# Input Redundancy under Input and State Constraints
(Extended version of [1])

Jean-François Trégouët  Univ Lyon, INSA Lyon, Université Claude Bernard Lyon
1, Ecole Centrale de Lyon, CNRS, Ampère UMR5005, F-69621 Villeurbanne, France
<EMAIL_ADDRESS>and  Jérémie Kreiss  Université de
Lorraine, CNRS, CRAN, F-54000 Nancy, France<EMAIL_ADDRESS>
###### Abstract.

For a given unconstrained dynamical system, input redundancy has been recently
redefined as the existence of distinct inputs producing identical output for
the same initial state. By directly referring to signals, this definition
readily applies to any input-to-output mapping. As an illustration of this
potentiality, this paper tackles the case where input and state constraints
are imposed on the system. This context is indeed of foremost importance since
input redundancy has been historically regarded as a way to deal with input
saturations. An example illustrating how constraints can challenge redundancy
is offered right at the outset. A more complex phenomenology is highlighted.
This motivates the enrichment of the existing framework on redundancy. Then, a
sufficient condition for redundancy to be preserved when imposing constraints
is offered in the most general context of arbitrary constraints. It is shown
that redundancy can be destroyed only when input and state trajectories lie on
the border of the set of constraints almost all the time. Finally, those
results are specialized and expanded under the assumption that input and state
constraints are linear.

###### Key words and phrases:

Input redundant systems; Control allocation; Over actuated systems; Control of
constrained systems; Measurement and actuation

This document is an extended version of the following paper accepted for
publication in Automatica:

Jean-François Trégouët and Jérémie Kreiss. Input redundancy under input and
state constraints. Automatica , vol. 159, p. 111344, 2024, ISSN 0005-1098,
https://doi.org/10.1016/j.automatica.2023.111344.

Additional material are printed in blue.

##  1\. Introduction

This paper is a follow-up of [2], where new definitions, taxonomy and
characterizations are proposed around the notion of input redundancy
(IR).111Throughout this paper, IR stands either for “input redundancy” or for
“input redundant” depending on the context. This paper expands on this topic,
by tackling the case where the system is affected by input and state
constraints.

Many systems are equipped by more actuators than strictly needed to meet the
control objectives. Such a design has many technological advantages: Examples
include state-of-health and/or thermal management, resilience to failure,
enhanced control capabilities, see [3, 4, 5, 6, 7] to cite a few. Those
systems are over-actuated. They belong to the class of IR systems.

Since the early nineties, literature devoted to IR systems have kept growing,
see [8, 9, 10, 11, 6, 12] and references therein. Many control designs have
been proposed in various fields, including aerospace and aeronautics [5, 13],
marine vessels [14] and, more recently, power electronics [15, 3, 4, 16]. In
addition to that, significant contributions have been obtained on the
methodological side [17, 18, 11, 19, 2]. The goal is to enrich control theory
by generalizing and formalizing ad hoc achievements.

Along this last line, the efforts toward a clear definition of IR, together
with tractable characterizations, are of foremost importance. This topic is
indeed crucial, as it impacts not only how dynamical systems are classified
but also how the related control problem is tackled [2]. This paper
contributes in this direction. It is concerned with the linear, proper and
time-invariant dynamical system $\Sigma$ governed by the following equations

(1a) $\displaystyle\dot{x}(t)$ $\displaystyle=Ax(t)+Bu(t),\;x(0)\eqqcolon
x_{0},$ (1b) $\displaystyle y(t)$ $\displaystyle=Cx(t)+Du(t),$

for some quadruple $(A,B,C,D)$ of appropriate dimensions. Here, vectors
$u(t)\in{}^{m}$, $x(t)\in{}^{n}$ and $y(t)\in{}^{p}$ are the input, the state
and the output at time $t$, respectively.

For the time being, let us assume that $\Sigma$ is not affected by any
constraints. For this class of system, there exist various definitions of IR.
A brief overview of them is now offered, starting from the observation that
the literature is divided into two branches. Each one has its own definition
of IR.

  1. (i)

_Input-to-state methods_ focus on (1a) and apply for non injective matrix $B$.
The first step is to decompose $Bu$ as $B_{\tau}\tau$ with $\tau=B_{u}u$ where
$B_{\tau}$ is injective, i.e. $B$ is factorized as $B_{\tau}B_{u}$ [10].
Signal $\tau$ usually refers to the overall contribution of the inputs. Then,
a two-layers control scheme is designed: The high level controller delivering
$\tau$ feeds the low-level allocator which select the optimal input $u$ among
the ones that satisfy $\tau=B_{u}u$. The allocator is as follows:

(2) $\tau\mapsto
u(\tau)\in\operatorname*{arg\,min}_{\hat{u}}J(\hat{u})\;\text{s.t.}\;B_{u}\hat{u}=\tau,$

where $J$ is a real-valued cost function. Usually, no closed-form expression
of $\tau\mapsto u(\tau)$ exists, so that computation of $u$ is performed
online, either by solving (2) parametrized by $\tau$ at each instant time [6],
or by implementing a controller converging to the optimal solution [19]. In
this framework, a system is IR if this strategy can be implemented, that is,
if $B$ is non-injective.

  2. (ii)

_Input-to-output methods_ enlarge the scope of the analysis by dealing with
the input-to-output mapping, i.e. (1b) is now taken into account [17, 18, 11,
2]. From the control point of view, an important achievement of this line of
research is [17], where the output regulation problem is revisited by adding
an optimizer to the classical control structure. This optimizer selects the
best steady-state among those achieving exact output regulation. Various
definitions of IR underpin those contributions. Ultimately, it is proposed in
[2] to define IR as the negation of left-invertibility, i.e. IR means that the
input-to-output mapping is _not_ injective for some $x_{0}$.222Interested
reader is referred to [2, Sec. 5] for a comprehensive comparison of the main
definitions of IR coexisting in this line of research.

To sum up, IR is defined as non injectivity of either the input-to-state or
the input-to-output mappings. For strictly proper system, note that the latter
is more general than the former: If the input-to-state mapping is non
injective, then the input-to-output mapping also enjoys the same property, a
fortiori. To put it in another way, the utmost merit of the input-to-output
methods is to consider the case of distinct inputs leading to the same output
for the same initial state, even if the induced state trajectories are
distinct.

So far, it is assumed that $\Sigma$ is free from input or state constraints.
However, from the very beginning, IR has been considered as a way to deal with
input constraints by redistributing overall control effort among actuators to
avoid saturation. Earliest input-to-state methods add the following condition
to the set of constraints of (2), see [8, 20, 9] and the survey [6]:

(3a) $u(t)\in\mathcal{U},$ where $\mathcal{U}\subseteq{}^{m}$ is a given set.
By doing so, IR is implicitly characterized by the fact that the feasible set
$\mathcal{F}(t)\coloneqq\\{\hat{u}:B\hat{u}=\tau(t)\\}\cap\mathcal{U}$ does
not reduce to an one-point set, at least for some $t$. Subsequently, crucial
questions are the followings: How to predict a priori that this condition
holds? If one can prove that it does not, can this condition be achieved by
redesigning the high level controller delivering $\tau(t)$? Or, by considering
either a different reference signal to be tracked and/or a different initial
condition? Answers to those questions are rather involved. Firstly, because
the high level controller is usually assumed to be given and is therefore out
of the scope of the analysis. Secondly, because of the interplay between the
two levels of the controller.

Some of the input-to-output methods also treat the input constraints. In [11],
the so-called allocator block injects an additional input signal which is
invisible from the output and such that the resulting input vector remains
within the saturation limits. Authors of [21, 22] offer methods for the
computation of the zero-error steady-state solutions of the regulation
problem, under input constraints. For the same problem, an online optimizer
scheme is proposed in [17], with the goal of promoting steady-state inputs
with the smallest infinity norm. In the discrete-time case, a model predictive
control scheme is offered in [23], to ensure that the internal dynamics comply
with state and input constraints. All those contributions are valuable
attempts toward control design handling input and state constraints. They
share the following common pattern, though: First, IR is defined by referring
to the _unconstrained_ context and, second, the control methodology is exposed
by referring to input constraints. As a result, it cannot be ensured that the
proposed methods apply to the class of IR systems characterized beforehand.

This is in stark contrast with [2] where IR is redefined as non injectivity of
the input-to-output mapping, i.e. an output $y$ together with an initial state
$x_{0}$ does not uniquely determine corresponding input $u$. Unlike the other
definitions, this new formulation refers to signals with the aim of
facilitating extensions out of the class of unconstrained linear time-
invariant systems. This paper illustrates this potentiality, by considering
not only input but also state constraints, i.e. both (3a) and the following
condition hold for all $t\in{}_{\geq 0}$:

(3b) $x(t)\in\mathcal{X},$

where $\mathcal{X}\subseteq{}^{n}$ is a given set. In the sequel, definitions
proposed in [2] are applied _verbatim_ to the constrained context.

Then, an immediate question is to ask whether this non uniqueness of the input
remains valid when considering other pairs $(x_{0},y)$? In the unconstrained
context where $\mathcal{U}={}^{m}$ and $\mathcal{X}={}^{n}$ hold, the answer
is positive, regardless of $(x_{0},y)$ (see [2, Prop. 2.1]). In the
constrained context, the following example explicitly shows that the answer
can be negative, so that constraints can challenge redundancy.

######  Example 1.1.

Consider the following system

$\dot{x}=-x+\begin{bmatrix}1&1\end{bmatrix}u,\quad
y=x+\begin{bmatrix}1&0\end{bmatrix}u,$

for which each input is enforced to be non-negative, whereas state trajectory
is unconstrained, i.e. $u(t)\in\mathcal{U}={}_{\geq 0}^{2}$ and
$x(t)\in\mathcal{X}=\real$ for all $t\in{}_{\geq 0}$. Observe that input-to-
output mapping is fully captured via the following equation on which $y$ acts
as a parameter and $u\eqqcolon[u_{a},u_{b}]^{\intercal}$ are the unknown
functions:

(4) $\dot{y}+y-u_{b}=\dot{u}_{a}+2u_{a},$

with $u_{a}(0)=y(0)-x(0)$. Consider the following case study:

  1. (i)

Define $x_{0,1}\coloneqq-1$ and $y_{1}:t\mapsto-e^{-t}$. In this case,
$u_{1}\eqqcolon[u_{1,a},u_{1,b}]^{\intercal}=\mathbf{0}$ is the unique input
satisfying $u(t)\in\mathcal{U}$ leading to $y_{1}$ when $x(0)=x_{0,1}$.
Indeed, (4) reduces to $-u_{b}=\dot{u}_{a}+2u_{a}$ with $u_{a}(0)=0$, so that
strictly positive value of $u_{b}(t)$ induces violation of constraint
$u_{a}(t)\geq 0$. Hence, $u_{1}(t)$ must equal $\mathbf{0}$ at all time.

  2. (ii)

Keep $x(0)=x_{0,1}$ and choose $y_{2}=\mathbf{0}$. Clearly, either
$u_{2}:t\mapsto[e^{-2t},0]^{\intercal}$ or
$u_{3}:t\mapsto[e^{-3t},e^{-3t}]^{\intercal}$ produce $y_{2}$ and are
compatible with the constraints.

  3. (iii)

Define $x_{0,2}=0$ and let $y_{2}=\mathbf{0}$ be unchanged. For the same
reason as in (i), output $y_{2}$ can only be produced by $u_{1}=\mathbf{0}$
for $x(0)=x_{0,2}$.

To sum up, when $x(0)=x_{0,1}$, output $y_{2}$ can be produced by multiple
inputs. On the contrary, substituting initial state by $x_{0,2}$ or output by
$y_{1}$ makes the input unique. Therefore, the ability of designing different
inputs giving rise to the same output depends on both output trajectory $y$
and initial condition $x_{0}$.

This example shows that constraints give rise to unseen phenomena when
instantaneous value of input and state vectors are free, as in [2].

From the above discussion, IR and input constraints are intrinsically related
research fields. If the former is still at its infancy by many aspects, the
latter benefits from solid results on classical notions of control theory.
Together with stabilizability, the concept of controllability have probably
monopolized most of the attention of the control community working on input
constrained dynamical systems, see [24] and any standard textbook on this
topic like e.g. [25, 26, 27]. Roughly speaking, this notion is related to the
existence of a suitable input trajectory. On the contrary, IR deals with
uniqueness of this input. The key point is that the question of existence
received much more attention than the one of uniqueness, by far.

The bottom line of the previous discussion is that, to our best knowledge, IR
has been neither defined nor characterized for systems subject to input or
state constraints. In this paper, this challenge is tackled head-on. By doing
so, it brings closer the literature dedicated to IR and the one devoted to
input and state constraints.

Main contributions of this paper are now exposed. 1) Framework introduced in
[2] is first enriched by new definitions to handle the more complex
phenomenology on IR in the constrained context, as partially highlighted by
Ex. 1.1. 2) Then, a sufficient condition for $(x_{0},y)$ to be compatible with
distinct inputs is derived, see Th. 4.1. This can be considered as the main
achievement of this paper. To arrive at this result, a rigorous incremental
point of view is adopted. Specifically, the inputs $u_{1}$ and $u_{2}$ leading
to the same output are described as $u_{1}$ and $u_{1}+\tilde{u}$ with
$\tilde{u}=u_{2}-u_{1}$, respectively. Let us emphasize that this analysis is
conducted for _arbitrary_ input and state constraint sets $\mathcal{U}$ and
$\mathcal{X}$ (e.g. they can be neither convex nor connected). 3) The case
where $\mathcal{U}$ and $\mathcal{X}$ are linear is then treated as a
byproduct of this analysis. In this case, a comprehensive characterization of
IR as well as its taxonomy is derived. The concept of degree of IR, as defined
in [2], is also generalized. In a nutshell, the overall contribution of this
paper is an extensive discussion on how $\mathcal{U}$ and $\mathcal{X}$ impact
properties of IR associated with the corresponding unconstrained system.

Section II sets the stage of this study. Section III enriches conceptual
framework on IR introduced in [2]. Main results are offered in Section IV, in
the general context where $\mathcal{U}$ and $\mathcal{X}$ are arbitrary.
Section V focuses on the particular case where input and state constraints are
linear.

#### Notations

Symbols $\wedge$, $\vee$ and $\lnot$ stand for logical operators “and”, “or”
and “not”, respectively. Symbol $\mathbf{0}$ stands for anything that is not a
real number and is zero (a vector, matrix, map, or subspace), according to
context. Identity matrix is denoted by $\mathrm{\mathbf{I}}$. Set $\sigma(A)$
is the spectrum of square matrix $A$. Cardinality of a set $S$, denoted by
$|S|$, is said to be greater than or equal to $\alpha\in\mathbb{N}$ if $S$
admits at least $\alpha$ distinct members. In particular, if $\mathrm{\bf
S}\subseteq\\{u:\real\rightarrow{}^{n}\\}$ is a function space, then
$|\mathrm{\bf S}|>\alpha$ means that there exist
$u_{1},\ldots,u_{\alpha}\in\mathrm{\bf S}$ such that every $u_{i},u_{j}$
differ on a strictly positive measure set, i.e.
$\textstyle\int{\left\|u_{i}(\tau)-u_{j}(\tau)\right\|d\tau}>0$ for all $i\neq
j\in\\{1,\ldots,\alpha\\}$. Signals $u_{1},u_{2}$ are equal if
$|\\{u_{1},u_{2}\\}|=1$ so that $u_{1}(t)=u_{2}(t)$ holds for almost all $t$.
Subsets (not necessarily vector/linear spaces) of Euclidean space are denoted
by script symbols, e.g. $\mathcal{U}$ or $\mathcal{X}$.
$\text{int}(\mathcal{X})$ refers to the interior of $\mathcal{X}$. Apart from
$\mathrm{\mathbf{I}}$, most of the time capital bold letters refer to function
space like e.g. $\mathrm{\bf U}=\\{u:{}_{\geq 0}\rightarrow\mathcal{U}\\}$.
Denote by $\mathrm{\bf C}(I,\mathcal{C})$ (resp. $\mathrm{\bf
PC}(I,\mathcal{C})$) the set of continuous functions (resp. piecewise
continuous functions) from $I\subseteq\real$ to $\mathcal{C}$. Set $S-a$ reads
$\\{s-a:s\in S\\}$, so that $a\in S$ is equivalent to $\mathbf{0}\in S-a$.
Given set $\mathcal{A}\subseteq{}^{n}$ and matrix $B$ (not necessarily
invertible or square) with $n$ lines, set $B^{-1}\mathcal{A}$ reads
$\\{u:Bu\in\mathcal{A}\\}$. With slight abuse of notation, relationship
$x\in\mathcal{X}$, where $x(\cdot)$ is a function of time, means
$x(t)\in\mathcal{X}$ for all $t$ in the domain of $x(\cdot)$. Laplace
transform of $u$ is denoted by $s\mapsto\mathfrak{L}[u](s)$.

