an analytic submanifold of \(\mathbb {R}^n\) if, and only if, for each \(x_0\in M\) there is an open neighbourhood \(U\subseteq \mathbb {R}^n\) of \(x_0\), an open set \(V\subseteq \mathbb {R}^n\) and an analytic diffeomorphism \(\varphi \in \textrm{Diff}^\omega (U,V)\), i.e. a real analytic bijective function \(\varphi :U\rightarrow V\) whose inverse function \(\varphi ^{-1}:V\rightarrow U\) is also analytic, with the property \(\varphi (M\cap U) = V\cap (\mathbb {R}^m\times \left\{ 0_{n-m} \right\} )\). The pair \((U,\varphi )\) is a local submanifold chart. Any collection \((U_i,\varphi _i)_{i\in I}\) of local submanifold charts with \(\bigcup _{i\in I} U_i\supset M\) is a submanifold atlas. This atlas induces the family \((U_i\cap M,\varphi _i\big \vert _{U_i\cap M})_{i\in I}\), which is a atlas of local manifold charts for M, the induced atlas.

Proposition 2.8

Let V be an analytic submanifold of \(\mathbb {R}^n\), \(n\in \mathbb {N}^*\), so that an induced atlas is countable and S is relative generic in V. Then \(S^c\cap V\) is a nullset with respect to the Riemann-Lebesgue measure on V.

Proof

Since V is a submanifold of the Riemannian maniold \(\mathbb {R}^n\) (with the standard scalar product as Riemannian metric), it is a Riemannian manifold itself and allows therefore the definition of a Riemann-Lebesgue measure. We show that each algebraic variety \(\mathbb {V}\in \mathscr {V}^{\text {prop}}(\mathbb {R}^n)\) which fullfills the condition (RG2) is a nullset with respect to the Riemann-Lebesgue measure on V. First, we recall a characterization of nullsets with respect to the Riemann-Lebesgue measure on a Riemannian manifold M: A set \(A\subseteq M\) is a nullset if, and only if, for each chart \((\varphi ,U)\) of a countable atlas of M the set \(\varphi (A\cap U)\) is a Lebesgue nullset. Since we require for the induced atlas of V to be countable, it suffices to show that the image of \(S^c\cap V\) under the charts of this atlas is a nullset. These charts however are analytic diffeomorphisms and therefore the image of \(S^c\cap V\) under those the preimage of zero under analytic mappings. The requirement that \(S^c\cap V\) is nowhere dense implies that these analytic mappings are nonzero. Federer [6, p. 240] proved that the preimage of zero under nonzero analytic functions is a Lebesgue nullset. Therefore, we conclude that \(S^c\cap V\) is indeed a nullset with respect to the Riemann-Lebesgue measure on V. \(\square \)

We would like to relate the previous proposition to our reference set \(\Sigma _{\ell ,n,m}^{\le r}\). This set is an algebraic variety. Assume that the ideal of polynomials which vanish on \(\Sigma _{\ell ,n,m}^{\le r}\) is generated by \(p_1,\ldots ,p_k\). Recall that \((E,A,B)\in \Sigma _{\ell ,n,m}^{\le r}\) is called nonsingular [15, p. 97] if the rank of the Fréchet derivative of \((p_1,\ldots ,p_k)\) at (E, A, B) is maximal. Using the implicit function theorem as in [16, pp. 88-89] it can be shown that each nonsingular (E, A, B) has an open neighbourhood U so that \(V = \Sigma _{\ell ,n,m}^{\le r}\cap U\) is an analytic submanifold with a single chart. Therefore, the induced atlas is especially countable and we can apply Proposition 2.8 to find that relative genericity in \(\Sigma _{\ell ,n,m}^{\le r}\) yields Riemann-Lebesgue-almost-surely near nonsingular points. It can be shown that the nonsingular points are contained in \(\Sigma _{\ell ,n,m}^{= r}\). This implies that the previous statement holds true for \(\Sigma _{\ell ,n,m}^{= r}\), too. Further research in this direction, however, is not part of the present note. The purpose of the previous discussion was to show how a measure theoretic interpretation of relative genericity could be attempted.

