# Comment on "Rotating Spin and Giant Splitting: Unoccupied Surface Electronic
Structure of Tl/Si(111)"

A. F. Campos, K. Wang, A. Tejeda  Laboratoire de Physique des Solides, CNRS,
Université Paris-Saclay, 91405 Orsay, France.

††preprint: AIP/123-QED

The Letter Stolwijk  _et al._ (2013) studies the unoccupied states of
Tl/Si(111) by spin-resolved inverse photoemission (IPES) using a rotatable
electron source Stolwijk  _et al._ (2014). Authors claim that the spin
polarization rotates from an in-plane Rashba polarization at
$\overline{\Gamma}$ to the surface normal at
$\overline{K}(\overline{K^{\prime}})$ valleys. This Comment first indicates
that the in-plane polarization does not vanish at the valleys despite the
claims and the data representation. Second, it highlights that the out-of-
plane polarization cannot be straightforwardly measured. Hypothesis and/or
unspecified data treatment are needed to experimentally conclude on the
polarization rotation and the completely out-of-plane spin-polarized valleys.

The Letter claims an in-plane polarization vanishing around $\overline{K}$ and
"completely out-of-plane spin-polarized valleys in the vicinity of the Fermi
level". These claims seem to be supported by the absence of experimental
points near the $\overline{K}$ valley in Fig. 3a of Ref. Stolwijk  _et al._ ,
2013. This is at odds with the spectra around $\overline{K}$
($60^{\circ}\leq\theta\leq 70^{\circ}$) showing differences in spin up/down
(Fig. 2a of Ref.Stolwijk  _et al._ , 2013, from wich Fig. 3a is extracted),
i.e. a non-vanishing in-plane polarization. More importantly, these points
with non-vanishing in-plane polarization are absent in Fig. 3a, so Fig. 3a
suggests that the in-plane component vanishes at $\overline{K}$. Note that
both panels in Fig. 2 have the same angular range (0 to 70∘) and when $\theta$
is converted to $k$, the same wavevector range should be found in Fig. 3
panels. However $k^{max}_{\bot}<1.2$ Å-1 (Fig. 3b) while $k^{max}_{||}<1.1$
Å-1 (Fig. 3a), i.e. before $\overline{K}$.

Figure 1: (a) Electron source configuration for in-plane polarization spectra
along $\overline{\Gamma}\overline{K}$ by varying $\theta$. (b) The
configuration in (a) provides an in-plane electron beam polarization,
perpendicular to $\overline{\Gamma}\overline{K}$ and with constant surface
projection when varying $\theta$. (c) IPES spectra in (a) configuration,
strictly sensitive to in-plane polarization. (d) Electron source configuration
for some sensitivity to the out-of-plane polarization. (e) The configuration
in (d) measures IPES spectra coupling in- and out-of-plane components of the
electron beam polarization. The polarization is projected along
$\overline{\Gamma}\overline{K}$ and its out-of-plane component varies with
$\theta$. (f) A deconvolution is needed to obtain the out-of-plane component
from measurements in configuration (d). Assumptions and/or a delicate analysis
of spectral intensities are needed.

Besides the unclear data representation, the setup cannot directly measure the
out-of-plane polarization. In-plane polarization IPES spectra correspond to
the situation shown in our Fig. 1 (top). Rotating the electron source (Fig. 1,
bottom), provides spectra with some out-of-plane sensitivity. It can be seen
that the pristine out-of-plane spectra could only be measured at strictly
grazing incidence (Fig. 1e), which is experimentally unreachable. The
difficulty of measuring the out-of-plane component near $\overline{\Gamma}$
($k=\theta=0$) is also evident. Instead, out-of-plane spectra and dispersions
close to $\overline{\Gamma}$ are shown (Fig. 2 and 3 in Ref. Stolwijk  _et
al._ , 2013).