##  2\. Context of the study

This section inherits and enriches the notations and the framework of [2]. We
refer the reader to [2] for details, bearing in mind that sets $\mathrm{\bf
U}$, $\mathrm{\bf Q}$ and $\mathrm{\bf W}$ of [2] are here renamed as
$\mathrm{\bf U}_{\Sigma}$, $\mathrm{\bf Q}_{\Sigma}$ and $\mathrm{\bf
W}_{\Sigma}$, respectively.

###  2.1. Linear system with constraints

From (1a), the input-to-state relationship is concisely captured via
$\mathrm{H}_{x}[x_{0};\cdot]$ which maps an input trajectory $u(\cdot)$ to the
state trajectory $x(\cdot)$ produced by the system when excited by $u(\cdot)$
with an initial condition $x(0)=x_{0}\in{}^{n}$. The input-to-output mapping
$\mathrm{H}[x_{0};\cdot]:u\mapsto C\mathrm{H}_{x}[x_{0};u]+Du$ derives from
(1b).

Throughout this paper, inputs $u$ are assumed to belong to $\mathrm{\bf
U}_{\Sigma}$, the set of causal, piecewise continuous and exponentially
bounded signals.333By causal signal, we mean a signal which is zero for all
strictly negative time instant. This ensures that the Laplace transform of
$u$, as well as that of corresponding $x=\mathrm{H}_{x}[x_{0};u]$ and
$y=\mathrm{H}[x_{0};u]$, exist, whatever is $x_{0}$. Since $x$ and $y$ are
continuous and piecewise continuous, respectively, one can define $\mathrm{\bf
X}_{\Sigma}\coloneqq\mathrm{\bf C}({}_{\geq 0},{}^{n})$ and $\mathrm{\bf
Y}\coloneqq\mathrm{\bf PC}({}_{\geq 0},{}^{p})$ as the codomain of
$\mathrm{H}_{x}[x_{0};\cdot]$ and $\mathrm{H}[x_{0};\cdot]$, respectively.

Let $x_{0}\in{}^{n}$ be a given initial condition. The set of all triples
$(u,x,y)$ (resp. pairs $(u,y)$) _compatible for $x_{0}$_ is denoted by
$\mathrm{\bf Q}_{\Sigma}(x_{0})$ (resp. $\mathrm{\bf W}_{\Sigma}(x_{0})$),
i.e.

$\displaystyle\mathrm{\bf Q}_{\Sigma}(x_{0})$
$\displaystyle\coloneqq\\{(u,x,y)\mid\mathrm{H}_{x}[x_{0};u]=x,\;\mathrm{H}[x_{0};u]=y\\},$
$\displaystyle\mathrm{\bf W}_{\Sigma}(x_{0})$
$\displaystyle\coloneqq\\{(u,y)\mid\exists x:(u,x,y)\in\mathrm{\bf
Q}_{\Sigma}(x_{0})\\}.$

Let us already introduce Fig. 1 supporting forthcoming definitions and
discussions. This picture should be regarded as a “running” informal sketch
the reader can refer to, to understand concepts introduced throughout the rest
of this section. For didactical purpose, assume that every triple compatible
with $\Sigma$ is depicted by Fig. 1, so that $\mathrm{\bf Q}_{\Sigma}(x_{0})$
has finite cardinality. The reader is also referred to Tab. 1, where main
notations are collected together. Note that some of them are introduced in the
sequel.

Figure 1. Graphical illustration of input/state/output triples associated with
$\Sigma$ and $\mathfrak{S}$ for some fixed $x_{0}\in\mathcal{X}$. It holds
$\mathrm{\bf
Q}_{\Sigma}(x_{0})=\\{(u_{i},x_{i},y_{i})_{i\in\\{1,\ldots,8\\}}\\}$.

In contrast with [2], this paper deals with dynamical system $\mathfrak{S}$
deriving from $\Sigma$ by imposing input and state constraints. For this
reason, input and state signals of $\mathfrak{S}$ belong to sets $\mathrm{\bf
U}$ and $\mathrm{\bf X}$ defined as follows:

$\displaystyle\mathrm{\bf U}$ $\displaystyle\coloneqq\\{u:{}_{\geq
0}\rightarrow\mathcal{U}\\}\cap\mathrm{\bf U}_{\Sigma},$
$\displaystyle\mathrm{\bf X}$ $\displaystyle\coloneqq\\{x:{}_{\geq
0}\rightarrow\mathcal{X}\\}\cap\mathrm{\bf X}_{\Sigma}.$

As a result, the set $\mathrm{\bf Q}(x_{0})$ of triple $(u,x,y)$ compatible
with $\mathfrak{S}$ derives from $\mathrm{\bf Q}_{\Sigma}(x_{0})$ by excluding
trajectories which violate constraints, i.e.

(5) $\mathrm{\bf Q}(x_{0})\coloneqq\mathrm{\bf
Q}_{\Sigma}(x_{0})\cap(\mathrm{\bf U}\times\mathrm{\bf X}\times\mathrm{\bf
Y}).$

Thus, $(u,x,y)\in\mathrm{\bf Q}(x_{0})$ implies that
$(u(t),x(t))\in\mathcal{U}\times\mathcal{X}$ holds for all non-negative time
$t$. Similarly, denote by $\mathrm{\bf W}(x_{0})$ the set of input-to-output
pair compatible for $x_{0}$ with $\mathfrak{S}$, i.e. $(u,y)\in\mathrm{\bf
W}(x_{0})$ if there exists $x$ such that $(u,x,y)\in\mathrm{\bf Q}(x_{0})$.

Table 1.  Nomenclature | $\Sigma$ | $\mathfrak{S}$
---|---|---
Input and state constraints | ${}^{m}\times{}^{n}$ | $\mathcal{U}\times\mathcal{X}$
Input-output pairs | $\mathrm{\bf W}_{\Sigma}(x_{0})$ | $\mathrm{\bf W}(x_{0})$
Input-state-output triples | $\mathrm{\bf Q}_{\Sigma}(x_{0})$ | $\mathrm{\bf Q}(x_{0})$
Admissible inputs | $\mathrm{\bf U}_{\Sigma}$ | $\mathrm{\bf U}_{a}(x_{0})$
Admissible initial condition | n | $\mathcal{X}_{c}$
Admissible pairs $(x_{0},y)$ | $\mathcal{X}\times\mathrm{\bf Y}$ | $\mathrm{\bf A}$
Input-to-output mapping | $\mathrm{H}[x_{0};\cdot]$ | $\mathfrak{H}[x_{0};\cdot]$
Input-to-state mapping | $\mathrm{H}_{x}[x_{0};\cdot]$ | $\mathfrak{H}_{x}[x_{0};\cdot]$

###  2.2. Admissible signals

In general, constraints (3) induce specific difficulties, like the one
highlighted by the following example.

######  Example 2.1.

Consider the dynamical system $\dot{x}=x+u$ with
$\mathcal{X}=\mathcal{U}=[0,1]$. Unique solution reads $x:t\mapsto
x_{0}e^{t}+\int^{t}_{0}e^{t-\tau}u(\tau)\mathrm{d}\tau$. Therefore, if
$0<x_{0}\leq 1$ holds, then $x(t)$ will inevitably escape from $\mathcal{X}$,
whatever is $u\in\mathrm{\bf U}$. Hence, there is no $u\in\mathrm{\bf U}$ such
that $x\in\mathrm{\bf X}$ holds. On the contrary, selecting $x_{0}=0$ and
$u=\mathbf{0}$ yields $x=\mathbf{0}\in\mathrm{\bf X}$.

As illustrated by the previous example, depending on initial state $x_{0}$ and
set $\mathcal{U}\times\mathcal{X}$, it might happen that no state trajectory
lying identically in $\mathcal{X}$ exists, whatever is $u\in\mathrm{\bf U}$.
For this reason, denote by $\mathcal{X}_{c}$ the set of all initial conditions
$x_{0}$ for which there exists at least one compatible input-to-output pair:

$\mathcal{X}_{c}\coloneqq\\{x_{0}\in{}^{n}:|\mathrm{\bf W}(x_{0})|\geq
1\\}\subseteq\mathcal{X}.$

For Ex. 2.1, $\mathcal{X}_{c}$ equals $\\{\mathbf{0}\\}$, a proper subset of
$\mathcal{X}$.  Note that $\mathcal{X}_{c}\subseteq\mathcal{X}$ necessarily
holds since $x_{0}\notin\mathcal{X}$ will violate inclusion
$x(t)\in\mathcal{X}$ for $t=0$.

An input signal $u$ (resp. output signal $y$) is said _admissible for $x_{0}$_
with $\mathfrak{S}$ if there exists an output $y$ (resp. an input $u$) for
which $(u,y)$ is compatible for $x_{0}$ with $\mathfrak{S}$. Let $\mathrm{\bf
U}_{a}(x_{0})\subseteq\mathrm{\bf U}_{\Sigma}$ be the set of admissible input
trajectories for $x_{0}$ with $\mathfrak{S}$, i.e. those which comply with
input _and_ state constraints:

$\mathrm{\bf U}_{a}(x_{0})\coloneqq\\{u\in\mathrm{\bf U}\mid\exists
y:(u,y)\in\mathrm{\bf W}(x_{0})\\}.$

Clearly, $\mathrm{\bf U}_{a}(x_{0})$ is non empty if and only if (iff) $x_{0}$
belongs to $\mathcal{X}_{c}$. For Ex. 2.1, $\mathrm{\bf U}_{a}(x_{0})$ equals
$\varnothing$ (resp. $\\{\mathbf{0}\\}$) for $x_{0}\in]0,1]$ (resp. $x_{0}=0$,
since every input distinct from $\mathbf{0}$ will drive $x(t)$ out of
$\mathcal{X}$).

Let $\mathrm{\bf A}$ be the set of _admissible pair_
$(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$ for which $y$ is admissible for
$x_{0}$, i.e.

$\mathrm{\bf A}\coloneqq\\{(x_{0},y)\in\mathcal{X}\times\mathrm{\bf
Y}\mid\exists u:(u,y)\in\mathrm{\bf W}(x_{0})\\}.$

Unless otherwise specified, compatibility and admissibility shall be
understood “ _with_ $\mathfrak{S}$”.

Finally, input-to-state mapping $\mathfrak{H}_{x}[x_{0};\cdot]$ (resp. input-
to-output mapping $\mathfrak{H}[x_{0};\cdot]$) associated with $\mathfrak{S}$
and parametrized by $x_{0}$, is naturally defined as the restriction of
$\mathrm{H}_{x}[x_{0};\cdot]$ (resp. $\mathrm{H}[x_{0};\cdot]$) to
$\mathrm{\bf U}_{a}(x_{0})$, i.e.

$\displaystyle\mathfrak{H}_{x}[x_{0};\cdot]$
$\displaystyle\coloneqq\mathrm{H}_{x}[x_{0};\cdot]\mid\mathrm{\bf
U}_{a}(x_{0}),$ $\displaystyle\mathfrak{H}[x_{0};\cdot]$
$\displaystyle\coloneqq\mathrm{H}[x_{0};\cdot]\mid\mathrm{\bf U}_{a}(x_{0}).$

###  2.3. Graphical illustration of input/state/output mappings

First comments about Fig. 1 are as follows. (i) $\mathrm{\bf U}$ (resp.
$\mathrm{\bf U}_{a}(x_{0})$) derives from $\mathrm{\bf U}_{\Sigma}$ by ruling
out signals which do not comply with input constraints (resp. input and state
constraints), so that $\mathrm{\bf U}_{a}(x_{0})\subseteq\mathrm{\bf
U}\subseteq\mathrm{\bf U}_{\Sigma}$ holds regardless of $x_{0}\in\mathcal{X}$.
(ii) It holds $(u_{i},x_{i},y_{i})\in\mathrm{\bf Q}(x_{0})$ for all $i$ but
$i=1$ since $x_{1}\not\in\mathrm{\bf X}$ which means that state constraints
are violated when exerting input $u_{1}$. (iii) For the same reason, the image
of $\mathrm{\bf U}$ by $\mathrm{H}_{x}[x_{0};\cdot]$ is not contained in
$\mathrm{\bf X}$ in general. (iv) $\mathrm{H}[x_{0};\mathrm{\bf U}]$ is, in
general, a proper subset of $\mathrm{\bf Y}$, as illustrated by the existence
of $\hat{y}\in\mathrm{\bf Y}\setminus\mathrm{H}[x_{0};\mathrm{\bf U}]$, i.e.
there is no $u$ such that $(u,\hat{y})\in\mathrm{\bf W}(x_{0})$. Existence of
$\hat{y}$ is possible since $\Sigma$ is not assumed to be right-invertible and
input space $\mathrm{\bf U}_{\Sigma}$ does not contain distributions (see e.g.
[28, Chap. 8]). (v) For analogous reasons, $\mathrm{H}_{x}[x_{0};\mathrm{\bf
U}]$ does not contain $\mathrm{\bf X}$ in general. (vi) Inequality $y_{3}\neq
y_{4}$ holds even if $x_{3}$ equals $x_{4}$. This situation can be
encountered, whenever $\Sigma$ is proper but not strictly proper.

##  3\. Enriched conceptual framework on IR

In this section, we enrich definitions and taxonomy associated to IR proposed
in [2], in order to cope with the more delicate context of constrained
dynamics where redundancy depends on both the output $y$ and the initial state
$x_{0}$, see Ex. 1.1.

###  3.1. IR pairs

######  Definition (IR pair).

A pair $(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$ is _IR_ if output $y$ can
be produced by (at least) two distinct inputs for initial condition $x_{0}$,
i.e.

(6) $\begin{array}[]{r}\exists(u_{1},y_{1}),(u_{2},y_{2})\in\mathrm{\bf
W}(x_{0}):\\\ u_{1}\neq u_{2},\;y_{1}=y_{2}=y.\end{array}$

Let $\mathrm{\bf S}\subseteq\mathcal{X}\times\mathrm{\bf Y}$ be the set of IR
pairs, i.e.

${\bf S}\coloneqq\\{(x_{0},y)\in\mathcal{X}\times\mathrm{\bf
Y}:\eqref{eq:singPair}\\}.$

The following inclusion chain holds:

$\mathrm{\bf S}\;\subseteq\;\mathrm{\bf
A}\;\subseteq\;\mathcal{X}\times\mathrm{\bf Y}.$

It suggests (i) that $\mathcal{X}\times\mathrm{\bf Y}$ might contain non
admissible pairs $(x_{0},y)$ (in this case, $\mathrm{\bf A}$ is a proper
subset of $\mathcal{X}\times\mathrm{\bf Y}$) and (ii) that an admissible pair
$(x_{0},y)\in\mathrm{\bf A}$ might be non IR (in this case, $\mathrm{\bf S}$
is a proper subset of $\mathrm{\bf A}$). Finally, inclusion $\mathrm{\bf
S}\subseteq\mathrm{\bf A}$ means that every IR pair $(x_{0},y)$ is admissible.

Let us now introduce state trajectories into (6): Given a pair
$(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$, define the following
relationships

(9)
$\displaystyle\begin{array}[]{r}\exists(u_{1},x_{1},y_{1}),(u_{2},x_{2},y_{2})\in\mathrm{\bf
Q}(x_{0}):\\\ u_{1}\neq u_{2},x_{1}=x_{2},y_{1}=y_{2}=y,\end{array}$ (12)
$\displaystyle\begin{array}[]{r}\exists(u_{1},x_{1},y_{1}),(u_{2},x_{2},y_{2})\in\mathrm{\bf
Q}(x_{0}):\\\ u_{1}\neq u_{2},x_{1}\neq x_{2},y_{1}=y_{2}=y.\end{array}$

Observe that if $(x_{0},y)$ is IR then at least one of the above relationships
holds.

Making use of (9) and (12), instead of (6), allows for investigating origin of
IR. This gives rise to a taxonomy distinguishing IR pairs.

######  Definition (IR pair of the $k$-th kind).

Let $(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$ be an IR pair. It is of:

  * •

The _1st kind_ if (9) holds but (12) does not, i.e. if the following
implication

(13) $\left.\begin{array}[]{r}u_{1}\neq u_{2}\\\
y=y_{1}=y_{2}\end{array}\right\\}\Rightarrow x_{1}=x_{2}$

holds for all $(u_{1},x_{1},y_{1}),(u_{2},x_{2},y_{2})\in\mathrm{\bf
Q}(x_{0})$;

  * •

The _2nd kind_ if (9) does not hold but (12) does, i.e. the following
implication

(14) $\left.\begin{array}[]{r}u_{1}\neq u_{2}\\\
y=y_{1}=y_{2}\end{array}\right\\}\Rightarrow x_{1}\neq x_{2}$

holds for all $(u_{1},x_{1},y_{1}),(u_{2},x_{2},y_{2})\in\mathrm{\bf
Q}(x_{0})$;

  * •

The _3rd kind_ if both (9) and (12) hold, i.e. neither (13) nor (14) is valid
for all $(u_{1},x_{1},y_{1}),(u_{2},x_{2},y_{2})\in\mathrm{\bf Q}(x_{0})$.