3 Controllability

In this section, we first recall the well-known definitions and characterizations of freely initializable, impulse controllable, behavioural controllable, completely controllable and strongly controllable systems. The algebraic criteria are then used to derive necessary and sufficient conditions for the sets of all matrix triples so that the corresponding DAE system (1) is controllable (in one of the five mentioned ways) is relative generic in the reference set \(\Sigma _{\ell ,n,m}^{\le r}\).

We fix, for the remainder of this note, \(\ell ,n,m\in \mathbb {N}^*\) and \(r\in \mathbb {N}\).

The following controllability definitions rely on the notion of behaviour used for (1) and defined as

$$\begin{aligned} \mathfrak {B}_{(E,A,B)} := \left\{ (x,u)\in {\mathscr {W}^{1,1}_{\text {loc}}(\mathbb {R},\mathbb {R}^n)} \times \mathscr {L}^1_{\text {loc}}(\mathbb {R},\mathbb {R}^m)\left| {\begin{array}{l} Ex \quad \text {is absolutely continuous} \\ \text {and for a.a.}~t\in \mathbb {R} \ :\\ \frac{\text { d}}{\text { d}t}(Ex)(t) = Ax(t) + Bu(t) \end{array}}\right\} \right. \end{aligned}$$

where \(\mathscr {L}^1_{\text {loc}}(\mathbb {R},\mathbb {R}^d)\) denotes the set of locally integrable functions \(f:\mathbb {R}\rightarrow {\mathbb {R}}^d\), and \(\mathscr {W}^{1,1}_{\text {loc}}(\mathbb {R},\mathbb {R}^d)\) is the Sobolev space of all functions \(f\in \mathscr {L}^1_{\text {loc}}(\mathbb {R},\mathbb {R}^d)\) with \(f^{(1)}\in \mathscr {L}^1_{\text {loc}}(\mathbb {R},\mathbb {R}^d)\). Note that any \(f\in \mathscr {W}^{1,1}_{\text {loc}}(\mathbb {R},\mathbb {R}^d)\) is continuous.

Proposition 3.1

For any \((E,A,B)\in \Sigma _{\ell ,n,m}\) the following controllability definitions associated to the system (1) are algebraically characterized as follows:

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

Moreover, the followoing equivalences are valid:

(a)

\((E,A,B)~\text {compl.~contr.}\)

\(\iff \)

\((E,A,B)~\text {freely~initial.~and~beh.~contr.}\)

(b)

\((E,A,B)~\text {str.~contr.}\)

\(\iff \)

\( (E,A,B)~\text {imp.~contr.~and~beh.~contr.}\)

Proof

All algebraic characterizations are proved in the survey article by Berger and Reis [4, Cor. 4.3]. The equivalences in (a) and (b) are mentioned in [4, Rem. 4.5], they are used in [10], however some further explanations are warranted for the claims.

We show Assertion (a): “Completely controllability \(\Rightarrow \) free initializability” is an immediate consequence of the algebraic characterizations. “Completely controllability \(\Rightarrow \) behavioural controllability” follows from the implication that the mapping \(\lambda \mapsto \textrm{rk}\,_{\mathbb {C}}[\lambda E-A,B]\) is constant, that \(\textrm{rk}\,_{\mathbb {R}(s)}[sE-A,B] = \max _{\lambda \in \mathbb {C}}\textrm{rk}\,[\lambda E-A,B]\) in conjunction with the algebraic characterization of behavioural controllability. It remains to show that “completely controllability \(\Leftarrow \) free initializability and behavioural controllability”. The presuppositions give

$$\begin{aligned} \forall \lambda \in \mathbb {C}: \textrm{rk}\,_{\mathbb {R}}[E,B] = \textrm{rk}\,_{\mathbb {R}}[E,A,B]~\wedge ~\textrm{rk}\,_{\mathbb {R}(s)}[sE-A,B] = \textrm{rk}\,_{\mathbb {C}}[\lambda E-A,B]. \end{aligned}$$

By Lemma A.5 we have \(\textrm{rk}\,_{\mathbb {R}}[E,B] \le \textrm{rk}\,_{\mathbb {R}(s)}[sE-A,B] \le \textrm{rk}\,_{\mathbb {R}}[E,A,B]\) and thus

$$\begin{aligned} \forall \lambda \in \mathbb {C}: \textrm{rk}\,_{\mathbb {R}}[E,B] = \textrm{rk}\,_{\mathbb {R}}[E,A,B] = \textrm{rk}\,_{\mathbb {R}(s)}[sE-A,B] = \textrm{rk}\,_{\mathbb {C}}[\lambda E-A,B]. \end{aligned}$$

Now the claim follows from the algebraic characterization of complete controllability.