Fig. 1 shows the necessity of assumptions to estimate the out-of-plane
spectra. Measurements must be deconvolved, requiring information on the in-
plane spectra. One can: (1) assume a vanishing in-plane component, or (2)
introduce in-plane spectra as an input. (1) is not an experimental
determination of the out-of-plane contribution. (2) relies on a complex data
treatment: (i) in-plane spectra measurement (ii) subtraction of in-plane
spectra (vertical electron source) from raw spectrum with out-of-plane
sensitivity (horizontal electron source). This involves a delicate spectral
intensity comparison measured at different geometries, complicated by
intensity changes when the electron source is rotated by 90∘: (a) the
macroscopic rotation of the source displaces the electron beam from the
optical axis and it must be realignedStolwijk  _et al._ (2014). (b) the beam
polarization changes its direction with respect to surface orbitals and
$\overline{\Gamma}\overline{K}$ ($||$ to $\bot$, see Fig. 1). (c) the
experimental geometry affects the IPES matrix elements. Finally, surface
freshness evolves with time, affecting the spectral intensity of the
measurements in the two different configurations. There are thus many factors
affecting intensities when turning the electron source, when varying $\theta$
or P. Their impact on the intensity variation is difficult to quantify. The
Letter concludes on a constant spectral intensity instead.

In summary, conclusions on the spin polarization vector rotation and the
completely out-of-plane spin-polarized valleys should be carefully reviewed,
since rotatable spin-polarized electron sources cannot measure the out-of-
plane polarization of electronic states. Reliable results require sources
allowing to measure in- and out-of-plane spectra without assumptions or
without data treatment involving spectral intensities Campos  _et al._ (2022);
Campos, Duden, and Tejeda (2022); Campos  _et al._ (2023).

## References

  * Stolwijk  _et al._ (2013) S. D. Stolwijk, A. B. Schmidt, M. Donath, K. Sakamoto, and P. Krüger, Phys. Rev. Lett. 111, 176402 (2013).
  * Stolwijk  _et al._ (2014) S. Stolwijk, H. Wortelen, A. Schmidt, and M. Donath, Rev. Sci. Instrum. 85, 013306 (2014).
  * Campos  _et al._ (2022) A. Campos, P. Duret, S. Cabaret, T. Duden, and A. Tejeda, Rev. Sci. instrum. 93, 093904 (2022).
  * Campos, Duden, and Tejeda (2022) A. Campos, T. Duden, and A. Tejeda, EPJ Web of Conferences 273, 01010 (2022).
  * Campos  _et al._ (2023) A. Campos, K. Wang, T. Duden, and A. Tejeda, Rev. Sci. instrum. 94, XX (2023), doi:10.1063/5.0137762.


# On a connection between the reliability of multi-channel systems and the
notion of controlled-invariance entropy

Getachew K. Befekadu G. K. Befekadu is with the Department of Electrical
Engineering, University of Notre Dame, Notre Dame, IN 46556, USA.
E-mail<EMAIL_ADDRESS>
Version - February 25, 2013.

###### Abstract

The purpose of this note is to establish a connection between the problem of
reliability (when there is an intermittent control-input channel failure that
may occur between actuators, controllers and/or sensors in the system) and the
notion of controlled-invariance entropy of a multi-channel system (with
respect to a subset of control-input channels and/or a class of control
functions). We remark that such a connection could be used for assessing the
reliability (or the vulnerability) of the system, when some of these control-
input channels are compromised with an external “malicious” agent that may try
to prevent the system from achieving more of its goal (such as from attaining
invariance of a given compact state and/or output subspace).

###### Index Terms:

Invariance entropy, multi-channel system, topological feedback entropy,
reliability.

##  I Introduction

In the last decades, the notions of measure-theoretic entropy and topological
entropy have been intensively studied in the context of measure-preserving
transformations or continuous maps (e.g., see [21], [19] and [11] for the
review of entropy in ergodic theory as well as in dynamical systems). For
instance, Adler et al. (in the paper [1]) introduced the notion of topological
entropy as a topologically invariant conjugacy, which is an analogue to the
notion of measure-theoretic entropy, for measuring the rate at which a
continuous map in a compact topological space generates initial-state
information. Later, [10] and [5] gave a new, but equivalent, definition of
topological entropy for continuous maps that led to proofs for connecting the
topological entropy with that of measure-theoretic entropy.

In the recent paper [9], the authors have introduced the notion of invariance
entropy for continuous-time systems as a measure of information that is
necessary to achieve invariance of a given state (or output) subspace (i.e., a
measure of how open-loop control functions have to be updated in order to
achieve invariance of a given subspace of the state space). In the present
paper, we explore this concept, which is closely related to the notion of
topological feedback entropy (see [17] and [13]), for assessing the
reliability of a multi-channel system when there is an intermittent control-
channel failure that may occur between actuators, controllers and/or sensors
in the system. Specifically, we provide conditions on the minimum rate at
which the multi-channel system can generate information with respect to a
subset of control-input channels and/or a class of control functions when the
system states are restricted to a given controlled-invariant subspaces. Here,
it is important to note that the intermittent control-channel failures may not
necessarily represent any physical failures within the system. Rather, this
can also be interpreted as external “malicious” agent who is trying to prevent
the system from achieving more of its goal, i.e., from attaining invariance of
a given state (or output) subspace.

