# Equality in Degrees of Compactness: Schauder’s Theorem and s-numbers

Asuman Güven AKSOY, Daniel Akech Thiong  ∗Department of Mathematics, Claremont
McKenna College, 850 Columbia Avenue, Claremont, CA 91711, USA.
<EMAIL_ADDRESS>1Department of Mathematics, Claremont Graduate University, 710
N. College Avenue, Claremont, CA 91711, USA<EMAIL_ADDRESS>
###### Abstract.

We investigate an extension of Schauder’s theorem by studying the relationship
between various $s$-numbers of an operator $T$ and its adjoint $T^{*}$. We
have three main results. First, we present a new proof that the approximation
number of $T$ and $T^{*}$ are equal for compact operators. Second, for non-
compact, bounded linear operators from $X$ to $Y$, we obtain a relationship
between certain $s$-numbers of $T$ and $T^{*}$ under natural conditions on $X$
and $Y$. Lastly, for non-compact operators that are compact with respect to
certain approximation schemes, we prove results for comparing the degree of
compactness of $T$ with that of its adjoint $T^{*}$.

###### Key words and phrases:

s-numbers, approximation schemes, Schauder’s theorem

###### 2010 Mathematics Subject Classification:

Primary 47A16, 47B10; Secondary 47A68

##  1\. Introduction

In the following, we give a brief review of the background, notation, and
terminology that will be relevant to this paper. Let $\mathcal{L}(X,Y)$ denote
the normed vector space of all continuous operators from $X$ to $Y$, $X^{*}$
be the dual space of $X$, and $\mathcal{K}(X,Y)$ denote the collection of all
compact operators from $X$ to $Y$. Denote by
$T^{*}\in\mathcal{L}(Y^{*},X^{*})$ the adjoint operator of
$T\in\mathcal{L}(X,Y)$. The well known theorem of Schauder states that
$T\in\mathcal{K}(X,Y)$ if and only if $T^{*}\in\mathcal{K}(Y^{*},X^{*})$. The
proof of Schauder’s theorem that uses Arzel$\grave{a}$-Ascoli Theorem is
presented in most textbooks on functional analysis (see, e.g., [19]). A new
and simple proof which does not depend on Arzel$\grave{a}$-Ascoli can be found
in [20]. Recalling the fact that a class of operators
$\mathcal{A}(X,Y)\subset\mathcal{L}(X,Y)$ is called symmetric if
$T\in\mathcal{A}(X,Y)$ implies $T^{*}\in\mathcal{A}(Y^{*},X^{*})$, we note
that Schauder’s Theorem assures that the class $\mathcal{K}(X,Y)$ of compact
operators between arbitrary Banach spaces $X$ and $Y$ is a symmetric operator
ideal in $\mathcal{L}(X,Y)$.

In [18] F. Riesz proved compact operators have at most countable set of
eigenvalues $\lambda_{n}(T)$, which arranged in a sequence, tend to zero. This
result raises the question of what are the conditions on
$T\in\mathcal{L}(X,Y)$ such that $(\lambda_{n}(T))\in\ell_{q}$? Specifically,
what is the rate of convergence to zero of the sequence $(\lambda_{n}(T))$? To
answer these questions, in [15] and [17], A. Pietsch developed $s$-numbers
$s_{n}(T)$ (closely related to singular values), which characterize the degree
of compactness of $T$. The concept of s-numbers $s_{n}(T)$ is introduced
axiomatically in [15], their relationship to eigenvalues are given in detail
in [17].

######  Definition 1.1.

A map which assigns to every operator $T$ a scalar sequence, is said to be a
$s$-function if the following conditions are satisfied:

  1. (1)

$||T||=s_{1}(T)\geq s_{2}(T)\geq\dots\geq 0$ for $T\in\mathcal{L}(X,Y)$.

  2. (2)

$s_{m+n-1}(S+T)\leq s_{m}(T)+s_{n}(T)$ for $S,T\in\mathcal{L}(X,Y)$ .

  3. (3)

$s_{n}(RTK)\leq||R||s_{n}(T)||K||$
for $K\in\mathcal{L}(X_{0},X)$, $T\in\mathcal{L}(X,Y)$,
$R\in\mathcal{L}(Y,Y_{o})$.

  4. (4)

If rank $(T)<n$, then $s_{n}(T)=0$.

  5. (5)

$s_{n}(I_{n})=1$ where $I_{n}$ is the identity map of $\ell_{2}^{n}$.

We call $s_{n}(T)$ the n-th $s$-number of the operator $T$. Observe that
$s_{n}(T)$ depends on $T$ continuously since

$|s_{n}(S)-s_{n}(T)|\leq||S-T||.$

In [15] it is shown that there is only one $s$-function on the class of all
operators between Hilbert spaces. For example, if we let $T$ be a diagonal
operator acting on $\ell_{2}$ such that

$T(x_{n})=(\lambda_{n}x_{n}),\quad\mbox{where}\,\,\,\lambda_{1}\geq\lambda_{2}\geq\dots\geq
0\quad\mbox{then}\,\,\,s_{n}(T)=\lambda_{n}$

for every $s$-function.

However for Banach spaces there are several possibilities of assigning to
every operator $T:X\to Y$ a certain sequence of numbers $\\{s_{n}(T)\\}$ which
characterizes the degree of approximability or compactness of $T$. The main
examples of s-numbers to be used in this paper are approximation numbers,
Kolmogorov numbers, Gelfand numbers and symmetrized approximation numbers
which are all defined below.

First, for two arbitrary normed spaces $X$ and $Y$, we define the collection
of the finite-rank operators as follows:

$\mathcal{F}_{n}(X,Y)=\\{A\in\mathcal{L}(X,Y):\text{rank}(A)\leq
n\\},\quad\mbox{and}\quad\mathcal{F}(X,Y)=\bigcup_{n=0}^{\infty}\mathcal{F}_{n}(X,Y)$

which forms the smallest ideal of operators that exists.

######  Definition 1.2.

In the following we define the s-numbers we will use.

  1. (1)

The nth approximation number

$a_{n}(T)=\inf\\{||T-A||:A\in\mathcal{F}_{n}(X,Y)\\},\quad n=0,1,\dots$

Note that $a_{n}(T)$ provides a measure of how well T can be approximated by
finite mappings whose range is at most n-dimensional. It is clear that the
sequence $\\{a_{n}(T)\\}$ is monotone decreasing and
$\displaystyle\lim_{n\to\infty}a_{n}(T)=0$ if and only if $T$ is the limit of
finite rank operators. It is known that the largest $s$-number is the
approximation number. This is so because $a:S\to(a_{n}(S))$ is an $s$-function
and if we consider $S\in\mathcal{L}(X,Y)$ and if $L\in\mathcal{F}(X,Y)$ with
$\text{rank}(L)<n$, then

$s_{n}(S)\leq s_{n}(L)+||S-L||=||S-L||.$

Therefore $s_{n}(S)\leq a_{n}(S)$. See [7] or [15] for more details.

  2. (2)

The nth Kolmogorov diameter of $T\in\mathcal{L}(X)$ is defined by

$\delta_{n}(T)=\inf\\{||Q_{G}T||:\dim G\leq n\\}$

where the infimum is over all subspaces $G\subset X$ such that $\dim G\leq n$
and $Q_{G}$ denotes the canonical quotient map $Q_{G}:X\rightarrow X/G$.

  3. (3)

The nth Gelfand number of $T$, $c_{n}(T)$ is defined as:

$c_{n}(T)=\inf\\{\epsilon>0:||Tx||\leq\sup_{1\leq i\leq k}|\langle
x,a_{i}\rangle|+\epsilon||x||\\}$

where $a_{i}\in X^{*},1\leq i\leq k\text{ with }k<n$. It follows that an
operator $T$ is compact if and only if $c_{n}(T)\to 0$ as $n\to\infty$.

  4. (4)

The nth symmetrized approximation number $\tau_{n}(T)$ for any operator $T$
defined between arbitrary Banach spaces $X$ and $Y$ is defined as follows:

$\tau_{n}(T)=\delta_{n}(J_{Y}T)\quad\mbox{where}\quad
J_{Y}:Y\to\ell_{\infty}(B_{Y^{*}})$

is an embedding map. Note that above definition is equivalent to

$\tau_{n}(T)=a_{n}(J_{Y}TQ_{X})$

as well as to

$\tau_{n}(T)=c_{n}(TQ_{X}),$

where $Q_{X}:\ell_{1}(B_{X})\rightarrow X$ is a metric surjection onto $X$
given by $Q_{X}(\xi_{x})=\sum_{B_{X}}\xi_{x}x$ for
$(\xi_{x})\in\ell_{1}(B_{X})$ .

It is possible to compare various s-numbers such as
$a_{n}(T),\,\delta_{n}(T),\,c_{n}(T)$ if one imposes some mild restrictions on
$X$ and $Y$. With this purpose in mind we define well known concepts of
lifting and extension properties.

######  Definition 1.3.

In the following we introduce two well-known important properties of Banach
spaces. See [7] for details.

  1. (1)

We say that a Banach space $X$ has the lifting property if for every
$T\in\mathcal{L}(X,Y/F)$ and every $\epsilon>0$ there exists an operator
$S\in\mathcal{L}(X,Y)$ such that

$||S||\leq(1+\epsilon)||T||\quad\mbox{and}\,\,\,T=Q_{F}S,$

where $F$ is a closed subspace of the Banach space $Y$ and $Q_{F}:Y\rightarrow
Y/F$ denotes the canonical projection.

######  Example 1.4.

The Banach space $\ell_{1}(\Gamma)$ of summable number families
$\\{\lambda_{\gamma}\\}_{\gamma\in\Gamma}$ over an arbitrary index set
$\Gamma$, whose elements $\\{\lambda_{\gamma}\\}_{\gamma\in\Gamma}$ are
characterized by $\sum_{\gamma\in\Gamma}|\lambda_{\gamma}|<\infty$, has the
metric lifting property.

  2. (2)

A Banach space $Y$ is said to have the extension property if for each
$T\in\mathcal{L}(M,Y)$ there exists an operator $S\in\mathcal{L}(X,Y)$ such
that $T=SJ_{M}$ and $||T||=||S||$, where $M$ is a closed subspace of an
arbitrary Banach space $X$ and $J_{M}:M\rightarrow Y$ is the canonical
injection.

######  Example 1.5.

The Banach space $\ell_{\infty}(\Gamma)$ of bounded number families
$\\{\lambda_{\gamma}\\}_{\gamma\in\Gamma}$ over an arbitrary index set
$\Gamma$ has the metric extension property.

We mention a couple of facts to illustrate the importance of lifting and
extensions properties with respect to $s$-numbers. If $T$ is any map from a
Banach space with metric lifting property to an arbitrary Banach space, then
$a_{n}(T)=\delta_{n}(T)$ ([7], Prop. $2.2.3)$. It is also known that every
Banach space $X$ appears as a quotient space of an appropriate space
$\ell_{1}(\Gamma)$ (see [7], p.$52$). Furthermore, If $T$ is any map from an
arbitrary Banach space into a Banach space with metric extension property,
then $a_{n}(T)=c_{n}(T)$ ([7], Prop. $2.3.3$). Additionally, every Banach
space $Y$ can be regarded as a subspace of an appropriate space
$\ell_{\infty}(\Gamma)$ (see [7], p.$60$).

For non-compact operator $T\in\mathcal{L}(X,Y)$, we do not have too much
information about the relationship between $s_{n}(T)$ with $s_{n}(T^{*})$. In
this paper, by imposing certain natural conditions on $X$ and $Y$ we are able
to obtain a relationship between $s_{n}(T)$ with $s_{n}(T^{*})$ for certain
s-numbers. Moreover, using a new characterization of compactness due to Runde
[20] together with the Principle of Local Reflexivity, we give a different,
simpler proof of Hutton’s theorem [11] establishing that for any compact map
$T$,

$a_{n}(T)=a_{n}(T^{*})\quad\mbox{for all}\,\,\,n.$

Next we consider operators which are not compact but compact with respect to
certain approximation schemes Q. We call such operators as Q-compact and prove
that for any Q-compact operator $T$, one has $\tau_{n}(T)=\tau_{n}(T^{*})$.
This result answers the question of comparing the degree of compactness for
$T$ and its adjoint $T^{*}$ for non-compact operators $T$.

