# Building Temperature Control: A Distributed Escort Dynamical approach

M. Sawant, J. Moyalan, J. Koonamparampath, A. Sheikh, S. Wagh, and N. Singh
EED, VJTI, Mumbai, India M. Sawant, J. Moyalan, J. Koonamparampath, A. Sheikh,
S. Wagh and N. Singh are with the Electrical Engineering Department, Veermata
Jijabai Technological Institute, Mumbai 400019, INDIA
<EMAIL_ADDRESS>
###### Abstract

The constrained multi-agent optimization problem of distributed resource
allocation is addressed using the evolutionary game theoretic framework. The
issue of building temperature control is analyzed in which the controller is
to devise a scheme to distribute available scarce power to every room to
regulate their temperature as per the user’s comfort in the best possible
manner. The paper correlates the global constraint of fixed resource amount
with the constant population size. The respective population game is evaluated
by means of a dynamical model of the evolutionary game theory to find the
necessary control action. The robustness of optimal solution with respect to
minor fluctuations in the temperature distribution is characterized using
evolutionary stable strategy (ESS). Along with the global constraint, the
problem formulation constitutes local constraint over an individual control
unit located in every room. The classical dynamical models of evolutionary
game theory such as replicator dynamics, logit dynamics, etc. fail to
incorporate respective constraints. With the escort evolutionary dynamical
(ED) model it is possible to address these local constraints through the
concept of the intersection of simplices. However, evaluation of these
classical dynamics along with ED is driven by expected payoff obtained by the
overall population, which renders a centralized implementation approach. To
mitigate this central dependency a distributed version of the ED model
referred to as distributed escort dynamics (DED) is proposed. The control
action devised adopting DED approach is shown to provide smooth trajectory
tracking along with the low start-up transience when compared with distributed
interior point (DIP) method.

###### Index Terms:

Consensus problems, Distributed algorithm, Escort dynamics, Graph theory,
Resource allocation.

TABLE I: List of Symbols $N$ | Total number of zones within the building
---|---
( rooms and walls )
$\mathcal{R}$ | Set comprising of rooms
$\mathcal{W}$ | Set comprising of walls
$\mathcal{Z}$ | Set representing all the zones
$\alpha_{i,j}$ | Thermal condunctance between $i^{th}$ and $j^{th}$ entity
$u_{i}$ | Output power of $i^{th}$ actuator unit
$U$ | Total power generation at every instance
$k$ | Total number of rooms
$\boldsymbol{t^{set}}$ | Set point temperature profiles
$\boldsymbol{t^{a}}$ | Ambient temperature profile
$\boldsymbol{d^{i}}$ | Disturbance profile causing temperature deviations
at $i^{th}$ instance
$\boldsymbol{u^{lo}}$ | Lower bounds on the individual actuator outputs
( local constraints )
$\boldsymbol{u^{up}}$ | Upper bounds on the individual actuator outputs
( local constraints )
$\mathcal{G}$ | Graph ensuring information transfer among
the neighbouring actuator mechanisms
$\boldsymbol{t^{i}}$ | Actual temperature profiles of zones at $i^{th}$ instance
$\boldsymbol{x^{i}}$ | Population distribution at $i^{th}$ instance
$f_{j}^{i}$ | Payoff obtained for employing $j^{th}$ strategy at $i^{th}$ instance
$\boldsymbol{x^{lo}}$ | Upper bounds on the population proportions
$\boldsymbol{x^{up}}$ | Lower bounds on the population proportions
$\boldsymbol{\eta}$ | Escort function accommodating $\boldsymbol{x^{lo}}$
$\boldsymbol{\xi}$ | Escort function accommodating $\boldsymbol{x^{up}}$

##  I Introduction

The rapid speed of urbanization has resulted in an ever-increasing number of
commercial as well as residential building infrastructures. The energy
efficient operation of these buildings is very crucial not only from financial
but also from environmental perspective [1],[2]. The variety of building
maintenance and regulation processes are taken care of by using heating,
ventilation and air conditioning (HVAC) systems [3],[4]. Of these, the process
of thermal regulation results in the majority of energy consumption.

In building temperature control (BTC) formulation [5], the main goal of an
HVAC system is to regulate the temperature within the rooms as per respective
residents’ comfort, while optimizing total energy consumption. The limited
availability of the energy presents BTC issue as a multi-objective, multi-
agent resource allocation problem. In this, the control mechanism installed
within every room acts as a different agent whose objective is to regulate
respective room temperature around resident’s desired setpoint. Mainly there
are two ways to address such multi-agent optimization problems, i.e.
centralized and distributed.

Though centralised approach in which central entity gathers system information
and provides an exact allocation scheme, it is a very time-consuming process.
Also, the dependency on a central authority raises security and privacy
concerns. Moreover, it is not always possible to satisfy the requirement of
central authority in distributed scenario [6]. On the other hand, the
distributed approach, instead of depending on a global state, solves the
optimization problem based on the locally available information [7] of the
respective agent’s neighbourhood. The distributed method provides consensus-
based approach [5] in which a fair play is established ensuring equal payoff
to every agent.

The paper exploits the interacting multi-agent perspective to propose an
evolutionary game theory based escort dynamical (ED) approach [8] to address
BTC problem. Unlike classical evolutionary dynamics, the ED model is shown to
accommodate bounds on the individual control dynamics [9] along with global
resource constraint through the intersection of simplices. Moreover, based on
a distributed version of the classical evolutionary model discussed in [10], a
consensus-based distributed implementation of ED model referred to as
distributed escort dynamics (DED) model is proposed. This model with the
property of positive invariantness is shown to converge to an equilibrium
state corresponding to equal payoff to every agent.

The performance evaluation of DED implementation is carried out through
comparative analysis with widely used resource allocation protocol [5],
distributed interior point (DIP) method based BTC problem implementation. It
has been shown that DED approach provides smooth trajectory tracking along
with a low start-up transience as compared to the DIP approach. The major
contributions of the proposed research work are:

  * •

Implementation of an evolutionary game theoretical based approach to address
resource allocation problem of BTC.

  * •

Distributed version ED model has been proposed and convergence analysis of the
same has been provided.

The rest of the paper is structured as follows. Section II describes the BTC
process and provides the control objective. In section III, ED dynamics have
been introduced, also, the distributed version of ED, referred to as DED is
proposed. Section IV analyzes the BTC problem implementation using DED.
Representative case studies and concluding remarks are present thereafter.

##  II Mathematical Prerequisite and Notations

The vector quantities are represented using bold lowercase symbols. The
symbolic representation $x_{j}^{i}$, where $i,j$ belongs to class of whole
numbers, represents $j^{th}$ component of vector $\boldsymbol{x}$ at $i^{th}$
instance. The identity matrix is represented as $I$. The difference notation
$|\cdot|$ represents the Euclidean norm.

###  II-A Graph Theory

In the multi-agent system considered in this paper, modeling of the
communication network with the help of graph allows the agents to coordinate
their decisions as given in [11]. A graph is described by the triplet
$\mathcal{C}=(\mathcal{S},\mathcal{L},\mathcal{A})$.
$\mathcal{S}={\\{1,...,k}\\}$ represents the set of nodes,
$\mathcal{L}\subseteq\mathcal{S}\times\mathcal{S}$ represents the set of edges
connecting the nodes and matrix $\mathcal{A}$ represents a $k\times k$
nonnegative matrix whose elements satisfy: $\rho_{ij}=1$ if and only if
$(i,j)\in\mathcal{L}$, and $\rho_{ij}=0$ if and only if
$(i,j)\notin\mathcal{L}$. The nodes and edges of the graph corresponds to
agents and communication channels of the multi-agent system respectively. The
neighbours of node $i$ i.e. the set of nodes that are able to receive/send
information from/to node $i$ is represented by
$\mathcal{N}_{i}={\\{j\in\mathcal{S}:(i,j)\in\mathcal{L}}\\}$.

The graphical models utilized throughout the text are assumed to constitute
several properties, i.e.