With the emergence of networked control systems (e.g., see [2]), these notions
of entropy have found renewed interest in the research community (e.g., see
[17], [18] and [6]). Notably, in the paper [17], Nair et al. have introduced
the notion of topological feedback entropy, which is based on the ideas of
[1], to quantify the minimum rate at which deterministic discrete-time
dynamical systems generate information relevant to the control objective of
set-invariance. More recently, the notion of controlled-invariance entropy (as
well as the notion of almost invariance entropy) has been studied for
continuous-time control systems in [6], [8], [14] and [9] based on the metric-
space technique of [5]. It is noted that such an invariant entropy provides a
measure of the smallest growth rate for the number of open-loop control
functions that are needed to confine the states within an arbitrarily small
distance (in the sense of gap metric) from a given subspace. For discontinuous
systems, we also note that the notion of topological entropy has been
investigated with respect to piecewise continuous piecewise monotone
transformations (e.g., see [15] and [16]).

The paper is organized as follows. In Section II, we present preliminary
results on the invariance entropy of multi-channel systems with respect to a
set of control-input channels and a class of control functions. Section III
presents the main results – where we provide conditions on the information
this is necessary for achieving invariance of the multi-channel system states
in (or near) a given subspace.

##  II Preliminaries

###  II-A Notation

For $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times r}$ and a linear
space $\mathscr{X}$, the supremal $(A,B)$-invariant subspace contained in
$\mathscr{X}$ is denoted by
$\mathscr{V}^{*}\triangleq\sup\mathscr{V}\left(A,B;\mathscr{X}\right)$. For a
subspace $\mathscr{V}\subset\mathscr{X}$, we use $\langle
A\,|\,\mathscr{V}\rangle$ to denote the smallest invariant subspace containing
$\mathscr{V}$.

###  II-B Problem formulation

Consider the following generalized multi-channel system

$\displaystyle\dot{x}(t)$
$\displaystyle=Ax(t)+\sum_{j\in\mathcal{N}}B_{j}u_{j}(t),\quad
x(t_{0})=x_{0},\quad t\in[t_{0},+\infty),$ (1)

where $A\in\mathbb{R}^{n\times n}$, $B_{j}\in\mathbb{R}^{n\times r_{j}}$,
$x(t)\in\mathscr{X}\subset\mathbb{R}^{n}$ is the state of the system,
$u_{j}(t)\in\mathscr{U}_{j}\subset\mathbb{R}^{r_{j}}$ is the control input to
the $j$th-channel and $\mathcal{N}\triangleq\\{1,2,\ldots,N\\}$ represents the
set of controllers (or the set of control-input channels) in the system.

Let us introduce the following class of admissible controls that will be used
in the sequel

$\displaystyle\mathscr{U}\subseteq\biggm{\\{}u\in\prod_{i\in\mathcal{N}}L_{\infty}(\mathbb{R},\,\mathbb{R}^{r_{i}})\,\biggm{|}\,u_{\neg
j}(t)\in\mathscr{U}_{\neg j}\triangleq\prod_{i\in\mathcal{N}_{\neg
j}}\mathscr{U}_{i}~{}~{}\text{{\em for almost all}}$
$\displaystyle\,\,t\in[0,\,\infty)~{}\,\text{and}$ $\displaystyle\forall
j\in\mathcal{N}\cup\\{0\\}\biggm{\\}},$ (2)

where $u_{\neg 0}(t)=\bigl{(}u_{1}(t),\,u_{2}(t),\,\ldots\,u_{N}(t)\bigr{)}$
and $u_{\neg
i}(t)=\bigl{(}u_{1}(t),\,\dots\,u_{i-1}(t),\,u_{i+1}(t),\,\ldots\,u_{N}(t)\bigr{)}$
for $i\in\mathcal{N}$. Moreover, $\mathcal{N}_{\neg 0}\triangleq\mathcal{N}$
and $\mathcal{N}_{\neg j}\triangleq\mathcal{N}\setminus\\{j\\}$ for
$j=1,2,\ldots,N$.