##  2\.  Comparing $s_{n}(T)$ and $s_{n}(T^{*})$

Hutton in [11] used the Principle of Local Reflexivity (PLR) to prove that for
$T\in\mathcal{K}(X,Y)$ we have

$a_{n}(T)=a_{n}(T^{*})\quad\mbox{ for all}\,\,\,n.$

This result fails for non-compact operators. For example, if
$T=I:\ell_{1}\rightarrow c_{0}$ is the canonical injection and
$T^{*}:\ell_{1}\rightarrow\ell_{\infty}$ is the natural injection, then one
can show

$1=a_{n}(T)\neq a_{n}(T^{*})=\frac{1}{2}.$

On the other hand by considering the ball measure of non-compactness, namely,

$\gamma(T):=\inf\\{r>0:T(B_{X})\subset\bigcup_{k=1}^{n}A_{k},\,\,\displaystyle\text{max}_{1\leq
k\leq n}\text{ diam }(A_{k})<r,\,n\in\mathbb{N}\\}$

Astala in [4] proved that if $T\in\mathcal{L}(X,Y)$, where X and Y are
arbitrary Banach spaces with metric lifting and extension property,
respectively, then

$\gamma(T)=\gamma(T^{*}).$

Our first result is a different, simpler proof of Hutton’s Theorem. We use
only the characterization of compactness by Runde [20], together with the
Principle of Local Reflexivity. Lindenstrass and Rosenthal [13] discovered a
principle that shows that all Banach spaces are “locally reflexive” or said in
another way, every bidual $X^{**}$ is finitely representable in the original
space $X$. The following is a stronger version of this property called
Principle of Local Reflexivity (PLR) due to Johnson, Rosenthal and Zippin [9]:

######  Definition 2.1.

Let $X$ be a Banach space regarded as a subspace of $X^{**}$, let $E$ and $F$
be finite dimensional subspaces of $X^{**}$ and $X^{*}$ respectively and let
$\epsilon>0$. Then there exist a one-to-one operator $T:E\to X$ such that

  1. (1)

$T(x)=x$ for all $x\in X\cap E$

  2. (2)

$f(Te)=e(f)$ for all $e\in E$ and $f\in F$

  3. (3)

$||T||||T^{-1}||<1+\epsilon$.

PLR is an effective tool in Banach space theory. For example Oja and Silja in
[14] investigated versions of the principle of local reflexivity for nets of
subspaces of a Banach space and gave some applications to duality and lifting
theorems.

######  Lemma 2.2 (Lemma $1$ in [20]).

Let $X$ be a Banach space and let $T\in\mathcal{L}(X)$. Then
$T\in\mathcal{K}(X)$ if and only if, for each $\epsilon>0$, there is a finite-
dimensional subspace $F_{\epsilon}$ of $X$ such that
$||Q_{F_{\epsilon}}T||<\epsilon$, where $Q_{F_{\epsilon}}:X\rightarrow
X/F_{\epsilon}$ is the canonical projection.

######  Theorem 2.3.

Let $T\in\mathcal{K}(X)$. Then $a_{n}(T)=a_{n}(T^{*})$ for all $n$.

###### Proof.

Since one always has $a_{n}(T^{*})\leq a_{n}(T)$, if we have $a_{n}(T)\leq
a_{n}(T^{**})$, then $a_{n}(T^{**})\leq a_{n}(T^{*})$ would imply
$a_{n}(T)\leq a_{n}(T^{*})$. Thus we must verify $a_{n}(T)\leq a_{n}(T^{**})$.
To this end, suppose $T\in\mathcal{K}(X)$, by Schauder’s theorem, $T^{*}$ and
$T^{**}$ are compact. Let $\epsilon>0$, then by definition, there exists
$A\in\mathcal{F}_{n}(X^{**})$ such that $||T^{**}-A||<a_{n}(T^{**})+\epsilon$.

By Lemma 2.2, there are finite-dimensional subspaces $E_{\epsilon}$ of
$X^{**}$ and $F_{\epsilon}$ of $X^{*}$ such that
$||Q_{E_{\epsilon}}T^{**}||<\epsilon$, where
$Q_{E_{\epsilon}}:X^{**}\rightarrow X^{**}/E_{\epsilon}$ and
$||Q_{F_{\epsilon}}T^{*}||<\epsilon$, where $Q_{F_{\epsilon}}:X^{*}\rightarrow
X^{*}/F_{\epsilon}$.

By the Principle of Local Reflexivity (PLR), there exists a one-to-one linear
operator $S:E_{\epsilon}\rightarrow X$ such that $||S||||S^{-1}||<1+\epsilon$,
$y^{*}(Sx^{**})=x^{**}(y^{*})$ for all $x^{**}\in E_{\epsilon}$ and all
$y^{*}\in F_{\epsilon}$, and $S_{|E_{\epsilon}\cap X}=I$.

Let $J:X\to X^{**}$ be the canonical map. By the Hahn-Banach theorem, since
$E_{\epsilon}$ is a subspace of $X^{**}$, $S:E_{\epsilon}\rightarrow X$ can be
extended to a linear operator $\overline{S}:X^{**}\rightarrow X$.

We now have $T\in\mathcal{L}(X)$ and $\overline{S}AJ\in\mathcal{L}(X)$ and
$\text{rank }(\overline{S}AJ)=\text{rank}(A)<n$, and therefore

$a_{n}(T)\leq||T-\overline{S}AJ||.$

To get an upper bound for $||T-\overline{S}AJ||$ we estimate
$||Tx-\overline{S}AJ(x)||$ for $x\in B_{X}$ using an appropriate element
$z_{j}$ of the covering of the set $T(B_{X})$.

Indeed, the compactness of $T$ implies that $T(B_{X})$ is relatively compact
so that one can extract a finite-dimensional subset $Y_{\epsilon}\subset
T(B_{X})\subset X$ and let $z_{j}=Tx_{j}$ be the n elements forming a basis.

Let $x\in B_{X}$. Then we have

$\displaystyle a_{n}(T)$ $\displaystyle\leq|Tx-\overline{S}AJ(x)||\leq||Tx-
z_{j}||+||z_{j}-\overline{S}AJ(x)||$
$\displaystyle\leq\epsilon+||z_{j}-\overline{S}AJ(x)||=\epsilon+||\overline{S}z_{j}-\overline{S}AJ(x)||$
$\displaystyle\leq\epsilon+(1+\epsilon)||z_{j}-AJ(x)||<\epsilon+(1+\epsilon)(a_{n}(T^{*})+\epsilon)$

since

$\displaystyle||z_{j}-AJ(x)||$
$\displaystyle=||Jz_{j}-AJ(x)||\leq||Jz_{j}-JTx||+||JTx-AJ(x)||$
$\displaystyle\leq\epsilon+||JTx-AJx||=\epsilon+||T^{**}Jx-
AJx||\leq||T^{**}-A||$ $\displaystyle<a_{n}(T^{*})+\epsilon.$

It follows that $a_{n}(T)\leq a_{n}(T^{**})$, as promised. ∎

######  Theorem 2.4.

If $T\in\mathcal{L}(X,Y)$, where X and Y are arbitrary Banach spaces with
metric lifting and extension property, respectively, then
$\delta_{n}(T^{*})=\delta_{n}(T)$ for all $n$.

###### Proof.

It is known that if $T\in\mathcal{L}(X,Y)$, where X and Y are arbitrary Banach
spaces, then $\delta_{n}(T^{*})=c_{n}(T)$ ( [7], Prop. $2.5.5$). We also know
that if $T\in\mathcal{L}(X,Y)$, where X and Y are arbitrary Banach spaces with
metric lifting and extension property, respectively, then
$\delta_{n}(T)=a_{n}(T)=c_{n}(T)$. Hence,

$\delta_{n}(T^{*})=c_{n}(T)=a_{n}(T)=\delta_{n}(T).\qed$

######  Remark 2.5.

As stated before, Astala in [4] proved that if $T\in\mathcal{L}(X,Y)$, where X
and Y are arbitrary Banach spaces with metric lifting and extension property,
respectively, then $\gamma(T)=\gamma(T^{*})$, where $\gamma(T)$ denotes the
measure of non-compactness of $T$. In [1], it is shown that
$\displaystyle\lim_{n\to\infty}\delta_{n}(T)=\gamma(T)$. This relationship
between Kolmogorov diameters and the measure of non-compactness together with
Theorem 2.4 provide an alternative proof for the result of Astala.

######  Theorem 2.6.

If $T\in\mathcal{K}(X,Y)$, where X and Y are arbitrary Banach spaces with
metric lifting and extension property, respectively, then
$c_{n}(T^{*})=c_{n}(T)$ for all $n$.

###### Proof.

If $T\in\mathcal{K}(X,Y)$, then it is known that $\delta_{n}(T)=c_{n}(T^{*})$
([7], Prop. $2.5.6$). If X and Y are Banach spaces with metric lifting and
extension property, respectively, then we also have
$\delta_{n}(T)=a_{n}(T)=c_{n}(T)$. Thus, $c_{n}(T^{*})=c_{n}(T)$ for all $n$.
∎

######  Remark 2.7.

In [10] it is shown that if $X$ has the lifting property, then $X^{*}$ has the
extension property. However, if $Y$ has the extension property, then $Y^{*}$
has the lifting property if and only if $Y$ is finite-dimensional. Therefore
one can observe that if $X$ has the lifting property and $Y$ is finite-
dimensional with the extension property, then $Y^{*}$ has the lifting property
and $X^{*}$ has the extension property, so that we have

$\delta_{n}(T^{*})=a_{n}(T^{*})=c_{n}(T^{*}).$

##  3\. Compactness with Approximation schemes

Approximation schemes were introduced in Banach space theory by Butzer and
Scherer in $1968$ [5] and independently by Y. Brudnyi and N. Kruglyak under
the name of “approximation families” in [6]. They were popularized by Pietsch
in his 1981 paper [16], for later developments we refer the reader to [1, 2,
3]. The following definition is due to Aksoy and generalizes the classical
concept of approximation scheme in a way that allows using families of subsets
of $X$ instead of elements of $X$, which is useful when we deal with n-widths.

######  Definition 3.1 (Generalized Approximation Scheme).

Let $X$ be a Banach space. For each $n\in\mathbb{N}$, let $Q_{n}=Q_{n}(X)$ be
a family of subsets of $X$ satisfying the following conditions:

  * $(GA1)$

$\\{0\\}=Q_{0}\subset Q_{1}\subset\cdots\subset Q_{n}\subset\dots$.

  * $(GA2)$

$\lambda Q_{n}\subset Q_{n}$ for all $n\in N$ and all scalars $\lambda$.

  * $(GA3)$

$Q_{n}+Q_{m}\subseteq Q_{n+m}$ for every $n,m\in N$.

Then $Q(X)=(Q_{n}(X))_{n\in N}$ is called a _generalized approximation scheme_
on $X$. We shall simply use $Q_{n}$ to denote $Q_{n}(X)$ if the context is
clear.

We use here the term “generalized” because the elements of $Q_{n}$ may be
subsets of $X$. Let us now give a few important examples of generalized
approximation schemes.

######  Example 3.2.

  1. (1)

$Q_{n}=$ the set of all at-most-$n$-dimensional subspaces of any given Banach
space $X$.

  2. (2)

Let $E$ be a Banach space and $X=L(E)$; let $Q_{n}=N_{n}(E)$, where
$N_{n}(E)=$ the set of all $n$-nuclear maps on $E$ [15].