###### Assumption 1

The graph does not contain self loops, $\rho_{ii}=0\text{ }\forall
i\in\mathcal{S}$ i.e.

###### Assumption 2

The graph is undirected. $\rho_{ij}=\rho_{ji}\text{ }\forall
i,j\in\mathcal{S}$

###### Assumption 3

The graph is connected, i.e. for every pair of nodes in $\mathcal{S}$ there
exists either direct or indirect path which connects the two.

For the graph $\mathcal{C}$, the Laplacian matrix $L(\mathcal{C})$ is a
$k\times k$ dimensional matrix with individual entities defined as,

$l_{ij}=\begin{cases}\sum_{j\in\mathcal{S}}\rho_{ij},\quad&\text{if}\quad
i=j\\\ -\rho_{ij},\quad\quad&\text{if}\quad i\neq j\\\ \end{cases}$ (1)

##  III Building Temperature Control process

A building can be considered to be partitioned mainly into two sets, a set of
$k$ rooms $\mathcal{R}$ and $m$ walls $\mathcal{W}$ also referred to as zones
Fig. 1. Let $\mathcal{Z}$ defines the composition of sets $\mathcal{R}$ and
$\mathcal{W}$ i.e. the total number of zones. Every room has a desired
temperature profile $\boldsymbol{t^{set}}$ defined over a time period by its
resident referred to as the setpoint temperature profile. The temperature
within every room is controlled by a central thermal unit (CTU) [12]. In this
case, a ventilation system connected to CTU is spread throughout the building
and is also connected to every room. The CTU controls the airflow which
affects the room temperature through an actuator mechanism installed therein.

Figure 1: (a) Building architecture with sequential room arrangement (b) Top
view representation of two adjacent rooms surrounded with walls

###  III-A The Temperature Dynamics

The building temperature evaluated over a certain time duration exhibits a
dynamic behaviour. At any time instance, the temperature within any zone gets
directly affected by temperature of adjacent zones. Also, the ambient
temperature of the surrounding affects the overall building temperature [13]
by directly interfering with the temperature of zones which are in direct
contact with it. The effect is mainly dependent on the value of thermal
conductance between the two interacting bodies. Also, the impromptu opening
and closing of windows and doors, and resident’s body temperature affects the
temperature of the respective zone.

The combined effect of all the factors on the temperature of the zone is
represented through a dynamical equation as,

$\theta_{i}\dot{t}_{i}=\sum_{j=1}^{N}\alpha_{i,j}(t_{j}-t_{i})+\alpha_{i,a}(t_{i}^{a}-t_{i})+v_{i}(u_{i}+d_{i}),\forall
i\in\mathcal{Z}$ (2)

where, $\alpha_{i,j}$ is the thermal conductance between $i^{th}$ and $j^{th}$
zone. Its value is positive if respective zones are in direct contact with
each other, else it is zero. Similarly, $\alpha_{i,a}$ represents the thermal
conductance between outside environment and zone (walls) which is in direct
contact with it. $\theta_{i}$ represents the thermal capacitance value for the
$i^{th}$ zone while $N$ is the total number of zones.

The control parameter $u_{i}$ corresponds to the output of the actuator
installed in $i^{th}$ zone. The parameter $v_{i}$ is binary variable, with
possible values between $\\{0,1\\}$. Positive value of $v_{i}$ ensures that
the $i^{th}$ zone that is being evaluated in indeed a room. The thermal
disturbance such as impromptu opening and closing of windows and/or effect of
body temperature of resident with respect to the $i^{th}$ zone is
characterized as $d_{i}$. The detailed description of BTC process can be found
in [12].

###  III-B The Control Architecture

The CTU generates a fixed $U$ amount of power at every instance which is to be
distributed among actuator mechanisms so as to modulate respective rooms’
temperature around desired setpoints.

$\sum_{i=1}^{k}u_{i}=U$ (3)

However, in this case, because of an actuator output constraints, the
individual actuator output $u_{i}$ is bounded within upper and lower bounds.

$\displaystyle u_{i}^{lo}\leq u_{i}\leq u_{i}^{up},\quad\forall
i\in\mathcal{R}$ (4)

Depending on the constraints value (4), the (3) may or may not hold true. To
accommodate this uncertainty, a positive semidefinite variable $u_{k+1}$ is
introduced, so that,

$\sum_{i=1}^{k+1}u_{i}=U$ (5)

In the BTC framework, the variable $u_{k+1}$ corresponds to the unused power
at respective time instance.

###  III-C BTC Problem Definition

The building represents a global system which comprises of various subsystems
i.e. different zones. These zones interact with each other according to
dynamical model (2). The global objective here is to utilize optimal power to
regulate individual room temperature around its setpoint value.

$\text{minimize }\sum_{i=1}^{k}\frac{(t_{i}-t_{i}^{set})^{2}}{2},\quad\forall
i\in\mathcal{A}$ (6)

The global objective function (6) can be decoupled and solved locally at
individual subsystem level, i.e.

$\text{minimize }\frac{(t_{i}-t_{i}^{set})^{2}}{2},\quad\forall
i\in\mathcal{A}$ (7)

Hence, the task is to develop individual actuator implementation strategies to
attain the local objective (7), while satisfying the global (5) and local
constraints (4), respectively.

##  IV BTC as a Dynamics Distributed Resource Allocation problem

Unlike general resource allocation problem formulations where objective
function $f_{i}$ is real valued function of resource variable $x_{i}$, the
payoff formulation in BTC problem is dependent on the state (7) i.e.
temperature values of rooms. These states are driven through a dynamical model
(2) which is affected by the resource being allocated (8).

$D^{P}_{i}:\begin{cases}\dot{t}_{i}=g(\boldsymbol{t},u_{i})\\\
f_{i}=h(\boldsymbol{t},u_{i})\\\ \end{cases}$ (8)

Here $D^{P}_{i}$ represents the state dynamical model which is driven by
instantaneous state values $\boldsymbol{t}$ and control parameter which also
corresponds to resource $u_{i}$.

The system model is interconnected with the controller module $D^{C}_{i}$
which perform the resource allocation task according to a function $u$, (9).

$D^{C}_{i}:\dot{x}_{i}=q(f_{i},f_{j}),\forall j\in\mathcal{N}_{i}$ (9)

Fig. 2 provides the schematic representation of overall system. In DED
approach to BTC formulation, functions $g(\cdot),h(\cdot),\text{ and
}q(\cdot)$ corresponds to (2), (7), and (20), respectively.

Figure 2: General system schematic

The dynamical system model (8) is considered to be reached equilibrium point
when output consensus is attained [5]. When a set of subsystems is considered
to be reached output consensus if $lim_{t\rightarrow\inf}|f_{i}-f_{j}|=0$, for
all $i,j=0,1,2,...,k$, where $f_{i}$ is the output of $i^{t}h$ subsystem i.e.
room.

Hence, to obtain the steady state, the controller model (9) must drive the (8)
to the output consensus. Also, controller parameter should satisfy system
defined constraints such as global resource constraint (5) and local bounds
(4).

###  IV-A Convergence Analysis to the Output Consensus

Let $(\boldsymbol{t^{*}},\boldsymbol{u^{*}})$ be the rest point of the
dynamical model Fig. 2. To analyze the system with respect to its convergence
to an output consensus, the system dynamics (8), (9) are represented in error
coordinates with respect to $(\boldsymbol{t^{*}},\boldsymbol{u^{*}})$.