In the remainder of this subsection, we provide some results from geometric
control theory (e.g., see [3], [22], [4] and [20] for details about this
theory).

###### Definition 1

Let $\mathscr{V}_{j}\subset\mathscr{X}$ for $j\in\mathcal{N}\cup\\{0\\}$.

  1. (i)

If $\mathscr{V}_{j}$ is $(A)$ \- invariant, then
$A\mathscr{V}_{j}\subset\mathscr{V}_{j}$.

  2. (ii)

If $\mathscr{V}_{j}$ is $(A,\,B_{\neg j})$-invariant, then
$A\mathscr{V}_{j}\subset\mathscr{V}_{j}+\mathscr{B}_{\neg j}$, where $B_{\neg
0}\triangleq\bigl{[}\begin{array}[]{cccc}B_{1}&B_{2}&\ldots&B_{N}\end{array}\bigr{]}$,
$B_{\neg
j}\triangleq~{}\bigl{[}\begin{array}[]{ccc}B_{1}\,\ldots\,B_{j-1}&B_{j+1}\,\ldots\,B_{N}\end{array}\bigr{]}$
and $\mathscr{B}_{\neg j}\triangleq\operatorname{Im}B_{\neg j}$ for
$j\in\mathcal{N}$.111In this paper, we consider a case in which one of the
controllers is extracted due to an intermittent failure. However, following
the same discussion, we can also consider when the fault is associated with at
most two or more possible controllers in the system.

The following lemma, which is a well-known result, will be stated without
proof (e.g., see [22] or [4]).

###### Lemma 1

Suppose $\mathfrak{I}_{j}\left(A,B_{\neg j};\mathscr{F}\right)$ is a family of
$(A,B_{\neg j})$-invariant subspaces for $j\in\mathcal{N}$. Then, every
subspace $\mathscr{F}\subset\mathscr{X}$ contains a unique supremal
$(A,B_{\neg j})$-invariant subspace which is given by
$\mathscr{V}_{j}^{*}=\sup\mathfrak{I}_{j}\left(A,B_{\neg
j};\mathscr{F}\right)$ for each $j\in\mathcal{N}$.

Then, we state the following result which is a direct application of Lemma 1.

###### Theorem 1

Let $\mathscr{V}_{j}\subset\mathscr{X}$ for each $j\in\mathcal{N}$. Then,
$\mathscr{V}_{j}$ is a member of the subspace families
$\mathfrak{I}_{j}(A,B_{\neg j};\mathscr{X})$ that preserves the property of
$(A,B_{\neg j})$-invariant (i.e.,
$\mathscr{V}_{j}\in\mathfrak{I}_{j}(A,B_{\neg j};\mathscr{X})$), if and only
if

$\displaystyle A\mathscr{V}_{j}\subset\mathscr{V}_{j}+\mathscr{B}_{\neg
j},~{}~{}\forall j\in\mathcal{N}.$ (3)

###### Proof:

The proof follows the same lines of argument as that of Wonham (see [22] p.
88). Suppose $\mathscr{V}_{j}\in\mathfrak{I}_{j}(A,B_{\neg j};\mathscr{X})$
and let $v^{j}\in\mathscr{V}_{j}$ for $j\in\mathcal{N}$, then
$(A,\sum_{i\in\mathcal{N}_{\neg j}}B_{i}K_{i})v^{j}=w^{j}$ for some
$w^{j}\in\mathscr{V}_{j}$, i.e.,

$\displaystyle Av^{j}=w^{j}-\sum_{i\in\mathcal{N}_{\neg
j}}B_{i}K_{i}v^{j}\in\mathscr{V}_{j}+\mathscr{B}_{\neg j},$ (4)

On the other hand, let $\\{v_{1}^{j},v_{2}^{j},\ldots,v_{\mu}^{j}\\}$ be a
basis for $\mathscr{V}_{j}$ for $j=1,2,\ldots,N$. Suppose that (3) holds true.
Then, there exist $w_{k}^{j}\in\mathscr{V}_{j}$ and $u_{k}^{\neg
j}\in\mathscr{U}_{\neg j}$ for $k\in\\{1,2,\dots,\mu^{j}\\}$ such that

$\displaystyle Av_{k}^{j}=w_{k}^{j}-B_{\neg j}u_{k}^{\neg j},\qquad
k\in\\{1,2,\dots,\mu^{j}\\}.$ (5)

If we further define the following mapping $K_{\neg
j}^{0}\colon\mathscr{V}_{j}\to\mathscr{U}_{\neg j}$