  3. (3)

Let $a^{k}=(a_{n})^{1+\frac{1}{k}},$ where $(a_{n})$ is a nuclear exponent
sequence. Then $Q_{n}$ on $X=L(E)$ can be defined as the set of all
$\Lambda_{\infty}(a^{k})$-nuclear maps on $E$ [8].

######  Definition 3.3 (Generalized Kolmogorov Number).

Let $B_{X}$ be the closed unit ball of $X$, $Q=Q(X)=(Q_{n}(X))_{n\in N}$ be a
_generalized approximation scheme_ on $X$, and $D$ be a bounded subset of $X$.
Then the $n^{\text{th}}$ _generalized Kolmogorov number_ $\delta_{n}(D;Q)$ of
$D$ with respect to $Q$ is defined by

$\delta_{n}(D;Q)=\inf\\{r>0:D\subset rB_{X}+A\text{ for some }A\in
Q_{n}(X)\\}.$ (3.1)

Assume that $Y$ is a Banach space and $T\in\mathcal{L}(Y,X)$. The
$n^{\text{th}}$ Kolmogorov number $\delta_{n}(T;Q)$ of $T$ is defined as
$\delta_{n}(T(B_{Y});Q)$.

It follows that $\delta_{n}(T;Q)$ forms a non-increasing sequence of non-
negative numbers:

$\|T\|=\delta_{0}(T;Q)\geq\delta_{1}(T;Q)\geq\cdots\geq\delta_{n}(T;Q)\geq 0.$
(3.2)

We are now able to introduce $Q$-compact sets and operators:

######  Definition 3.4 ($Q$-compact set).

Let $D$ be a bounded subset of $X$. We say that $D$ is $Q$-_compact_ if
$\displaystyle\lim_{n}\delta_{n}(D;Q)=0$.

######  Definition 3.5 ($Q$-compact map).

We say that $T\in L(Y,X)$ is a $Q$-_compact map_ if $T(B_{Y})$ is a
$Q$-compact set,

$\displaystyle\lim_{n}\delta_{n}(T;Q)=0.$

$Q$-compact maps are a genuine generalization of compact maps since there are
examples of $Q$-compact maps which are not compact in the usual sense.

In the following we present two examples of Q-compact maps which are not
compact. First of this examples is known (see [1]) and it involves a
projection $P:L_{p}[0,1]\to R_{p}$ where $R_{p}$ denotes the closure of the
span of the space of Rademacher functions. Second example is new and
illustrates the fact that if $B_{w}$ is a weighted backward shift on
$c_{0}(\mathbb{N})$ with $w=(w_{n})_{n}$ a bounded sequence not converging to
0, then $B_{w}$ is $Q$-compact operator which is not compact.

######  Example 3.6.

Let $\\{r_{n}(t)\\}$ be the space spanned by the Rademacher functions. It can
be seen from the Khinchin inequality [12] that

$\ell_{2}\approx\\{r_{n}(t)\\}\subset L_{p}[0,1]\text{ for all }1\leq
p\leq\infty.$ (3.3)

We define an approximation scheme $A_{n}$ on $L_{p}[0,1]$ as follows:

$A_{n}=L_{p+\frac{1}{n}}.$ (3.4)

$L_{p+\frac{1}{n}}\subset L_{p+\frac{1}{n+1}}$ gives us $A_{n}\subset
A_{n+1}$. for $n=1,2,\dots,$ and it is easily seen that $A_{n}+A_{m}\subset
A_{n+m}$ for $n,m=1,2,\dots,$ and that $\lambda A_{n}\subset A_{n}$ for all
$\lambda$. Thus $\\{A_{n}\\}$ is an approximation scheme.

It can be shown that for $p\geq 2$ the projection $P:L_{p}[0,1]\to R_{p}$ is a
non-compact $Q$-compact map, where $R_{p}$ denotes the closure of the span of
$\\{r_{n}(t)\\}$ in $L_{p}[0,1]$. (See [1] for details)

Next we give another example is an $Q$-operator which is not compact .

######  Example 3.7.

Consider the weighted backward shift

$B(x_{1},x_{2},x_{3},\dots)=(w_{2}x_{2},w_{3}x_{3},w_{4}x_{4},\dots)$

where $w=(w_{n})_{n}$ is a sequence of non-zero scalars called a _weight
sequence_. Any weighted shift is a linear operator and is bounded if and only
if $w$ is a bounded sequence.

Let $w=(w_{n})_{n}$ be a bounded sequence of positive real numbers. The
unilateral weighted shift on $c_{0}(\mathbb{N})$ is defined by

$B_{w}(e_{1})=0\quad\mbox{ and }\quad B_{w}(e_{n})=w_{n}e_{n-1}\quad\mbox{ for
all }\quad n\geq 2.$

######  Proposition 3.8.

Suppose the approximation scheme $Q=(A_{n})_{n=1}^{\infty}$ of
$c_{0}(\mathbb{N})$ is defined as $A_{n}=\ell_{n}(\mathbb{N})$ for all $n$.
Then any bounded weighted shift on $c_{0}$ is $Q$-compact

###### Proof.

Let $B_{w}$ be any bounded and linear weighted shift on $c_{0}$, then
$w=(w_{n})_{n}$ is a bounded weight. Let $m\geq 1$. Consider,

$\displaystyle\delta_{m}(B_{w}(U_{c_{0}}),(A_{n})_{n})$
$\displaystyle=\inf\\{r>0:B_{w}(U_{c_{0}})\subseteq rU_{c_{0}}+\ell_{m}\\}$
$\displaystyle=\inf\\{r>0:\forall x\in U_{c_{0}},\exists y\in
U_{c_{0}},\exists z\in\ell_{m}\mbox{ with }B_{w}(x)=ry+z\\}.$

Let $x=(x_{n})_{n\geq 1}\in U_{c_{0}}$. Let us define $y=(y_{n})_{n\geq 1}\in
U_{c_{0}}$ and $z=(z_{n})_{n\geq 1}\in\ell_{1}\subseteq\ell_{m}$ such that
$B_{w}(x)=\frac{1}{2^{m}}y+z$.

Let $A:=\\{n\geq 1:2^{m}\left\lvert x_{n}w_{n}\right\rvert>1\\}$. The set $A$
is finite, otherwise $(w_{n})_{n}$ is unbounded. Set,

$\left\\{\begin{array}[]{ll}x_{n}w_{n}=z_{n-1}\\\ y_{n-1}=0,&\quad\forall n\in
A.\\\ \end{array}\right.$

Observe that $(w_{n}x_{n})_{n\in\mathbb{N}\setminus A}\in c_{0}$, hence there
exists a subsequence $(n_{k})_{k}$ such that $\sum_{k=1}^{\infty}\left\lvert
w_{n_{k}}x_{n_{k}}\right\rvert<\infty$. Set,

$\left\\{\begin{array}[]{ll}x_{n_{k}}w_{n_{k}}=z_{n_{k}-1}\\\
y_{n_{k}-1}=0,&\quad\forall k\geq 1.\\\ \end{array}\right.$

Finally, set

$\left\\{\begin{array}[]{ll}2^{m}x_{n}w_{n}=y_{n-1}\\\ z_{n-1}=0,&\quad\forall
n\in\mathbb{N}\setminus\\{(n_{k})_{k}\cup A\\}.\\\ \end{array}\right.$

Hence, $x_{n}w_{n}=\frac{1}{2^{m}}y_{n-1}+z_{n-1}$, for all $n\geq 2$. In
other words, $B_{w}(x)=\frac{1}{2^{m}}y+z$. Note that $y\in U_{c_{0}}$ and
$z\in\ell_{1}\subset\ell_{m}$. In conclusion,
$\delta_{m}(B_{w}(U_{c_{0}}),(A_{n})_{n})\leq\frac{1}{2^{m}}$. As $m$ goes to
$\infty$, we obtain that $\delta_{m}(B_{w}(U_{c_{0}}),\ (A_{n})_{n})$ goes to
$0$ and $B_{w}$ is $Q$-compact. ∎

It is well-known that $B_{w}$ is compact if and only if $w=(w_{n})_{n}$ is a
null-sequence.

######  Corollary 3.9.

Let $B_{w}$ be a weighted backward shift on $c_{0}(\mathbb{N})$ with
$w=(w_{n})_{n}$ a bounded sequence not converging to 0. Consider the
approximation schemes on $c_{0}(\mathbb{N})$ as $Q=(A_{n})_{n=1}^{\infty}$
with $A_{n}=\ell_{n}(\mathbb{N})$ for all $n$. Then, $B_{w}$ is a non-compact
$Q$-compact operator.

Our next objective here is to ascertain whether or not Schauder’s type of
theorem is true for $Q$-compact maps. For this purpose we use symmetrized
approximation numbers of $T$. For our needs, we choose the closed unit ball
$B_{Z}$ of the Banach space $Z$ as an index set $\Gamma$. Our proof of the
Schauder’s theorem for Q-compact operators will depend on the fact that
$\ell_{1}(B_{Z})$ has the lifting property and $\ell_{\infty}(B_{Z})$ has the
extension property. First we recall the following proposition.

######  Proposition 3.10 (Refined version of Schauder’s theorem [7], p.$84$).

An operator $T$ between arbitrary Banach spaces $X$ and $Y$ is compact if and
only if

$\lim_{n\to\infty}\tau_{n}(T)=0$

and moreover,

$\tau_{n}(T)=\tau_{n}(T^{*}).$

Motivated by this, we give the definition of Q-compact operators using the
symmetrized approximation numbers.

######  Definition 3.11.

We say $T$ is Q-symmetric compact if and only if

$\lim_{n\to\infty}\tau_{n}(T,Q)=0.$

######  Remark 3.12.

We need the following simple facts for our proof, for details we refer the
reader to [7] Prop. $2.5.4-6$.

  1. a)

Recall that $\tau_{n}(T,Q)=c_{n}(TQ_{X},Q)$, where
$Q_{X}:\ell_{1}(B_{X})\rightarrow X$.

  2. b)

We will also abbreviate the canonical embedding

$K_{\ell_{1}(B_{Y^{*}})}:\ell_{1}(B_{Y^{*}})\rightarrow\ell_{\infty}(B_{Y^{*}})^{*}$

by $K$ so that $Q_{Y^{*}}=J^{*}_{Y}K$.

  3. c)

Denote by $P_{0}:\ell_{\infty}(B_{X^{**}})\rightarrow\ell_{\infty}(B_{X})$ the
operator which restricts any bounded function on $B_{X^{**}}$ to the subset
$K_{X}(B_{X})\subset B_{X^{**}}$ so that $Q^{*}_{X}=P_{0}J_{X^{*}}$.

  4. d)

The relations (b) and (c) are crucial facts for the estimates of
$\delta_{n}(T^{*},Q^{*})$ and $c_{n}(T^{*},Q^{*})$. In particular, we have
$c_{n}(T^{*},Q^{*})\leq\delta_{n}(T,Q)$.

We now state and prove the following theorem which states that the degree of
Q-compactness of $T$ and $T^{*}$ is the same in so far as it is measured by
the symmetrized approximation numbers $\tau_{n}$.

######  Theorem 3.13 (Schauder’s theorem for Q-compact operators).

Let $T\in\mathcal{L}(X,Y)$ with $X,Y$ are arbitrary Banach spaces, and let
$Q=(Q_{n}(X))$ be a generalized approximation scheme on $X$. Then

$\tau_{n}(T^{*},Q^{*})=\tau_{n}(T,Q)$

for all $n$.

