Correction to: Relative genericity of controllability and stabilizability for differential-algebraic systems

Correction to: Math. Control Signals Syst. https://doi.org/10.1007/s00498-022-00332-3

The reasoning in Step (iv).4 in the proof of [1, Prop. 2.6] is erroneous: In line 4 of the step, we say “The induction assumption implies that ...”; however, this does not hold if $E\in V_{\rho +1}\setminus V_\rho$ since the latter is an open subset of $V_{\rho +1}$. The overall claim of Step (iv).4 is

$$\begin{aligned} \forall r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\} \ \ \forall \rho \in \left\{ 0,\ldots , r \right\} : \ \textrm{dom}\,T^\rho \text { is Euclidian dense in }V_\rho . \end{aligned}$$

(1)

Instead of fixing the error and proving (1) directly, we show Proposition 2 which is the ever so slightly more general result encompassing (1) and which might be worth knowing in its own right.

For the formulation of our result, we need to define some matrix actions and some sets related to the Gaussian algorithm of elementary row operations.

FormalPara Definition 1

Define, for $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$, the sets

$$\begin{aligned} V_{\le r}:= \left\{ E\in \mathbb {R}^{\ell \times n}\,\Big \vert \,\textrm{rk}\,E\le r \right\} \qquad \text {and} \qquad V_{= r}:= \left\{ E\in \mathbb {R}^{\ell \times n}\,\Big \vert \,\textrm{rk}\,E = r \right\} , \end{aligned}$$

and, for $*\in \left\{ \le ,= \right\}$ and $i\in \left\{ 0,\ldots ,r \right\}$, the ith step of the Gaussian algorithm on $V_{*r}$ is given by

$$\begin{aligned} i=0 : \ D_0^{*r} = V_{*r},~~\tau _0^{*r} \equiv I_\ell ,~~T_0^{*r}:= \textrm{id}\,_{V_{*r}} \ \text {(the identity function on }V_{*r}\text {)} \end{aligned}$$

and, for $i\ge 1$,

$$\begin{aligned} D_i^{* r}&:= \left\{ E\in V_{*r}\,\big \vert \,E_{i,i}\ne 0 \right\} \\ \tau _i^{* r}&: D_i^{*r}\rightarrow \textbf{Gl}(\mathbb {R}^\ell ),\quad E\mapsto \begin{bmatrix} 1\\ &{} \ddots \\ &{} &{} 1\\ &{} &{} -\frac{E_{i+1,i}}{E_{i,i}} &{} 1\\ &{} &{} -\frac{E_{i+2,i}}{E_{i,i}} &{} &{} 1\\ &{} &{} \vdots &{} &{} &{} \ddots \\ &{} &{} -\frac{E_{\ell ,i}}{E_{i,i}} &{} &{} &{} &{} 1 \end{bmatrix},\\ T_i^{*r}&: D_i^{*r}\rightarrow V_{*r},\quad E\mapsto \tau _{i}^{*r}(E)E. \end{aligned}$$

Then, the Gaussian algorithm without normalization and without interchange of rows up to step $i\in \left\{ 0,\ldots ,r \right\}$ is recursively defined by:

$$\begin{aligned} i=0: \ D_{*r}^0:= D_0^{*r},\quad \tau _{*r}^0:= \tau _0^{*r}\quad T_{*r}^0:= T_0^{*r} = \textrm{id}\,_{V_{*r}}, \end{aligned}$$

and

$$\begin{aligned} D_{*r}^{i+1}&:= \left\{ E\in D_{*r}^i\,\Big \vert \, \big (T_{*r}^i E\big )_{i+1,i+1}\ne 0 \right\} ,\\ \tau _{*r}^{i+1}&: D_{*r}^{i+1}\rightarrow \mathscr{G}\mathscr{L}(\mathbb {R}^\ell ),\quad E\mapsto \tau ^{*r}_{i+1}\big (T_{*r}^i(E)\big )\tau _{*r}^i(E),\\ T_{*r}^{i+1}&:= T_{i+1}^{*r}\circ T_{*r}^i: D_{*r}^{i+1}\rightarrow V_{*r},\quad E\mapsto \tau _{*r}^{i+1}(E)E. \end{aligned}$$

      $\diamond$

We are now ready to formulate the main result. That is, the Gaussian algorithm requires generically no permutation of rows if applied to the set of matrices $E\in \mathbb {R}^{\ell \times n}$ with $\textrm{rk}\,E \le r$, for some fixed $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$, and produces a matrix of the form

$$\begin{aligned} E\leadsto \begin{bmatrix} D &{} M\\ 0 &{} 0 \end{bmatrix}, \end{aligned}$$

where $D\in \mathbb {R}^{r\times r}$ is a regular upper triangular matrix and $M\in \mathbb {R}^{r\times (n-r)}$. In formal terms, this means:

FormalPara Proposition 2

Let $r\in \left\{ 0,\ldots ,\min \left\{ \ell ,n \right\} \right\}$. Then, the set $D_{\le r}^r$ is relative generic in $V_{\le r}$.