Assertion (b) can be shown similarly to Assertion (a) by replacing [E, B] with [E, AZ, B] for some \(Z\in \mathbb {R}^{n\times (n-\textrm{rk}\,E)}\) with \(\textrm{im}\,Z = \ker E\). This isomitted. \(\square \)

We are now ready to prove our main result, that is the characterization of relative genericity of the different controllability concepts in terms of the system dimensions. To this end we introduce the notation

$$\begin{aligned} S_{\text {controllable}} \ := \ \left\{ (E,A,B)\in \Sigma _{\ell ,n,m}\ \big \vert \ (1)~\text {is controllable} \right\} \end{aligned}$$

where ‘controllable’ stands for one of the controllability concepts. Using these algebraic characterizations, we find necessary and sufficient conditions on \(\ell ,n,m\in {\mathbb {N}}^*\) and \(r\in {\mathbb {N}}\) so that \(S_{\text {controllable}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{\le r}\). As we already discussed after Proposition 2.6, it is not very surprising that we recover, unless \(r = 0\), for impulse controllability, behavioural controllability and strong controllability, the necessary and sufficient conditions for the unrestrained case, see [10, Theorem 2.3].

Recall that by Corollary 2.5 the following statements do also hold for the reference set \(\Sigma _{\ell ,n,m}^{= r}\).

Theorem 3.2

The following equivalences hold:

(a)

\(S_{\text {freely initializable}}\)

is rel. gen. in   \(\Sigma _{\ell ,n,m}^{\le r}\)

\(\iff \)

\(\ell \le r+m\),

(b)

\(S_{\text {impulse controllable}}\)

is rel. gen. in   \(\Sigma _{\ell ,n,m}^{\le r}\)

\(\iff \)

\(\ell \le n+m\vee r = 0\),

(c)

\(S_{\text {behavioural controllable}}\)

is rel. gen. in   \(\Sigma _{\ell ,n,m}^{\le r}\)

\(\iff \)

\(\ell \ne n+m\vee r = 0\),

(d)

\(S_{\text {completely controllable}}\)

is rel. gen. in   \(\Sigma _{\ell ,n,m}^{\le r}\)

\(\iff \)

\(\ell <\min \left\{ r+1,n \right\} +m\),

(e)

\(S_{\text {strongly controllable}}\)

is rel. gen. in   \(\Sigma _{\ell ,n,m}^{\le r}\)

\(\iff \)

\(\ell <n+m\vee r = 0\),

If a condition in one of the equivalences is not met, then the respective complement of \(S_{\text {controllable}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{\le r}\).

Proof

We consider in the following proofs the reference set \(\Sigma _{\ell ,n,m}^{=r}\) instead of \(\Sigma _{\ell ,n,m}^{\le r}\).

  1. (a)

    The sets \(S_{(i)}^r\) and \(S_{(ii)}^r\) are relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) by Proposition 2.6, and therefore Proposition 2.3(c) yields that \(S_{(i)}^r\cap S_{(ii)}^r\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Now Proposition 2.3(b) and (c) give that \(S_{\text {freely~init.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(S_{\text {freely~init.}}\cap S_{(i)}^r\cap S_{(ii)}^r\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Proposition 3.1 yields

    $$\begin{aligned} S_{\text {freely~init.}}\cap S_{(i)}^r\cap S_{(ii)}^r = {\left\{ \begin{array}{ll} S_{(i)}^r\cap S_{(ii)}^r, &{} \min \left\{ \ell ,r+m \right\} = \min \left\{ \ell ,r+n+m \right\} ,\\ \emptyset , &{} \text {else}. \end{array}\right. } \end{aligned}$$

    Therefore, and since \(\emptyset \) is not relative generic in \(\Sigma _{\ell ,n,m}^{= r}\not = \emptyset \), we have that \(S_{\text {freely~init.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(\min \left\{ \ell ,r+m \right\} = \min \left\{ \ell ,r+n+m \right\} \) or, equivalently, \(\ell \le r+m\). If \(\ell >r+m\), then \(S_{(i)}^r\cap S_{(ii)}^r\subseteq S_{\text {freely~init.}}^c\) and Proposition 2.3(b) yield that \(S_{\text {freely~init.}}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\).