###### Proof.

Let us show that $\tau_{n}(T^{*},Q^{*})=\tau_{n}(T,Q)$. By Remark 3.12 parts
(a) and (b) we have the following estimates:

$\displaystyle\tau_{n}(T^{*},Q^{*})$
$\displaystyle=c_{n}(T^{*}Q_{Y^{*}},Q^{*})=c_{n}(T^{*}J_{Y}^{*}K,Q^{*})$
$\displaystyle\leq
c_{n}((J_{Y}T)^{*},Q^{*})\leq\delta_{n}(J_{Y}T,Q)=t_{n}(T,Q)$

Conversely, we have by using Remark 3.12 parts (c) and (d):

$\displaystyle t_{n}(T,Q)$
$\displaystyle=c_{n}(TQ_{X},Q)=\delta_{n}(TQ_{X})^{*},Q^{*})$
$\displaystyle=\delta_{n}(Q_{X}^{*}T^{*},Q^{*})=\delta_{n}(P_{0}J_{X^{*}}T^{*},Q^{*})$
$\displaystyle\leq\delta_{n}(J_{X^{*}}T^{*},Q^{*})=t_{n}(T^{*},Q^{*}).$

∎

Next we define approximation numbers with respect to a given scheme as
follows:

######  Definition 3.14.

Given an approximation scheme $\\{Q_{n}\\}$ on $X$ and $T\in\mathcal{L}(X)$,
the n-th approximation number $a_{n}(T,Q)$ with respect to this approximation
scheme is defined as:

$a_{n}(T,Q)=\inf\\{||T-B||:\,\,B\in\mathcal{L}(X),\,\,\,B(X)\subseteq
Q_{n}\\}$

Let $X^{*}$ and $X^{**}$ be the dual and second dual of $X$. Note that if we
let $J:X\rightarrow X^{**}$ be the canonical injection and let $(X,Q_{n})$ be
an approximation scheme, then $(X^{**},J(Q_{n}))$ is an approximation scheme.

Let $\\{Q_{n}\\}$ and $\\{Q^{**}_{n}\\}:=\\{J(Q_{n})\\}$ denote the subsets of
$X$ and $X^{**}$ respectively.

######  Definition 3.15.

We say $(X,Q_{n})$ has the Extended Local Reflexivity Property (ELRP) if for
each countable subset $C$ of $X^{**}$ , for each $F\in Q_{n}^{**}$ for some
$n$ and each $\epsilon>0$, there exists a continuous linear map

$P:\mbox{span}(F\cup C)\to X\quad\mbox{such that}\,\,$

  1. (1)

$||P||\leq 1+\epsilon$

  2. (2)

$P\restriction_{C\cap X}=I(Identity)$

Note that ELRP is an analogue of local reflexivity principle which is
possessed by all Banach spaces.

######  Theorem 3.16.

Suppose $(X,Q_{n})$ has ELRP and $T\in\mathcal{L}(X)$ has separable range.
Then for each $n$ we have $a_{n}(T,Q)=a_{n}(T^{*},Q^{*})$.

###### Proof.

Since one always have $a_{n}(T^{*},Q^{*})\leq a_{n}(T,Q)$ we only need to
verify $a_{n}(T,Q)\leq a_{n}(T^{**},Q^{**})$. Let $J:X\to X^{**}$ be the
canonical map and $U_{X}$ be the unit ball of $X.$ Given $\epsilon>0$, choose
$B\in\mathcal{L}(X^{**})$ such that $B(X^{**})\in Q^{**}_{n}$ and

$||B-T^{**}||<\epsilon+a_{n}(T^{**},Q^{**}_{n}).$

Let $\\{z_{j}\\}$ be a countable dense set in $T(X)$, thus $Tx_{j}=z_{j}$
where $x_{j}\in X$. Consider the set

$K=\mbox{span}\\{(JTx_{j})_{1}^{\infty}\cup B(X^{**})\\}$

applying ELRP of $X$ we obtain a map

$P:K\to X\,\,\mbox{such that}\,\,\,||P||\leq
1+\epsilon\,\,\,\,\mbox{and}\,\,\,P\restriction_{(JTx_{j})_{1}^{\infty}\cap
X}=I$

For $x\in U_{X}$, consider

$\displaystyle||Tx-PBJx||$ $\displaystyle\leq||Tx-z_{j}||+||z_{j}-PBJx||$
$\displaystyle\leq\epsilon+||PJTx_{j}-PBJx||$
$\displaystyle\leq\epsilon+(1+\epsilon)||JTx_{j}-BJx||$
$\displaystyle\leq\epsilon+(1+\epsilon)[||JTx_{j}-JTx||+||JTx-BJx||]$
$\displaystyle\leq\epsilon+(1+\epsilon)[a_{n}(T^{**},Q^{**}_{n})+2\epsilon]$

and thus

$a_{n}(T,Q)\leq a_{n}(T^{**},Q^{**}_{n}).\qed$

## References

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Decentralized PI-control and Anti-windup in Resource Sharing Networks

We consider control of multiple stable first-order systems which have a control coupling described by an M-matrix. These agents are subject to incremental sector-bounded nonlinearities. We show that such plants can be globally asymptotically stabilized to a unique equilibrium using fully decentralized proportional integral anti-windup-equipped controllers subject to local tuning rules. In addition, we show that when the nonlinearities correspond to the saturation function, the closed-loop asymptotically minimizes a weighted 1-norm of the agents state mismatch. The control strategy is finally compared to other state-of-the-art controllers on a numerical district heating example.

Energy Systems, Constrained Control, Decentralized Control

§ INTRODUCTION

In this paper we consider the control of agents sharing a central distribution system with limited capacity as in [1]. We investigate systems where the positive action of one agent negatively impacts others. This type of competitive structure can arise in many domains, for instance internet congestion control [2, 3] and district heating systems [4]. In the district heating scenario, the structure arises because of the hydraulic constraints of the grid. If one agent (building) locally decides to increase their heat demand by opening their control valves, this will lead to higher flow rates and greater frictional pressure losses. These losses make it so that other agents now receive lower flow rates [4]. We consider a simple description of such systems as follows. Let $x_i$ be the state of each agent $i$. Then

\begin{equation}
\dot{x}_i = -a_i x_i + \sum_{j=1}^n B_{i,j} \sat{u_j} + w_i
\label{eq: simplified system description}
\end{equation}

where $a_i \in \R$ and $a_i > 0$, $u_i \in \R$ is the control action of each agent $i$, and $B \in \R^{n \times n}$ having elements $B_{i,j}$ represents the network interconnection between the agents. $w_i \in \R$ is a disturbance on agent $i$. The saturation function $\sat{\cdot}$ represents the limitations of the network, but in the later analysis we consider a richer class of nonlinear functions. To capture the resource-sharing aspect of the system we consider the case where $B$ is an M-matrix satisfying $B_{i,i} > 0$ and $B_{i,j} \leq 0$ when $i \neq j$.

In this work, we analyze (<ref>) under a fully decentralized PI (proportional-integral) control strategy. Semi-decentralized control strategies for multi-agent systems have been considered in the following works. In [5], each networked agent is equipped with a local controller that receives the control input of its neighbors. In [6], semi-decentralized anti-windup was considered for stable SISO plants that are decentralized in the linear domain, but become coupled during saturation. This is demonstrated on unmanned aerial vehicles. These and other works focus on stabilization when the disturbance $w$ in plant (<ref>) is energy bounded. In this work we focus instead on the asymptotic properties of plant (<ref>), which become important when $w$ is expected to vary slower than the system and can be assumed constant. We show that our strategy minimizes asymptotic costs of the form $\sum_{i=1}^n c_i|x_i|$. Previous works considering asymptotic optimality for plants of the form (<ref>) are [7] and [1]. In [7], it was shown that the static controller $u = -B^\top x$ asymptotically minimizes the cost $x^\top Ax + v^\top v$ where $A = \text{diag}(a_1, a_2, \cdots, a_n)$ and $v = \sat{u}$. This result also extends to the case when $B$ is not an M-matrix. In [1], it was shown that, when $B$ is an M-matrix and $w$ fulfills certain criteria, decentralized PI-controllers with a rank-one coordinating anti-windup scheme can minimize the cost $\max_i |x_i|$. Both of these control strategies maintain certain scalability properties: With $u = -B^\top x$ [7], any sparsity structure in the $B$-matrix is maintained and the rank-one coordination scheme of [1] admits scalable implementations. However, the most scalable control solution is one that is fully decentralized. In general, it is non-trivial that decentralized PI-controllers are stabilizing, let alone fulfill any optimality criterion. In this paper we manage to not only show the aforementioned asymptotic optimality properties. We also show that the resulting equilibrium is globally asymptotically stable under decentralized controller tuning rules.

The paper is organized as follows. Section <ref> presents the considered plant and control strategy. Section <ref> presents the main results of the paper, namely equilibrium existence and uniqueness, global asymptotic stability, and equilibrium optimality for our considered closed-loop. A motivating numerical example consisting in the flow control of a simplified district-heating network is subsequently given in section <ref>. The proofs of the main results are presented in sections <ref>, <ref>, and <ref> respectively. Conclusions and future work are covered in section <ref>.

Notation: $v_i$ denotes element $i$ of vector $v \in \R^n$, $A_i$ denotes row $i$ of matrix $A \in \R^{n \times m}$, and $A_{i, j}$ denotes its $(i,j)$-th element. A matrix $A$ is strictly diagonally row-dominant if $|A_{i,i}| > \sum_{j \neq i} |A_{i,j}|$ for all $i$. $A$ is strictly diagonally column-dominant if $A^\top$, denoting the transpose of $A$, is strictly diagonally row-dominant. Let the 2-norm of a vector $x \in \R^n$ be given by $\twonorm{x} = (\sum_{i=1}^n x_i^2)^{1/2}$. Let the 1-and-infinity-norms of a vector $x \in \R^n$ be given by $\onenorm{x} = \sum_{i=1}^n |x_i|$ and $\infnorm{x} = \max_i |x_i|$ respectively. Let the norm $\twotwonorm{A}$ of a matrix $A$ be the induced 2-norm. Let $\ones \in \R^n$ be a vector of all ones, where $n$ is taken in context.
We say that a function $f : \R \to \R$ is increasing (non-decreasing) if $y > x$ implies that $f(y) > f(x)$ ($f(y) \geq f(x)$).

§ PROBLEM DATA AND PROPOSED CONTROLLER

We consider control of plants of the form

\begin{equation}
\Dot{x} = -Ax + Bf(u) + w
\label{eq:system description}
\end{equation}

where vector $x\in \mathbb{R}^{n}$ gathers the states $x_i$ of each agent, $A \in \mathbb{R}^{n \times n}$ is a diagonal positive definite matrix, and $w\in \mathbb{R}^{n}$ is a constant disturbance acting on the plant. The input nonlinearity $f:\R^n \to \R^n$ satisfies Assumption <ref> introduced below and $B\in \mathbb{R}^{n \times n}$ is an M-matrix, as characterized next.

If the matrix $B$ has strictly positive diagonal elements and non-positive off-diagonal elements, then the following statements are all equivalent[8]: $B$ is an M-matrix. There is a diagonal positive definite matrix $Q$ such that . There is a diagonal positive definite matrix $U$ such that $U B$ and $U B U\inv$ are strictly column-diagonally dominant. $DB$ is an M-matrix for any positive definite diagonal $D$. For a more extensive list of equivalent statements, refer to [8].

For (<ref>), we focus on a certain class of functions $f : \R^n \to \R^n$ characterized by the following assumption.

$f(x) = \left[ f_1(x_1), f_2(x_2), \dots, f_n(x_n)\right]^\top$ has components $f_i$ satisfying $f_i(0)=0$ and incrementally sector-bounded in the sector $\left[0, 1\right]$, namely satisfying $0 \leq \left( f_i(y)-f_i(x) \right)/\left(y-x\right) \leq 1$ for all $x \in \R$, $y \in \R$, $x \neq y$.

Note that Assumption <ref> implies that $f$ is non-decreasing and Lipschitz with Lipschitz constant 1. Since $f(0)=0$, $f$ also enjoys a sector $\left[0, 1\right]$ condition.