The definition of “relative genericity” and other notions can be found in [1].

Before proving Proposition 2, we show that Step (iv).4 in the proof of [1, Prop. 2.6] is a corollary of it.

FormalPara Proof of (1)

First verify that, for all $r\in \{0,\ldots ,\min \{\ell ,n\}\}$ and $\rho \in \{0,\ldots ,\rho \}$,

$$\begin{aligned} V_{\le \rho } \ \cap \ \textrm{dom}\,T^\rho \ = \ D_{\le \rho }^\rho , \end{aligned}$$

where $\textrm{dom}\,T^\rho$ is defined in [1, (18)]. Then Proposition 2 yields that $V_{\le \rho } \cap \textrm{dom}\,T^\rho$ is relative generic in $V_{\le \rho }$. By [1, Lem. 2.1], the set $V_{\le \rho } \cap \textrm{dom}\,T^\rho$ is Euclidean dense in $V_{\le \rho }$, whence $\textrm{dom}\,T^\rho$ is Euclidean dense in $V_{\le \rho }$. Finally, in view of $V_\rho = V_{\le \rho }$ for $V_\rho$ defined in [1, p. 69, l. 4], the claim in (1) holds. $\square$

In the remainder of this erratum we prove Proposition 2. To do this, the following lemma is helpful.

FormalPara Lemma 3

$V_{= r}$ is relative generic in $V_{\le r}$. Therefore, a set $S\subseteq \mathbb {R}^{n\times m}$ is relative generic in $V_{\le r}$ if, and only if, S is relative generic in $V_{=r}$.

FormalPara Proof

In [1, Lemma 2.4], we have shown that $V_{=r}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ is relative generic in $V_{\le r}\times \mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$, $m\in \mathbb {N}^*$. The proof goes through analogously if the Cartesian factors $\mathbb {R}^{\ell \times n}\times \mathbb {R}^{\ell \times m}$ are omitted. This gives the first statement.

The second statement is a consequence of [1, Prop. 2.3 (f)]. Please note that [1, Prop. 2.3 (f)] contains a typo: “$V'$” is supposed to be ‘$S_2$’. $\square$

For convenience, we introduce a particular elementary matrix. For $n\in \mathbb {N}^*$, $i,j\in {\underline{n}}$ and $\lambda \in \mathbb {R}$ let the matrix $E^{i,j,n}\in \mathbb {R}^{n\times n}$ be defined by

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

where $\delta _{\alpha ,\beta }$ denotes the Kronecker delta symbol. The action of the elementary matrix

$$\begin{aligned} T_{i\rightarrow i+\lambda j}&:= I_n + \lambda E^{i,j,n} = \begin{bmatrix} I_{i-1}\\ &{} 1\\ &{} &{} I_{j-i-1}\\ &{} \lambda &{} &{} 1\\ &{} &{} &{} &{} I_{n-i-j} \end{bmatrix}, \end{aligned}$$

where the structural sketch depicts the case $j>i$, on $\mathbb {R}^{n\times m}$ from the left is the addition of the $\lambda$-multiple of the jth row to the ith row; the action of $T_{i\rightarrow i+\lambda j}$ on $\mathbb {R}^{m\times n}$ from the right is addition of the $\lambda$-multiple of the jth column to the ith column.

Finally we are ready to prove the main result.

FormalPara Proof of Proposition 2

By Lemma 3, it suffices to show that $D_{\le r}$ is relative generic in $V_{= r}$. By [1, Prop. 2.3 (c)], the latter holds if, and only if, $D_{=r}^r = D_{\le r}^r\cap V_{=r}$ is relative generic in $V_{=r}$. To prove this, we show by induction

$$\begin{aligned} \forall \rho \in \left\{ 0,\ldots ,r \right\} : \ D_{=r}^{\rho } \quad \text {is relative generic in }V_{=r}. \end{aligned}$$

(2)

In Step (iv).3 in the proof of [1, Prop. 2.6], it is verified that $\textrm{dom}\,T^\rho$ is a relative Zariski-open subset of $V_{\le r}$. Since $V_{=r}\subseteq V_{\le r}$, this yields that $D_{=r}^{\rho }= V_{=r} \cap \textrm{dom}\,T^\rho$ is a relative Zariski-open subset of $V_{=r}$, and to show (2), it remains to show that

$$\begin{aligned} \forall \rho \in \left\{ 0,\ldots ,r \right\} : \ \quad D_{=r}^{\rho }\quad \text {is dense in }V_{=r}. \end{aligned}$$

(3)