Previous definition induces that different kinds are mutually exclusive: No
pair $(x_{0},y)$ can be simultaneously of different kinds. This motivates
partitioning of $\mathrm{\bf S}$ as follows:

$\mathrm{\bf S}=\mathrm{\bf S}_{1}\cup\mathrm{\bf S}_{2}\cup\mathrm{\bf
S}_{3},$

where $\mathrm{\bf S}_{k}$ gathers IR pairs of the $k$-th kind and satisfies
$\mathrm{\bf S}_{k}\cap\mathrm{\bf S}_{j}=\varnothing$ for all $k\neq j$.

Let us provide a didactical illustration of those concepts via Fig. 1. Observe
that the set $\mathrm{\bf S}$ of all IR pairs of $\mathfrak{S}$ equals
$\\{(x_{0},y_{5}),(x_{0},y_{7})\\}$. It also comes out $\mathrm{\bf
S}_{1}=\\{(x_{0},y_{5})\\}$ and $\mathrm{\bf S}_{2}=\\{(x_{0},y_{7})\\}$. Note
that pair $(x_{0},y_{1})$ is not IR, even if $y_{1}=y_{2}$ and $u_{1}\neq
u_{2}$ hold. Indeed, $x_{1}\not\in\mathrm{\bf X}$ holds so that
$(u_{1},y_{1})$ does not belong to $\mathrm{\bf W}(x_{0})$.

######  Remark 1 (Alternative formulation of (6)).

Let $\mathfrak{H}^{-1}[x_{0};y]$ refers to preimage of $y\in\mathrm{\bf Y}$ by
$\mathfrak{H}[x_{0};\cdot]$, i.e.

(15) $\mathfrak{H}^{-1}[x_{0};y]\coloneqq\\{u\in\mathrm{\bf
U}:y=\mathfrak{H}[x_{0};u]\\}\\\ =\\{u\in\mathrm{\bf U}:(u,y)\in\mathrm{\bf
W}(x_{0})\\}.$

The following relationship

(16) $|\mathfrak{H}^{-1}[x_{0};y]|\geq 2$

gives an alternative way to define IR of
$(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$. This reformulation highlights
the fact that existence of IR is equivalent to _non-injectivity_ of
$\mathfrak{H}[x_{0};\cdot]$ for some $x_{0}$.

###  3.2. IR systems as defined in [2]

Definitions and taxonomy introduced in [2, Sec. 2] apply _verbatim_ for
$\mathfrak{S}$.

######  Definition (IR).

System $\mathfrak{S}$ is _IR_ if there exists an IR pair
$(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$, i.e. $\mathrm{\bf
S}\neq\varnothing$.

Whenever every IR pair enjoys the same qualifying term, system $\mathfrak{S}$
itself shall inherit from the proposed taxonomy, previously coins for IR
pairs.

######  Definition (IR of the $k$-th kind).

System $\mathfrak{S}$ is _IR of the $k$-th kind_ if (i) it is IR and (ii)
every IR pair $(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$ is of the $k$-th
kind, i.e. $\mathrm{\bf S}_{k}=\mathrm{\bf S}\neq\varnothing$.

###  3.3. Uniform IR

To prove that $\mathfrak{S}$ is IR, it suffices to find a single pair
$(x_{0},y)$ which is IR. Yet, for some system, this IR might occur _for all_
admissible pair $(x_{0},y)\in(\mathcal{X}\times\mathrm{\bf Y})$. The adverb
“uniformly” is used to qualify this situation.

######  Definition (Uniform IR).

System $\mathfrak{S}$ is _uniformly_ IR if (i) it is IR and (ii) every
admissible pair $(x_{0},y)\in(\mathcal{X}\times\mathrm{\bf Y})$ is IR, i.e.
$\mathrm{\bf A}=\mathrm{\bf S}\neq\varnothing$ holds.

This property of uniformity can be combined with the proposed taxonomy of
system $\mathfrak{S}$.

######  Definition (Uniform IR of the $k$-th kind).

System $\mathfrak{S}$ is _uniformly IR of the $k$-th kind_ if (i) it is
uniformly IR and (ii) every IR pair $(x_{0},y)\in\mathcal{X}\times\mathrm{\bf
Y}$ is of the $k$-th kind, i.e. $\mathrm{\bf A}=\mathrm{\bf S}=\mathrm{\bf
S}_{k}\neq\varnothing$.

In view of Fig. 1, $\mathfrak{S}$ is neither IR of any kind (both $\mathrm{\bf
S}_{1}$ and $\mathrm{\bf S}_{2}$ are non empty) nor uniformly IR ($\mathrm{\bf
A}\ni(x_{0},y_{2})\not\in\mathrm{\bf S}$).

Tab. 2 can be referred to as a summary of the introduced definitions.

Table 2.  Definitions associated to IR IR | $\mathrm{\bf S}\neq\varnothing$
---|---
IR of the $k$-th kind | $\mathrm{\bf S}_{k}=\mathrm{\bf S}\neq\varnothing$
Uniform IR | $\mathrm{\bf A}=\mathrm{\bf S}\neq\varnothing$
Uniform IR of the $k$-th kind | $\mathrm{\bf A}=\mathrm{\bf S}=\mathrm{\bf S}_{k}\neq\varnothing$

##  4\. General results

In addition of being part of the description of $\mathfrak{S}$, system
$\Sigma$ can be regarded as the unconstrained version of $\mathfrak{S}$.
Therefore, comparison between those two systems allows to evaluate how
constraints impact IR. This is the goal of this section.

Note that most of the proofs of this section and the next one are postponed in
the appendices. Among them, Appendix B proposes a key analysis of input-to-
output mapping associated with $\mathfrak{S}$ by adopting an incremental view
point. This analysis is instrumental in achieving tractable characterization
of definitions associated with IR.

###  4.1. Sufficient condition for IR to be preserved

Via Ex. 1.1, it has been observed that constraints can destroy IR of a pair
$(x_{0},y)$, in the sense that $(x_{0},y)$ can be IR for $\Sigma$ but not for
$\mathfrak{S}$. Following theorem, proved in Appendix B, offers a sufficient
condition to prevent this situation to occur, i.e. for IR to be preserved in
the constrained context.

######  Theorem 4.1.

Define the following integer:

$\rho\coloneqq\dim(\mathrm{Ker}\left\\{B\right\\}\cap\mathrm{Ker}\left\\{D\right\\}),$

as well as the following relationships:

(17a) $\displaystyle u(t)$ $\displaystyle\in\mathrm{int}(\mathcal{U}),$ (17b)
$\displaystyle x(t)$ $\displaystyle\in\mathrm{int}(\mathcal{X}),$

parametrized by $t\in{}_{\geq 0}$, $u\in\mathrm{\bf U}$ and $x\in\mathrm{\bf
X}$. Assume that the unconstrained system $\Sigma$ is IR. Then, admissible
pair $(x_{0},y)\in\mathrm{\bf A}$ is IR (for $\mathfrak{S}$) if there exist
$(u,x)$ and $0\leq t_{0}<t_{f}$ such that $(u,x,y)\in\mathrm{\bf Q}(x_{0})$
holds and:

  * •

if $\rho>0$, (17a) holds for all $t\in]t_{0},t_{f}[$;

  * •

if $\rho=0$, (17a) and (17b) hold for all $t\in]t_{0},t_{f}[$.

When $\mathcal{U}$ is open, (17a) (resp. (17b)) is valid for all
$u\in\mathrm{\bf U}$ (resp. for all $x\in\mathrm{\bf X}$) and for all
$t\in{}_{\geq 0}$. This leads to the following corollary.

######  Corollary 1.

Assume that the unconstrained system $\Sigma$ is IR. If one of the following
conditions is valid:

  * •

$\rho>0$ and $\mathcal{U}$ is open,

  * •

$\rho=0$ and both $\mathcal{U}$ and $\mathcal{X}$ are open,

then, $\mathfrak{S}$ is uniformly IR, i.e. $\mathrm{\bf S}=\mathrm{\bf
A}=\varnothing$.

Contraposition of Th. 4.1 is also of major importance. In the context of Th.
4.1, assume that $\rho>0$ (resp. $\rho=0$) holds and that $(x_{0},y)$ is
admissible but _not_ IR, so that there is a unique pair $(u,x)$ satisfying
$(u,x,y)\in\mathrm{\bf Q}(x_{0})$. Then for all non empty
$]t_{0},t_{f}[\subseteq{}_{\geq 0}$, condition (17a) (resp. (17a) and (17b))
is violated for some $t=t_{i}\in]t_{0},t_{f}[$, i.e.
$u(t_{i})\in\mathcal{U}\setminus\text{int}(\mathcal{U})$ holds (resp. either
$u(t_{i})\in\mathcal{U}\setminus\text{int}(\mathcal{U})$ or
$x(t_{i})\in\mathcal{X}\setminus\text{int}(\mathcal{X})$ hold). As formally
proved in Appendix C, this observation together with piecewise continuity of
$u$ and continuity of $x$ lead to the following corollary.

######  Corollary 2.

Define the following relationships:

(18a) $\displaystyle u(t)$
$\displaystyle\in\mathcal{U}\setminus\mathrm{int}(\mathcal{U}),$ (18b)
$\displaystyle x(t)$
$\displaystyle\in\mathcal{X}\setminus\mathrm{int}(\mathcal{X}),$

parametrized by $t\in{}_{\geq 0}$, $u\in\mathrm{\bf U}$ and $x\in\mathrm{\bf
X}$. Assume that the unconstrained system $\Sigma$ is IR. Then, admissible
pair $(x_{0},y)\in\mathrm{\bf A}$ _is not IR (for $\mathfrak{S}$) only if_ the
unique $(u,x)$ satisfying $(u,x,y)\in\mathrm{\bf Q}(x_{0})$ is such that:

  * •

if $\rho>0$, then (18a) holds for all $t\in\mathcal{C}$;

  * •

if $\rho=0$, then (18a) or (18b) holds for all $t\in\mathcal{C}$.

Here, $\mathcal{C}$ is the largest open subset of ≥0 where $u\in\mathcal{U}$
is continuous.

If $\mathcal{U}$ and $\mathcal{X}$ are closed and $\Sigma$ is IR, then Cor. 2
proves that admissible pairs $(x_{0},y)$ which are not IR are necessarily
associated with input and state trajectories that lie alternatively on the
boundary of $\mathcal{U}$ and $\mathcal{X}$ for all $t$ where
$u\in\mathcal{U}$ is continuous.

######  Example 4.1.

Consider the parallel interconnection of two buck converters feeding a single
resistive load of magnitude $R$ and fed by a single voltage source of
magnitude $V$. Interested reader is referred to [3] for details about this
application.

This system can be controlled via a pulse width modulation strategy, so that
the duty cycles of each converter are the components of the input vectors
$u(t)\in\mathcal{U}=[0,1]^{2}$. Resulting averaged dynamics can be modeled via
(1) with

$A=\begin{bmatrix}0&0&-\frac{1}{L}\\\ 0&0&-\frac{1}{L}\\\
\frac{1}{C}&\frac{1}{C}&-\frac{1}{RC}\end{bmatrix},B=\frac{V}{L}\begin{bmatrix}1&0\\\
0&1\\\ 0&0\end{bmatrix},C=\begin{bmatrix}0\\\ 0\\\
1\end{bmatrix}^{\intercal},D=\mathbf{0},$

where $L$ and $C$ denote the inductances and the output capacitor magnitudes,
respectively. Components of the state vector $x(t)\in\mathcal{X}={}^{3}$ are
the two currents flowing through the inductances, followed by the voltage at
the load. This voltage is the output signal.

For this example, $\rho$ equals $0$ and one can prove that $\Sigma$ is IR, see
[29] where it is shown that $\mathcal{R}(\Sigma)$ equals
$\mathrm{span}\left(\left[\begin{smallmatrix}1&-1&0\end{smallmatrix}\right]^{\intercal}\right)$.
To demonstrate that $\mathfrak{S}$ is also IR, first pick any
$x_{0}\in\mathcal{X}={}^{3}$. Then, define input $u_{1}$ as follows:

$u_{1}:t\mapsto\left\\{\begin{array}[]{ll}\begin{bmatrix}1-t&t\end{bmatrix}^{\intercal},&(t\in[0,1]),\\\
\begin{bmatrix}0&1\end{bmatrix}^{\intercal},&(t>1).\end{array}\right.$

Let output $y_{1}$ satisfy $(u_{1},y_{1})\in\mathrm{\bf W}(x_{0})$. Since
$u_{1}(t)$ belongs to $\mathrm{int}(\mathcal{U})$ for all $t\in]0,1[$ and
$\mathrm{int}(\mathcal{X})=\mathcal{X}$ so that (17b) holds for all $t\geq 0$,
Th. 4.1 ensures that $(x_{0},y_{1})$ is an IR pair for $\mathfrak{S}$. Indeed,
input $u_{2}:t\mapsto\left[\begin{smallmatrix}1\\\ 0\end{smallmatrix}\right]$
leads to output $y_{2}=\mathrm{H}[x_{0};u_{2}]$ which is equal to $y_{1}$.
This fact can be highlighted via the following equation capturing the input-
output mapping:

(19)
$C\ddot{y}+\frac{1}{R}\dot{y}+\frac{2}{L}y=\frac{E}{L}\begin{bmatrix}1&1\end{bmatrix}u.$

Indeed, observe that exerting either $u_{1}$ or $u_{2}$ lead to identical
right-hand side of (19), so that resulting outputs $y_{1,2}$ are the same.
More generally, observe that a given output trajectory $y$ imposes the
expression of $\left[\begin{smallmatrix}1&1\end{smallmatrix}\right]u$ via
(19), so that any signal of the form $\tilde{u}=\left[\begin{smallmatrix}1\\\
-1\end{smallmatrix}\right]w$ with $w:{}_{\geq 0}\rightarrow\real$ can be added
to the input $u$ leading to $y$ without affecting this output, i.e.
$\mathrm{H}[x_{0};u+\tilde{u}]=y$.

Consider now the input $u_{3}=\mathbf{0}$ leading to the output
$y_{3}=\mathbf{0}$ for zero initial state $x_{0}=\mathbf{0}$. The pair
$(x_{0},y_{3})$ is _not_ IR. To see this, observe that (19) together with
$(x_{0},y)=(\mathbf{0},\mathbf{0})$ imply
$\left[\begin{smallmatrix}1&1\end{smallmatrix}\right]u=0$. This last equation
admits a unique solution lying identically in $\mathcal{U}$, that is
$u_{3}=\mathbf{0}$. Note that the fact that $(x_{0},y_{3})$ is not IR agree
with the statement of Th. 4.1: (17a) is never valid, whatever is the selection
of the interval $]t_{0},t_{f}[\subseteq{}_{\geq 0}$.

######  Example 4.2 (Ex. 1.1 continued).

Case (ii) proves that $\mathfrak{S}$ and, in turn, $\Sigma$ are IR. For cases
(i) and (iii), unique input $u_{1}=\mathbf{0}$ making pairs
$(x_{0,1},y_{1}),(x_{0,2},y_{2})$ admissible satisfies
$u_{1}(t)\in\mathcal{U}\setminus\mathrm{int}(\mathcal{U})=(\\{0\\}\oplus{}_{\geq
0})\cup({}_{\geq 0}\oplus\\{0\\})$ for all $\mathcal{C}={}_{\geq 0}$, as
predicted by Cor. 2.

######  Remark 2.

Assume that $\Sigma$ is IR. Clearly, Cor. 1 ensures that $\mathfrak{S}$ is
uniformly IR if both $\mathcal{U}$ and $\mathcal{X}$ are open, _regardless of
$\rho$_. In the same spirit, Cor. 2 proves that $(x_{0},y)\in\mathrm{\bf
A}\setminus\mathrm{\bf S}$ holds only if the unique $(u,x)$ satisfying
$(u,x,y)\in\mathrm{\bf Q}(x_{0})$ is such that (18a) or (18b) holds for all
$t\in\mathcal{C}$, _whatever is the value of $\rho$_. Similarly, if there
exist $(u,x,y)\in\mathrm{\bf Q}(x_{0})$ and $0\leq t_{0}<t_{f}$ such that both
(17a) and (17b) hold for all $t\in]t_{0},t_{f}[$, then $(x_{0},y)$ is IR for
$\mathfrak{S}$ _regardless of $\rho$_, by virtue of Th. 4.1.

###  4.2. How constraints impact the kind of IR ?

Assume that the unconstrained system $\Sigma$ is IR of the $k$-th kind. Given
an IR pair $(x_{0},y)\in\mathrm{\bf S}$. Then, a natural question is to ask:
Does $(x_{0},y)$ belong to $\mathrm{\bf S}_{k}$ ? Saying it differently, do
the constraints can modify the kind of IR of a pair ?

For $k=3$, the answer to the last question might be positive even when the
constraints are linear, as shown by the next example.