Stability properties for feedback with incrementally sector-bounded nonlinearities has long been considered in the literature. As far back as [9] it was used for input-output stability analysis. Both [10] and [11] consider the type of diagonally partitioned incrementally sector-bounded functions that we consider here, whereas [12, 13, 14] consider a richer class of incremental sector-bound constraints of the form $\left(f(x)-f(y)-S_1(x-y)\right)^\top \left( f(x)-f(y)-S_2(x-y)\right) \leq 0$ for all $x \in \R^n$, $y \in \R^n$ where $S_1$ and $S_2$ are real symmetric matrices with $0 \preceq S_1 \prec S_2$.

We will consider function pairs $f(\cdot)$, $h(\cdot)$ where $f(x) + h(x) = x$. These pairs fulfill the following property, the proof of which is in the appendix.

Let $f: \R^n \to \R^n$ satisfy Assumption <ref>. Then $h(u) = u - f(u)$ also satisfies Assumption <ref>.

The considered class of function pairs is well motivated by the common case $f(x) = \sat{x}$ where $\sat{x} = \max \left( \min \left( x, \ones \right), -1\ones \right)$ and $h(x) = \dz{x} = x - \sat{x}$.

We propose controlling the plant (<ref>) with fully decentralized PI controllers having decentralized anti-windup for each agent $i = 1,\dots,n$.

\begin{align}
\dot{z_i} = x_i + s_i h_i(u_i) \label{eq:controller z}\\
u_i = - p_i x_i - r_i z_i \label{eq:controller u}
\end{align}

where $z_i$ is the integral state, $u_i$ is the controller output, $p_i > 0$ and $r_i > 0$ are proportional and integral controller gains respectively, $s_i > 0$ is an anti-windup gain, and $h(u) = u - f(u)$ is an anti-windup signal. Note that while the notation $h$ is not needed (indeed we could equivalently replace $h(u)$ with $u-f(u)$), we will use the pair $f$, $h$ both to simplify the exposition and to highlight that $f$ is the nonlinearity acting on the plant while $h$ is the nonlinearity acting on the controller. We assume that the closed-loop system satisfies the following assumption.

$A$ is a diagonal positive definite matrix, $B$ is an M-matrix, and $w$ is a constant disturbance. The controller parameters $p_i$, $r_i$, and $s_i$, for $i=1,\dots,n$, are all positive.

§ MAIN RESULTS

In this section we will cover the main results of this paper. In particular, we will consider the proposed control law eq:controller zeq:controller u for the plant (<ref>). We will show that this closed-loop system admits an equilibrium for any constant disturbance $w$. We will additionally show that this equilibrium is globally asymptotically stable and enjoys a notion of optimality. We will leave the proofs for Sections <ref> to <ref>.

Let us first consider the existence of an equilibrium, which corresponds to well-posedness of the equations eq:system descriptioneq:controller u with $\dot{x} = \dot{z} = 0$.
(Equilibrium Existence and Uniqueness)
Let $f$ satisfy Assumption <ref> and let Assumption <ref> hold. Then for each constant $w \in \R^n$, closed-loop eq:system descriptioneq:controller u has a unique equilibrium $(x^0$, $z^0)$, inducing input $u^0$ from (<ref>), which satisfies eq:system descriptioneq:controller u with $\dot{x}=\dot{z}=0$.

In addition to the existence of the unique equilibrium $(x^0$, $z^0)$, we can also show that it is globally asymptotically stable under the following assumption on the tuning of the control parameters.

Assume that $a_i p_i > r_i$ and $p_i s_i < 1$ for all $i$, where $a_i$ are the diagonal elements of $A$ in (<ref>) and $p_i$, $r_i$, and $s_i$ are the controller gains in eq:controller zeq:controller u.

(Global Asymptotic Stability)
Let $f$ satisfy Assumption <ref> and let $f(u)+h(u)=u$. Let Assumptions <ref> and <ref> hold. Then there is a globally asymptotically stable equilibrium for the closed-loop eq:system descriptioneq:controller u.

The tuning rules of Assumption <ref> are fully decentralized. Each agent $i$ can tune their own controller gains to satisfy $r_i < a_i p_i$ and $s_i < 1/p_i$.

Let us now focus on the case where the function pair $f(\cdot)$ and $h(\cdot)$ are given by the pair $\sat{\cdot}$ and $\dz{\cdot}$ respectively, motivated by classical anti-windup for saturating controllers. Let $\gamma_i$ be positive scalar weights, and consider the problem of minimizing the weighted sum of all state errors $\sum_{i=1}^n \gamma_i |x_i|$. We can define this problem through the optimization problem
\beginmini!
x,v∑_i=1^n γ_i |x_i| = Γx

-Ax+Bv+w = 0
\endmini!
where $\Gamma = \text{diag}\{\gamma_1, \dots, \gamma_n \}$. The inequalities (<ref>) are considered componentwise. This problem can be motivated by a district heating example. Let $w$ be the outdoor temperature, $x_i$ be the deviation from the comfort temperature for each agent $i$, and let $B v$ denote the heat provided to the agents, limited by (<ref>). Then if $\Gamma = I$, this corresponds to minimizing the total discomfort experienced by all agents. One could consider $\gamma_i$ to be a cost describing the severity of agent $i$ deviating from the comfort temperature, where $\gamma_i$ would be high for e.g. a hospital. Note that this cost does not capture the notion of fairness as considered in [1]. For instance, with $\Gamma = I$, $x = \left[ n, 0, \dots, 0 \right]^\top$, and $y = \left[ 1, 1, \dots, 1 \right]^\top$ we achieve the same costs $\onenorm{\Gamma x} = \onenorm{\Gamma y}$. With the problem (<ref>) defined, the following holds.

Let Assumption <ref> hold and let $\Gamma A\inv B$ be a strictly diagonally column-dominant M-matrix. Let $f(u) = \sat{u}$ and $h(u) = \dz{u} = u - \sat{u}$. Let $(x^0$, $z^0)$, be an equilibrium for the closed-loop system in eq:system descriptioneq:controller u, associated with input $u^0$. Then $x^* = x^0$ and $v^* = f(u^0)$ solves (<ref>).

Note that, since $B$ is an M-matrix, $A\inv B$ is also an M-matrix. In this scenario there always exists a diagonal, positive definite $\Gamma$ such that $\Gamma A\inv B$ is strictly diagonally column-dominant. This means that there is always a selection of weights $\gamma_i$ for which the control law eq:controller zeq:controller u stabilizes an optimal point.

§ NUMERICAL EXAMPLE

This motivating example compares three different control strategies on a simplified, linear model of 10 buildings connected in a district heating grid. The compared strategies are the same as the ones considered in [1]. Each building $i$ has identical thermodynamics on the form

\begin{equation}
\dot{x}_i =  - \frac{a_i}{C_i} (x_c + x_i - T_\text{ext}(t)) + \frac{1}{C_i}\dot{Q}_i(u),
\end{equation}

where $x_i$ denotes agent $i$'s indoor temperature deviation from the comfort temperature $x_c$, $C_i$ is the heat capacity of each building and $T_\text{ext}$ is the outdoor temperature. Here the assumption of a constant disturbance $w$ is replaced with the slowly time-varying disturbance  $T_\text{ext}$. $\dot{Q}_i$ is the heat supplied to building $i$. This heat supply is given by

\begin{equation}
\Dot{Q} = B \sat{u},
\end{equation}

where $B$ represents the network interconnection. The simulation was conducted with $a_i = 0.167\left[ \text{kW/C}^\circ\right]$, $C_i = 2.0 \left[ \text{kWh/C}^\circ\right]$, $p_i = 2.5\left[ \text{1/C}^\circ\right]$, $r_i = 0.2\left[ \text{1/C}^\circ\text{h}\right]$, and $s_i=2.0\left[ \text{C}^\circ\right]$ for all $i$. The parameters $a_i$, $C_i$ are chosen close to the values found in [15] which discusses parameter estimation for a single-family building. Matrix $B$ is selected as

\begin{equation}
\scalebox{0.6}{$
B = \begin{bmatrix}
12.0 & -0.15 & -0.15 & -0.15 & -0.15 & -0.15 & -0.15 & -0.15 & -0.15 & -0.15 \\
-0.15 & 12.0 & -0.3 & -0.3 & -0.3 & -0.3 & -0.3 & -0.3 & -0.3 & -0.3 \\
-0.15 & -0.3 & 12.0 & -0.45 & -0.45 & -0.45 & -0.45 & -0.45 & -0.45 & -0.45 \\
-0.15 & -0.3 & -0.45 & 12.0 & -0.6 & -0.6 & -0.6 & -0.6 & -0.6 & -0.6 \\
-0.15 & -0.3 & -0.45 & -0.6 & 12.0 & -0.75 & -0.75 & -0.75 & -0.75 & -0.75 \\
-0.15 & -0.3 & -0.45 & -0.6 & -0.75 & 12.0 & -0.9 & -0.9 & -0.9 & -0.9 \\
-0.15 & -0.3 & -0.45 & -0.6 & -0.75 & -0.9 & 12.0 & -1.05 & -1.05 & -1.05 \\
-0.15 & -0.3 & -0.45 & -0.6 & -0.75 & -0.9 & -1.05 & 12.0 & -1.2 & -1.2 \\
-0.15 & -0.3 & -0.45 & -0.6 & -0.75 & -0.9 & -1.05 & -1.2 & 12.0 & -1.35 \\
-0.15 & -0.3 & -0.45 & -0.6 & -0.75 & -0.9 & -1.05 & -1.2 & -1.35 & 12.0
\end{bmatrix}
\end{equation}

with elements $B_{i,j}$ in $\left[ \text{kW}\right]$. Matrix $B$ is constructed such that building 1 is closest to a production facility and thus has the best ability to extract heat, whereas building 10 is the farthest away. In addition, the entries of $B$ are such that fully opened control valves at $\sat{u} = \ones$ gives $\dot{Q}$ representing a reasonable peak heat demand for small houses. We simulate the system using the  toolbox in Julia[16], for an outdoor temperature scenario given by data from the city of Gävle, Sweden in October 2022 during which the temperature periodically drops to almost -20$^\circ$C. The data is gathered from the Swedish Meteorological and Hydrological Institute (SMHI). We compare three different controllers and three different cost functions. The first controller is the fully decentralized PI-controller considered in this paper. Secondly the coordinating controller consists of the same PI-controllers as the decentralized case, but with the coordinating rank-1 anti-windup signal $\dot{z}_i = x_i + \beta \ones^\top \dz{u}$ considered in [1]. Finally, the static controller is given by $u = -B^\top C\inv x$ as considered in [7], where $C$ is the diagonal matrix of all heat capacities $C_i$.

xlabel=time [h],
ylabel=x [$^\circ$C],
no markers,
every axis plot/.append style=blue, solid,
ıin 1,...,10
[blue, solid] table [x=t, y=xı, col sep=comma] data/x-diagonal.csv;

axis y line*=right,
axis x line=none,
yticklabel style=color=black,
ylabel style=color=black,
[black, no markers, thick, dashed] table [x=t, y=t_out, col sep=comma] data/t.csv;

Decentralized controller.

xlabel=time [h],
ylabel=x [$^\circ$C],
no markers,
every axis plot/.append style=blue, solid,
ıin 1,...,10
[blue, solid] table [x=t, y=xı, col sep=comma] data/x-coordinating.csv;

axis y line*=right,
axis x line=none,
yticklabel style=color=black,
ylabel style=color=black,
[black, no markers, thick, dashed] table [x=t, y=t_out, col sep=comma] data/t.csv;

Coordinating controller.

xlabel=time [h],
ylabel=x [$^\circ$C],
no markers,
every axis plot/.append style=blue, solid,
ıin 1,...,10
[blue, solid] table [x=t, y=xı, col sep=comma] data/x-static.csv;

axis y line*=right,
axis x line=none,
yticklabel style=color=black,
ylabel style=color=black,
[black, no markers, thick, dashed] table [x=t, y=t_out, col sep=comma] data/t.csv;

Static controller.