$ErrD^{P}_{i}:\begin{cases}\dot{e}_{t_{i}}=e_{g}(\boldsymbol{e_{t}},\boldsymbol{e_{u}})\\\
e_{f_{i}}=e_{h}(\boldsymbol{e_{t}},\boldsymbol{e_{u}})\end{cases}$ (10)
$ErrD^{C}_{i}:\dot{e_{u_{i}}}=e_{q}(\boldsymbol{e_{f}},\boldsymbol{e_{u}})$
(11)

The feedback interconnection Fig. 2 is considered to satisfy the uniqueness
assumption. The statement is as follows;

###### Assumption 4

For a feedback interconnection of plant (8) and controller (9) dynamics
represents in error coordinates with respect to
$(\boldsymbol{t^{*}},\boldsymbol{u^{*}})$ as (10), (11), if
$e_{g}(0,\boldsymbol{e_{u}})=0,$ then $e_{u_{i}}=0,\forall i=1,2,...,k.$

The Assumption 4 ensures the uniqueness of the rest point
$(\boldsymbol{t^{*}},\boldsymbol{u^{*}})$ i.e. feedback interconnection
dynamics Fig. 2 will only have one convergence point. For the feedback
interconnection of multi-agent systems satisfying the Assumption 4, the
Theorem 3.2.1 from [5] provides a set of sufficient conditions which ensures
the convergence of feedback interconnection Fig. 2 to output consensus. The
reformulated theorem is stated as,

###### Theorem 1

For a feedback interconnection of system dynamics (8), (9) satisfying the
Assumption 4, let $(\boldsymbol{t^{*}},\boldsymbol{u^{*}})$ be an equilibrium
point, where control law $q(\cdot)$ is of the form (9), if following
conditions hold true:

  1. C1.

The communication graph $\mathcal{G}$ for multi-agent system (9) is connected.

  2. C2.

The plant dynamics (8) expressed in error coordinates with respect to
$(\boldsymbol{t^{*}},\boldsymbol{u^{*}})$, (10), is strictly passive from the
input $e_{u}$ to the output $e_{f}$ with radially unbounded storage function.

  3. C3.

$u^{*}$ and $u(0)$ satisfy the global resource constraint (5).

Then, (8) reaches output consensus.

The condition C1 ensures that there always exists a path between two distinct
agents to facilitate information transfer. The passive nature of plant
dynamics is ensured by the fulfillment of condition C2. Moreover, the
interconnection of two passive systems results in the convergence to an
equilibrium point. In order to utilize this property of passive
interconnection not only plant dynamics (8) but also controller dynamics (9)
must be passive. According to the proposition provided in [5], the passive
nature of controller dynamics is ensured by the fulfillment of the condition
C3. The proposition is reformulated as follows,

###### Proposition 1

Given that the $\boldsymbol{u^{*}}$ satisfies the constant population
requirement (5), if $\boldsymbol{u(0)}$ satisfies (5) and graph $\mathcal{G}$
is connected, then multi-agent dynamics in error coordinates (11) is passive
and lossless from the input $\boldsymbol{e_{f}}$ to the output
$\boldsymbol{-e_{u}}$.

In this paper, an escort dynamics based distributed algorithm whose
interconnection with the BTC dynamics (2) as shown in Fig. 2 satisfies the
conditions enlisted in  Theorem 1, providing convergence to an output
consensus.

##  V Escort Evolutionary Game Dynamics

The evolutionary game theory analyzes the interactions among the large number
of players with comparatively less number of strategies. Instead of analyzing
individual player dynamics, it evaluates the evolution of the spread of
available strategies over the entire population. During this evolution
process, every player within the population chooses a certain strategy
according to some classical dynamical revision protocols such as, replicator
dynamics, logit dynamics, projection dynamics, etc. These models, also
referred to as classical dynamical models are of the form.

$\dot{x}_{i}^{next}=\mathcal{D}(x_{i}^{present},f_{i}^{present})$ (12)

These dynamical models drives the population proportions towards best
population proportion, such that,

$x_{i}^{*}f_{i}^{*}>x_{i}f_{i}\text{ for all }i=1,2,...,k$ (13)

where, $\boldsymbol{x}=[x_{1},x_{2},...,x_{k}]^{T}$ represents population
state vector. $x_{i}$ corresponds to the proportion of population employing
$i^{th}$ pure strategy. Similarly, $f_{i}$ corresponds to the payoff received
for employing $i^{th}$ strategy. $\boldsymbol{x^{*}}$ and $\boldsymbol{f^{*}}$
represent best possible population proportion vector and payoff vector,
respectively.

The normalized population proportion vector $\boldsymbol{x}$ can be considered
as a population distribution on a $k-1$ dimensional statistical simplex
manifold.

$\Delta_{k}=\\{\boldsymbol{x}\in\mathds{R}^{k}|\sum_{i=1}^{k}{x_{i}}=1,x_{i}\geq
0\\}$ (14)

hence, the evolution towards best possible population distribution corresponds
to the motion of initial population state vector $\boldsymbol{x_{0}}$ over
$(k-1)$ dimensional manifold of $\Delta_{k}$, (14), as shown in Fig. 3.

Figure 3: Simplex geometry in three dimensional Euclidean space

In the perspective of BTC process, the population proportion vector
$\boldsymbol{x}$ corresponds to the actuator output profile $\boldsymbol{u}$.
Also, the global resource constraint (5) gets reflected in simplex formulation
(14). In evolutionary game theoretical framework, the local bounds on the
actuator mechanism (4) are correlated as lower and upper bounds on individual
strategy proportion (15).

$\Gamma=\\{x_{i}^{lo}\leq x_{i}\leq x_{i}^{up}\\},\text{ for all }i=1,2,...,k$
(15)

Then the task is to determine best possible population distribution
corresponding to the best output profile for individual actuator mechanism.

However, the classical dynamical models of the form (12) do not have any
provisions to accommodate local individual proportion constraints (15). In
[14], the mixture of classical dynamical models is considered to incorporate
such proportion bound. In this paper, the evolutionary game theoretical model
of ED [8] has been used to address this issue.

###  V-A Escort Dynamics Features

The ED dictates the evolution of normalised population distribution over $k$
pure strategies under the influence of payoff achieved [8]. The continuous
time representation of the ED is,

$\dot{x_{i}}=\phi(x_{i})(f_{i}(\boldsymbol{x})-f_{\phi})\text{ for all
}i=1,2,...,k$ (16)

where, $\phi(x_{i})$ is the positive semidefinite function of population
proportion, also referred to as escort function. Also, $f_{i}(\boldsymbol{x})$
and $f_{\phi}$ represents the payoff obtained for employing $i^{th}$ pure
strategy and weighted average payoff, respectively.

The weighted average payoff $f_{\phi}$ is computed as,

$f_{\phi}=\frac{1}{\Phi(\boldsymbol{x})}\sum_{i=1}^{k}\phi_{k}(x_{i})f_{i}(\boldsymbol{x})$
(17)

where,
$\Phi(\boldsymbol{x})=\sum_{i=1}^{k}{\phi_{i}(x_{i})f_{i}(\boldsymbol{x})}$.
The presence of $f_{\phi}$ within dynamical model (16) renders centralised
implementation approach. For positive escort functions, (17) can be considered
as the expected value of the payoff function over a probability distribution
defined by escort functions as,

$\hat{\phi}(\boldsymbol{x})=\frac{1}{\Phi{\boldsymbol{x}}}[\phi(x_{1}),\phi(x_{2}),...,\phi(x_{k})]$
(18)

summation of (16) over entire population for all $k$ pure strategies is
computed as

$\sum_{i=1}^{k}{\dot{x}_{i}}=\sum_{i=1}^{k}{x_{i}f_{i}(\boldsymbol{x})}-f_{\phi}\sum_{i=1}^{k}{x_{i}}=0$
(19)

that is initially defined population size, $\sum_{i=1}^{k}{x_{i}}=1$, remains
constant throughout the evolution for $t>0$. Hence, the definition of escort
function is very crucial to constrain population proportion within more
restrictive region.

However, the presence of weighted average term $f_{\phi}$ within ED (16)
demands that at every revision instance the player must know the payoff
received by every other player belonging to population. This center oriented
arrangement is quite restrictive in nature. Also, corresponding real life
implementation will require more communicational as well as computational
infrastructure. To mitigate this central dependency the paper proposed a graph
theoretic based distributed version of ED referred to as DED.