For $\rho = 0$, the property (3) is obviously fulfilled. Assume that $D_{=r}^{\rho }$ is dense in $V_{=r}$ for $\rho \in \left\{ 0,\ldots ,r-1 \right\}$. We show that the set $D_{=r}^{\rho +1}$ is dense in $V_{= r}$. For later purposes, we choose the maximum norm

$$\begin{aligned} \Vert A\Vert \ {}:= & {} \ \max \left\{ |A_{i,j}| \ \big | \ i \in \left\{ 1,\ldots ,a \right\} , \ j \in \left\{ 1,\ldots ,b \right\} \right\} ,\\{} & {} \text {where }A\in \mathbb {R}^{a\times b}, a,b\in \mathbb {N}^*. \end{aligned}$$

This norm is not sub-multiplicative. Due to the definition of the matrix multiplication and the triangle inequality and submultiplicativity for the absolute value, however, it satisfies

$$\begin{aligned} \forall \, A \in \mathbb {R}^{a\times b} \ \forall \, B \in \mathbb {R}^{b\times c}: \ \Vert AB\Vert \le b \ \Vert A\Vert \, \Vert B\Vert . \end{aligned}$$

(4)

In these terms we show

$$\begin{aligned} \forall \, \varepsilon >0 \ \forall \, E^\star \in V_{ = r} \ \exists \, {\widehat{E}} \in D_{=r}^{\rho +1} \cap V_{ = r}: \ \Vert E^\star - {\widehat{E}} \Vert < \varepsilon . \end{aligned}$$

(5)

Since $D_{=r}^\rho$ is dense in $V_{=r}$ by assumption, there exists $E \in D_{=r}^{\rho }$ so that $\left\| E^\star - E \right\| <\frac{\varepsilon }{2}$.

We show that

$$\begin{aligned} \exists \ \iota \in \left\{ \rho +1,\ldots ,\ell \right\} \ \exists \ \kappa \in \left\{ \rho +1,\ldots ,n \right\} : \ \big (T_{=r}^{\rho }(E)\big )_{\iota ,\kappa }\ne 0. \end{aligned}$$

(6)

If, on the contrary, (6) is false, then regularity of $\tau _{=r}^{\rho }(E)$ gives

$$\begin{aligned} r = \textrm{rk}\,E = \textrm{rk}\,\tau _{=r}^{\rho }(E) E = \textrm{rk}\,T^{\rho }_{=r}(E) \le \rho , \end{aligned}$$

which contradicts the choice of $\rho$.

Choose $\delta >0$ so that

$$\begin{aligned} \delta <\min \left\{ 1,\frac{\varepsilon }{6\ell \left\| T^{\rho }(E) \right\| \left\| \tau ^{\rho }_{=r}(E)^{-1} \right\| } \right\} \end{aligned}$$

and

$$\begin{aligned} \big (T^{\rho }_{=r}(E)\big )_{\rho +1,\kappa } + \big (T^{\rho }_{=r}(E)\big )_{\iota ,\rho +1} + \delta \big (T^{\rho }_{=r}(E)\big )_{\iota ,\kappa }\ne 0 \end{aligned}$$

and choose the matrix

$$\begin{aligned} {\widehat{E}} := \big (\tau _{=r}^{\rho }(E)\big )^{-1} T_{(\rho +1)\rightarrow (\rho +1)+\delta \iota }\tau _{=r}^{\rho }(E) E T_{(\rho +1)\rightarrow (\rho +1)+\delta \kappa }. \end{aligned}$$

Then

Furthermore, $\tau _{=r}^{\rho }(E)$ depends by construction only on the upper left $\rho \times \rho$-submatrix and therefore we have $\tau _{=r}^{\rho }({\widehat{E}}) = \tau _{=r}^{\rho }(E)$. Since

$$\begin{aligned} \big (\tau _{=r}^{\rho }({\widehat{E}}){\widehat{E}}\big )_{\rho +1} = \delta \left( \big (T^{\rho }_{=r}(E)\big )_{\rho +1,\kappa } + \big (T^{\rho }_{=r}(E)\big )_{\iota ,\rho +1} + \delta \big (T^{\rho }_{=r}(E)\big )_{\iota ,\kappa }\right) \ne 0, \end{aligned}$$

we conclude that ${\widehat{E}}\in D_{=r}^{\rho +1}$. Furthermore,

$$\begin{aligned} \textrm{rk}\,{\widehat{E}} = \textrm{rk}\,\big (\tau _{=r}^{\rho }(E)\big )^{-1} T_{(\rho +1)\rightarrow (\rho +1)+\delta \iota }\tau _{=r}^{\rho }(E) E T_{(\rho +1)\rightarrow (\rho +1)+\delta \kappa } = \textrm{rk}\,E = r \end{aligned}$$

which shows that ${\widehat{E}} \in V_{ = r}$. Finally, the inequality

$$\begin{aligned} \Vert E^\star -{\widehat{E}} \Vert \le \Vert E^\star -E\Vert + \Vert E -{\widehat{E}} \Vert < \varepsilon \end{aligned}$$

shows that (5) holds. Therefore, the proof of the proposition is complete. $\square$