Simulation results showing temperature deviations $x$ for each control strategy. The blue lines (with values on the left axis) are the temperature deviations $x$. The dotted, black line (with values on the right axis) is the outdoor temperature. At about hour 100, the temperature becomes critically low. This induces saturation in the controllers, leading to a drop of the indoor temperatures.

Figure <ref> shows the resulting deviations $x$ during the simulations. At around hour 100, the outdoor temperature is critically low. At this time, the buildings do not receive sufficient heat, regardless of the control strategy. Figure <ref> shows that with the decentralized strategy, the worst deviations become larger than with the coordinating strategy (Figure <ref>). However, not all buildings experience temperature deviations, whereas with the coordinating strategy, all the buildings share the discomfort. Lastly, the static controller has large deviations experienced by many buildings. Even when the outdoor temperature is manageable, the static controller has a constant offset from the comfort temperature, highlighting the usefulness of the integral action.

bar width=0.3cm,
symbolic x coords=decentralized, coordinating, static,
xticklabel style=rotate=20,
ymin=0, ymax=0.45,
nodes near coords,
nodes near coords style=rotate=90, anchor=west
table[col sep=comma, x=strategies, y=infnorm]data/costs.csv;

The cost $J_\infty$ is minimized by the coordinating strategy.

bar width=0.3cm,
symbolic x coords=decentralized, coordinating, static,
xticklabel style=rotate=20,
ymin=0, ymax=3.0,
nodes near coords,
nodes near coords style=rotate=90, anchor=west
table[col sep=comma, x=strategies, y=onenorm]data/costs.csv;

The cost $J_1$ is minimized by the decentralized strategy.

bar width=0.3cm,
symbolic x coords=decentralized, coordinating, static,
xticklabel style=rotate=20,
ymin=0, ymax=5.0,
nodes near coords,
nodes near coords style=rotate=90, anchor=west
table[col sep=comma, x=strategies, y=twonorm]data/costs.csv;

The strategies incur similar costs $J_2$, minimized by static feedback.

Cost functions evaluated for each control strategy.

We evaluate the performance through the three cost functions

\begin{align}
J_1 &= \frac{1}{T}\int_0^T \onenorm{x(t)} dt, \\
J_\infty &= \frac{1}{T}\int_0^T \infnorm{x(t)} dt, \\
J_2 &= \frac{1}{T}\int_0^T x(t)^\top L x(t) + \sat{u(t)}^\top \sat{u(t)} dt.
\end{align}

where $T$ is the simulation time and $L$ is a diagonal matrix where each element is given by $l_i = \frac{q_i}{C_i}$. The cost $J_1$ mimics the optimality notion considered in this paper, $J_\infty$ mimics the optimality notion considered in [1], and $J_2$ mimics the optimality considered in [7]. Figure <ref> shows the resulting evaluations. Figure <ref> shows that the coordinating controller gives minimal worst-case deviations $J_\infty$, but Figure <ref> shows that the total discomfort $J_1$ is minimized in the decentralized strategy. This result, i.e. that the total discomfort is minimized by decentralized control but the worst-case discomfort is minimized by coordination, is found also in [4] where a nonlinear model of the grid hydraulics and a 2-state model of building dynamics is employed. On the weighted cost $J_2$ as seen in Figure <ref>, all controllers provide similar performance. The static controller slightly outperforms the other two in this scenario, but it is outperformed in every other measure.

§ PROOF OF EQUILIBRIUM EXISTENCE AND UNIQUENESS

We will now prove Theorem <ref> through the use of Banach's fixed-point theorem [17]. This proof requires the following two lemmas, the proofs of which are found in the appendix.

Let $f: \R^n \to \R^n$ and $h: \R^n \to \R^n$ where $h(x) = x - f(x)$ satisfy Assumption <ref>. Then $\Tilde{f}: \R^n \to \R^n$ and $\Tilde{h}: \R^n \to \R^n$ given by $\Tilde{f}(x) = f(x + x^0) - f(x^0)$ and $\Tilde{h}(x) = h(x + x^0) - h(x^0)$ for some $x^0 \in \R^n$ also satisfy Assumption <ref> and $\Tilde{h}(x) + \Tilde{f}(x) = x$.

Let $f: \R^n \to \R^n$ and $h: \R^n \to \R^n$ where $h(x) = x - f(x)$ satisfy Assumption <ref>. Then $\Tilde{f}: \R^n \to \R^n$ and $\Tilde{h}: \R^n \to \R^n$ given by $\Tilde{f}(x) = D f(D\inv x)$ and $\Tilde{h}(x) = D h(D\inv x)$ where $D$ is a diagonal positive definite matrix also satisfy Assumption <ref> and $\Tilde{h}(x) + \Tilde{f}(x) = x$.

(of Theorem <ref>)
Denoting by $S$ a diagonal positive definite matrix gathering the anti-windup gains $s_i$, $i=1,\dots n$, we can rearrange eq:system descriptioneq:controller u by imposing $\dot{x} = \dot{z} = 0$, which yields

\begin{equation}
0 = h(u^0) + S\inv A\inv B f(u^0) + S\inv A\inv w.
\label{eq: stationary condition}
\end{equation}

If there is a unique $u^0$ solving (<ref>) then $x^0=A\inv\left(Bf(u^0)+w\right)$ and $z^0 = R\inv(-Px^0 - u^0)$ are uniquely determined by (<ref>) and (<ref>) respectively, where $R=\text{diag}\{r_1,\dots,r_n\}$ is invertible by Assumption <ref>. Hence we need only show that there is a unique $u^0$ solving (<ref>) for the proof to be complete. Let $D$ be a diagonal positive definite matrix such that $D  S\inv A\inv B D\inv$ is strictly diagonally column-dominant. Note that such a $D$ always exists because $A$ and $S$ are diagonal positive definite and $B$ is an M-matrix. Left-multiply $\eqref{eq: stationary condition}$ by $D$ and insert multiplication by $I = D\inv D$ before $f(u^0)$ to obtain

\begin{equation}
0 = Dh(u^0) + D S\inv A\inv B D\inv D f(u^0) + D S\inv A\inv w.
\label{eq:banach mid-proof change of vars}
\end{equation}

Introduce the change of variables $\hat{B} = D S\inv A\inv B D\inv$,
$\zeta = D u^0$, and . Then (<ref>) yields

\begin{equation}
0 = Dh(D\inv \zeta) + \hat{B} D f(D\inv \zeta) + \hat{w}.
\label{eq: equilibrium proof middle-equation}
\end{equation}

Here we can use Lemma <ref> to replace $f(\cdot)$, $h(\cdot)$ with $\hat{f}(\cdot)$, $\hat{h}(\cdot)$, which satisfy Assumption <ref> and $\hat{f}(\zeta) + \hat{h}(\zeta) = \zeta$. Introduce a scalar $k$ satisfying $k > \max( 1, 2 \underset{i}{\max} \hat{B}_{i,i} )$. Divide (<ref>) by $-k$, add $\zeta$ to the left-hand side, and $\zeta = \hat{f}(\zeta) + \hat{h}(\zeta)$ to the right-hand side of (<ref>) to obtain

\begin{equation}
\zeta = -\frac{1}{k}\left( (1-k)\hat{h}(\zeta) + (\hat{B} - k I)\hat{f}(\zeta) + \hat{w} \right). \label{eq:banach fixed-point condition}
\end{equation}

We define the right-hand side of this expression as $T_w(\zeta)$, defined for a specific $w$. By showing that $T_w$ is a contractive mapping for any $\hat{w}$, we can use Banach's fixed point theorem [17] to show that there is a unique solution $\zeta = T_w(\zeta)$ (and thus a unique $u^0=D\inv \zeta$) for any $\hat{w}$ (and thus any $w = A S D\inv \hat{w}$). Consider any $\alpha \in \R^n$, $\beta \in \R^n$. Then

\begin{equation}
\begin{array}{rl}
T_w(\alpha)-T_w(\beta) &=   \frac{-1 + k}{k}\left(\hat{h}(\alpha) - \hat{h}(\beta)\right)  \\
&  + \frac{-\hat{B} + k I}{k}\left(\hat{f}(\alpha) - \hat{f}(\beta)\right) .
\end{array}
\end{equation}

Here we use Lemma <ref> to introduce $\Tilde{h}(\alpha-\beta) = \hat{h}(\alpha) - \hat{h}(\beta)$ and $\Tilde{f}(\alpha-\beta) = \hat{f}(\alpha) - \hat{f}(\beta)$. Denote $\Delta = \alpha - \beta$ and $\Delta^+ = T_w(\alpha) - T_w(\beta)$. Then

\begin{equation}
|\Delta^+_i| \leq \frac{k - 1}{k}|\Tilde{h}_i(\Delta_i)| + \frac{k - \hat{B}_{i,i}}{k} |\Tilde{f}_i(\Delta_i)| + \sum_{j \neq i}\frac{|\hat{B}_{i,j}|}{k}|\Tilde{f}_j(\Delta_j)|.
\end{equation}

\begin{align}
\onenorm{\Delta^+} &= \sum_{i=1}^n |\Delta_i^+| \leq \sum_{i=1}^n \bigg( \frac{k - 1}{k}|\Tilde{h}_i(\Delta_i)| \nonumber \\
& + \frac{k - \hat{B}_{i,i}}{k} |\Tilde{f}_i(\Delta_i)| + \sum_{j \neq i}\frac{|\hat{B}_{j,i}|}{k}|\Tilde{f}_i(\Delta_i)| \bigg).
\end{align}

Due to the diagonal column-dominance of $\hat{B}$ and the definition of $k$, it holds that $k > \hat{B}_{i,i} > \sum_{j \neq i}|\hat{B}_{j,i}|$. Thus, selecting $\lambda = \frac{k - 1}{k} < 1$, $\mu_i = \frac{k- \left(\hat{B}_{i,i} - \sum_{j \neq i}|\hat{B}_{j,i}|\right)}{k} < 1$, $\gamma_i = \max(\lambda, \mu_i) < 1$, and $\Bar{\gamma} = \max_i \gamma_i < 1$, we obtain

\begin{align}
\onenorm{\Delta^+} &\leq \sum_{i=1}^n \lambda |\Tilde{h}_i(\Delta_i)| + \mu_i |\Tilde{f}_i(\Delta_i)| \nonumber \\
&\leq \sum_{i=1}^n \gamma_i \left(|\Tilde{h}_i(\Delta_i)| + |\Tilde{f}_i(\Delta_i)| \right) \nonumber \\
&\leq \sum_{i=1}^n \Bar{\gamma}|\Delta_i| = \gamma \onenorm{\Delta}.
\end{align}

Note that $|\Tilde{h}_i(\Delta_i)| + |\Tilde{f}_i(\Delta_i)| = | \Delta_i|$ since $\Tilde{f}_i(\Delta_i)$ and $\Tilde{h}_i(\Delta_i)$ always have the same sign by Assumption <ref>, and sum to $\Delta_i$. This proves that $T_w$ is a contraction mapping with respect to the metric $\onenorm{\cdot}$. Thus, by Banach's fixed point theorem, for each $w$ and the ensuing $\hat{w} = D S\inv A\inv w$ there is a unique $\zeta$ such that (<ref>) holds, and thus a $u^0 = D \inv \zeta$ such that (<ref>) holds, which completes the proof.