$\displaystyle K_{\neg j}^{0}v_{k}^{j}=u_{k}^{\neg j},\qquad
k\in\\{1,2,\dots,\mu^{j}\\},$ (6)

and then by letting $K_{\neg j}$ to be any extension of $K_{\neg j}^{0}$ to
$\mathscr{X}$. We, therefore, have $(A+\sum_{i\in\mathcal{N}_{\neg
j}}B_{i}K_{i})v_{k}^{j}=w_{k}^{j}\in\mathscr{V}$, i.e.,
$(A+\sum_{i\in\mathcal{N}_{\neg
j}}B_{i}K_{i})\mathscr{V}_{j}\subset\mathscr{V}_{j}$, so that the controlled-
invariant subspace $\mathscr{V}_{j}$ satisfies

$\displaystyle\mathscr{V}_{j}\in\mathfrak{I}_{j}(A,B_{\neg
j};\mathscr{X}),\quad\forall j\in\mathcal{N}.$ (7)

$\Box$

###### Corollary 1

Let the subspace $\mathscr{V}\subset\mathscr{X}$ be $(A,\,B_{\neg j})$ \-
invariant for each $j\in\mathcal{N}\cup\\{0\\}$, then there exists a class of
maps $\mathcal{K}\ni K:\mathscr{X}\to\mathscr{U}$ that satisfies

$\displaystyle\mathcal{K}\subseteq\biggm{\\{}\underbrace{\bigm{(}K_{1},K_{2},\ldots,K_{N}\bigm{)}}_{\triangleq
K}\in\prod_{j\in\mathcal{N}}\mathbb{R}^{r_{j}\times
n}\,\Bigl{\lvert}\,\biggm{(}A+\sum_{i\in\mathcal{N}_{\neg
j}}B_{i}K_{i}\biggm{)}\mathscr{V}\subset\mathscr{V},~{}~{}\forall
j\in\mathcal{N}\cup\\{0\\}\biggm{\\}}.$ (8)

###### Remark 1

Note that the controlled-invariant subspace $\mathscr{V}$, which is given in
the aforementioned corollary, is also a subspace of $\mathscr{V}^{*}$ (see
also Equation (9) below)

Next, we introduce the following theorem on the family of supremal controlled-
invariant subspaces that will be useful for our work in the next section.

###### Theorem 2

Let $\mathfrak{V}\triangleq\\{\mathscr{V}_{j}^{*}\\}_{j\in\mathcal{N}}$ be a
set of supremal controlled-invariant subspaces with respect to the family of
systems $\left\\{\bigl{(}A,B_{\neg j}\bigr{)}\right\\}_{j\in\mathcal{N}}$.
Then, the set $\mathfrak{V}$ forms a lattice of controlled-invariant
subspaces. Moreover, there exists a unique (nonempty) subspace that satisfies

$\displaystyle\mathscr{V}^{*}=\bigcap_{j\in\mathcal{N}}\mathscr{V}_{j}^{*}\in\left\\{\bigcap_{j\in\mathcal{N}}\mathscr{V}_{j}~{}\biggm{|}~{}\mathscr{V}_{j}\subset\sup\mathfrak{I}_{j}(A,B_{\neg
j};\mathscr{X}),~{}\forall j\in\mathcal{N},~{}~{}\exists\,u_{\neg
j}\in\mathscr{U}_{\neg j}\right\\},$ (9)

where
$\mathscr{V}_{j}^{*}=\sup\mathfrak{I}_{j}\left((A+\sum_{i\in\mathcal{N}_{\neg
j}}B_{i}K_{i}),\mathscr{B}_{j}\right)$ and
$\mathscr{B}_{j}\triangleq\operatorname{Im}B_{j}$ for all $j\in\mathcal{N}$.