######  Example 4.3.

Define dynamical system $\Sigma$ as follows

$\displaystyle\dot{x}(t)$ $\displaystyle=\begin{bmatrix}-1&0&0\\\ 0&-1&0\\\
0&0&-1\end{bmatrix}x(t)+\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\
0&0&1&1\end{bmatrix}u(t),$ $\displaystyle y(t)$
$\displaystyle=\begin{bmatrix}0&0&1\end{bmatrix}x(t)$

In view of [2, Th. 3.2], one can prove that $\Sigma$ is IR of the 3rd
kind.444Indeed, it holds
$\mathcal{R}(\Sigma)=\mathrm{span}\left(\left[\begin{smallmatrix}\mathrm{\mathbf{I}}_{2}\\\
\mathbf{0}\end{smallmatrix}\right]\right)$ and $\rho=1$. As an illustration,
define $u_{1}=\mathbf{0}$, $u_{2}:t\mapsto[0,0,1,-1]^{\intercal}$ and
$u_{3}:t\mapsto[1,0,0,0]^{\intercal}$. Denoting
$(u_{k},x_{k},y_{k})\in\mathrm{\bf Q}(x_{0}=\mathbf{0}),(k\in\\{1,2,3\\})$,
observe that $y_{1}=y_{2}=y_{3}=\mathbf{0}$. Since inputs $u_{k}$ are all
distinct and $x_{1}=x_{2}\neq x_{3}$, pair $(x_{0},y)=(\mathbf{0},\mathbf{0})$
is IR of the 3rd kind for $\Sigma$.

Let $\mathfrak{S}$ derive from $\Sigma$ by adding following linear constraints

$\displaystyle\mathcal{U}$
$\displaystyle=\left\\{[u_{a},u_{b},u_{c},u_{d}]^{\intercal}\in{}^{4}~{}|~{}u_{c}-u_{d}=0\right\\},$
$\displaystyle\mathcal{X}$
$\displaystyle=\left\\{[x_{a},x_{b},x_{c}]^{\intercal}\in{}^{3}~{}|~{}x_{b}+x_{c}=0\right\\}.$

State $x(t)$ belongs identically in $\mathcal{X}$ iff
$Ax(t)+Bu(t)\in\mathcal{X}\Leftrightarrow-(x_{b}(t)+x_{c}(t))+u_{b}(t)+u_{c}(t)+u_{d}(t)=0$
for all $t$ and for all $x(t)\in\mathcal{X}$. Together with constraint
$u(t)\in\mathcal{U}$, this is equivalent to $u_{c}=u_{d}=-u_{b}/2$. In this
case, dynamical equations reduce to

$\displaystyle\dot{x}_{a}(t)$ $\displaystyle=-x_{a}(t)+u_{a}(t),$
$\displaystyle\dot{y}(t)$ $\displaystyle=-y(t)-u_{b}(t).$

Clearly, any pair $(x_{0},y=x_{c}=-x_{b})$ uniquely defines $u_{b}$ and, in
turn, $u_{c}$ and $u_{d}$. But, it let $u_{a}$ free. This proves that any
admissible pair $(x_{0},y)$ is IR of the 2nd kind since distinct $u_{a}$ leads
to distinct $x_{a}$. This proves that $\mathfrak{S}$ is IR of the 2nd kind,
whereas $\Sigma$ is IR of the 3rd kind.

For $k\in\\{1,2\\}$, the answer to the last question is always negative, i.e.
constraints cannot change IR of the 1st or of the 2nd kind. To see this, let
$(x_{0},y)\in\mathrm{\bf S}$. Assume that $\Sigma$ is IR of the 1st (resp.
2nd) kind. Then $u_{1}\neq u_{2}$ together with $y_{1}=y_{2}=y$ implies
$x_{1}=x_{2}$ (resp. $x_{1}\neq x_{2}$) for all
$(u_{1},x_{1},y_{1}),(u_{2},x_{2},y_{2})\in\mathrm{\bf Q}_{\Sigma}(x_{0})$.
Since $\mathrm{\bf Q}(x_{0})\subseteq\mathrm{\bf Q}_{\Sigma}(x_{0})$, this
proves that $(x_{0},y)$ belongs to $\mathrm{\bf S}_{1}$ (resp. to $\mathrm{\bf
S}_{2}$).

######  Lemma 4.2.

Let $\Sigma$ be IR of the $k$-th kind, with $k\in\\{1,2\\}$. If $\mathfrak{S}$
is IR, then $\mathfrak{S}$ is IR of the $k$-th kind, i.e. $\mathrm{\bf
S}=\mathrm{\bf S}_{k}$.

This lemma proves that if $\Sigma$ is IR of the $k$-th kind, with
$k\in\\{1,2\\}$, then all IR pairs of $\mathfrak{S}$ (if any) is also of the
$k$-th kind. This is in contrast with the case where $k=3$. In this situation,
not only $\mathfrak{S}$ can be of the 1st or 2nd kind (see Ex. 4.3), but also
IR pairs of different kinds can coexist for $\mathfrak{S}$, as shown by the
following example.

######  Example 4.4.

Let $\Sigma$ be defined as follows:

$\displaystyle\dot{x}(t)$ $\displaystyle=\begin{bmatrix}1&0\\\
0&1\end{bmatrix}x(t)+\begin{bmatrix}1&0&0\\\ 0&1&1\end{bmatrix}u(t)$
$\displaystyle y(t)$ $\displaystyle=\begin{bmatrix}0&1\end{bmatrix}x(t).$

Constrained system $\mathfrak{S}$ derives from $\Sigma$ by imposing
$\mathcal{U}=[-1;+\infty[^{3}$ and $\mathcal{X}=[0;1]\times[0;2]$. Let us show
that $\mathfrak{S}$ admits IR pairs of each kind:

  * •

Pair $(x_{0,1},y_{1})\coloneqq(\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right],\mathbf{0})$ is of the 1st kind. Indeed, first input
necessarily equals $t\mapsto-1$ for first state not to escape from $[0;1]$.
Together with $y=\mathbf{0}$, this implies that $x$ is constant and equals
$t\mapsto x_{0,1}$. Thus, $y_{1}$ can be produced by any input
$u\in\mathrm{\bf U}_{\Sigma}$ satisfying
$u(t)\in\\{-1\\}\times(\mathrm{Ker}\left\\{[1,1]\right\\}\cap[-1;+\infty[^{2})$
for all $t\in{}_{\geq 0}$.

  * •

Pair $(x_{0,2},y_{2})\coloneqq(\left[\begin{smallmatrix}0.5\\\
2\end{smallmatrix}\right],t\mapsto 2)$ is of the 2nd kind. This time, last two
inputs must equal $t\mapsto-1$ for $y(t)=2$ to hold for all $t\in{}_{\geq 0}$.
Thus, $y_{2}$ originates from any input $u\in\mathrm{\bf U}_{\Sigma}$ such
that (i) $u(t)$ belongs to $[-1;+\infty[\times\\{-1\\}^{2}$ for all
$t\in{}_{\geq 0}$ and (ii) first state belongs to $[0;1]$. In particular,
$\eta_{\alpha}:t\mapsto-\alpha e^{(1-\alpha)t}/2$ is a suitable trajectory for
first input, whatever is $\alpha\in[1;2]$. To see this more easily, note that
$\eta_{\alpha}$ can be interpreted as the feedback of gain $-\alpha$ between
first state and first input. Furthermore, any of those input candidates leads
to distinct state trajectories.

  * •

Pair $(x_{0,3},y_{3})\coloneqq(\left[\begin{smallmatrix}0.5\\\
0\end{smallmatrix}\right],\mathbf{0})$ is of the 3rd kind. This can be proved
by constructing triples $(u_{4},x_{4},y_{4})$, $(u_{5},x_{5},y_{5})$,
$(u_{6},x_{6},y_{6})\in\mathrm{\bf Q}(x_{0,3})$ satisfying
$y_{4}=y_{5}=y_{6}=y_{3}$, $x_{4}=x_{5}\neq x_{6}$ and
$u_{k},(k\in\\{4,5,6\\})$, all distinct. Using distinct
$\alpha_{1},\alpha_{2}\in[1;2]$, it can be verified that
$u_{4}=[\eta_{\alpha_{1}},\mathbf{0},\mathbf{0}]^{\intercal}$,
$u_{5}=(t\mapsto[\eta_{\alpha_{1}}(t),1,-1]^{\intercal})$ and
$u_{6}=[\eta_{\alpha_{2}},\mathbf{0},\mathbf{0}]^{\intercal}$ comply with
those constraints.

As a last comment, note that constraints can completely destroy redundancy. A
trivial two inputs example is the case where $\Sigma$ is IR of the 1st kind,
$\mathrm{Ker}\left\\{\left[\begin{smallmatrix}B\\\
D\end{smallmatrix}\right]\right\\}=\mathrm{span}\left(\left[\begin{smallmatrix}0\\\
1\end{smallmatrix}\right]\right)$,
$\mathcal{U}=\\{\left[\begin{smallmatrix}x_{a}\\\
x_{b}\end{smallmatrix}\right]:x_{b}=0\\}$ and $\mathcal{X}={}^{n}$.

##  5\. Linear spaces

Let us turn our attention to the case where both $\mathcal{U}$ and
$\mathcal{X}$ are linear. The following assumption is considered valid
throughout this section.

######  Assumption 1 (Linear spaces).

Sets $\mathcal{U}\subseteq{}^{m}$ and $\mathcal{X}\subseteq{}^{n}$ are linear
over the field .

By specializing the results of the incremental analysis (see Appendix B)
supporting results exposed in the previous section, the following theorem can
be obtained, see Appendix B for the proof. Roughly speaking, this theorem
states that “linearity implies uniformity”.

######  Theorem 5.1.

Under Hyp. 1, if $\mathfrak{S}$ admits an IR pair
$(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$ of the $k$-th kind, then
$\mathfrak{S}$ is uniformly IR of the $k$-th kind, i.e. $\mathrm{\bf
A}=\mathrm{\bf S}=\mathrm{\bf S}_{k}\neq\varnothing$.

As compared with [2, Prop. 2.1] where $\mathcal{U}={}^{m}$ and
$\mathcal{X}={}^{n}$ hold, this theorem is more general since arbitrary linear
spaces are considered. It also emphasizes that $\mathrm{\bf S}$ equals
$\mathrm{\bf S}_{k}$, i.e. Hyp. 1 prevents pairs of distinct kind to coexist
for the same system. This is in stark contrast with the general non linear
case, as shown by Ex. 4.4.

This section heavily relies on geometric control theory whose essential
aspects are summarized in Appendix A. Reader is also referred to [30, Chap. 0]
or [28, Sec. 2.2]. Following [28], and relatively to system $\Sigma$, the
weakly unobservable subspace and the controllable weakly unobservable subspace
are denoted by $\mathcal{V}(\Sigma)$ and $\mathcal{R}(\Sigma)$, respectively.

###  5.1. Description of $\mathrm{\bf Q}(x_{0})$ via trajectories of an
unconstrained system

As an initial step, let us get rid of constraints associated with
$\mathcal{U}$ and $\mathcal{X}$ by constructing a quadruple whose
_unconstrained_ trajectories can be gathered into a set isomorphic to
$\mathrm{\bf Q}(x_{0})$. To this end, this subsection summarizes, formalizes
(via Lem. 5.2) and illustrates existing material collected in Appendix A.6.
Let $R$ denote the insertion of $\mathcal{U}$ in m. Define
$B_{\mathcal{U}}\coloneqq B\mid\mathcal{U}=BR$ (resp.
$D_{\mathcal{U}}\coloneqq D\mid\mathcal{U}=DR$) as the domain restriction of
$B$ (resp. $D$) to $\mathcal{U}$. Let
$\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}})$ (or
$\mathcal{V}^{*}(\mathcal{X})$ for short), be the largest
$(A,B_{\mathcal{U}})$-controlled invariant subspace contained in
$\mathcal{X}$. Pick any friend $F$ in
$\underline{F}(\mathcal{V}^{*}(\mathcal{X}))$, the set of friend of
$\mathcal{V}^{*}(\mathcal{X})$. Select $L$ as any injective linear map such
that

(20)
$\mathrm{Im}\left\\{L\right\\}=B_{\mathcal{U}}^{-1}\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}}).$

This allows to define _unconstrained_ system $\Sigma_{F}$, characterized by
quadruple $(A_{F},B_{F},C_{F},D_{F})$ where
$A_{F}\coloneqq(A+BRF)\mid\mathcal{V}^{*}(\mathcal{X})$,
$B_{F}\coloneqq\mathcal{V}^{*}(\mathcal{X})\mid(BRL)$,
$C_{F}\coloneqq(C+DRF)\mid\mathcal{V}^{*}(\mathcal{X})$ and $D_{F}\coloneqq
DRL$.

######  Example 5.1 (Ex. 4.3 continued).

Select following injective matrix $R$ satisfying
$\mathrm{Im}\left\\{R\right\\}=\mathcal{U}$:

$R=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1/2\\\ 0&0&1/2\\\ \end{bmatrix},$

so that $B_{\mathcal{U}}=BR=\mathrm{\mathbf{I}}_{3}$ and
$D_{\mathcal{U}}=DR=\mathbf{0}$ hold. Since $A\mathcal{X}$ equals
$\mathcal{X}$, we trivially have $\mathcal{V}^{*}(\mathcal{X})=\mathcal{X}$
and $\mathbf{0}\in\underline{F}(\mathcal{V}^{*}(\mathcal{X}))$. Apply input
$u=R(Fx+Lw)$ with $F=\mathbf{0}$ and

$L=\begin{bmatrix}1&0\\\ 0&1\\\ 0&-1\end{bmatrix},$

which satisfies (20). This leads to
$(A_{F},B_{F},C_{F},D_{F})=(-\mathrm{\mathbf{I}}_{2},\mathrm{\mathbf{I}}_{2},[0,-1],\mathbf{0})$.
Indeed, change of variable $\left[\begin{smallmatrix}\eta\\\
\varpi\end{smallmatrix}\right]=P^{-1}x$ with

$P^{-1}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&1&1\end{bmatrix}\Leftrightarrow
P=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&-1&1\end{bmatrix}$

gives rise to the following dynamics, with $w$ as new input:

$\displaystyle\begin{bmatrix}\dot{\eta}\\\ \dot{\varpi}\end{bmatrix}$
$\displaystyle=P^{-1}(A+B_{\mathcal{U}}F)P\begin{bmatrix}\eta\\\
\varpi\end{bmatrix}+P^{-1}B_{\mathcal{U}}Lw$
$\displaystyle=-\mathrm{\mathbf{I}}_{3}\begin{bmatrix}\eta\\\
\varpi\end{bmatrix}+\begin{bmatrix}\mathrm{\mathbf{I}}_{2}\\\
\mathbf{0}\end{bmatrix}w$

and

$\displaystyle y$ $\displaystyle=(C+D_{\mathcal{U}}F)P\begin{bmatrix}\eta\\\
\varpi\end{bmatrix}+D_{\mathcal{U}}Lw$
$\displaystyle=\begin{bmatrix}0&-1&1\end{bmatrix}\begin{bmatrix}\eta\\\
\varpi\end{bmatrix}.$

Since first two columns of $P$ span $\mathcal{V}^{*}(\mathcal{X})$,
$\Sigma_{F=\mathbf{0}}$ is governed by following equation

(21)
$\dot{\eta}(t)=-\mathrm{\mathbf{I}}_{2}\eta(t)+\mathrm{\mathbf{I}}_{2}w(t),\quad
y(t)=\begin{bmatrix}0&-1\end{bmatrix}\eta(t),$

corresponding to restriction of the dynamics to
$\mathcal{V}^{*}(\mathcal{X})$.

Let $x_{0}\in\mathcal{X}_{c}$. By construction, it is possible to link any
triple $(u,x,y)\in\mathrm{\bf Q}(x_{0})$ with some element of the set
$\mathrm{\bf Q}_{F}(\eta_{0})$ which gathers all input $w$, state $\eta$ and
output $\varphi$ trajectories originating from initial condition
$\eta(0)=\eta_{0}$ and compatible with $\Sigma_{F}$: Such a relationship is
established via the following mapping:

$\displaystyle E(F,\eta_{0};\cdot):\mathrm{\bf Q}_{F}(\eta_{0})$
$\displaystyle\rightarrow\mathrm{\bf Q}(T\eta_{0})$
$\displaystyle(w,\eta,\varphi)$
$\displaystyle\mapsto(RLw+RFT\eta,T\eta,\varphi)$

where $T$ denote insertion of $\mathcal{V}^{*}(\mathcal{X})$ in n. This
mapping is parametrized by $F$ and $\eta_{0}\in{}^{l}$ where

$l\coloneqq\dim{\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}})}.$

######  Lemma 5.2.

Assume that Hyp. 1 and $l>0$ hold. For all $\eta_{0}\in{}^{l}$ and
$F\in\underline{F}(\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}}))$:

  1. (i)

mapping $E(F,\eta_{0};\cdot)$ is linear and bijective, so that $\mathrm{\bf
Q}_{F}(\eta_{0}){\coloneqq E^{-1}(F,\eta_{0};\mathrm{\bf Q}(T\eta_{0}))}$ and
$\mathrm{\bf Q}(T\eta_{0})$ are isomorphic;

  2. (ii)

for all $(w,\eta,\varphi)\in\mathrm{\bf Q}_{F}(\eta_{0})$, input $w$ is
causal, piecewise continuous and exponentially bounded.