§ PROOF OF GLOBAL ASYMPTOTIC STABILITY

Given the existence of an equilibrium $(x^0, z^0)$ and the associated input $u^0$, consider the change of variables $\Tilde{z} = - R (z - z^0)$, $\Tilde{u} = u - u^0$, $\Tilde{f}(\Tilde{u}) = f(u^0 + \Tilde{u}) - f(u^0)$, and $\Tilde{h}(\Tilde{u}) = h(u^0 + \Tilde{u}) - h(u^0)$. Due to Lemma <ref>, $\Tilde{f}(\cdot)$, $\Tilde{h}(\cdot)$ satisfy Assumption <ref>, and $\Tilde{f}(\Tilde{u})+\Tilde{h}(\Tilde{u}) = \Tilde{u}$. This allows rewriting the eq:system descriptioneq:controller u as

\begin{equation}
\scalebox{0.8}{
\begin{bmatrix}
\dot{\Tilde{z}} \\
\dot{\Tilde{u}}
\end{bmatrix} = \begin{bmatrix}
-R P\inv & R P\inv \\
A - R P\inv & -A + R P\inv
\end{bmatrix} \begin{bmatrix}
\Tilde{z} \\
\Tilde{u}
\end{bmatrix}
- \begin{bmatrix}
0 \\
\end{bmatrix} \Tilde{f}(\Tilde{u})
- \begin{bmatrix}
R S \\
R S
\end{bmatrix} \Tilde{h}(\Tilde{u})
\label{eq:transformed system}
\end{equation}

where $P$, $R$, and $S$ are diagonal matrices gathering the controller parameters $p_i$, $r_i$, and $s_i$. Stabilizing this system to $\Tilde{z} = \Tilde{u} = 0$ is equivalent to stabilizing the original system system to the equilibrium $x = x^0$, $z = z^0$, and $u = u^0$. We will therefore now prove Theorem <ref> with a Lyapunov-based argument considering system (<ref>).

(of Theorem <ref>)
Consider the Lyapunov function candidate

\begin{align}
V(\Tilde{z}, \Tilde{u}) &= \sum_{i=1}^n \int_{0}^{\Tilde{z}_i} q_i (a_i \frac{p_i}{r_i} - 1)\left( \Tilde{f}_i(\zeta) + \epsilon \zeta) \right) d\zeta \nonumber \\
& + \sum_{i=1}^n \int_{0}^{\Tilde{u}_i} q_i \left( \Tilde{f}_i(\zeta) + \epsilon \zeta \right) d \zeta
\end{align}

where scalars $q_i > 0$ and $\epsilon > 0$ are parameters to be fixed later. For any such choice of parameters, $V$ is positive definite and radially unbounded because $\Tilde{f}_i(\zeta) + \epsilon \zeta$ is increasing in $\zeta$ and zero at zero. Also $a_i \frac{p_i}{r_i} - 1 > 0$ due to Assumption <ref>. The time derivative of $V$ along the trajectories of system (<ref>) is given by

\begin{align}
\Dot{V}(\Tilde{z}, \Tilde{u}) = &-\left(\Tilde{f}(\Tilde{z})+\epsilon \Tilde{z}-\Tilde{f}(\Tilde{u})-\epsilon \Tilde{u}\right)^\top \Tilde{D}(\Tilde{z}-\Tilde{u}) \label{eq: Vdot 1} \\
& -\left(\Tilde{f}(\Tilde{z})+\epsilon \Tilde{z}\right)^\top \Tilde{D}PS \Tilde{h}(\Tilde{u}) \label{eq: Vdot 2} \\
& -\left(\Tilde{f}(\Tilde{u})+\epsilon \Tilde{u}\right)^\top QRS \Tilde{h}(\Tilde{u}) \label{eq: Vdot 3} \\
& -\left(\Tilde{f}(\Tilde{u})+\epsilon \Tilde{u}\right)^\top QPB \Tilde{f}(\Tilde{u}) \label{eq: Vdot 4}
\end{align}

where $\Tilde{D}$ is a diagonal positive definite matrix gathering the elements $q_i\left(a_i - r_i/p_i\right)$ and $Q$ is a diagonal positive definite matrix gathering the elements $q_i$. To simplify this expression, we split it into

\begin{equation}
\Dot{V}(\Tilde{z}, \Tilde{u}) = \Dot{V}_1(\Tilde{z}, \Tilde{u}) + \Dot{V}_2(\Tilde{z}, \Tilde{u})
\end{equation}

where $\Dot{V}_1(\Tilde{z}, \Tilde{u})$ corresponds to the terms eq: Vdot 1eq: Vdot 2 and $\Dot{V}_2(\Tilde{z}, \Tilde{u})$ corresponds to the terms eq: Vdot 3eq: Vdot 4. Since $\Tilde{D}$ and $\Tilde{D}PS$ are diagonal, $\dot{V}_1$ can be analyzed for each $i$ individually. $\Tilde{f}_i(\zeta_i) + \epsilon \zeta_i$ is increasing in $\zeta_i$, therefore $\text{sign}\left(\Tilde{f}_i(\Tilde{z}_i)+\epsilon \Tilde{z}_i-\Tilde{f}_i(\Tilde{u}_i)-\epsilon \Tilde{u}_i\right) = \text{sign}\left(\Tilde{z}_i - \Tilde{u}_i\right)$ and thus (<ref>) is negative semi-definite. If $\Tilde{z}_i$ and $\Tilde{u}_i$ have the same sign, (<ref>) contributes negatively to $\dot{V}_1$. If they have opposite signs the contribution is positive, but in this case (<ref>) only comprises negative terms because  . Indeed, since $p_i s_i < 1$ from Assumption <ref> and $|\Tilde{h}_i(\Tilde{u}_i)| \leq |\Tilde{u}_i|$ from Assumption <ref>, then (<ref>) as developed above dominates (<ref>) which is upper bounded by , and thus $\dot{V}_1$ is negative semidefinite. We now turn our attention to $\dot{V}_2$. Note that $\Tilde{u}$, $\Tilde{f}(\Tilde{u})$, and $\Tilde{h}(\Tilde{u})$ elementwise have the same sign and $QRS$ is diagonal, positive definite. Thus

\begin{align}
\scalebox{0.8}{$
\begin{aligned}
&\left(\Tilde{f}(\Tilde{u})+\epsilon \Tilde{u}\right)^\top QRS \Tilde{h}(\Tilde{u}) = \left(\Tilde{f}(\Tilde{u})+\epsilon \Tilde{f}(\Tilde{u}) + \epsilon\Tilde{h}(\Tilde{u})\right)^\top QRS \Tilde{h}(\Tilde{u}) \\
&= (1 + \epsilon)\Tilde{f}(\Tilde{u})^\top QRS \Tilde{h} + \epsilon \Tilde{h}(\Tilde{u})^\top QRS \Tilde{h}(\Tilde{u}) \geq \epsilon \beta \twonorm{\Tilde{h}(\Tilde{u})}^2
\end{aligned}
\label{eq:luca lol}
\end{align}

where $\beta$ is the minimum diagonal element of $QRS$. Note also that

\begin{align}
\left(\Tilde{f}(\Tilde{u})+\epsilon \Tilde{u}\right)^\top QPB \Tilde{f}(\Tilde{u}) &= (1+\epsilon)\Tilde{f}(\Tilde{u})^\top QPB \Tilde{f}(\Tilde{u}) \nonumber\\
&\quad + \epsilon \Tilde{h}(\Tilde{u})^\top QPB \Tilde{f}(\Tilde{u}). \label{eq:Vdot 4 partitioned}
\end{align}

Fix now the weights $q_i$ in such a way that $QPB + B^\top P Q$ is positive definite. This is possible because $B$ is an M-matrix according to Assumption <ref>. Therefore $\exists \alpha > 0$ such that $QPB + B^\top P Q \succ 2 \alpha I$. Thus the first term of (<ref>) satisfies

\begin{equation}
(1+\epsilon)\Tilde{f}(\Tilde{u})^\top QPB \Tilde{f}(\Tilde{u}) \geq (1+\epsilon)\alpha \twonorm{\Tilde{f}(\Tilde{u})}^2.
\label{eq:Vdot 4 bound on fBf}
\end{equation}

We also note that the second term in (<ref>) satisfies

\begin{equation}
\epsilon\Tilde{h}(\Tilde{u})^\top QPB \Tilde{f}(\Tilde{u}) \geq -\epsilon\gamma \twonorm{\Tilde{f}(\Tilde{u})}\twonorm{\Tilde{h}(\Tilde{u})}
\label{eq:Vdot 4 bound on hBf}
\end{equation}

where $\gamma = \twotwonorm{QPB}$. Thus, combining the bounds in (<ref>), (<ref>) and (<ref>) within eq: Vdot 3eq: Vdot 4, we obtain

\begin{equation}
\scalebox{0.8}{
\Dot{V}_2(\Tilde{z}, \Tilde{u}) &\leq -(1+\epsilon)\alpha\twonorm{\Tilde{f}(\Tilde{u})}^2 - \epsilon \beta \twonorm{\Tilde{h}(\Tilde{u})}^2 + \epsilon \gamma \twonorm{\Tilde{f}(\Tilde{u})}\twonorm{\Tilde{h}(\Tilde{u})} \\
\begin{pmatrix}
\twonorm{\Tilde{f}(\Tilde{u})} \\ \twonorm{\Tilde{h}(\Tilde{u})}
\end{pmatrix}^\top
\begin{pmatrix}
-(1+\epsilon)\alpha & \frac{1}{2}\epsilon \gamma \\
\frac{1}{2}\epsilon \gamma & - \epsilon \beta
\end{pmatrix}
\begin{pmatrix}
\twonorm{\Tilde{f}(\Tilde{u})} \\ \twonorm{\Tilde{h}(\Tilde{u})}
\end{pmatrix}.
\end{aligned}$}
\label{eq:Vdot_2}
\end{equation}

We may now select the Lyapunov function parameter $\epsilon$ sufficiently small such that $\left(\alpha + \epsilon\alpha - \frac{\epsilon \gamma^2}{4 \beta}\right) > 0$. This makes the quadratic form (<ref>) negative definite. Thus $\Dot{V}_2(\Tilde{z}, \Tilde{u})=0$ if and only if $\Tilde{f}(\Tilde{u}) = \Tilde{h}(\Tilde{u}) = 0$, i.e. if and only if $\Tilde{u} = 0$. In this case, $\Dot{V}_1(\Tilde{z}, \Tilde{u})$ is clearly negative definite in $\Tilde{z}$. Thus $\Dot{V}(\Tilde{z}, \Tilde{u})$ is negative definite, which implies that the origin is globally asymptotically stable for system (<ref>). Equivalently, the equilibrium $(x^0,z^0)$, with input $u^0$, is therefore globally asymptotically stable for the original system eq:system descriptioneq:controller u.

§ PROOF OF EQUILIBRIUM OPTIMALITY

Here we prove Theorem <ref>.

Firstly, it is clear that $v^* = \sat{u^0}$ and $x^*_i = x^0_i = -s_i\dz{u^0_i}$ for all $i$ satisfies (<ref>) due to $x^0$, $z^0$ being an equilibrium, and satisfies (<ref>) because $\sat{\cdot}$ is bounded in the range $\left[-1,1\right]$. Consider, for establishing a contradiction, that there exists $\mu \neq 0$ such that $v^\dagger = v^* + \mu$ and $x^\dagger = A\inv Bv^\dagger + A\inv w = x^*+A\inv B \mu$ is the optimal solution to (<ref>) with a smaller cost (<ref>) than the one obtained by $x^*$, $v^*$. Then $\mu$ solves the optimization problem
\beginmini!
μ∑_i=1^n |γ_i x^*_i + B_i μ|

-≤v^* + μ≤.
\endmini!
where $\Tilde{B}_i$ is row $i$ of the matrix $\Tilde{B} = \Gamma A \inv B$. The equilibrium of (<ref>) implies $x^*_i = -s_i\dz{u^0_i}$. Therefore we can leverage (<ref>) to see that $x^*_i > 0 \implies u^0_i < -1 \implies v_i = -1 \implies \mu_i \geq 0$ and conversely $x^*_i < 0 \implies u^0_i > 1 \implies v_i = 1 \implies \mu_i \leq 0$. Combining this with $\Gamma$ and $A$ both being diagonal, positive definite and the fact that $B$ is an M-matrix which implies that $\Tilde{B}_{i,i} > 0$, we obtain $|\gamma_i x_i + \Tilde{B}_{i,i} \mu_i| = |\gamma_i x_i| + |\Tilde{B}_{i,i} \mu_i|$ for all $i$. Thus (<ref>) can be expanded as follows

\begin{equation}
\scalebox{0.8}{
\sum_{i=1}^n |\gamma_i x_i + \Tilde{B}_i \mu| &\geq \sum_{i \neq j} \left(|\gamma_i x_i + \Tilde{B}_{i,i} \mu_i| - |\sum_{i\neq j} \Tilde{B}_ {i,j} \mu_j|\right) \\
&\geq \sum_{i=1}^n \left(|\gamma_i x_i| + |\Tilde{B}_{i,i}| |\mu_i|\right) - \sum_{i=1}^n\sum_{j \neq i} |\Tilde{B}_{i,j}| |\mu_j| \\
& = \sum_{i=1}^n |\gamma_i x_i| + \sum_{k=1}^n\left(|\Tilde{B}_{k,k}| - \sum_{j\neq k} |\Tilde{B}_{j,k}|\right)|\mu_k|.
\end{aligned}$
\end{equation}

Since $\Tilde{B}$ is diagonally column-dominant, then $|\Tilde{B}_{k,k}| - \sum_{j\neq k} |\Tilde{B}_{j,k}|$ is strictly positive for all $k$. Thus this expression is minimized by $\mu = 0$, which completes the proof.