###  V-B Distributed Escort Dynamics formulation

The central dependency of ED (16) model is removed through a graph theoretical
based approach. In this case, at every revision instance the $i^{th}$ strategy
proportion evolution is governed by the local information obtained from its
neighbourhood $\mathcal{N}_{i}$. The proposed DED model is represented as,

$\dot{x}_{i}=\phi(x_{i})\sum_{j\in\mathcal{N}_{i}}{\phi(x_{j})[f_{j}(\boldsymbol{x})-f_{i}(\boldsymbol{x})]}\text{
for all }i=1,2,...,k$ (20)

Here, $\phi(\cdot)$ is the escort function. Instead of depending on the
expected average payoff as in (16), the DED dynamics $i^{th}$ proportion (20)
is driven by the payoffs obtained by neighbourhood proportion. The DED model
can be considered as consensus based algorithm. i.e. the DED dynamics (20)
reaches steady state either if respective $\phi(\cdot)$ becomes zero or if the
payoff values obtained by respective proportion is same as that of its
neighbouring proportions.

The DED dynamics (20) can also be represented as,

$\dot{x}_{i}=\sum_{j=1}^{k}\rho_{i,j}[f_{j}(\boldsymbol{x})-f_{i}(\boldsymbol{x})]\text{
for all }i=1,2,...,k$ (21)

where, $\rho_{i,j}$ governs the connectivity between the two distinct
population proportions. Its value represents the weight of the links between
two adjacent population proportions. Here, the connectivity graph
$\mathcal{G}$ is assumed to be undirected, hence $\rho_{i,j}$ equals
$\rho_{j,i}$. The link weightage $\rho_{i,j}$ is formulated as,

$\displaystyle\rho_{i,j}$ $\displaystyle=\phi(x_{i})\phi(x_{j})$
$\displaystyle\text{ if }(i,j)\in\Sigma$ (22) $\displaystyle\rho_{i,j}$
$\displaystyle=0$ Otherwise

####  V-B1 The Positive Invariantness of DED

Similar to (19), the summation of (21) over all strategy proportion is
evaluated as,

$\displaystyle\sum_{i=1}^{k}\dot{x}_{i}=\sum_{i=1}^{k}\bigg{[}\sum_{j=1}^{k}\rho_{i,j}[f_{j}(\boldsymbol{x})-f_{i}(\boldsymbol{x})]\bigg{]}=$
(23)
$\displaystyle\rho_{1,1}(f_{1}-f_{1})+\rho_{1,2}(f_{2}-f_{1})+...+\rho_{1,k}(f_{k}-f_{1})$
$\displaystyle+\rho_{2,1}(f_{1}-f_{2})+\rho_{2,2}(f_{2}-f_{2})+...+\rho_{2,k}(f_{k}-f_{2})$
$\displaystyle:$
$\displaystyle+\rho_{k,1}(f_{1}-f_{k})+\rho_{k,2}(f_{2}-f_{k})+...+\rho_{k,k}(f_{k}-f_{k})$

As the connectivity graph $\mathcal{G}$ is undirected the expression (23)
equates to zero. i.e.

$\sum_{i=1}^{k}\dot{x}_{i}=0$ (24)

equation (24) ensures that the population size remains constant throughout the
evolution process. i.e. if for the initial distribution
$\sum_{i=1}^{k}x_{i}=1$ then it will remain the same for all $t>0$. This
property is referred to as positive invariantness.

The property of positive invariantness limits the possible population
distribution vectors within the manifold corresponding to
$\sum_{i=1}^{k}x_{i}=1$. However, to restrict the population dynamics within
more constrained bounds (15) the escort formulation is very crucial. In order
to accommodate system defined local constraints (4) on actuator output which
correlates into bounded nature of individual proportion coefficient $x_{i}$
for all $i=1,2,...,k$, (15) the escort function formulation is discussed.

###  V-C Escort function formulation for BTC problem

The geometric representation of individual strategy proportion bounds (15)
corresponds to the intersection of two $k-1$ dimensional simplices,
$\Delta_{k}^{lo}$ and $\Delta_{k}^{up}$, where

$\displaystyle\Delta_{k}^{lo}$
$\displaystyle=\\{\boldsymbol{x}\in\mathds{R}^{k}|x_{i}\geq{x_{i}^{lo}},\sum_{i=1}^{k}x_{i}=1\\}$
(25) $\displaystyle\Delta_{k}^{up}$
$\displaystyle=\\{\boldsymbol{x}\in\mathds{R}^{k}|x_{i}\leq{x_{i}^{up}},\sum_{i=1}^{k}x_{i}=1\\}$
(26)

Structures $\Delta_{k}^{lo}$ and $\Delta_{k}^{up}$ along with $\Delta_{k}$ are
convex in nature which allow them to represent just with the help of their
vertices. Let $S^{lo}$ and $S^{up}$ be $k\times k$ dimensional matrices whose
column vectors represent vertices of $\Delta_{k}^{lo}$ and $\Delta_{k}^{up}$,
respectively. Construction of $S^{lo}$ and $S^{up}$ is given as,

$\displaystyle S^{lo}$ $\displaystyle=X^{lo}+\sigma^{lo}I$ (27) $\displaystyle
S^{up}$ $\displaystyle=X^{up}+\sigma^{up}I$ (28)

where column spaces of matrices $X^{lo}$ and $X^{up}$ are spanned by
$\boldsymbol{x^{lo}}$ and $\boldsymbol{x^{up}}$ respectively. $\sigma^{lo}$
and $\sigma^{up}$ are scalars which are multiplied with identity matrix $I$
and added with column vectors of $X^{lo}$ and $X^{up}$ so as to form $S^{lo}$
and $S^{up}$. This structure of matrices, $S^{lo}$ and $S^{up}$ provides an
alternative way to define constraint simplices, $\Delta_{k}^{lo}$ and
$\Delta_{k}^{up}$ as,

$\displaystyle\Delta_{k}^{lo}$
$\displaystyle=\\{\boldsymbol{x}=S^{lo}\boldsymbol{\eta},\text{
}\boldsymbol{\eta}\in\mathds{R}^{k}|\sum_{i=1}^{k}\eta_{i}=1,\eta_{i}\geq
0\\}$ (29) $\displaystyle\Delta_{k}^{up}$
$\displaystyle=\\{\boldsymbol{x}=S^{up}\boldsymbol{\xi},\text{
}\boldsymbol{\xi}\in\mathds{R}^{k}|\sum_{i=1}^{k}\xi_{i}=1,\xi_{i}\geq 0\\}$
(30)

Hence it is possible to represent known population state $\boldsymbol{x}$ onto
$\Delta_{k}^{lo}$ and $\Delta_{k}^{up}$ in terms of $\boldsymbol{\eta}$ and
$\boldsymbol{\xi}$ respectively.

$\displaystyle\boldsymbol{\eta}$
$\displaystyle=(S^{lo})^{-1}\boldsymbol{x}=\frac{1}{\sigma^{lo}}\\{I-X^{lo}\\}$
(31) $\displaystyle\boldsymbol{\xi}$
$\displaystyle=(S^{up})^{-1}\boldsymbol{x}=\frac{1}{\sigma^{up}}\\{I-X^{up}\\}$
(32)

As $\sum_{i=1}^{k}x_{i}=1$, (31) and (32) can be interpreted in simplified
form as,

$\displaystyle\eta_{i}=\frac{1}{\sigma^{lo}}(x_{i}-x_{i}^{lo})$ (33)
$\displaystyle\xi_{i}=\frac{1}{\sigma^{up}}(x_{i}-x_{i}^{up})$ (34)

Here, $\boldsymbol{\eta}$ which correlates with lower bounds
$\boldsymbol{x^{lo}}$ is increasing in nature while $\boldsymbol{\xi}$ is
monotonically decreasing in nature which corresponds to upper bounds
$\boldsymbol{x^{up}}$. It is so because $\sigma^{lo}>0$ and $\sigma^{up}<0$,
which ensures that intersection of $\Delta_{k}^{lo}$ and $\Delta_{k}^{up}$ is
not empty, [15].