###### Proof:

Note that $\mathscr{V}_{j}+\mathscr{V}_{\neg j}\in\mathfrak{V}$ and
$\mathscr{V}_{j}\cap\mathscr{V}_{\neg j}\in\mathfrak{V}$ for all
$j\in\mathcal{N}$. Moreover, if we define the gap metric
$\varrho_{j}(\mathscr{V}_{0},\mathscr{V}_{j})$ between the controlled-
invariant subspaces $\mathscr{V}_{0}$ and $\mathscr{V}_{j}$ as

$\displaystyle\varrho_{j}(\mathscr{V}_{0},\mathscr{V}_{j})=\|P_{\mathscr{V}_{0}}-P_{\mathscr{V}_{j}}\|,~{}~{}\forall
j\in\mathcal{N},$ (10)

where $\mathscr{V}_{0}=\sup\mathfrak{I}_{0}(A,B_{\neg 0};\mathscr{X})$,
$P_{\mathscr{V}_{0}}$ and $P_{\mathscr{V}_{j}}$ are orthogonal projectors on
$\mathscr{V}_{0}$ and $\mathscr{V}_{j}$, respectively. Then, the set of all
controlled-invariant subspaces in $\mathscr{X}$ forms a compact metric state-
space with respect to the above gap metric (see also [12]). On the other hand,
let us define the following family of subspaces

$\displaystyle\tilde{\mathscr{V}}=\left\\{\bigcap_{j\in\mathcal{N}}\mathscr{V}_{j}~{}\biggm{|}~{}\mathscr{V}_{j}\subset\sup\mathfrak{I}_{j}(A,B_{\neg
j};\mathscr{X}),~{}\forall j\in\mathcal{N},~{}~{}\exists\,u_{\neg
j}\in\mathscr{U}_{\neg j}\right\\}.$ (11)

Suppose the subspace $\mathscr{V}^{*}$ exists, then it is a unique member of
the family that is defined in (11), i.e.,

$\displaystyle\mathscr{V}^{*}=\bigcap_{j\in\mathcal{N}}\mathscr{V}_{j}^{*}\in\tilde{\mathscr{V}},$
(12)

with
$\mathscr{V}_{j}^{*}=\sup\mathfrak{I}_{j}\left((A+\sum_{i\in\mathcal{N}_{\neg
j}}B_{i}K_{i}),\mathscr{B}_{j}\right)$ for all $j\in\mathcal{N}$. Note that we
have $\mathscr{V}_{j}=\langle A+\sum_{i\in\mathcal{N}_{\neg
j}}B_{i}K_{i}|\mathscr{B}_{j}\rangle$ which also implies that
$\operatorname{Im}B_{j}\subset\mathscr{V}_{j}^{*}$.222We remark that the
induced continuous maps in $\mathscr{X}/\mathscr{V}_{j}^{*}$ and
$\mathscr{X}/\mathscr{V}_{\neg j}^{*}$ admit an enveloping lattice for the
family of controlled-invariant subspaces $\mathscr{V}_{j}^{*}$, $\forall
j\in\mathcal{N}$ (e.g., see [12]). $\Box$

###  II-C Properties of (controlled)-invariance entropy

In the following, we start by giving the definition of (controlled)-invariance
entropy for the multi-channel system in (1) with respect to the subset of
control-input channels and that class of control functions.

###### Definition 2

For a given subspace
$\mathscr{F}\subset\tilde{\mathscr{V}}^{*}\in\tilde{\mathscr{V}}$ with
nonempty interior and $T$, $\epsilon>0$, the class of control functions
$\mathscr{C}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})\subset\mathscr{U}$
is called $(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$-spanning, if
there exits $u\in\mathscr{C}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$
for almost all $t\in[0,\,T)$ such that

$\displaystyle\max_{\begin{subarray}{c}j\in\mathcal{N}\end{subarray}}\sup_{\begin{subarray}{c}t\in[0,\,T]\end{subarray}}\inf_{\begin{subarray}{c}y\in\tilde{\mathscr{V}}^{*}\end{subarray}}\|\phi_{\neg
j}(t,x_{0},u_{\neg j}(t))-y\|\leq\epsilon,\quad\forall x_{0}\in\mathscr{F}.$
(13)

###### Remark 2

In the aforementioned definition, we use the notation $\phi_{\neg
j}(t,x_{0},u_{\neg j}(t))$ to denote the unique solution of the multi-channel
system with initial condition $x_{0}\in\mathscr{F}$ and control $u_{\neg
j}\in\mathscr{U}_{\neg j}$, i.e.,

$\displaystyle x(t)$ $\displaystyle=\phi_{\neg j}(t,x_{0},u_{\neg j}(t)),$
$\displaystyle\triangleq\exp
A\bigl{(}t-t_{0}\bigr{)}x_{0}+\sum_{i\in\mathcal{N}_{\neg
j}}\int_{t_{0}}^{t}\exp
A\bigl{(}t-s\bigr{)}B_{i}u_{i}(s)ds,\quad\forall[t_{0},\,t]\in[0,\,T],$ (14)

for each $j\in\mathcal{N}\cup\\{0\\}$.