###### Proof.

$E(F,\eta_{0};\cdot)$ is surjective by construction. Its injectivity comes
from that of $RL$. Linearity is obvious. This proves (i). As far as (ii) is
concerned, pick any $(u,x,y)\coloneqq(RLw+RFT\eta,T\eta,\varphi)$ in
$\mathrm{\bf Q}(T\eta_{0})\subseteq\mathrm{\bf U}_{\Sigma}\times\mathrm{\bf
C}({}_{\geq 0},{}^{n})\times\mathrm{\bf Y}$. Since $u\in\mathrm{\bf
U}_{\Sigma}$ holds, $x=\mathrm{H}_{x}[T\eta_{0};u]$ is causal, continuous and
exponentially bounded. Matrix $T$ being injective, $\eta$ enjoys the same
property. Due to equality $RLw=u-RFT\eta$ and injectivity of $RL$, $w$
inherits (a) piecewise continuity from $u$ and (b) exponential boundedness and
causality from $u$ and $\eta$. ∎

Let us emphasize that any triple $(w,\eta,\varphi)$ satisfying differential-
algebraic equations analogous to (1) (but associated with $\Sigma_{F}$)
belongs to $\mathrm{\bf Q}_{F}(x_{0})$. This is in stark contrast with triple
$(u,x,y)\in\mathrm{\bf Q}(x_{0})$ which satisfies (1) and, at the same time,
belongs to $\mathrm{\bf U}\times\mathrm{\bf X}\times\mathrm{\bf Y}$ (see (5)).
Notwithstanding, $E(F,\eta_{0};\cdot)$ maps $\mathrm{\bf Q}_{F}(x_{0})$ to
$\mathrm{\bf Q}(x_{0})$ in a bijective way. To arrive at this result, input
and state constraints associated to $\mathfrak{S}$ have been somehow
_structurally embed_ into quadruple of $\Sigma_{F}$.

######  Example 5.2 (Ex. 4.3 continued).

Exerting input $w:t\mapsto[1,1]^{\intercal}$ to $\Sigma_{F=\mathbf{0}}$ for
$\eta_{0}=[1,0]^{\intercal}$ produces $\eta:t\mapsto[1,1-e^{-t}]$ and, in
turn, $y=\varphi:t\mapsto e^{-t}-1$. Since $T$ can be selected as the first
two columns of $P$, one can check that

$(u,x,y)=E(\mathbf{0},\eta_{0};(w,\eta,\varphi))\\\
=(t\mapsto\left[\begin{smallmatrix}1\\\ 1\\\ -1/2\\\
-1/2\end{smallmatrix}\right],t\mapsto\left[\begin{smallmatrix}1\\\ 1-e^{-t}\\\
e^{-t}-1\end{smallmatrix}\right],t\mapsto e^{-t}-1)$

belongs to $\mathrm{\bf Q}(x_{0}=[1,0,0]^{\intercal})$, i.e. both (1) and
$(u(t),x(t))\in\mathcal{U}\times\mathcal{X}$ hold for all $t\in{}_{\geq 0}$.

###  5.2. Characterizations of IR

In this subsection, it is shown how the results of [2] can be extrapolated to
the context of current Section 5.

Assume that $l>0$ hold, and select any $\eta_{0}\in{}^{l}$ and
$F\in\underline{F}(\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}}))$. Let
$x_{0}\coloneqq T\eta_{0}$. For $k\in\\{1,2\\}$, pick any
$(w_{k},\eta_{k},\varphi_{k})\in\mathrm{\bf Q}_{F}(\eta_{0})$ such that
$\varphi_{1}=\varphi_{2}$ and $w_{1}\neq w_{2}$ hold. Define
$(u_{k},x_{k},y_{k})\coloneqq E(F,\eta_{0};(w_{k},\eta_{k},\varphi_{k}))$.
Then, (i) $y_{1}=y_{2}$ holds, by definition of $E(F,\eta_{0};\cdot)$, and
(ii) $u_{1}\neq u_{2}$ holds, since $u_{1}=u_{2}\Rightarrow x_{1}=x_{2}$ which
would contradict injectivity $E(F,\eta_{0};\cdot)$. This proves that if
$(\eta_{0},\varphi_{1})$ is an IR pair for $\Sigma_{F}$, then $(x_{0},y_{1})$
is an IR pair for $\mathfrak{S}$. Using the same reasoning, one can prove that
the opposite implication is also valid, i.e. if $(x_{0},y_{1})$ is an IR pair
for $\mathfrak{S}$, then $(\eta_{0},\varphi_{1})$ is an IR pair for
$\Sigma_{F}$. Besides, equality $\eta_{1}=\eta_{2}$ is equivalent to
$x_{1}=x_{2}$, by definition of $E(F,\eta_{0};\cdot)$ and from injectivity of
$T$. This allows to conclude that $(\eta_{0},\varphi_{1})$ is an IR pair for
$\Sigma_{F}$ of the $k$-th kind iff $(x_{0},y_{1})$ is an IR pair for
$\mathfrak{S}$ of the same kind.

From this discussion and Th.5.1, one can readily extrapolate the results of
[2] since Lem. 5.2 ensures that $w_{k}$ are causal, piecewise continuous and
exponentially bounded. For the sake of completeness, the conclusions drawn in
this way are now exposed.

######  Theorem 5.3.

Assume that $l>0$and Hyp. 1 hold. Select any friend $F$ of
$\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}})$. Define integers $\rho_{F}$,
$\nu_{F}$ and subspace $\mathcal{N}_{F}\subseteq\mathcal{U}$ as follows:

(22) $\displaystyle\rho_{F}$
$\displaystyle\coloneqq\dim(\mathrm{Ker}\left\\{B_{F}\right\\}\cap\mathrm{Ker}\left\\{D_{F}\right\\}),$
(23) $\displaystyle\mathcal{N}_{F}$ $\displaystyle\coloneqq
B_{F}^{-1}\mathcal{V}(\Sigma_{F})\cap\mathrm{Ker}\left\\{D_{F}\right\\},$ (24)
$\displaystyle\nu_{F}$
$\displaystyle\coloneqq\dim(\mathcal{N}_{F}/\mathrm{Ker}\left\\{\left[\begin{smallmatrix}B_{F}\\\
D_{F}\end{smallmatrix}\right]\right\\})=\dim(\mathcal{N}_{F})-\rho_{F}.$

Then, the following statements are equivalent:

  1. (i)

System $\mathfrak{S}$ is IR;

  2. (ii)

$\nu_{F}>0$ or $\rho_{F}>0$;

  3. (iii)

Transfer matrix $G_{F}(s)\coloneqq
C_{F}(s\mathrm{\mathbf{I}}-A_{F})^{-1}B_{F}+D_{F}$ of $\Sigma_{F}$ is not
left-invertible,

  4. (iv)

System matrix
$P_{F}(s)\coloneqq\left[\begin{smallmatrix}s\mathrm{\mathbf{I}}-A_{F}&-B_{F}\\\
C_{F}&D_{F}\end{smallmatrix}\right]$ of $\Sigma_{F}$ is not left-invertible.

Furthermore, the kind of redundancy of $\mathfrak{S}$ is characterized by
$\rho_{F}$ and $\nu_{F}$, as in the following table:

IR | $\rho_{F}$ | $\nu_{F}$
---|---|---
1st kind | $>0$ | $=0$
2nd kind | $=0$ | $>0$
3rd kind | $>0$ | $>0$

Besides, it holds

$\dim\mathcal{R}(\Sigma_{F})>0\;\Leftrightarrow\;\nu_{F}>0.$

######  Example 5.3 (Ex. 4.3 continued).

For this example, one gets
$\mathcal{R}(\Sigma_{F})=\mathcal{V}(\Sigma_{F})=\mathcal{N}_{F}=\mathrm{Im}\left\\{\left[\begin{smallmatrix}1\\\
0\end{smallmatrix}\right]\right\\}$ and $\rho_{F}=0$. Thus, $\mathfrak{S}$ is
IR of the 2nd kind with degree $(0,1)$.

Let us exemplify discussion conducted in [2, Subsec. 4.1] on $\Sigma_{F}$.
First note that original input and state basis of $\Sigma_{F}$ are already
adapted to $\mathcal{N}_{F}$ and $\mathcal{R}(\Sigma_{F})$, respectively.
Thus, $\Sigma_{F}$ enjoys a cascaded structure without regular feedback. Here,
this cascade actually degenerates into two decoupled subsystems. Indeed, (21)
can be rewritten as follows:

$\dot{\eta}_{a}(t)=-\eta_{a}(t)+w_{a}(t),\quad\dot{y}(t)=-y(t)-w_{b}(t),$

using notations $w\coloneqq[w_{a},w_{b}]^{\intercal}$ and
$\eta\coloneqq[\eta_{a},\eta_{b}]^{\intercal}$. The crucial point is the
following: If $w_{b}$ is uniquely defined by initial condition
$x(0)=[\eta^{\intercal}(0),y(0)]^{\intercal}$ and output $y$, first input
$w_{a}$ does not impact $y$ and, hence, can be arbitrarily selected. For
instance, pick $w_{1}:t\mapsto[1,0]$ and $w_{2}=\mathbf{0}$. Those inputs
produce $\eta_{1}:t\mapsto[1-e^{-t},0]$ and $\eta_{2}=\mathbf{0}$ from zero
initial condition, respectively. By $E$, one derives corresponding triples: It
holds
$(u_{1},x_{1},y_{1})=(t\mapsto[1,0,0,0]^{\intercal},t\mapsto[1-e^{-t},0,0],\mathbf{0})$
and $(u_{2},x_{2},y_{2})=(\mathbf{0},\mathbf{0},\mathbf{0})$, so that both
admissible $u_{1}$ and $u_{2}$ produce admissible state trajectories and
identical outputs.

To conclude, note that the concept of degree of IR defined in [2, Subsec. 4.2]
can be readily generalized in this context.

######  Definition (Degree of IR).

Assume that Hyp. 1 and $l>0$ hold. Pair $(\rho_{F},\nu_{F})$, defined via (22)
and (24), is the _degree of redundancy_ of system $\mathfrak{S}$.

###  5.3. Additional remarks

######  Remark 3 (If $l=0$).

Consider the degenerate case where
$\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}})$ equals $\\{\mathbf{0}\\}$,
i.e. $l=0$. In such a situation, state trajectory must start from the origin
and cannot escape from this point without violating input or state
constraints. System $\mathfrak{S}$ is somehow over-constrained and state-space
of $\Sigma_{F}$ collapses to $\\{\mathbf{0}\\}$. Therefore, set $\mathrm{\bf
U}_{a}(x_{0})$ of admissible inputs for $x_{0}$ is non empty iff
$x_{0}=\mathbf{0}$. In this case, $\mathrm{\bf U}_{a}(x_{0})$ reads
$\mathrm{\bf U}_{\Sigma}\cap\\{u:{}_{\geq
0}\rightarrow\mathcal{U}\cap\mathrm{Ker}\left\\{B\right\\}\\}$. Furthermore,
$\Sigma_{F}$ degenerates into static input-output map $w\mapsto\varphi=Mw$
where $M\coloneqq D\mid(\mathcal{U}\cap\mathrm{Ker}\left\\{B\right\\})$. It
should be clear that $\mathfrak{S}$ is IR iff
$\dim{\mathcal{U}\cap\mathrm{Ker}\left\\{\left[\begin{smallmatrix}B\\\
D\end{smallmatrix}\right]\right\\}}=\dim\mathrm{Ker}\left\\{M\right\\}>0$. In
such a case, every IR pair is of the 1st kind.

######  Remark 4 (Dependency w.r.t. $F$).

From discussion above, one concludes that $T\mathcal{V}(\Sigma_{F})$ is the
weakly unobservable subspace of quadruple
$(A,B_{\mathcal{U}},\left[\begin{smallmatrix}C\\\
X^{\perp}\end{smallmatrix}\right],\left[\begin{smallmatrix}D_{\mathcal{U}}\\\
\mathbf{0}\end{smallmatrix}\right])$, where $X^{\perp}$ satisfies
$\mathrm{Ker}\left\\{X^{\perp}\right\\}=\mathcal{X}$. It follows that
$\mathcal{V}(\Sigma_{F})$ is independent of
$F\in\mathcal{V}^{*}(\mathcal{X})$. Observe that $B_{F}$, $D_{F}$ and, in
turn, $\rho_{F}$ do not depend on $F$ either.555Subscript $F$ of $B_{F}$ and
$D_{F}$ aims distinguishing those matrices from $B$ and $D$. As a result,
$\mathcal{N}_{F}$ and $\nu_{F}$ too are independent on the selection of $F$.
In the same vein, note that friend
$F\in\underline{F}(\mathcal{V}^{*}(\mathcal{X}))$ can be arbitrarily selected
in Th. 5.3. This proves that left-invertibility of $G_{F}(s)$ and $P_{F}(s)$
are independent of $F$.

##  6\. Conclusions

This paper investigates how input and state constraints affect IR. It is
perhaps the most natural extension of results obtained in [2], and the most
desirable from the control application point of view.

The general case where $\mathcal{U}$ and $\mathcal{X}$ are arbitrary is first
investigated. If those sets are closed, then Cor. 2 proves that admissible
pairs $(x_{0},y)$ losing IR due to the constraints are necessarily associated
with trajectories that lie on the boundary of $\mathcal{U}$ and $\mathcal{X}$.
It is also shown that constraints might (i) change the kind of IR of the
system (see Ex. 4.3) and (ii) give rise to IR pairs of different kinds
coexisting for the same system (see Ex. 4.4). Lem. 4.2 also proves that such a
phenomena can be observed only if $\Sigma$ is IR of the 3rd kind. Otherwise,
the kind of redundancy is preserved, provided that redundancy itself is not
destroyed by the constraints.

Whenever both input and state spaces are linear, admissible trajectories of
$\mathfrak{S}$ are uniquely associated to that of an _unconstrained_ auxiliary
system $\Sigma_{F}$, by way of some bijective mapping. This key feature allows
to extrapolate results derived in [2].

In the general case, condition for admissible $(x_{0},y)$ to be IR is only
sufficient (see Th. 4.1). Refinement of this condition as well as derivation
of a necessary counterpart might improve understanding of how geometry of
constraints impacts IR.

Given an IR pair $(x_{0},y)$, so that there exist $u_{1}\neq u_{2}$ satisfying
$(u_{1},y),(u_{2},y)\in\mathrm{\bf W}(x_{0})$. In the linear case, it is
implicitly proved that $u_{1,2}$ can be selected as “different” as desired. In
practice this feature is of major interest since it is expected that
“insufficiently different” inputs might be useless for redundancy to be
exploited. Since this property is destroyed in the non linear case, a
criterion ensuring that $(x_{0},y)$ is “IR enough” would be desirable.

Another possible extension of this work is the refinement of results of Sect.
4 in the case where $\mathcal{U}$ and $\mathcal{X}$ enjoy specific properties
like e.g. convexity. In this case, it is expected that tighter approximation
of $\mathrm{\bf S}$ can be derived.

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##  Appendix A Background on geometric control theory

Let us summarized some essential aspects of geometric control theory for
system $\Sigma$ governed by (1).

###  A.1. Controlled invariance

A subspace $\mathcal{W}$ is said to be an $(A,B)$-_controlled invariant_
subspace (or simply _controlled invariant_ subspace) if, for any initial state
$x_{0}\in\mathcal{W}$, there exists an input function $u$ such that the state
trajectory generated by the system remains identically in $\mathcal{W}$.
Subspace $\mathcal{W}$ enjoys this property iff there exists a matrix $F$ such
that

(25) $(A+BF)\mathcal{W}\subseteq\mathcal{W}$

holds. Matrix $F$ is called a _friend_ of $\mathcal{W}$ and the set of such
matrices is denoted by
$\underline{F}(\mathcal{W})\coloneqq\\{F:\eqref{eq:charFriend}\\}$.

Consider any $x_{0}\in\mathcal{W}$, $F\in\underline{F}(\mathcal{W})$ and $L$
such that $\mathrm{Im}\left\\{L\right\\}\in B^{-1}\mathcal{W}$. Then, state
trajectory remains in $\mathcal{W}$ iff there exists a function $w$ such that
corresponding input $u$ reads

(26) $u:t\mapsto\kappa_{w}(t,x(t)),$

where time-varying feedback $\kappa_{w}$ is defined as follows

(27) $\kappa_{w}:(t,x)\mapsto Fx+Lw(t),$

and is parametrized by $w$ [28, Th. 4.3]. In particular, selecting
$w=\mathbf{0}$ proves that simple feedback $u(t)=Fx(t)$ makes $\mathcal{W}$
controlled invariant. This input function admits implicit form
$u=F\mathrm{H}_{x}[x_{0};u]$ from which closed-form expression $u:t\mapsto
Fe^{t(A+BF)}x_{0}$ can be derived by means of (1a).

###  A.2. Weak unobservability

Following [28, p.159], a point $x_{0}\in{}^{n}$ is called _weakly
unobservable_ if there exists an input function $u$ such that corresponding
output is identically null. The set of all weakly unobservable point is
denoted by $\mathcal{V}(\Sigma)$, i.e.