To incorporate the both lower and upper bounds (15) the final escort function
formulation consists multiplication of (33) and (34).

$\phi(x_{i})=\eta_{i}\times\xi_{i},\text{ for all }i=1,2,...,k$ (35)

The escort function formulation (35) ensures that if the initial distribution
estimate belongs to the interior of the constraints (15) then the individual
strategy proportion will always remain bounded during the evolution process.

##  VI Escort Dynamical approach for BTC problem

The BTC temperature dynamics (2) with objective function (7) depicts the plant
dynamics (10) whereas the DED dynamics (20) represents the controller model
$q(\cdot)$ in (9).

The Assumption 4 demands controller dynamics (9) to subdue as system state
dynamics (8) attains equilibrium point. If $\boldsymbol{e_{t}}=0$, i.e.
$\boldsymbol{t}-\boldsymbol{t^{*}}=0$, then according to (7) the respective
output vector $\boldsymbol{f}=0$. This in turn will diminish the controller
dynamics given by (20), i.e. $e_{x_{i}}=0$ for all $i=1,2,...,k$. Hence, BTC
dynamics (2) controlled by proposed DED protocol (20) satisfy Assumption 4.

The connectivity Assumption 3 of the system inherent communication topology
ensures the fulfillment of C1 in Theorem 1. The dynamics (2) is shown to be
satisfying condition C2 in [5]. Moreover, according to the positive
invariantness property (23) the DED dynamics (20) satisfies C3. The
fulfillment of C3 ensures that the DED model is passive in nature. Hence, the
interconnection of two passive dynamics (2) and (20) results in the stable
rest point $(t^{*},x^{*})$ which corresponds to the attainment of output
consensus.

Hence, according to theorem 1 the BTC temperature dynamics (2) for individual
room with respective objective (7), driven by consensus like DED dynamic (20)
will reach equilibrium point obtaining output consensus i.e. welfare scheme
ensuring equal payoff to every room.

The BTC process is a dynamic in nature where at every instance the temperature
values gets affected by thermal disturbance $\boldsymbol{d}$ and surrounding
temperature $\boldsymbol{t^{a}}$ according to (2). This temperature is then
regulated around setpoint temperature $\boldsymbol{t^{set}}$ using actuator
mechanism which is driven by DED dynamics (20).The algorithmic implementation
of BTC problem using DED dynamics is provided in Algorithm 1.  1\. BTC using
DED approach

1: Initialize:
$\boldsymbol{t^{set}},\boldsymbol{t^{a}},\boldsymbol{t^{1}},\boldsymbol{W_{z}},\boldsymbol{W_{a}},\boldsymbol{X},\boldsymbol{x^{1}},$
$\boldsymbol{x^{lo}},\boldsymbol{x^{up}},x^{total}$

2: Define: $\sigma^{lo}=x^{total}-\sum_{j=1}^{k}{x^{lo}_{j}}$
$\sigma^{up}=x^{total}-\sum_{j=1}^{k}{x^{up}_{j}}$

3: Compute: $\eta_{j}=\frac{1}{\sigma^{lo}}(x_{j}-x^{lo}_{j}),$
$\xi_{j}=\frac{1}{\sigma^{up}}(x_{j}-x^{up}_{j}),$
$\phi(x_{j})=\eta_{j}\times\xi_{j}$ for all $j=1,2,...,k$

4: Compute: $\boldsymbol{f^{i}}=[f_{1}^{i},f_{2}^{i},...,f_{k}^{i}]^{T}$
where, $f_{j}^{i}=t_{j}^{i}-t_{j}^{set_{i}}$

5: Evaluate:
$\boldsymbol{\dot{x}^{i}}=[\dot{x}_{1}^{i},\dot{x}_{2}^{i},...,\dot{x}_{k}^{i}]$
where, $\dot{x}_{j}^{i}$ is evaluated according to (20)

6: Compute: $\boldsymbol{x^{i+1}}=[x_{1}^{i+1},x_{2}^{i+1},...,x_{k}^{i+1}]$
where, $x_{j}^{i+1}=x_{j}^{i}+\dot{x}_{j}^{i}$

7: Evaluate:
$\boldsymbol{\dot{t}^{i}}=[\dot{t}_{1}^{i},\dot{t}_{2}^{i},...,\dot{t}_{N}^{i}]$
where, $\dot{t}_{j}^{i}$ is evaluated according to (2)

8: Compute: $\boldsymbol{t^{i+1}}=[t_{1}^{i+1},t_{2}^{i+1},...,t_{N}^{i+1}]$
where, $t_{j}^{i+1}=t_{j}^{i}+\dot{t}_{j}^{i}$

9: Assign: $i\leftarrow{i+1}$

##  VII REPRESENTATIVE CASE STUDY AND RESULTS

The BTC issue formulated in Section II has been addressed using the DED
approach. For comparison purpose the same problem has been implemented using
consensus-based resource allocation protocol of DIP, refer Appendix-A for its
mathematical formulation.

###  VII-A Operational Scenario

For implementation purpose building with $50$ rooms surrounded by the
comparatively cold environment is analyzed over a period of an entire day,
i.e. $k=50$. The ambient temperature of the surrounding is varying as shown in
Fig. 4(a). Fig. 4(b) depicts the desired temperature trajectories for the room
temperatures throughout the day, where $50$ rooms are sectioned into $3$
groups viz., $1-17,18-34,35-50$. It is possible to define a separate
temperature profile for each individual room however to neatly display the
performance the rooms are divided into $3$ groups. Initially, every room is at
$13^{0}C$ temperature whereas initial actuator output is considered to be
$0.5\text{ kWh}$. Every actuator output is locally constrained within the
range of $0-3.25\text{ kWh}$ while global constrained is obtained by
restricting the cumulative output of all actuators $130\text{ kWh}$ at every
given instance.

(a)

(b)

Figure 4: Operational conditions (a) Ambient temperature profile (b) Desired
room temperature profiles

###  VII-B Observations

Observations include the comparison between desired trajectories and actually
obtained trajectories of 50 rooms sectioned in 3 groups. The colour notations
used to represent these trajectories is shown in Fig. 5.

Results obtained for implementation of building temperature control using DED
are represented in Fig. 6(a), 7(a), and 8(a). In the initial operational phase
fast increase in actuator outputs is observed which results in on average
$2^{0}C$ temperature increment above desired set points. Corresponding
oscillations can also be observed in payoff values when payoff values converge
to zero, temperature profiles at respective interval attain desired
temperature values, Fig. 8(a). Within one damped oscillatory cycle temperature
deviations are reduced and desired trajectories are traced thereafter Fig.
6(a). After initial perturbations, actuator profiles display smooth
variations, Fig. 7(a).

Fig. 6(b), 7(b), and 8(b) represent observations regarding implementation of
the DIP method for the same problem. Like DED, DIP displayed oscillatory
behaviour in the initial phase, also referred to as startup transience, but it
persisted for more than one oscillation cycles, Fig. 7(b). Similar
fluctuations are reflected in initial temperature profiles, as temperature
varies back and forth of the desired value, Fig. 6(b). Peak temperature, as
well as actuator output overshoots in case of DIP, are higher than that of
DED, which results in larger temperature deviations shown in Fig. 8(b).

Figure 5: Colour notations of actual and desired room parameters’ trajectories

(a)

(b)

Figure 6: Desired temperature trajectories plotted against temperature
trajectories obtained by implementing (a) DED approach (b) DIP approach

(a)

(b)

Figure 7: Actuator trajectories obtained by implementing (a) DED approach (b)
DIP approach

(a)

(b)

Figure 8: Payoff trajectories computed for (a) DED approach (b) DIP approach

Also, temperature trajectories utilizing DED approach are closer to desired
temperature trajectories than that of trajectories implementing DIP. When
ambient temperature of the surrounding is above the average value the
corresponding actuator outputs are low, Fig. 7. It has been observed that the
initial peak overshoot is directly proportional to the initial difference
between the desired and actual temperature.