Moreover, the relation $\phi_{\neg j}(t+t_{0},x_{0},u_{\neg j}(t))=\phi_{\neg
j}(t,\phi_{\neg j}(t,x_{0},u_{\neg j}(t)),u_{\neg j}(t_{0}+.))$ will also hold
for all $j\in\mathcal{N}\cup\\{0\\}$.

Let $r_{\rm inv}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$ be the
smallest cardinality of
$\mathscr{C}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$-spanning sets.
Then, we have the following properties for $r_{\rm
inv}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$.

  1. (i)

Clearly $r_{\rm
inv}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})\in[0,\,\infty)$.333The
value of $r_{\rm inv}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$ could
be an infinity.

  2. (ii)

If $\epsilon_{1}<\epsilon_{2}$, then $r_{\rm
inv}(T,\epsilon_{1},\mathscr{F},\tilde{\mathscr{V}}^{*})\geq r_{\rm
inv}(T,\epsilon_{2},\mathscr{F},\tilde{\mathscr{V}}^{*})$.

###### Definition 3

The (controlled)-invariance entropy of the multi-channel system in (1) (i.e.,
with respect to the subset of control-input channels and/or the class of
control functions) is given by

$\displaystyle{h_{\rm
inv}}(\mathscr{F},\tilde{\mathscr{V}}^{*})=\lim_{\begin{subarray}{c}\epsilon\searrow
0\end{subarray}}\biggm{\\{}\varlimsup_{\begin{subarray}{c}T\to\infty\end{subarray}}\frac{1}{T}\log
r_{\rm inv}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})\biggm{\\}}.$ (15)

###### Remark 3

We remark that the existence of such a limit in the aforementioned definition
for ${h_{\rm inv}}(\mathscr{F},\tilde{\mathscr{V}}^{*})$ follows directly from
the monotonicity of $r_{\rm
inv}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$ with respect to
$\epsilon$. Moreover, such an invariance entropy ${h_{\rm
inv}}(\mathscr{F},\tilde{\mathscr{V}}^{*})$ equals to the minimum amount of
information that is required to render $\tilde{\mathscr{V}}^{*}$-invariant
subspace by using a causal coding and/or control law (see [7] for discussion
on single control-channel systems).

Then, we have the following properties for ${h_{\rm
inv}}(\mathscr{F},\tilde{\mathscr{V}}^{*})$.

  1. (i)

${h_{\rm
inv}}(\mathscr{F},\tilde{\mathscr{V}}^{*})\in[0,\,\infty)\cup\\{\infty\\}$.

  2. (ii)

If $\mathscr{F}\triangleq\bigcup_{l\in\\{1,2,\ldots,L\\}}\mathscr{F}_{l}$ with
compact $\mathscr{F}_{l}$, $\forall l\in\\{1,2,\ldots,L\\}$, then ${h_{\rm
inv}}(\mathscr{F},\tilde{\mathscr{V}}^{*})=\max_{\begin{subarray}{c}l\in\\{1,2,\ldots,L\\}\end{subarray}}{h_{\rm
inv}}(\mathscr{F}_{l},\tilde{\mathscr{V}}^{*})$.

In the following, we state the main problem of this paper – where we establish
a connection between the invariance entropy of a multi-channel system and the
reliability of a multi-channel system.

Problem: Find a condition on the minimum amount of “information” (with respect
to the subset of control-input channels and/or the class of control functions)
that is necessary to keep the states of the multi-channel system in a given
subspace $\tilde{\mathscr{V}}^{*}$.

##  III Main Results

In this section, we present our main results – where we provide a connection
between the invariance entropy (as a measure of “information” needed with
respect to the subset of control-input channels to keep the system in (or
near) this compact subspace) and the reliability of the multi-channel system
(when there is an intermittent channel failure that may occur between
actuators, controllers and/or sensors in the system).