$\mathcal{V}(\Sigma)\coloneqq\\{x_{0}\in{}^{n}\mid\exists u:{}_{\geq
0}\rightarrow{}^{m}:\mathrm{H}[x_{0};u]=\mathbf{0}\\}.$

It can be proved that $\mathcal{V}(\Sigma)$ is a linear subspace of n.
Referring to the following inclusion parametrized by subspace
$\mathcal{W}\subseteq{}^{n}$,

(28) $\mathcal{W}\subseteq\mathrm{Ker}\left\\{C+DF\right\\},$

set $\mathcal{V}(\Sigma)$ is the largest subspace $\mathcal{W}$ for which
there exists $F$ such that (25) and (28) hold [28, Th. 7.10]. Such a matrix
$F$ is called a friend of $\mathcal{V}(\Sigma)$ and the set of such friends is
denoted by $\underline{F}(\mathcal{V}(\Sigma))$.666As in [28], two definitions
of $\underline{F}(\mathcal{W})$ coexist depending if $\mathcal{W}$ is
$\mathcal{V}(\Sigma)$ or any other subspace. The context should clarify which
one is referred to, though.

Consider any $x_{0}\in\mathcal{V}(\Sigma)$,
$F\in\underline{F}(\mathcal{V}(\Sigma))$ and $L$ satisfying

(29) $\mathrm{Im}\left\\{L\right\\}\in
B^{-1}\mathcal{V}(\Sigma)\cap\mathrm{Ker}\left\\{D\right\\}.$

Then, input $u:{}_{\geq 0}\rightarrow{}^{m}$ satisfies
$\mathrm{H}[x_{0};u]=\mathbf{0}$ iff it reads (26) where $\kappa_{w}$ is given
by (27) and is parametrized by some function $w$ [28, Th. 7.11]. Selecting
$w=\mathbf{0}$ yields $u(t)=Fx(t)$ so that output $y$ is governed by
$\dot{x}=(A+BF)x,\;y=(C+DF)x$ (see (1)). From previous characterization of
$\mathcal{V}(\Sigma)$ via (25) and (28), this input not only makes output
identically zero
($\mathcal{V}(\Sigma)\subseteq\mathrm{Ker}\left\\{C+DF\right\\}$) but also
enforces state trajectory to remain in $\mathcal{V}(\Sigma)$
($(A+BF)\mathcal{V}(\Sigma)\subseteq\mathcal{V}(\Sigma)$). This explains
alternative naming of $\mathcal{V}(\Sigma)$ as the largest output nulling
controlled invariant subspace [31].

Note that if $D$ equals $\mathbf{0}$, then $\mathcal{V}(\Sigma)$ reduces to
$\mathcal{V}^{*}(\mathrm{Ker}\left\\{C\right\\},A,B)$, the largest
$(A,B)$-controlled invariant subspace contained in the kernel of $C$.

###  A.3. Controllability and weak unobservability

From [28, p.163], a point $x_{0}\in{}^{n}$ is called _controllable weakly
unobservable_ if there exist $T>0$ and $u:[0,T]\rightarrow{}^{m}$ such that
the following conditions hold with $x_{f}=\mathbf{0}$:

(30a) $\displaystyle\mathrm{H}[x_{0};u](t)$
$\displaystyle=\mathbf{0},\;\forall t\in[0,T],$ (30b)
$\displaystyle\mathrm{H}_{x}[x_{0};u](T)$ $\displaystyle=x_{f}.$

The set of all controllable weakly unobservable points is denoted by
$\mathcal{R}(\Sigma)$. As suggested by its definition, $\mathcal{R}(\Sigma)$
is included in $\mathcal{V}(\Sigma)$. It can also be proved that
$\mathcal{R}(\Sigma)$ is a linear subspace of n which satisfies (25) and (28)
with $\mathcal{W}=\mathcal{R}(\Sigma)$ and any $F$ in
$\underline{F}(\mathcal{V}(\Sigma))$.

###  A.4. A matrix view point

Pick any $F\in\underline{F}(\mathcal{V}(\Sigma))$ and $L$ satisfying (29).
Choose a basis of state space n adapted to $\mathcal{R}(\Sigma)$ and
$\mathcal{V}(\Sigma)$, i.e. define invertible matrices $T=[T_{a},T_{b},T_{c}]$
satisfying $\mathrm{Im}\left\\{T_{a}\right\\}=\mathcal{R}(\Sigma)$ and
$\mathrm{Im}\left\\{[T_{a},T_{b}]\right\\}=\mathcal{V}(\Sigma)$. Apply input
$u$ given by (26) with (27) to $\Sigma$. Resulting closed-loop system is such
that new input $w$ drives output $y$ via quadruple $(A+BF,BL,C+DF,DL)$. From
the above discussion, dynamics in the new basis reads:

(31a) $\displaystyle\begin{bmatrix}\dot{\phi}_{a}\\\ \dot{\phi}_{b}\\\
\dot{\phi}_{c}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}A_{11}&A_{12}&A_{13}\\\
\mathbf{0}&A_{22}&A_{23}\\\
\mathbf{0}&\mathbf{0}&A_{33}\end{bmatrix}\begin{bmatrix}\phi_{a}\\\
\phi_{b}\\\ \phi_{c}\end{bmatrix}+\begin{bmatrix}B_{1}\\\ \mathbf{0}\\\
\mathbf{0}\end{bmatrix}w,$ (31b) $\displaystyle y$
$\displaystyle=\begin{bmatrix}\mathbf{0}&\mathbf{0}&C_{3}\end{bmatrix}\begin{bmatrix}\phi_{a}\\\
\phi_{b}\\\ \phi_{c}\end{bmatrix}.$

Here, $\phi_{i}$ denotes coordinates of $x$ in
$\mathrm{Im}\left\\{T_{i}\right\\}$, so that
$[\phi_{a}^{\intercal},\phi_{b}^{\intercal},\phi_{c}^{\intercal}]^{\intercal}=T^{-1}x$
is the new state vector. One recovers that for any $w$ and
$x_{0}\in\mathcal{V}(\Sigma)$ (so that $\phi_{c}(0)=\mathbf{0}$), state
trajectory remains identically in $\mathcal{V}(\Sigma)$ (since substate
$\phi_{c}$ equals $\mathbf{0}$), yielding $y=\mathbf{0}$.

###  A.5. Facts about controllability

From (30), $\mathcal{R}(\Sigma)$ gathers initial states which are output-
nulling “controllable-to-the-origin”. It actually coincides with the set of
output-nulling reachable states (reachability being “controllability-from-the-
origin”) [28, p.170], i.e. $x_{f}$ belongs to $\mathcal{R}(\Sigma)$ iff there
exist $T>0$ and $u:[0,T]\rightarrow{}^{m}$ such that (30a) and (30b) hold with
$x_{0}=\mathbf{0}$. Among other things, this proves that pair $(A_{11},B_{1})$
is controllable. To sees this, let $x_{0}=\mathbf{0}$, pick any $\phi_{a,f}$
and define $x_{f}\coloneqq T_{a}\phi_{1,f}\in\mathcal{R}(\Sigma)$. Since (30a)
holds for some $T$ and $u$, so that the latter reads (26) with (27) and
resulting dynamics is governed by (31). In the new coordinates, (30b) reads
$\phi_{a}(T)=\phi_{a,f}$ which proves controllability of $(A_{11},B_{1})$
since $T$ and $\phi_{a,f}$ are arbitrary.

Standard result on controllability (see e.g. [32, Cor. 2.14]) ensures that
substate $\phi_{a}$ can be transferred from any $\phi_{a,0}$ to any
$\phi_{a,f}$ in arbitrary finite time $T>0$ by way of input defined in (32),
i.e. $\phi_{a}(T)=\phi_{a,f}$ when $\phi_{a}(0)=\phi_{a,0}$ and $w$ reads as
follows

(32) $w:t\mapsto
B_{1}^{\intercal}e^{A_{11}^{\intercal}(T-t)}W_{r}^{-1}(T)(\phi_{a,f}-e^{A_{11}T}\phi_{a,0}),$

where
$W_{r}:T\mapsto\textstyle\int_{0}^{T}e^{(T-\tau)A_{11}}B_{1}B_{1}^{\intercal}e^{(T-\tau)A_{11}^{\intercal}}\text{d}\tau$
is the reachability Gramian from $0$ to $T$. Translated in the original
coordinates, this discussion ensures that $x$ can be transferred from any
$x_{0}\in\mathcal{R}(\Sigma)$ to any $x_{f}\in\mathcal{R}(\Sigma)$ in
arbitrary finite time $T>0$ and in such a way that both
$x(t)\in\mathcal{R}(\Sigma)$ and $y(t)=\mathbf{0}$ hold for all $t\in[0,T]$.
Input $u:t\mapsto FT_{a}\phi_{a}(t)+Lw(t)$ achieves such a result. Here,
$F\in\underline{F}(\mathcal{V}(\Sigma))$ holds, $\phi_{a}$ satisfies
$\dot{\phi}_{a}=A_{11}\phi_{a}+B_{1}w$ and $w$ is given by (32) where
$\phi_{a,f}$ and $\phi_{a,0}$ are such $x_{f}=T_{a}\phi_{1,f}$ and
$x_{0}=T_{a}\phi_{a,0}$ hold. From this expression, it comes out that $u$ is
continuous on $[0,T]$. Indeed, $\phi_{a}$ inherits continuity from $w$ and,
together, they prove that $u$ enjoys this property as well.

###  A.6. Linear input and state constraints

Consider now the case where following constraints is imposed on the system for
all $t\in{}_{\geq 0}$ :

$(u(t),x(t))\in\mathcal{U}\times\mathcal{X},$

where $\mathcal{U}\subseteq{}^{m}$ and $\mathcal{X}\subseteq{}^{n}$ are given
_linear_ sets.

Let $R$ denote the insertion of $\mathcal{U}$ in m. Define
$B_{\mathcal{U}}\coloneqq B\mid\mathcal{U}=BR,$ the domain restriction of $B$
to $\mathcal{U}$. Pick any $x_{0}\in\mathcal{X}_{c}$ and any $(u,x,y)$ in
$\mathrm{\bf Q}(x_{0})$. In this case, state trajectory $x$ lies in the
largest set contained in $\mathcal{X}$ which can be made invariant by way of
some input $u\in\mathrm{\bf U}$. This set inherits linearity property of
$\mathcal{X}$ and corresponds to
$\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}})$ (or
$\mathcal{V}^{*}(\mathcal{X})$ for short), the largest
$(A,B_{\mathcal{U}})$-controlled invariant subspace contained in
$\mathcal{X}$. The set of input $u$ producing such a dynamics can be
parametrized by way of any matrix $F$ in
$\underline{F}(\mathcal{V}^{*}(\mathcal{X}))$, the set of friend of
$\mathcal{V}^{*}(\mathcal{X})$. Indeed, in this context, [28, Th. 4.3 and Th.
4.5] apply and ensures that $u$ produces $x$ in $\mathrm{\bf X}$ iff $x(0)$
belongs to $\mathcal{V}^{*}(\mathcal{X})$ and $u$ equals $R(Fx+Lw)$ for some
signal $w$ of appropriate dimension. Here, $L$ is an injective linear map such
that

(33)
$\mathrm{Im}\left\\{L\right\\}=B_{\mathcal{U}}^{-1}\mathcal{V}^{*}(\mathcal{X},A,B_{\mathcal{U}}).$

Using this expression of $u$ transforms quadruple $\Sigma$ into
$(A+B_{\mathcal{U}}F,B_{\mathcal{U}}L,C+D_{\mathcal{U}}F,D_{\mathcal{U}}L)$
where $D_{\mathcal{U}}\coloneqq D\mid\mathcal{U}=DR$ is the domain restriction
of $D$ to $\mathcal{U}$. Let $\Sigma_{F}$ be the restriction of dynamics
induced by this new quadruple to $\mathcal{V}^{*}(\mathcal{X})$, i.e.
quadruple $(A_{F},B_{F},C_{F},D_{F})$ of $\Sigma_{F}$ satisfies
$A_{F}\coloneqq(A+B_{\mathcal{U}}F)\mid\mathcal{V}^{*}(\mathcal{X})$,
$B_{F}\coloneqq\mathcal{V}^{*}(\mathcal{X})\mid(B_{\mathcal{U}}L)$,
$C_{F}\coloneqq(C+D_{\mathcal{U}}F)\mid\mathcal{V}^{*}(\mathcal{X})$ and
$D_{F}\coloneqq D_{\mathcal{U}}L$.

######  Example A.1 (When $\mathcal{V}^{*}(\mathcal{X})$ is a proper subet of
$\mathcal{X}$).

Let us clarify that $\mathcal{V}^{*}(\mathcal{X})$ can be strictly included in
$\mathcal{X}$, so that $\dim{\mathcal{V}^{*}(\mathcal{X})}<\dim{\mathcal{X}}$.
As an example, consider following dynamical system

$\displaystyle\dot{x}=\begin{bmatrix}0&-1&0\\\ 1&0&0\\\
0&0&1\end{bmatrix}x+\begin{bmatrix}0\\\ 0\\\ 1\end{bmatrix}u$

associated with constraints $\mathcal{U}=\real$ and
$\mathcal{X}=\\{0\\}\oplus{}^{2}$, translating the fact that first state is
enforced to be zero. In this case, input cannot modify first two states, whose
dynamics are coupled. As a result, second state must be identically null and
$\mathcal{V}^{*}(\mathcal{X})$ reduces to $\\{0\\}^{2}\oplus\real$.

Let $T$ denote insertion of $\mathcal{V}^{*}(\mathcal{X})$ in n. We have just
proved that (i) $\mathrm{\bf Q}(x_{0})$ is non empty iff $x_{0}$ belongs to
$\mathcal{V}^{*}(\mathcal{X})$, i.e. $x_{0}=T\eta_{0}$ for some $\eta_{0}$,
and (ii) $(u,x,y)$ belongs to $\mathrm{\bf Q}(T\eta_{0})$ iff there exists
$(w,\eta,\varphi)\in\mathrm{\bf Q}_{F}(\eta_{0})$. Here, set $\mathrm{\bf
Q}_{F}(\eta_{0})$ gathers all input $w$, state $\eta$ and output $\varphi$
trajectories originating from initial condition $\eta(0)=\eta_{0}$ and
compatible with system $\Sigma_{F}$. More precisely, if $l>0$, then the
following mapping is surjective for all
$\underline{F}(\mathcal{V}^{*}(\mathcal{X}))$ and $\eta_{0}$:

$\displaystyle E(F,\eta_{0};\cdot):\mathrm{\bf Q}_{F}(\eta_{0})$
$\displaystyle\rightarrow\mathrm{\bf Q}(T\eta_{0})$
$\displaystyle(w,\eta,\varphi)$
$\displaystyle\mapsto(RLw+RFT\eta,T\eta,\varphi)$

######  Remark 5 (About $\mathcal{X}_{c}$ and $\mathrm{\bf U}_{a}(x_{0})$).

Note that $\mathcal{X}_{c}$ is nothing but $\mathcal{V}^{*}(\mathcal{X})$.
Furthermore, if $l>0$ and
$x_{0}\in\mathcal{X}_{c}\coloneqq\\{x_{0}\in{}^{n}:|\mathrm{\bf W}(x_{0})|\geq
1\\}\subseteq\mathcal{X}$, then $\mathrm{\bf
U}_{a}(x_{0})\coloneqq\\{u\in\mathrm{\bf U}\mid\exists y:(u,y)\in\mathrm{\bf
W}(x_{0})\\}$ reads

$\\{RLw+RFT\eta\mid\\\ RLw\in\mathrm{\bf
U}_{\Sigma},\;\dot{\eta}=A_{F}\eta+B_{F}w,\;T\eta(0)=x_{0}\\},$

for any $F\in\underline{F}(\mathcal{V}^{*}(\mathcal{X}))$.

##  Appendix B Incremental analysis

This appendix aims analyzing input-to-output mapping associated with
$\mathfrak{S}$ by adopting an incremental view point: Inputs pair
$(u_{1},u_{2})$ are described via $(u_{1},\tilde{u})$ where
$\tilde{u}=u_{2}-u_{1}$ so that $u_{2}=u_{1}+\tilde{u}$ can be expressed
_relatively_ to $u_{1}$.

Instrumental are the following well-known relationships which hold for all
$x_{0},\tilde{x}_{0}\in{}^{n}$, $u,\tilde{u}\in\mathrm{\bf U}_{\Sigma}$ and
$\alpha\in\real$, due to linearity of (1):

(34a)
$\displaystyle\mathrm{H}_{x}[x_{0};u]+\mathrm{H}_{x}[\tilde{x}_{0};\tilde{u}]$
$\displaystyle=\mathrm{H}_{x}[x_{0}+\tilde{x}_{0};u+\tilde{u}],$ (34b)
$\displaystyle\mathrm{H}_{x}[\alpha x_{0};\alpha u]$
$\displaystyle=\alpha\mathrm{H}_{x}[x_{0};u].$

The same two properties also hold for $\mathrm{H}$ in place of
$\mathrm{H}_{x}$.