###  VII-C Analysis

In an ideal implementation algorithm, real-time trajectories should
convergence fast to desired temperature trajectories without any overshoots.
DED, DIP being a step size dependent first-order dynamical protocols show some
initial oscillatory behaviour.

####  VII-C1 Presence of startup transience

The oscillatory response for DED approach is less as compared to DIP approach,
Fig. 9 and 10. In DIP approach, control parameters i.e. actuator outputs are
constrained by means of a barrier function, (38), which actively participates
in manipulating actuator parameter, keeping it in the centre of the predefined
range. When actuator value reaches toward one of the boundaries the payoff
value changes drastically which in turn force it towards the centre. This
operation introduces oscillatory behaviour over a couple of cycles, Fig. 9(b)
and 10(b). On the other hand, the inclusion of constraint directly in the
dynamical equation (16) using escort function guarantees a comparatively
smooth transition from initial conditions to the desired ones, without
violating the constraints, Fig. 10(a) and 10(a). This is because payoff values
are not directly affected by changes in actuator output.

(a)

(b)

Figure 9: Desired temperature trajectories plotted against temperature
trajectories obtained by implementing (a) DED approach (b) DIP approach

(a)

(b)

Figure 10: Actuator trajectories obtained by implementing (a) DED approach (b)
DIP appraoch

####  VII-C2 Desired trajectory tracking

In DED approach, deviations in temperature value is defined as a payoff. As
operations continue every sub-system (room) reach on a common consensus value
of payoff function. The negative difference between actual and required
temperature value increases the actuator output value and vice versa. Hence
algorithm implementing DED tracks desired trajectory closely, Fig. 6(a). While
in DIP approach along with temperature deviations cost function also includes
barrier function. Hence when DIP reaches consensus, the consensus value gets
biased by the barrier function value, which introduces a marginal error
between actual temperature trajectories and desired trajectories, Fig. 6(b).

Lower actuator output value in comparatively warm surrounding environment,
Fig. 7 corresponds to efficient energy utilization.

##  VIII Conclusion

The issue of BTC in the resource allocation framework has been analyzed using
evolutionary game theory. The distributed version of evolutionary dynamics of
ED has been utilized to address resource allocation problem within system-
defined constraints. The local bounds on the individual actuator bounds are
incorporated within the DED framework through the concept of the intersection
of simplices. The consensus like DED approach is shown to attain output
consensus for dynamic resource allocation problem with global resource
constraint. The performance analysis on the metric of smooth trajectory
tracking and low startup transience of BTC mechanism implementation through
DED approach and DIP approach is carried out. Unlike DED where local bounds on
individual actuator output are accommodated through barrier formulation, the
DED approach restricts the dynamics within the dynamics through escort
function formulation. This escort function formulation is shown to have better
performance as compared to DIP approach in terms of mechanism longevity
through low startup transience than that of DIP.

This research work can be extended to dynamic resource allocation problems
where the constraint set is time-varying. This can be incorporated through
varying domains of the intersection of simplices.

## Appendix

###  VIII-A Distributed Interior Point method

Considering the BTC problem described in Section II, DIP protocol used to
obtain next control vector values, is given as,

$\dot{v}_{i}=\sum_{j\in\mathcal{M}_{i}}[r_{j}(\boldsymbol{v})-r_{i}(\boldsymbol{v})];$
(36)

where, $(1\times k)$ dimensional payoff vector is constructed as

$r_{i}(\boldsymbol{v})=f_{i}(\boldsymbol{v})+b_{i}(v_{i})\text{ for
}i=1,2,...,k$ (37)

here, $b_{i}(\boldsymbol{v}_{i})$ is defined as the derivative of barrier
function $\mathcal{B}(\boldsymbol{v})$ with respect to $v_{i}$ is added with
original objective of the $i^{th}$ sub-system to form payoff function for
respective sub-system.

Barrier function restricts the control vector values within a predefined
constraints. Logarithmic barrier function is one of the barrier function, [5],
given as

$\mathcal{B}(\boldsymbol{v})=-\epsilon[in(v_{i}-v_{i}^{lo})+ln(v_{i}^{up}-v_{i})],\forall
i=1,2,...,k$ (38)

## References

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# On the Cesàroro operator on the Hardy space in the upper half-plane

Valentin V. Andreev,
Department of Mathematics,
Lamar University, Beaumont, TX 77710, USA
Miron B. Bekker111Corresponding author,
Department of Mathematics
University of Pittsburgh at Johnstown, Johnstown, PA 15904, USA
Joseph A. Cima, Department of Mathematics,
The University of North Carolina at Chapel Hill
CB 3250, 329 Phillips Hall, Chapel Hill, NC, 27599, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We discuss the Cesàro operator on the Hardy space in the upper half-plane. We
provide a new simple proof of the boundedness of this operator, prove that
this operator is equal to the sum of the identity operator and a unitary
operator, which implies its normality.

Keywords: Cesàro operator, Hardy space, reproducing kernel, unitary
operator.
MSC 2020: 47B38, 30H10, 47B32

##  1 Introduction

In a classical paper by A. Brown, P. R. Halmos and A. L. Shields [2] were
investigated the Cesáro operators on different spaces ($l^{2}$, $L^{2}(0,1)$
$L^{2}(0,\infty)$). In particular, in [2] was proved that $I-C_{\infty}^{*}$
is a bilateral shift of multiplicity one, where $C_{\infty}$ is the Cesàro
operator on $L^{2}(0,\infty)$. From that time study of Cesàro operators
attracted a lot of attention and many important results were obtained. We
mention the result of T. L. Kriete and David Trutt [6] who proved that the
Cesàro operator on $l^{2}$ is subnormal.

Study of the Cesàro operators on spaces of analytic functions is currently an
active field of research on intersection of operator theory and complex
analysis. Many aspects of of the theory of Cesàro operators in different
settings, as well as some generalizations of Cesàro operators, are presented
in the survey article by W. Ross [7].

In 2013 A. Arvanitidis and A. Siskasis [1] introduced the Cesàro operator on
the Hardy spaces $H^{p}({\mathbb{C}}_{+})$ in the upper half-plane
${\mathbb{C}}_{+}$. They proved that the operator is bounded for
$1<p\leq\infty$ and determined its spectrum for $1<p<\infty$. These results
were obtained using profound theorems from the theory of semigroups, in
particular, the semigroups of composition operators. Application of this
technique to the investigation of invariant subspaces of the Cesàro operator
on the Hardy space $H^{2}({\mathbb{D}})$ can be found in the recent paper [5].

In the present paper we give an alternative elementary proof of the
boundedness of the Cesàro operator on $H^{p}({\mathbb{C}}_{+})$. For $p=2$ we
proof in an elementary way that the Cesàro operator is the sum of the identity
operator and a unitary operator, from which it follows, in particular, that
the Cesàro operator on $H^{2}({\mathbb{C}}_{+})$ is normal. In our approach we
use the fact that $H^{2}({\mathbb{C}}_{+})$ is a reproducing kernel Hilbert
space. Note that the representation of the Cesàro operator as the sum of the
identity operator and a unitary operator can be obtained by further refinement
of the methods that were used in [1].