  2. (b)

    By Proposition 2.6 and Proposition 2.3(c), \(S_{(i)}^r\cap S_{(iv)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Now Proposition 2.3(b) and (c) give that \(S_{\text {imp.~contr.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(S_{\text {imp.~contr.}}\cap S_{(i)}^r\cap S_{(iv)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). By Proposition 3.1, we have the identity

    $$\begin{aligned} S_{\text {imp.~contr.}}\cap S_{(i)}^r\cap S_{(iv)} = {\left\{ \begin{array}{ll} S_{(i)}^r\cap S_{(iv)}, &{} \min \left\{ \ell ,n+m \right\} = \min \left\{ \ell ,r+n+m \right\} \\ \emptyset , &{} \text {else.} \end{array}\right. } \end{aligned}$$

    Similar to (a), we conclude that \(S_{\text {imp.~contr.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(\min \left\{ \ell ,n+m \right\} = \min \left\{ \ell ,r+n+m \right\} \) or, equivalently, \(\ell \le n+m\) or \(r = 0\). If \({}^{\lnot } \big ( \ell \le n+m \ \vee \ r = 0\big )\), then \(S_{(i)}^r\cap S_{(iv)}\subseteq S_{\text {imp.~contr.}}^c\) and Proposition 2.3(b) yields that \(S_{\text {imp.~contr.}}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\).

  3. (c)

    Let \(\big ( \ell \ne n+m \ \vee \ r = 0\big )\). Then Proposition 2.6 yields that \(S_{(iii)}\) and \(S_{(v)}^c\) are relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Now Proposition 2.3(ii) and (iii) gives that \(S_{\text {beh.~contr.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(S_{\text {beh.~contr.}}\cap S_{(v)}^c\cap S_{(iii)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). By Proposition 3.1,

    $$\begin{aligned} S_{\text {beh.~contr.}}\cap S_{(v)}^c\cap S_{(iii)} = S_{(v)}^c\cap S_{(iii)} \end{aligned}$$

    holds and therefore \(S_{\text {beh.~contr.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Let \({}^{\lnot } \big ( \ell \ne n+m \ \vee \ r = 0\big ) \equiv \big ( \ell = n+m \ \wedge \ r > 0\big ) \). Then Proposition 2.6 yields that \(S_{(iii)}\) and \(S_{(v)}^c\) are relative generic in \(\Sigma _{\ell ,n,m}^{= r}\), and thus, in view of Proposition 2.3(c), \(S_{(v)}^c\cap S_{(iii)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Now Proposition 2.3(b) yields that \(S_{\text {beh.~contr.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(S_{\text {beh.~contr.}}\cap S_{(v)}^c\cap S_{(iii)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). However, Proposition 3.1 gives \(S_{\text {beh.~contr.}}\cap S_{(v)}^c\cap S_{(iii)} = \emptyset \), and since the empty set is not relative generic, it follows that \(S_{\text {beh.~contr.}}\) is not relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). If \( \big ( \ell = n+m \ \wedge \ r > 0\big ) \) does not hold, then we have seen in the previous paragraph that \(S_{\text {beh.~contr.}}\cap S_{(v)}^c\cap S_{(iii)} = \emptyset \) and so \(S_{(v)}^c\cap S_{(iii)}\subseteq S_{\text {beh.~contr.}}^c\). Since \(S_{(v)}^c\cap S_{(iii)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\), we may apply Proposition 2.3(b) to conclude that \(S_{\text {beh.~contr.}}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\).