###### Theorem 3

Suppose that Theorem 2 holds true and let $\mathscr{F}$ be a subspace of
$\tilde{\mathscr{V}}^{*}$. For every $x_{0}\in\mathscr{F}$, if there exists
$u(t)\in\mathscr{C}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$ for
almost all $t\in[0,\,T]$ that renders $\tilde{\mathscr{V}}^{*}$-invariant,
then the (controlled)-invariance entropy of the multi-channel system is given
by

$\displaystyle h_{\rm
inv}(\mathscr{F},\tilde{\mathscr{V}}^{*})=\lim_{\begin{subarray}{c}\epsilon\searrow
0\end{subarray}}\biggm{\\{}\varlimsup_{\begin{subarray}{c}T\to\infty\end{subarray}}\frac{1}{T}\log
r_{\rm inv}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})\biggm{\\}}.$ (16)

###### Proof:

For any subspace $\mathscr{F}\subset\tilde{\mathscr{V}}^{*}$, suppose there
exists $u(t)\in\mathscr{C}(T,\epsilon,\mathscr{F},\tilde{\mathscr{V}}^{*})$
for almost all $t\in[0,\,T]$ such that

$\displaystyle\sup_{\begin{subarray}{c}t\in[0,\,T]\end{subarray}}\inf_{\begin{subarray}{c}y\in\tilde{\mathscr{V}}^{*}\end{subarray}}\|\phi_{\neg
j}(t,x_{0},u_{\neg j}(t))-y\|\leq\epsilon,~{}~{}\forall
j\in\mathcal{N},~{}~{}\forall x_{0}\in\mathscr{F},$

with some finite-positive number $\epsilon>0$. Then, we see that (16) (i.e.,
the (controlled)-invariance entropy of the multi-channel system in (1)) will
directly follow. Moreover, the supremum is taken overall the set of all
admissible controls that renders $\tilde{\mathscr{V}}^{*}$-invariant. $\Box$

###### Remark 4

We remark that the aforementioned theorem essentially states that the set of
admissible controls $\mathscr{U}$ renders $\tilde{\mathscr{V}}^{*}$-invariant,
even if there is an intermittent failure in any one of the control-input
channels. Note that this quantity (which is also the minimum growth rate for
the number of open-loop control functions with respect to intermittently
faulty channel) provides a condition on the minimum amount of “information”
that is necessary to keep the system states in (or near) this subspace.

We conclude this section with the following result which is an immediate
corollary of Theorem 3.

###### Corollary 2

Suppose there exists a finite-positive number $\epsilon_{\mathcal{K}}>0$ such
that

$\displaystyle\max_{\begin{subarray}{c}j\in\mathcal{N}\end{subarray}}\sup_{\begin{subarray}{c}t\in[0,\,T]\end{subarray}}\inf_{\begin{subarray}{c}y\in\tilde{\mathscr{V}}_{\mathcal{K}}^{*}\end{subarray}}\|\phi_{\neg
j}(t,x_{0},u_{\neg j}(t))-y\|\leq\epsilon,\quad\forall x_{0}\in\mathscr{F},$
(17)

where

$\displaystyle\tilde{\mathscr{V}}_{\mathcal{K}}^{*}=\sup\left\\{\bigcap_{j\in\mathcal{N}}\mathscr{V}_{j}~{}\biggm{|}~{}\mathscr{V}_{j}\subset\sup\mathfrak{I}_{j}(A,B_{\neg
j};\mathscr{X}),~{}\forall
j\in\mathcal{N},~{}~{}\exists\,K\in\mathcal{K}\right\\}\supset\mathscr{F}.$
(18)

Then, the (controlled)-invariance entropy of the multi-channel system in (1)
is given by

$\displaystyle h_{\rm
inv}(\mathscr{F},\tilde{\mathscr{V}}_{\mathcal{K}}^{*})=\lim_{\begin{subarray}{c}\epsilon_{\mathcal{K}}\searrow
0\end{subarray}}\biggm{\\{}\varlimsup_{\begin{subarray}{c}T\to\infty\end{subarray}}\frac{1}{T}\log
r_{\rm
inv}(T,\epsilon_{\mathcal{K}},\mathscr{F},\tilde{\mathscr{V}}_{\mathcal{K}}^{*})\biggm{\\}}.$
(19)

###### Remark 5

Note that the bounds for $h_{\rm inv}(\mathscr{F},\tilde{\mathscr{V}}^{*})$
and $h_{\rm inv}(\mathscr{F},\tilde{\mathscr{V}}_{\mathcal{K}}^{*})$ are
different, since they may depend on their respective classes of control
functions. Moreover, the following $h_{\rm
inv}(\mathscr{F},\tilde{\mathscr{V}}^{*})\leq h_{\rm
inv}(\mathscr{F},\tilde{\mathscr{V}}_{\mathcal{K}}^{*})$ also holds true.

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