##  2 Boundedness of the Cesàro operator

We denote by ${\mathbb{C}}_{+}$ the upper half-plane,
${\mathbb{C}}_{+}=\\{z=x+iy:y>0\\}$. For $1\leq p\leq\infty$ we denote by
$H^{p}({\mathbb{C}}_{+})$ the Hardy space in ${\mathbb{C}}_{+}$, i.e. a
function $f\in H^{p}({\mathbb{C}}_{+})$ ($1\leq p<\infty$) if $f$ is
holomorphic in ${\mathbb{C}}_{+}$ and

$\sup_{y>0}\int\limits_{-\infty}^{\infty}|f(x+iy)|^{p}dx<\infty.$ (1)

Also,

$\|f\|_{p}=\left[\sup_{y>0}\int\limits_{-\infty}^{\infty}|f(x+iy)|^{p}dx\right]^{1/p},$
(2)

and for $p=\infty$, $\|f\|_{\infty}=\sup_{z\in{\mathbb{C}}_{+}}|f(z)|$. For
$p=2$ the Hardy space $H^{2}({\mathbb{C}}_{+})$ is a Hilbert space and for
$f,g\in H^{2}({\mathbb{C}}_{+})$ the inner product $<f,g>$ is defined by the
formula

$<f,g>=\int\limits_{-\infty}^{\infty}f(x)\overline{g(x)}dx$ (3)

where $f(x)=\lim_{z\to x}f(z)$ and the limit exists non-tangentially for
almost all $x$ with respect to the Lebesgue measure. The boundary function
$f(x)$ belongs to $L^{2}(-\infty,\infty)$.

The Cesàro operator $C$ is defined on $H^{p}({\mathbb{C}}_{+})$ by the
following expression

$(Cf)(z)=\frac{1}{z}\int\limits_{0}^{z}f(\zeta)d\zeta.$ (4)

######  Theorem 1.

For $1<p\leq\infty$ the Cesàro operator is bounded on
$H^{p}({\mathbb{C}}_{+})$.

###### Proof.

The statement about $H^{\infty}({\mathbb{C}}_{+})$ is obvious, and
$\|C\|_{\infty}\leq 1$. Here $\|C\|_{\infty}$ is the norm of the Cesàro
operator on $H^{\infty}({\mathbb{C}}_{+})$.

Consider now the case $1<p<\infty$. Put in (4) $\zeta=zs$, $0\leq s\leq 1$.
Then (4) takes the form

$(Cf)(z)=\int\limits_{0}^{1}f(zs)ds.$

Consequently,

$\displaystyle|Cf(z)|\leq\int\limits_{0}^{1}|f(zs)|s^{\beta}s^{-\beta}ds\leq$
$\displaystyle\left[\int\limits_{0}^{1}s^{\beta
p}|f(zs)|^{p}ds\right]^{1/p}\left[\int\limits_{0}^{1}s^{-\beta
q}ds\right]^{1/q}=$ $\displaystyle\frac{1}{(1-\beta
q)^{1/q}}\left[\int\limits_{0}^{1}s^{\beta p}|f(zs)|^{p}ds\right]^{1/p}.$

Here $1/p+1/q=1$ and $0<\beta<1/q$. Therefore we have

$\displaystyle\int\limits_{-\infty}^{\infty}|(Cf)(x+iy)|^{p}dx\leq\frac{1}{(1-\beta
q)^{p/q}}\int\limits_{0}^{1}\left[\int\limits_{-\infty}^{\infty}s^{\beta
p}|f(sx+isy)|^{p}dx\right]ds=$ $\displaystyle\frac{1}{(1-\beta
q)^{p/q}}\int\limits_{0}^{1}s^{\beta
p-1}\left[\int\limits_{-\infty}^{\infty}|f(t+isy)|^{p}dt\right]ds.$

Since for each $s\geq 0$ and $f\in H^{p}({\mathbb{C}}_{+})$

$\int\limits_{-\infty}^{\infty}|f(t+isy)|^{p}dt\leq\|f\|^{p}_{p}$

we obtain

$\int\limits_{-\infty}^{\infty}|(Cf)(x+iy)|^{p}dx\leq\frac{\|f\|^{p}_{p}}{(1-\beta
q)^{p/q}}\int_{0}^{1}s^{\beta p-1}ds=\frac{1}{(1-\beta q)^{p/q}\beta
p}\|f\|_{p}^{p}.$

Last inequality means

$\|Cf\|_{p}\leq\frac{1}{(1-\beta q)^{1/q}(\beta p)^{1/p}}\|f\|_{p},$ (5)

that is the Cesàro operator is bounded. Moreover, $\|C\|\leq\inf\\{(1-\beta
q)^{-1/q}(\beta p)^{-1/p}:0<\beta<(p-1)/p\\}.$ A simple calculation gives

$\|C\|_{p}\leq\frac{p}{p-1}.$

In [1] it was proved that $\|C\|_{p}=p/(p-1)$.
Simple examples show the theorem does not hold for $p=1$. ∎

######  Remark 1.

In [8] it was proved that the class $H^{p}({\mathbb{C}}_{+})$ coincides with
the class of functions $f$ which are holomorphic in the upper half-plane and
such that

$(\|f\|_{p}^{*})^{p}=\sup\limits_{0<t<\pi}\int\limits_{0}^{\infty}|f(re^{it})|^{p}dr<\infty,\quad
0<p<\infty$

and $\|f\|_{p}^{*}$ is equivalent to the usual $H^{p}$ norm, that is there are
constants $0<A_{p}\leq B_{p}<\infty$ which depend only on $p$ such that
$A_{p}\|f\|_{p}\leq\|f\|_{p}^{*}\leq B_{p}\|f\|_{p}$ (for $p=2$ this class of
functions was considered in [3]). This fact can also be used to prove the
boundedness of the Cesàro operator on $H^{p}({\mathbb{C}}_{+})$ in the same
way as above.

######  Remark 2.

The same arguments that were used in the proof of Theorem 1 allow to prove the
following slightly more general statement:
Let $\operatorname{Re}\mu>-\dfrac{p-1}{p}$, $1<p<\infty$. Then the operator
$C_{\mu}$ defined by

$(C_{\mu}f)(z)=\frac{1}{z^{\mu+1}}\int_{0}^{z}\zeta^{\mu}f(\zeta)d\zeta$

is bounded on $H^{p}({\mathbb{C}}_{+})$.

##  3 Cesàro operator on $H^{2}({\mathbb{C}}_{+})$

In this section we assume that $p=2$ and use the Hilbert space structure of
$H^{2}({\mathbb{C}}_{+})$, in particular, the fact that
$H^{2}({\mathbb{C}}_{+})$ is a reproducing kernel Hilbert space.

######  Lemma 1.

The operator $C^{*}$ on $H^{2}({\mathbb{C}}_{+})$ is defined by the formula

$(C^{*}f)(z)=\frac{1}{2\pi
i}\int\limits_{-\infty}^{\infty}f(t)\frac{1}{t}\log{\left(1-\frac{t}{z}\right)}dt$
(6)

###### Proof.

Recall that for any $f\in H^{2}({\mathbb{C}}_{+})$ the boundary value
$\lim_{y\downarrow 0}f(x+iy)$ exists for almost all $x$ and the boundary
function belongs to $L^{2}({\mathbb{R}})$. We use the same letter $f$ for the
boundary function. The values of function $f$ in the upper half-plane are
recovered from the boundary function by the Cauchy formula

$f(z)=\frac{1}{2\pi i}\int\limits_{\infty}^{\infty}\frac{f(t)}{t-z}dt.$ (7)

(see, for example [4], Theorem 11.8). Formula (7) means that the function

$k_{z}(w)=\frac{1}{2\pi i}\frac{1}{\bar{z}-w}$ (8)

is the reproducing kernel for $H^{2}({\mathbb{C}}_{+})$, that is

$f(z)=<f,k_{z}>.$

Observe that

$(Ck_{z})(\zeta)=\frac{1}{2\pi
i\zeta}\int_{0}^{\zeta}\frac{1}{\bar{z}-w}dw=-\frac{1}{2\pi
i\zeta}\log{\left(1-\frac{\zeta}{\bar{z}}\right)}$ (9)

Since

$(C^{*}f)(z)=<C^{*}f,k_{z}>=<f,Ck_{z}>$

formula (6) follows. ∎

Later on we will need a special case of formula (6).
Because
$(C^{*}k_{z})(\zeta)=<C^{*}k_{z},k_{\zeta}>=\overline{<Ck_{\zeta},k_{z}>}$ it
follows that

$(C^{*}k_{z})(\zeta)=\frac{1}{2\pi
i}\frac{1}{\bar{z}}\log{\left(1-\frac{\bar{z}}{\zeta}\right)}.$ (10)

######  Theorem 2.

For the Cesàro operator on $H^{2}({\mathbb{C}}+)$ the following equality is
valid

$C=I+U$ (11)

where $I$ is the identity operator and $U$ is a unitary operator. In
particular, the Cesàro operator on $H^{2}({\mathbb{C}}_{+})$ is normal.