  4. (d)

    Proposition 3.1 yields

    $$\begin{aligned} S_{\text {compl.~contr.}} = S_{\text {freely init.}}\cap S_{\text {beh.~contr.}}. \end{aligned}$$

    Therefore, Proposition 2.3(b) and (c) give that \(S_{\text {compl.~contr.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, both \(S_{\text {freely init.}}\) and \(S_{\text {beh.~contr.}}\) are relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Since we have already proved the statements (a) and (c) of the present theorem, it remains to prove

    $$\begin{aligned} \ell <\min \left\{ r+1,n \right\} +m \quad \iff \quad \big ( \ell \le r +m\big ) \ \wedge \ \big ( \ell \ne n+m \ \vee \ r = 0\big ),\nonumber \\ \end{aligned}$$
    (6)

    which is easily verified. If \(\ell <\min \left\{ r+1,n \right\} +m\) does not hold true, then (6) together with the characterizations in (a) and (c) yield that \(S_{\text {freely~init.}}^c\) or \(S_{\text {beh.~contr.}}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Suppose \(S_{\text {freely~init.}}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Then

    $$\begin{aligned} S_{\text {freely~init.}}^c \subseteq S_{\text {freely~init.}}^c \cup S_{\text {beh.~contr.}}^c = S_{\text {compl.~contr.}}^c \end{aligned}$$

    together with Proposition 2.3(b) gives relative genericity of \(S_{\text {compl.~contr.}}^c\) in \(\Sigma _{\ell ,n,m}^{= r}\). If \(S_{\text {beh.~contr.}}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\), then the same arguments give that \(S_{\text {compl.~contr.}}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\).

  5. (e)

    By Proposition 3.1 (b), the equality

    $$\begin{aligned} S_{\text {strongly controllable}} = S_{\text {impulse controllable}}\cap S_{\text {beh.~contr.}} \end{aligned}$$

    holds. Analogously to the proof of statement (d) of the present theorem we conclude that the former set is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, both latter sets are relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) and otherwise \(S_{str. contr.}^c\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Using the results of the statements (b) and (c) of the present theorem, it remains to verify

    $$\begin{aligned} \ell <n+m\ \vee \ r = 0 ~\iff ~ (\ell \le n+m~\vee ~r = 0)\ \wedge \ (\ell \ne n+m~\vee ~r = 0), \end{aligned}$$

    which is straightforward. This completes the proof of the proposition.

\(\square \)

4 Stabilizability

In the present section we recall the well-known definitions and characterizations of completely stabilizable, strongly stabilizable and behavioral stabilizable systems. Using these algebraic characterizations in combination with some results from the “Appendix A”, we derive necessary and sufficient criteria in terms of systems dimensions when stabilizability of (1) holds relative generically in \(\Sigma _{\ell ,n,m}^{\le r}\).

Proposition 4.1

Let \(\ell ,n,m\in \mathbb {N}^*\). We use, as for controllability, the abbreviation

$$\begin{aligned} S_{{\text {stabilizable}}} := \left\{ (E,A,B)\in \Sigma _{\ell ,n,m}\,\big \vert \,~(1)~{\text {stabilizable}} \right\} . \end{aligned}$$

Considering the concepts of completely stabilizable, strongly stabilizable and behavioural stabilizable systems, the following characterizations can be found

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

Proof

For the first two definitions see [4, Def. 2.1 (h) and (k)]; the corresponding algebraic criteria are proved in [4, Corollary 4.3]. The definition of behavioural stabilizability in [4, Def. 2.1 (e)] contains two typos. We have therefore adopted [14, Def. 5.2.29]. This definition in conjunction with the normal form under feedback equivalence [4, Thm. 3.3] yields the algebraic characterization claimed in the proposition. \(\square \)

Similar as for controllability, Proposition 2.6 in conjunction with the algebraic criteria from Proposition 4.1 give necessary and sufficient conditions on \(\ell ,n,m\) and r so that \(S_{{\textit{stabilizable}}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{\le r}\). Unless \(r = 0\), we recover, as discussed for controllability, the necessary and sufficient conditions for the unrestrained case, see [10, Theorem 3.3].

Theorem 4.2

Let \(\ell ,n,m\in \mathbb {N}^*\), \(r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\} \) and consider, as in Theorem 3.2, the reference set \(\Sigma _{\ell ,n,m}^{\le r}\). Then the following equivalences are valid:

(a)

 \(S_{\text {completely stabilizable}}\)

is rel. gen. in \(\Sigma _{\ell ,n,m}^{\le r}\)

 \(\iff \)

 \(\ell \le \min \left\{ r,n-1 \right\} +m\),

(b)

 \(S_{\text {strongly stabilizable}}\)

is rel. gen. in \(\Sigma _{\ell ,n,m}^{\le r}\)

 \(\iff \)

 \(\ell \le n+m+\min \left\{ r,1 \right\} \),

(c)

 \(S_{\text {behavioural stabilizable}}\)

is rel. gen. in \(\Sigma _{\ell ,n,m}^{\le r}\)

 \(\iff \)

 \(\ell \ne n+m~\vee ~r = 0\).