§ CONCLUSIONS

In this paper we considered fully decentralized PI-control for a class of interconnected systems subject to incrementally sector-bounded nonlinearities. We showed that for systems where the input matrix is an M-matrix, fully decentralized PI-controllers globally asymptotically stabilize a specific equilibrium. Furthermore, this equilibrium is optimal in that it minimizes costs of the form $\sum_{i=1}^n \gamma_i| x_i|$.
The proposed control strategy was employed in a numerical example of a simplified district heating system model. The example showed that, with our decentralized strategy, the total discomfort in the system is minimized, at the cost of higher worst-case discomforts when compared with a alternative coordinated control strategies. We have thus demonstrated that a fully decentralized and easily tuned control law constitutes a relevant design for a large class of systems.

Open questions include analysis of the transient response, and finding controller tuning rules accordingly. Furthermore, to better capture the district heating application, a richer class of systems should be considered: Multi-state models for each individual building, as well as more complex, nonlinear models of the interconnection $B$ should be considered.

[1]
F. Agner, P. Kergus, A. Rantzer, S. Tarbouriech, and L. Zaccarian,
“Anti-windup coordination strategy around a fair equilibrium in resource
sharing networks,” IEEE Control Systems Letters, 2023.

[2]
S. Low, F. Paganini, and J. Doyle, “Internet congestion control,” IEEE
Control Systems Magazine, vol. 22, no. 1, pp. 28–43, 2002.

[3]
F. Kelly, “Fairness and stability of end-to-end congestion control,” European Journal of Control, vol. 9, no. 2, pp. 159–176, 2003.

[4]
F. Agner, P. Kergus, R. Pates, and A. Rantzer, “Combating district heating
bottlenecks using load control,” Smart Energy, vol. 6, 2022.

[5]
L. D. Col, I. Queinnec, S. Tarbouriech, and L. Zaccarian, “Regional
${H}_{\infty}$ synchronization of identical linear multiagent systems under
input saturation,” IEEE Transactions on Control of Network Systems,
vol. 6, no. 2, pp. 789–799, 2019.

[6]
N. A. Ofodile and M. C. Turner, “Decentralized approaches to antiwindup design
with application to quadrotor unmanned aerial vehicles,” IEEE
Transactions on Control Systems Technology, vol. 24, no. 6, pp. 1980–1992,

[7]
D. Bauso, F. Blanchini, L. Giarré, and R. Pesenti, “The linear saturated
decentralized strategy for constrained flow control is asymptotically
optimal,” Automatica, vol. 49, no. 7, pp. 2206–2212, 2013.

[8]
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis.
Cambridge University Press, 1991.

[9]
G. Zames, “On the input-output stability of time-varying nonlinear feedback
systems part one: Conditions derived using concepts of loop gain, conicity,
and positivity,” IEEE Transactions on Automatic Control, vol. 11,
no. 2, pp. 228–238, 1966.

[10]
F. Zhang, H. L. Trentelman, and J. M. Scherpen, “Robust synchronization of
lur'e networks with incremental nonlinearities,” in 52nd IEEE
Conference on Decision and Control, pp. 3439–3444, 2013.

[11]
P. DeLellis and M. di Bernardo, “Adaptive pinning control of complex networks
of lur'e systems,” in 2012 IEEE 51st IEEE Conference on Decision and
Control (CDC), pp. 6060–6064, 2012.

[12]
M. Giaccagli, V. Andrieu, S. Tarbouriech, and D. Astolfi, “LMI conditions
for contraction, integral action, and output feedback stabilization for a
class of nonlinear systems,” Automatica, vol. 154, p. 111106, 2023.

[13]
F. Zhang, H. L. Trentelman, and J. M. Scherpen, “Fully distributed robust
synchronization of networked lur’e systems with incremental
nonlinearities,” Automatica, vol. 50, no. 10, pp. 2515–2526, 2014.

[14]
M. Giaccagli, V. Andrieu, S. Tarbouriech, and D. Astolfi, “Infinite gain
margin, contraction and optimality: An LMI-based design,” European
Journal of Control, vol. 68, p. 100685, 2022.
2022 European Control Conference Special Issue.

[15]
P. Bacher and H. Madsen, “Identifying suitable models for the heat dynamics of
buildings,” Energy and Buildings, vol. 43, pp. 1511–1522, July 2011.

[16]
C. Rackauckas and Q. Nie, “Differentialequations.jl–a performant and
feature-rich ecosystem for solving differential equations in Julia,” Journal of Open Research Software, vol. 5, no. 1, 2017.

[17]
P. Agarwal, M. Jleli, and B. Samet, Fixed Point Theory in Metric Spaces:
Recent Advances and Applications.
Singapore: Springer Singapore, 1 ed., 2018.
See p. 5 for reference.

[1]
F. Agner, P. Kergus, A. Rantzer, S. Tarbouriech, and L. Zaccarian,
“Anti-windup coordination strategy around a fair equilibrium in resource
sharing networks,” IEEE Control Systems Letters, 2023.

[2]
S. Low, F. Paganini, and J. Doyle, “Internet congestion control,” IEEE
Control Systems Magazine, vol. 22, no. 1, pp. 28–43, 2002.

[3]
F. Kelly, “Fairness and stability of end-to-end congestion control,” European Journal of Control, vol. 9, no. 2, pp. 159–176, 2003.

[4]
F. Agner, P. Kergus, R. Pates, and A. Rantzer, “Combating district heating
bottlenecks using load control,” Smart Energy, vol. 6, 2022.

[5]
L. D. Col, I. Queinnec, S. Tarbouriech, and L. Zaccarian, “Regional
${H}_{\infty}$ synchronization of identical linear multiagent systems under
input saturation,” IEEE Transactions on Control of Network Systems,
vol. 6, no. 2, pp. 789–799, 2019.

[6]
N. A. Ofodile and M. C. Turner, “Decentralized approaches to antiwindup design
with application to quadrotor unmanned aerial vehicles,” IEEE
Transactions on Control Systems Technology, vol. 24, no. 6, pp. 1980–1992,

[7]
D. Bauso, F. Blanchini, L. Giarré, and R. Pesenti, “The linear saturated
decentralized strategy for constrained flow control is asymptotically
optimal,” Automatica, vol. 49, no. 7, pp. 2206–2212, 2013.

[8]
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis.
Cambridge University Press, 1991.

[9]
G. Zames, “On the input-output stability of time-varying nonlinear feedback
systems part one: Conditions derived using concepts of loop gain, conicity,
and positivity,” IEEE Transactions on Automatic Control, vol. 11,
no. 2, pp. 228–238, 1966.

[10]
F. Zhang, H. L. Trentelman, and J. M. Scherpen, “Robust synchronization of
lur'e networks with incremental nonlinearities,” in 52nd IEEE
Conference on Decision and Control, pp. 3439–3444, 2013.

[11]
P. DeLellis and M. di Bernardo, “Adaptive pinning control of complex networks
of lur'e systems,” in 2012 IEEE 51st IEEE Conference on Decision and
Control (CDC), pp. 6060–6064, 2012.

[12]
M. Giaccagli, V. Andrieu, S. Tarbouriech, and D. Astolfi, “LMI conditions
for contraction, integral action, and output feedback stabilization for a
class of nonlinear systems,” Automatica, vol. 154, p. 111106, 2023.

[13]
F. Zhang, H. L. Trentelman, and J. M. Scherpen, “Fully distributed robust
synchronization of networked lur’e systems with incremental
nonlinearities,” Automatica, vol. 50, no. 10, pp. 2515–2526, 2014.

[14]
M. Giaccagli, V. Andrieu, S. Tarbouriech, and D. Astolfi, “Infinite gain
margin, contraction and optimality: An LMI-based design,” European
Journal of Control, vol. 68, p. 100685, 2022.
2022 European Control Conference Special Issue.

[15]
P. Bacher and H. Madsen, “Identifying suitable models for the heat dynamics of
buildings,” Energy and Buildings, vol. 43, pp. 1511–1522, July 2011.

[16]
C. Rackauckas and Q. Nie, “Differentialequations.jl–a performant and
feature-rich ecosystem for solving differential equations in Julia,” Journal of Open Research Software, vol. 5, no. 1, 2017.

[17]
P. Agarwal, M. Jleli, and B. Samet, Fixed Point Theory in Metric Spaces:
Recent Advances and Applications.
Singapore: Springer Singapore, 1 ed., 2018.
See p. 5 for reference.

§ APPENDIX

We prove here suitable properties of the function class characterized by Assumption <ref>, as stated in Lemmas 1, 2 and 3. To simplify the exposition, we drop the index $i$.
(of Lemma <ref>)

Clearly, $h(0) = 0 - f(0) = 0$. Additionally,

\begin{align}
\frac{h(y)-h(x)}{y-x}&= \frac{y-f(y) - x + f(x)}{y-x} \nonumber \\
&= 1 - \frac{f(y)-f(x)}{y-x} \in \left[0, 1\right]
\end{align}

which clearly shows that if $x \neq y$ then , thus concluding the proof.

(of Lemma <ref>)

Clearly, $\Tilde{f}(0) = f(x^0)-f(x^0)=0$. In addition,

\begin{equation}
\frac{\Tilde{f}(y)-\Tilde{f}(x)}{y-x}= \frac{f(y+x^0)-f(x+x^0)}{(y+x^0)-(x+x^0)} \in \left[0, 1\right]
\end{equation}

which clearly shows that if $x \neq y$ then $0 \leq (\Tilde{f}(y)-\Tilde{f}(x))/(y-x) \leq 1$. Finally $\Tilde{f}(x) + \Tilde{h}(x) = f(x + x^0) - f(x^0) + h(x + x^0) - h(x^0) = x + x^0 - x^0 = x$, thus concluding the proof.

(of Lemma <ref>)

$\Tilde{f}(0) = D\inv f(0) = 0$. Additionally,

\begin{align}
\frac{\Tilde{f}(y)-\Tilde{f}(x)}{y-x} &= \frac{df(y/d)-df(x/d)}{y-x} \nonumber \\
&= \frac{f(y/d)-f(x/d)}{y/d-x/d} \in \left[0, 1\right]
\end{align}

which clearly shows that if $x \neq y$ then $0 \leq (\Tilde{f}(y)-\Tilde{f}(x))/(y-x) \leq 1$. Finally $\Tilde{f}(x) + \Tilde{h}(x) = D f( D\inv x) + D h(D\inv x)= D \left( f( D\inv x) + h(D\inv x)\right) = D D\inv x = x$, thus concluding the proof.