###  B.1. Incremental characterization of IR

Given a pair $(x_{0},y)\in\mathcal{X}\times\mathrm{\bf Y}$.  Recall that set
$\mathfrak{H}^{-1}[x_{0};y]$, defined in (15), gathers all admissible input
$u$ giving rise to output $y$ when state is initialized at $x_{0}$. By
definition, $\mathfrak{H}^{-1}[x_{0};y]$ is non empty iff
$(x_{0},y)\in\mathrm{\bf A}$ holds.

Pair $(x_{0},y)$ is IR iff $|\mathfrak{H}^{-1}[x_{0};y]|\geq 2$ holds. In
order to derive conditions under which this inequality holds, let us describe
$\mathfrak{H}^{-1}[x_{0};y]$ _relatively_ to some element $u$ of this set.
Specifically, given any $u\in\mathfrak{H}^{-1}[x_{0};y]$, define
$\widetilde{\mathrm{\bf U}}(x_{0},u)$ as follows:

(35) $\widetilde{\mathrm{\bf
U}}(x_{0},u)\coloneqq\mathfrak{H}^{-1}[x_{0};y]-u.$

Note that $\widetilde{\mathrm{\bf U}}$ is parametrized by
$(x_{0},u)\in\mathcal{X}\times\mathrm{\bf U}$, but not by $y$ which can be
uniquely777Recall that signals in $\mathrm{\bf U}\subseteq\mathrm{\bf
U}_{\Sigma}$ are piecewise continuous. recovered from $x_{0}$ and $u$. Putting
(35) in the other way around makes it easier to grasp: Equality
$\mathfrak{H}^{-1}[x_{0};y]=\widetilde{\mathrm{\bf U}}(x_{0},u)+u$ simply says
that $\widetilde{\mathrm{\bf U}}(x_{0},u)$ gathers signals $\tilde{u}:{}_{\geq
0}\rightarrow{}^{m}$ such that $u+\tilde{u}$ belongs to
$\mathfrak{H}^{-1}[x_{0};y]$ or, equivalently, satisfies
$(u+\tilde{u},y)\in\mathrm{\bf W}(x_{0})$. In this case, both inputs $u$ and
$u+\tilde{u}$ comply with the constraints and produce identical output $y$ for
the same initial state $x_{0}$.

Observing that $\mathbf{0}$ trivially belongs to $\widetilde{\mathrm{\bf
U}}(x_{0},u)$, this discussion leads to a new characterization of IR.

######  Proposition B.1.

Pair $(x_{0},y)\in\mathrm{\bf A}$ is IR iff the following condition holds

(36) $\forall u\in\mathfrak{H}^{-1}[x_{0};y]:|\widetilde{\mathrm{\bf
U}}(x_{0},u)|\geq 2.$

Further, set $\widetilde{\mathrm{\bf U}}(x_{0},u)$ reads

(37) $\widetilde{\mathrm{\bf U}}(x_{0},u)=\\{\tilde{u}:{}_{\geq
0}\rightarrow{}^{m}:\\\ (\tilde{u},\mathbf{0})\in\mathrm{\bf
W}(x_{0})-(u,\mathrm{H}[x_{0};u])\\}.$

###### Proof.

Pick any $u$ in non empty set $\mathfrak{H}^{-1}[x_{0};y]$ parametrized by
$(x_{0},y)\in\mathrm{\bf A}$. Equality between $|\mathfrak{H}^{-1}[x_{0};y]|$
and $|\widetilde{\mathrm{\bf U}}(x_{0},u)|$ follows from (35), so that IR of
$(x_{0},y)$ is equivalent to $|\widetilde{\mathrm{\bf U}}(x_{0},u)|\geq 2$.
Input $u$ being arbitrary, this proves (36).

To prove (37), observe that $(u+\tilde{u},y)=(u,y)+(\tilde{u},\mathbf{0})$
belongs to $\mathrm{\bf W}(x_{0})$ for all $\tilde{u}\in\widetilde{\mathrm{\bf
U}}(x_{0},u)$, which is equivalent to saying that $\tilde{u}$ satisfies
$(\tilde{u},\mathbf{0})\in\mathrm{\bf W}(x_{0})-(u,y)$. Since
$y=\mathrm{H}[x_{0};u]$, this proves (37). ∎

Consider the following technical result.

######  Lemma B.2.

Given arbitrary $x_{0,1},x_{0,2}\in\mathcal{X}_{c}$ and $(u,x,y)\in\mathrm{\bf
Q}(x_{0,2})$, set $\mathrm{\bf Q}(x_{0,1})-(u,x,y)$ reads

(38) $\\{(\tilde{u},\tilde{x},\tilde{y})\in\mathrm{\bf U}\times\mathrm{\bf
X}\times\mathrm{\bf Y}-(u,x,y):\\\
\mathrm{H}_{x}[x_{0,1}-x_{0,2};\tilde{u}]=\tilde{x},\mathrm{H}[x_{0,1}-x_{0,2};\tilde{u}]=\tilde{y}\\}.$

###### Proof.

Let us first demonstrate that $\mathrm{\bf Q}(x_{0,1})-(u,x,y)$ is included in
the set defined by (38). Pick arbitrary $x_{0,1},x_{0,2}\in\mathcal{X}_{c}$,
$(u,x,y)\in\mathrm{\bf Q}(x_{0,2})$ and
$(\tilde{u},\tilde{x},\tilde{y})\in\mathrm{\bf Q}(x_{0,1})-(u,x,y)$, i.e.
$(\tilde{u},\tilde{x},\tilde{y})=(\bar{u},\bar{x},\bar{y})-(u,x,y)$ for some
$(\bar{u},\bar{x},\bar{y})\in\mathrm{\bf Q}(x_{0,1})$. Observe that

$\displaystyle\mathrm{H}_{x}[x_{0,1}-x_{0,2};\tilde{u}]$
$\displaystyle=\mathrm{H}_{x}[x_{0,1}-x_{0,2};\bar{u}-u]$
$\displaystyle{\color[rgb]{0,0,1}\overset{\eqref{eq:lin1}}{=}}\mathrm{H}_{x}[x_{0,1};\bar{u}]+\mathrm{H}_{x}[-x_{0,2};-u]$
$\displaystyle{\color[rgb]{0,0,1}\overset{\eqref{eq:lin2}}{=}}\mathrm{H}_{x}[x_{0,1};\bar{u}]-\mathrm{H}_{x}[x_{0,2};u]$
$\displaystyle=\bar{x}-x=\tilde{x}$

holds and, similarly, $\mathrm{H}[x_{0,1}-x_{0,2};\tilde{u}]$ equals
$\tilde{y}$. Then, note that $\mathrm{\bf
Q}(x_{0,1})-(u,x,y)\subseteq\mathrm{\bf U}\times\mathrm{\bf
X}\times\mathrm{\bf Y}-(u,x,y)$ follows from (5).

Conversely, consider arbitrary $x_{0,1},x_{0,2}\in\mathcal{X}_{c}$ and triple
$(u,x,y)\in\mathrm{\bf Q}(x_{0,2})$ and pick any
$(\tilde{u},\tilde{x},\tilde{y})$ in the set defined in (38). Then, let us
prove that $(\tilde{u},\tilde{x},\tilde{y})$ also belongs to $\mathrm{\bf
Q}(x_{0,1})-(u,x,y)$. Define
$(\bar{u},\bar{x},\bar{y})\coloneqq(\tilde{u},\tilde{x},\tilde{y})+(u,x,y)$ so
that $(\bar{u},\bar{x},\bar{y})\in\mathrm{\bf U}\times\mathrm{\bf
X}\times\mathrm{\bf Y}$ holds. Then, observe that
$\mathrm{H}_{x}[x_{0,1};\bar{u}]=\mathrm{H}_{x}[x_{0,2};u]+\mathrm{H}_{x}[x_{0,1}-x_{0,2};\tilde{u}]=x+\tilde{x}=\bar{x}$
and, in an analogous way, $\mathrm{H}[x_{0,1};\bar{u}]=\bar{y}$. This proves
that $(\bar{u},\bar{x},\bar{y})\in\mathrm{\bf Q}(x_{0,1})$ so that
$(\tilde{u},\tilde{x},\tilde{y})\in\mathrm{\bf Q}(x_{0,1})-(u,x,y)$. ∎

Particularizing this result for $x_{0}=x_{0,1}=x_{0,2}$ allows to conclude
that $\mathrm{\bf W}(x_{0})-(u,y)$ reads

$\\{(\tilde{u},\tilde{y})\in\mathrm{\bf U}\times\mathrm{\bf Y}-(u,y):\\\
\mathrm{H}_{x}[\mathbf{0};\tilde{u}]\in\mathrm{\bf
X}-\mathrm{H}_{x}[x_{0};{u}],\;\mathrm{H}[\mathbf{0};\tilde{u}]=\tilde{y}\\},$

for any $(u,y)\in\mathrm{\bf W}(x_{0})$. This leads to a more explicit
expression of $\widetilde{\mathrm{\bf U}}(x_{0},u)$ than (37).

######  Corollary 3.

Given any $x_{0}\in\mathcal{X}_{c}$ and $u\in\mathrm{\bf U}_{a}(x_{0})$. Set
$\widetilde{\mathrm{\bf U}}(x_{0},u)$ reads

$\\{\tilde{u}\in\mathrm{\bf
U}-u:\mathrm{H}_{x}[\mathbf{0};\tilde{u}]\in\mathrm{\bf
X}-\mathrm{H}_{x}[x_{0};{u}],\;\mathrm{H}[\mathbf{0};\tilde{u}]=\mathbf{0}\\}.$

Let us summarize results of this section. Given $x_{0}\in\mathcal{X}_{c}$ and
$(u,x,y)\in\mathrm{\bf Q}(x_{0})$, _incremental_ input $u+\tilde{u}$ is
admissible for $x_{0}$ and leads to the same output $y$ iff
$\tilde{u}\in\widetilde{\mathrm{\bf U}}(x_{0},u)$. From Cor. 3, this means
that (i) $\tilde{u}\in\mathrm{\bf U}_{\Sigma}$, (ii) both
$\tilde{u}(t)\in\mathcal{U}-u(t)$ and
$\tilde{x}(t)\coloneqq\mathrm{H}_{x}[\mathbf{0};\tilde{u}](t)\in\mathcal{X}-x(t)$
hold for all $t\in{}_{\geq 0}$ and (iii)
$\tilde{y}\coloneqq\mathrm{H}[\mathbf{0};\tilde{u}]=\mathbf{0}$. An essential
point here is that $\tilde{x}$ and $\tilde{y}$ derive from $\tilde{u}$ and
_zero initial condition_ , i.e. $(\tilde{u},\tilde{x},\tilde{y})\in\mathrm{\bf
Q}_{\Sigma}(\mathbf{0})$.

######  Remark 6.

Just like $v_{1},v_{2}\in\mathcal{A}$ does not imply
$v_{1}-v_{2}\in\mathcal{A}$ for arbitrary (non linear) set $\mathcal{A}$, let
us stress that $\tilde{u}\in\widetilde{\mathrm{\bf U}}(x_{0},u)$ does not
imply that $\tilde{u}$ is an admissible input for $x_{0}$ in general.

###  B.2. Incremental characterization of the proposed taxonomy

By making use of set $\widetilde{\mathrm{\bf U}}(x_{0},u)$, next lemma offers
a new characterization of the taxonomy.

######  Lemma B.3.

Pair $(x_{0},y)\in\mathrm{\bf S}$ is IR of

  * •

the _1st kind_ iff it holds

(39) $\forall
u\in\mathfrak{H}^{-1}[x_{0};y],\;\forall\tilde{u}\in\widetilde{\mathrm{\bf
U}}(x_{0},u)\setminus\\{\mathbf{0}\\}:\\\
\mathrm{H}_{x}[\mathbf{0};\tilde{u}]=\mathbf{0}$

  * •

the _2nd kind_ iff it holds

(40) $\forall
u\in\mathfrak{H}^{-1}[x_{0};y],\;\forall\tilde{u}\in\widetilde{\mathrm{\bf
U}}(x_{0},u)\setminus\\{\mathbf{0}\\}:\\\
\mathrm{H}_{x}[\mathbf{0};\tilde{u}]\neq\mathbf{0}$

  * •

the _3rd kind_ iff neither (39) nor (40) is valid.

###### Proof.

By definition, $(x_{0},y)$ is IR of the 1st kind iff $u\neq u_{2}$ and
$y=y_{2}$ imply $x=x_{2}$ for all $(u,x,y),(u_{2},x_{2},y_{2})\in\mathrm{\bf
Q}(x_{0})$. Defining
$(\tilde{u},\tilde{x},\tilde{y})\coloneqq(u_{2},x_{2},y_{2})-(u,x,y)$, this is
equivalent to saying that $\tilde{u}\neq\mathbf{0}$ and $\tilde{y}=\mathbf{0}$
imply $\tilde{x}=\mathbf{0}$ for all $(u,x,y)\in\mathrm{\bf Q}(x_{0})$ and for
all $(\tilde{u},\tilde{x},\tilde{y})\in\mathrm{\bf Q}(x_{0})-(u,x,y)$.
Dropping state-trajectories and observing that
$(\tilde{u},\tilde{x},\tilde{y})$ are associated with zero initial state (see
Lem. B.2), this is equivalent to next condition

(41) $\forall(u,y)\in\mathrm{\bf
W}(x_{0}),\;\forall(\tilde{u},\tilde{y})\in\mathrm{\bf W}(x_{0})-(u,y):\\\
\left.\begin{array}[]{r}\tilde{u}\neq\mathbf{0}\\\
\tilde{y}=\mathbf{0}\end{array}\right\\}\Rightarrow\mathrm{H}_{x}[\mathbf{0};\tilde{u}]=\mathbf{0}.$

Eq. (37) and the fact that $(u,y)\in\mathrm{\bf W}(x_{0})$ is equivalent to
$u\in\mathfrak{H}^{-1}[x_{0};y]$ prove that (41) is equivalent to (39). The
fact that (40) characterizes IR of the 2nd kind can be proved in a similar
way. IR of the third refers to IR which is neither of the first nor of the 2nd
kind. ∎

###  B.3. Proof of Th. 4.1

Figure 2. Graphical illustration of the proof of Th. 4.1 for $m=2$ and
$\rho=0$. Components of $u$ are denoted by $u_{a}$ and $u_{b}$.

Given $x_{0}\in\mathcal{X}_{c}$ and $(u,x,y)\in\mathrm{\bf Q}(x_{0})$. By
Prop. B.1, $(x_{0},y)$ is IR if $|\widetilde{\mathrm{\bf U}}(x_{0},u)|\geq 2$.
From Cor. 3, recall that $\widetilde{\mathrm{\bf U}}(x_{0},u)$ gathers signals
$\tilde{u}\in\mathrm{\bf U}-u$ satisfying
$\mathrm{H}[\mathbf{0};\tilde{u}]=\mathbf{0}$ and
$\mathrm{H}_{x}[\mathbf{0};\tilde{u}]\in\mathrm{\bf X}-x$. Due to the fact
that $\mathbf{0}\in\widetilde{\mathrm{\bf U}}(x_{0},u)$ trivially holds, let
us prove IR by constructing any non zero measure signal in
$\widetilde{\mathrm{\bf U}}(x_{0},u)$.

Fig. 2 is a sketch the reader can refer to in order to ease figuring out
forthcoming technical developments.

Define closed hyper-ball $B_{k}(r)$ as follows:

$B_{k}(r)\coloneqq\\{\eta\in{}^{k}:\|\eta\|\leq r\\}.$

Eq. (17a) (resp. (17b)) is equivalent to the existence of $r_{u}(t)>0$ (resp.
$r_{x}(t)>0$) such that $u(t)+B_{m}(r_{u}(t))\subseteq\mathcal{U}$ (resp.
$x(t)+B_{n}(r_{x}(t))\subseteq\mathcal{X}$). As a preliminary step, let us
prove that $t\mapsto r_{u}(t)$ (resp. $t\mapsto r_{x}(t)$) admits a strictly
positive lower bound $\underline{r}_{u}$ (resp. $\underline{r}_{x}$) on some
interval $[t_{1},t_{2}]\subseteq]t_{0},t_{f}[$ satisfying $t_{1}<t_{2}$. First
note that $r_{u}$ inherits piecewise continuity from $u$. Thus, there exists
$]t_{1}^{\prime},t_{2}^{\prime}[\subseteq]t_{0},t_{f}[$, with
$t_{1}^{\prime}<t_{2}^{\prime}$ on which $r_{u}$ is continuous. Therefore, on
any closed interval $[t_{1},t_{2}]\subseteq]t_{1}^{\prime},t_{2}^{\prime}[$
with $t_{1}<t_{2}$, function $r_{u}$ admits a minimum $\underline{r}_{u}$
(necessarily strictly positive) by virtue of extreme value theorem. Since $x$
is continuous on $]t_{0},t_{f}[$, the same argument can be invoked to prove
that $\underline{r}_{x}\coloneqq\textstyle\min_{t\in[t_{1},t_{2}]}r_{x}(t)$
exists and is strictly positive.