###### Proof.

We claim that for any $s,t>0$ the following equalities are valid:

$<(C-I)k_{is},(C-I)k_{it}>=<k_{is},k_{it}>$ (12)
$<(C^{*}-I)k_{is},(C^{*}-I)k_{it}>=<k_{is},k_{it}>$ (13)

Indeed,

$\displaystyle<(C-I)k_{is},(C-I)k_{it}>=$
$\displaystyle<Ck_{is},Ck_{it}>-<Ck_{is},k_{it}>-<k_{is},Ck_{it}>+<k_{is},k_{it}>=$
$\displaystyle<Ck_{is},Ck_{it}>-<Ck_{is},k_{it}>-<C^{*}k_{is},k_{it}>+<k_{is},k_{it}>.$

Consider the inner product $<Ck_{is},Ck_{it}>$. Using (9) and integrating by
parts on the interval of the form $[-n,n]$ and letting $n$ tend to infinity
one obtains

$\displaystyle<Ck_{is},Ck_{it}>=\frac{1}{4\pi^{2}}\int\limits_{-\infty}^{\infty}\frac{1}{x^{2}}\log{\left(1-i\frac{x}{s}\right)}\log{\left(1+i\frac{x}{t}\right)}dx=$
$\displaystyle\frac{1}{4\pi^{2}}\left\\{-\frac{1}{x}\log{\left(1-i\frac{x}{s}\right)}\log{\left(1+i\frac{x}{t}\right)}|_{x=-\infty}^{\infty}+\int\limits_{-\infty}^{\infty}\frac{1}{x}\frac{d}{dx}\left[\log{\left(1-i\frac{x}{s}\right)}\log{\left(1+i\frac{x}{t}\right)}\right]dx\right\\}=$
$\displaystyle\frac{1}{4\pi^{2}}\int\limits_{-\infty}^{\infty}\left[\frac{1}{x}\frac{\log{\left(1+i\dfrac{x}{t}\right)}}{1-i\dfrac{x}{s}}\left(-\frac{i}{s}\right)+\frac{1}{x}\frac{\log{\left(1-i\dfrac{x}{s}\right)}}{1+i\dfrac{x}{t}}\left(\frac{i}{t}\right)\right]dx=$
$\displaystyle\frac{1}{4\pi^{2}}\int\limits_{-\infty}^{\infty}\frac{1}{x}\frac{\log{\left(1+i\dfrac{x}{t}\right)}}{x+is}dx+\frac{1}{4\pi^{2}}\int\limits_{-\infty}^{\infty}\frac{1}{x}\frac{\log{\left(1-i\dfrac{x}{s}\right)}}{x-it}dx$

Note that for both integrands the point $x=0$ is a regular point. Both
integrals are easily evaluated using the calculus of residue. For the first
integral we need to close the contour of integration in the lower half plane
and for the second integral we need to close the contour of integration in the
upper half plane. As result one obtains

$<Ck_{is},Ck_{it}>=\frac{1}{2\pi}\left[\frac{1}{s}\log{\left(1+\frac{s}{t}\right)}+\frac{1}{t}\log{\left(1+\frac{t}{s}\right)}\right]$
(14)

From (14) it follows that $\|Ck_{is}\|^{2}=\dfrac{\log{2}}{\pi s}$.

According to (9) and (10)

$\displaystyle<Ck_{is},k_{it}>=(Ck_{is})(it)=\frac{1}{2\pi
t}\log{\left(1+\frac{t}{s}\right)},$
$\displaystyle<C^{*}k_{is},k_{it}>=(C^{*}k_{is})(it)=\frac{1}{2\pi
s}\log{\left(1+\frac{s}{t}\right)}.$

Now (12) follows.

In order to prove (13) we evaluate $<C^{*}k_{is},C^{*}k_{it}>$. Using (10) one
obtains

$<C^{*}k_{is},C^{*}k_{it}>=\frac{1}{4\pi^{2}st}\int\limits_{-\infty}^{\infty}\log{\left(1+\frac{is}{x}\right)}\log{\left(1-\frac{it}{x}\right)}dx.$

Making change of variables $x\mapsto 1/x$ and then integrating by parts one
obtains

$\displaystyle<C^{*}k_{is},C^{*}k_{it}>=\frac{1}{4\pi^{2}st}\int\limits_{-\infty}^{\infty}\frac{1}{x^{2}}\log{(1+isx)}\log{(1-itx)}dx=$
$\displaystyle\frac{1}{4\pi^{2}st}\int\limits_{-\infty}^{\infty}\frac{is\log{(1-itx)}}{x(1+isx)}dx+\frac{1}{4\pi^{2}st}\int\limits_{-\infty}^{\infty}\frac{-it\log{(1+isx)}}{x(1-itx)}dx=$
$\displaystyle\frac{1}{4\pi^{2}st}\int\limits_{-\infty}^{\infty}\frac{\log{(1-itx)}}{x(x-\dfrac{i}{s})}dx+\frac{1}{4\pi^{2}st}\int\limits_{-\infty}^{\infty}\frac{\log{(1+isx)}}{x(x+\dfrac{i}{t})}dx.$

Using calculus of residue again as above one obtains

$<C^{*}k_{is},C^{*}k_{it}>=\frac{1}{2\pi}\left[\frac{1}{t}\log{\left(1+\frac{t}{s}\right)}+\frac{1}{s}\log{\left(1+\frac{s}{t}\right)}\right].$

In particular, we have proved that for any $s,t>0$

$<Ck_{is},Ck_{it}>=<C^{*}k_{is},C^{*}k_{it}>.$ (15)

Using again (9) and (10) we obtain equality (13). Therefore, for any vector
$f\in H^{2}({\mathbb{C}}_{+})$ of the form $f=\sum a_{j}k_{is_{j}}$ (the sum
is finite) we have

$\|(C-I)f\|^{2}=\|(C^{*}-I)f\|^{2}=\|f\|^{2}.$

Let $\\{s_{j}\\}_{j=1}^{\infty}$ be a set of positive distinct numbers which
has a limit point $s_{0}>0$. Then the set of reproducing kernels
$\\{k_{is_{j}}\\}$ is linearly independent and
$\vee\\{k_{is_{j}}:j=1,2,\ldots\\}=H^{2}({\mathbb{C}}_{+})$. Indeed, if $h\in
H^{2}({\mathbb{C}}_{+})$ and $h\perp k_{is_{j}}$ for all $j=1,2,\ldots$, then
$f(is_{j})=0$ and, since $f$ is holomorphic, $f=0$. Our previous consideration
implies that $\|(C-I)f\|^{2}=\|(C^{*}-I)f\|^{2}=\|f\|^{2}$ for any $f\in
H^{2}({\mathbb{C}}_{+})$, that is that is the operator $C-I$ is unitary. ∎

######  Corollary 1.

The operator $C+C^{*}$ is a positive self-adjoint operator.

Since the operator $C-I$ is unitary it follows that $(C^{*}-I)(C-I)=I$, that
is $C^{*}C=C+C^{*}$.

The next corollary is a byproduct of the proof of Theorem 2.

######  Corollary 2.

For any $s,t>0$

$\frac{1}{2}\left[\frac{1}{s}\log{\left(1+\frac{s}{t}\right)}+\frac{1}{t}\log{\left(1+\frac{t}{s}\right)}\right]\leq\frac{\log{2}}{\sqrt{st}}$

with equality if and only if $s=t$.

Acknowledgement. Authors are very thankful to Dr. Mark A. Nudel’man (Odessa,
Ukraine) for numerous useful discussions and comments.

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