All of the above holds true, if the reference set \(\Sigma _{\ell ,n,m}^{\le r}\) is replaced by \(\Sigma _{\ell ,n,m}^{=r}\).

Proof

The interchangeability of the reference set \(\Sigma _{\ell ,n,m}^{=r}\) and \(\Sigma _{\ell ,n,m}^{\le r}\) is shown in Corollary 2.5. We consider \(\Sigma _{\ell ,n,m}^{=r}\).

  1. (a)

    Suppose \(S_{\text {compl.~stab.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). In Proposition 2.6 we have shown that the sets \(S_{(i)}^r\) and \(S_{(ii)}^r\) are relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Hence Proposition 2.3(b) and (c) imply that \(S_{\text {compl.~stab.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(S_{\text {compl.~stab.}}\cap S_{(i)}^r\cap S_{(ii)}^r\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). In view of Proposition 4.1 and Proposition 2.6 we conclude

    $$\begin{aligned} \ell >r+m \quad \implies \quad S_{\text {compl.~stab.}}\cap S_{(i)}^r\cap S_{(ii)}^r = \emptyset . \end{aligned}$$

    Since \(\emptyset \) is not relative generic, we conclude that relative genericity of \(S_{\text {compl.~stab.}}\) yields \(\ell \le r+m\). By Propositions 4.1 and 2.6 give

    $$\begin{aligned} S_{\text {compl.~stab.}}\cap S_{(i)}^r\cap S_{(ii)}^r = S_{(vi)} \cap S_{(i)}^r\cap S_{(ii)}^r. \end{aligned}$$
    (7)

    Therefore, Proposition 2.3(b) and (c) ensure that \(S_{\text {compl.~stab.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(S_{(vi)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Now Proposition 2.6 gives that relative genericity of \(S_{\text {compl.~stab.}}\) implies

    $$\begin{aligned} \ell \le r+m~\wedge ~(\ell \ne n+m~\vee ~r = 0). \end{aligned}$$

    This condition is equivalent to the equivalent conditions

    $$\begin{aligned} \ell \le r+m~\wedge ~{}^{\lnot }(\ell = n+m~\wedge ~r = n) \iff \ell \le \min \left\{ r,n-1 \right\} +m. \end{aligned}$$
    (8)

    It remains to prove the converse implication. Assume that \(\ell \le \min \left\{ r,n-1 \right\} +m\) holds. Then (8) yields \(\ell \le r+m\) and we conclude (7). Since (8) yields \(\big ( \ell \ne n+m \ \vee \ r = 0\big )\), we may apply Proposition 2.6 to conclude that \(S_{(vi)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). Finally, me may apply, Proposition 2.3(b) and (c) to (7) to conclude that \(S_{\text {compl.~stab.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\).

  2. (b)

    Replace \(S_{(ii)}^r\) by \(S_{(iv)}\) in Step (a). Then a similar reasoning gives that \(S_{\text {str.~stab.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if,

    $$\begin{aligned} \ell \le n+m~\wedge ~(\ell \ne n+m~\vee ~r = 0). \end{aligned}$$

    This condition is equivalent to \(\ell < n+m+\min \left\{ r,1 \right\} .\)

  3. (c)

    From Proposition 2.6 we know that \(S_{(iii)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). By Proposition 2.3(b) and (c),  \(S_{\text {beh.~stab.}}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, the intersection \(S_{\text {beh.~stab.}}\cap S_{(iii)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\). In view of Proposition 4.1, the equality

    $$\begin{aligned} S_{\text {beh.~stab.}}\cap S_{(iii)} = S_{(vi)}\cap S_{(iii)} \end{aligned}$$

    holds. Therefore, Proposition 2.6 implies that \(S_{(vi)}\cap S_{(iii)}\) is relative generic in \(\Sigma _{\ell ,n,m}^{= r}\) if, and only if, \(\ell \ne n+m\) or \(r = 0\). This completes the proof of the theorem.

\(\square \)