Unconstrained system $\Sigma$ is IR by assumption. This implies that there
exists a non zero measure input $\hat{u}\in\mathrm{\bf U}_{\Sigma}$ (i)
producing output $\mathbf{0}$ from initial state $\mathbf{0}$ when exerted to
$\Sigma$ (i.e. $(\hat{u},\hat{x},\mathbf{0})\in\mathrm{\bf
Q}_{\Sigma}(\mathbf{0})$ for some $\hat{x}$) and such that (ii)
$\hat{u}(t)=\mathbf{0}$ holds for all $t\in{}_{\geq 0}\setminus[t_{1},t_{2}]$.
Indeed, [2, Th. 3.2] applies for $\Sigma$ and proves that
$\rho=\dim{\mathrm{Ker}\left\\{\left[\begin{smallmatrix}B\\\
D\end{smallmatrix}\right]\right\\}}$ or $\dim{\mathcal{R}(\Sigma)}$ is
strictly positive. If $\rho>0$, then $\hat{u}$ can be selected as a continuous
signal which is not identically zero and satisfies
$\hat{u}(t)\in\mathrm{Ker}\left\\{\left[\begin{smallmatrix}B\\\
D\end{smallmatrix}\right]\right\\}$ if $t\in[t_{1},t_{2}]$. Clearly, this
signal $\hat{u}$ belongs to $\mathrm{\bf U}_{\Sigma}$ and corresponding state
trajectory $\hat{x}$ equals $\mathbf{0}$. If $\rho=0$, so that
$\dim{\mathcal{R}(\Sigma)}>0$ holds, pick arbitrary
$\hat{x}_{i}\in\mathcal{R}(\Sigma)\setminus\\{\mathbf{0}\\}$ and any $t_{i}$
such that $t_{1}<t_{i}<t_{2}$. Then, there exists a piecewise continuous input
transferring state from $\mathbf{0}$ to $\hat{x}_{i}$ and then back to the
origin while maintaining output identically null (see Subsection A.5). Since
this motion can be performed arbitrarily fast, $\hat{u}$ can be selected in
such a way that $\hat{x}(t_{1})=\hat{x}(t_{2})=\mathbf{0}$ and
$\hat{x}(t_{i})=\hat{x}_{i}$ hold. Note that $\hat{x}(t)=\mathbf{0}$ holds for
all $t\in{}_{\geq 0}\setminus[t_{1},t_{2}]$ because $\hat{u}(t)$ equals zero
on the same interval. Also remark that $\hat{x}_{i}\neq\mathbf{0}$ implies
that input $\hat{u}$ is not identically zero. Once again,
$\hat{u}\in\mathrm{\bf U}_{\Sigma}$ holds in this case.

Let us show that there exists $\alpha>0$ such that
$\alpha\hat{u}\eqqcolon\tilde{u}\in\widetilde{\mathrm{\bf U}}(x_{0},u)$, which
proves IR since $\hat{u}$ has non zero measure. Define
$\alpha_{u}\coloneqq\underline{r}_{u}/b_{u}^{+}$ where $b_{u}^{+}$ is an upper
bound of $t\mapsto\|\hat{u}(t)\|$ on compact interval $[t_{1},t_{2}]$, which
necessarily exists since $\hat{u}$ is piecewise continuous. Then, it holds
$\alpha_{u}\leq r_{u}(t)/\|\hat{u}(t)\|$ and, in turn,
$\alpha_{u}\hat{u}(t)\in B_{m}(r_{u}(t))$ for all $t\in[t_{1},t_{2}]$. Bearing
in mind that $B_{m}(r_{u}(t))\subseteq\mathcal{U}-u(t)$ is valid for all
$t\in[t_{1},t_{2}]$ and that $\hat{u}(t)=\mathbf{0}\in\mathcal{U}-u(t)$ holds
for all $t\in{}_{\geq 0}\setminus[t_{1},t_{2}]$, this proves that
$\alpha\hat{u}\in\mathrm{\bf U}-u$ holds for all $0<\alpha\leq\alpha_{u}$. If
$\rho=0$, then $\hat{x}=\mathbf{0}$ so that $\alpha\hat{x}\in\mathrm{\bf X}-x$
trivially holds for all $0<\alpha\leq\alpha_{u}$. If $\rho>0$, continuity of
$\hat{x}$ is ensured for any $\hat{u}\in\mathrm{\bf U}_{\Sigma}$ which proves
existence of strictly positive
$\alpha_{x}\coloneqq\underline{r}_{x}/\textstyle\max_{t\in[t_{1},t_{2}]}\|\hat{x}(t)\|$.
Then, similar arguments as before ensure that $\alpha_{x}\hat{x}\in\mathrm{\bf
X}-x$ holds for all $0<\alpha\leq\alpha_{x}$. If $\rho=0$ (resp. $\rho>0$),
define $\alpha\coloneqq\alpha_{u}>0$ (resp.
$\alpha\coloneqq\min\\{\alpha_{u},\alpha_{x}\\}>0$). In both cases ($\rho=0$
or $\rho>0$), observe that (i)
$\mathrm{H}_{x}[\mathbf{0};\alpha\tilde{u}]=\alpha\hat{x}$ holds by virtue of
(34b), (ii) $\mathrm{H}[\mathbf{0};\alpha\hat{u}]=\mathbf{0}$ is valid for
similar reasons and (iii) $(\alpha\hat{u},\alpha\hat{x})\in(\mathrm{\bf
U}-u)\times(\mathrm{\bf X}-x)$ holds. We have just proved that non zero
measure input $\tilde{u}=\alpha\hat{u}$ belongs to $\widetilde{\mathrm{\bf
U}}(x_{0},u)$, so that $(x_{0},y)$ is IR.

###  B.4. Proof of Th. 5.1

Let us consider that Hyp. 1 is valid throughout this subsection. Th. 5.1 is
proved by specializing previous results in this context.

Clearly, function spaces $\mathrm{\bf U}$ and $\mathrm{\bf X}$ inherit
linearity property from $\mathcal{U}$ and $\mathcal{X}$. An essential
consequence of this feature is given by the following lemma.

######  Lemma B.4.

Assume that Hyp. 1 holds and consider any $x_{0,1},x_{0,2}\in\mathcal{X}_{c}$.
For any $(u,y)\in\mathrm{\bf W}(x_{0,2})$, it holds

(42) $\mathrm{\bf W}(x_{0,1})-(u,y)=\mathrm{\bf W}(x_{0,1}-x_{0,2})$

###### Proof.

Let $x$ be such that $(u,x,y)\in\mathrm{\bf Q}(x_{0,2})$. Since $\mathrm{\bf
Q}(x_{0,2})\subset\mathrm{\bf U}\times\mathrm{\bf X}\times\mathrm{\bf Y}$, it
holds $\mathrm{\bf U}\times\mathrm{\bf X}\times\mathrm{\bf
Y}-(u,x,y)=\mathrm{\bf U}\times\mathrm{\bf X}\times\mathrm{\bf Y}$ under Hyp.
1. Hence, $\mathrm{\bf W}(x_{0,1})-(u,y)$ reads
$\\{(\tilde{u},\tilde{y})\in\mathrm{\bf U}\times\mathrm{\bf
Y}:\mathrm{H}_{x}[x_{0,1}-x_{0,2};\tilde{u}]\in\mathrm{\bf
X},\;\mathrm{H}[x_{0,1}-x_{0,2};\tilde{u}]=\tilde{y}\\}$ (see (38)) which is
nothing but $\mathrm{\bf W}(x_{0,1}-x_{0,2})$. ∎

As a result, $\widetilde{\mathrm{\bf U}}(x_{0},u)$ reduces to

(43) $\widetilde{\mathrm{\bf U}}\coloneqq\\{\tilde{u}:{}_{\geq
0}\rightarrow{}^{m}\mid(\tilde{u},\mathbf{0})\in\mathrm{\bf W}(\mathbf{0})\\}$

whenever Hyp. 1 is valid (see (37)). This set gathers inputs which produce an
identically zero output from zero initial state and comply with the
constraints. As suggested by the notation, note that $\widetilde{\mathrm{\bf
U}}$ depends neither on $x_{0}$ nor on $u$ anymore.

Let us particularized Prop. B.1 and Lem. B.3 in this context.

######  Lemma B.5.

Assume that Hyp. 1 holds. Pair $(x_{0},y)\in\mathrm{\bf A}$ is IR iff it
holds:

(44) $|\widetilde{\mathrm{\bf U}}\setminus\\{\mathbf{0}\\}|\geq 1.$

Furthermore, $(x_{0},y)$ is of:

  * •

the _1st kind_ iff it holds

(45) $\forall\tilde{u}\in\widetilde{\mathrm{\bf
U}}\setminus\\{\mathbf{0}\\}:\mathrm{H}_{x}[\mathbf{0};\tilde{u}]=\mathbf{0}$

  * •

the _2nd kind_ iff it holds

(46) $\forall\tilde{u}\in\widetilde{\mathrm{\bf
U}}\setminus\\{\mathbf{0}\\}:\mathrm{H}_{x}[\mathbf{0};\tilde{u}]\neq\mathbf{0}$

  * •

the _3rd kind_ iff neither (45) nor (46) is valid.

Since (44), (45) and (46) depend neither on $x_{0}$ nor on $y$, previous
properties are independent of the choice of $(x_{0},y)$, provided that $y$ is
admissible for $x_{0}$ so that $\mathfrak{H}^{-1}[x_{0};y]$ is non empty.
Statement of Th. 5.1 follows immediately.

######  Remark 7 ($\mathrm{\bf W}(x_{0})$ and $\mathfrak{H}^{-1}{[}x_{0};y{]}$
are affine).

Under Hyp. 1, it can easily be checked that $\mathrm{\bf Q}(\mathbf{0})$ and,
in turn, $\mathrm{\bf W}(\mathbf{0})$ are linear spaces. In view of Lem. B.4,
this means that $\mathrm{\bf Q}(x_{0})$ and, in turn, $\mathrm{\bf W}(x_{0})$
are affine spaces, for all $x_{0}\in\mathcal{X}$. Indeed, set $\mathrm{\bf
Q}(x_{0})$ can be written as $\mathrm{\bf Q}(\mathbf{0})+(u,x,y)$ for any
$(u,x,y)\in\mathrm{\bf Q}(x_{0})$. It also follows that
$\widetilde{\mathrm{\bf U}}$ is linear, so that $\mathfrak{H}^{-1}[x_{0};y]$
is affine since this last set equals $\widetilde{\mathrm{\bf U}}+u$ for all
$u\in\mathfrak{H}^{-1}[x_{0};y]$ (see (35)).

######  Remark 8.

Note that Lem. 5.2 allows to construct an _admissible_ input $\tilde{u}$ with
$\mathfrak{S}$ enjoying properties analoguous to that of $w$: Denoting
$E(F,\mathbf{0};(w,\eta,\mathbf{0}))$ by $(\tilde{u},\tilde{x},\tilde{y})$, it
follows that $\tilde{u}=RLw+RFT\eta$ is admissible and produces
$\tilde{y}=\mathbf{0}$ for $x_{0}=\mathbf{0}$, i.e.
$\tilde{u}\in\widetilde{\mathrm{\bf U}}$ (see (43)). Existence of input
$\tilde{u}$ has crucial consequence since it proves IR.

######  Remark 9.

IR of $\mathfrak{S}$ can be alternatively qualified by means of sets
$\mathrm{\bf G}\coloneqq\mathrm{\bf Q}(\mathbf{0})\cap(\mathrm{\bf
U}\times\mathrm{\bf X}\times\\{\mathbf{0}\\})$ and $\mathrm{\bf
G}_{0}\coloneqq\mathrm{\bf Q}(\mathbf{0})\cap(\mathrm{\bf
U}\times\\{\mathbf{0}\\}\times\\{\mathbf{0}\\})$. Indeed, Lem. B.5 proves that
$\mathfrak{S}$ is IR (i) of the 1st kind if $\mathrm{\bf
G}\subseteq\mathrm{\bf G}_{0}$, (ii) of the 2nd kind if $\mathrm{\bf
G}\cap\mathrm{\bf G}_{0}=\\{\mathbf{0}\\}$ and (iii) of the 3rd kind if none
of those two relationships is valid.

##  Appendix C Proof of Cor. 2

As a preliminary step, consider following lemma.

######  Lemma C.1.

Given open set $\mathcal{Z}$ and signal $z:{}_{\geq 0}\rightarrow\mathcal{Z}$.
Let $\mathcal{C}_{z}\neq\varnothing$ be the largest open subset of ≥0 where
$z$ is continuous. Then,
$\mathcal{Q}_{z}\coloneqq\\{t\in\mathcal{C}_{z}:z(t)\in\mathcal{Z}\\}$ is
open.

###### Proof.

Pick any $t_{0}\in\mathcal{Q}_{z}$. Since $\mathcal{Z}$ is open, there exists
an open hyper-ball $B$ centered on $z(t_{0})$ such that
$B\subseteq\mathcal{Z}$. From $t_{0}\in\mathcal{C}_{z}$, there exists a
connected open set $\mathcal{T}\subseteq\mathcal{C}_{z}$ on which $z$ is
continuous and satisfying $t_{0}\in\mathcal{T}$. Observe that if
$z(\mathcal{T})\subset B$, then $\mathcal{T}\subseteq\mathcal{Q}_{z}$.
Otherwise, continuity of $z$ on $\mathcal{T}$ ensures existence of open set
$\hat{\mathcal{T}}\subseteq\mathcal{T}$ satisfying $t_{0}\in\hat{\mathcal{T}}$
and such that $z(\hat{\mathcal{T}})\subset B$ so that
$\hat{\mathcal{T}}\subseteq\mathcal{Q}_{z}$. ∎

Define the following sets:

$\displaystyle\mathcal{T}_{u}$ $\displaystyle\coloneqq\\{t\in{}_{\geq
0}:u(t)\in\mathcal{U}\setminus\text{int}(\mathcal{U})\\},$
$\displaystyle\mathcal{T}_{x}$ $\displaystyle\coloneqq\\{t\in{}_{\geq
0}:x(t)\in\mathcal{X}\setminus\text{int}(\mathcal{X})\\}.$

From discussion above Cor. 2, $\mathcal{T}_{u}$ (resp.
$\mathcal{T}_{u}\cup\mathcal{T}_{x}$) is dense in ≥0 if $\rho>0$ (resp.
$\rho=0$).

Assume that $\rho>0$ holds. Then, $\mathcal{T}_{u}\cap\mathcal{C}$ is also
dense in $\mathcal{C}$, so that closure of $\mathcal{T}_{u}\cap\mathcal{C}$,
denoted by $\overline{\mathcal{T}_{u}\cap\mathcal{C}}$, equals $\mathcal{C}$
holds.  Indeed, for all open set $\mathcal{O}\subseteq{}_{\geq 0}$, it holds
$\mathcal{O}\cap\mathcal{T}_{u}\neq\varnothing$, so that for all open set
$\mathcal{O}\subseteq\mathcal{C}\subseteq{}_{\geq 0}$, set
$\mathcal{O}\cap\mathcal{T}_{u}=(\mathcal{O}\cap\mathcal{T}_{u})\cap\mathcal{C}=\mathcal{O}\cap(\mathcal{T}_{u}\cap\mathcal{C})$
is distinct from $\varnothing$. Furthermore, observe that
$\mathcal{C}\setminus\mathcal{T}_{u}$ equals
$\\{t\in\mathcal{C}:u(t)\in\text{int}(\mathcal{U})\\}$ which is open, by
virtue of Lem. C.1. This, in turn, proves that
$\mathcal{T}_{u}\cap\mathcal{C}$ is a closed subset _of_ $\mathcal{C}$. As a
result,
$\mathcal{T}_{u}\cap\mathcal{C}=\overline{\mathcal{T}_{u}\cap\mathcal{C}}=\mathcal{C}$,
so that (18a) holds for all $t$ where $u$ is continuous.

The case $\rho=0$ can be handled similarly. It holds
$\overline{(\mathcal{T}_{u}\cup\mathcal{T}_{x})\cap\mathcal{C}}=\mathcal{C}$.
Set $\mathcal{C}\setminus(\mathcal{T}_{u}\cup\mathcal{T}_{x})$ equals

$\\{t\in\mathcal{C}:u(t)\in\text{int}(\mathcal{U})\wedge
x(t)\in\text{int}(\mathcal{X})\\}=\\\
\\{t\in\mathcal{C}:u(t)\in\text{int}(\mathcal{U})\\}\cap\\{t\in\mathcal{C}:x(t)\in\text{int}(\mathcal{X})\\}$

which can be proved to be open by Lem. C.1 since $x$ in continuous on
$\mathcal{C}$. This yields
$(\mathcal{T}_{u}\cup\mathcal{T}_{x})\cap\mathcal{C}=\overline{(\mathcal{T}_{u}\cup\mathcal{T}_{x})\cap\mathcal{C}}=\mathcal{C}$,
so that either (18a) or (18b) holds for all $t\in\mathcal{C}$.