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Northwest, Hammond, USA
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Kilpatrick, Arun Kumar, W. Li, B.P. Padley, R. Redjimi, J. Roberts, J. Rorie,
W. Shi, A.G. Stahl Leiton, Z. Tu, A. Zhang University of Rochester, Rochester,
USA
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Galanti, A. Garcia-Bellido, O. Hindrichs, A. Khukhunaishvili, E. Ranken, R.
Taus Rutgers, The State University of New Jersey, Piscataway, USA
B. Chiarito, J.P. Chou, A. Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart,
M. Heindl, E. Hughes, S. Kaplan, I. Laflotte, A. Lath, R. Montalvo, K. Nash,
M. Osherson, S. Salur, S. Schnetzer, S. Somalwar, R. Stone, S. Thomas
University of Tennessee, Knoxville, USA
H. Acharya, A.G. Delannoy, S. Spanier Texas A&M University, College Station,
USA
O. Bouhali78, M. Dalchenko, M. De Mattia, A. Delgado, R. Eusebi, J. Gilmore,
T. Huang, T. Kamon79, H. Kim, S. Luo, S. Malhotra, D. Marley, R. Mueller, D.
Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock,
USA
N. Akchurin, J. Damgov, F. De Guio, V. Hegde, S. Kunori, K. Lamichhane, S.W.
Lee, T. Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang,
A. Whitbeck Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij University of
Virginia, Charlottesville, USA
M.W. Arenton, P. Barria, B. Cox, G. Cummings, J. Hakala, R. Hirosky, M. Joyce,
A. Ledovskoy, C. Neu, B. Tannenwald, Y. Wang, E. Wolfe, F. Xia Wayne State
University, Detroit, USA
R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa University of Wisconsin
- Madison, Madison, WI, USA
K. Black, T. Bose, J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De
Bruyn, L. Dodd, C. Galloni, H. He, M. Herndon, A. Hervé, U. Hussain, A.
Lanaro, A. Loeliger, K. Long, R. Loveless, J. Madhusudanan Sreekala, A.
Mallampalli, D. Pinna, T. Ruggles, A. Savin, V. Sharma, W.H. Smith, D. Teague,
S. Trembath-reichert †: Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at UFMS, Nova Andradina, Brazil
6: Also at Universidade Federal de Pelotas, Pelotas, Brazil
7: Also at Université Libre de Bruxelles, Bruxelles, Belgium
8: Also at University of Chinese Academy of Sciences, Beijing, China
9: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
10: Also at Joint Institute for Nuclear Research, Dubna, Russia
11: Also at Helwan University, Cairo, Egypt
12: Now at Zewail City of Science and Technology, Zewail, Egypt
13: Also at Purdue University, West Lafayette, USA
14: Also at Université de Haute Alsace, Mulhouse, France
15: Also at Tbilisi State University, Tbilisi, Georgia
16: Also at Erzincan Binali Yildirim University, Erzincan, Turkey
17: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
18: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
19: Also at University of Hamburg, Hamburg, Germany
20: Also at Brandenburg University of Technology, Cottbus, Germany
21: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary,
Debrecen, Hungary
22: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
23: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary, Budapest, Hungary
24: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India
25: Also at Institute of Physics, Bhubaneswar, India
26: Also at G.H.G. Khalsa College, Punjab, India
27: Also at Shoolini University, Solan, India
28: Also at University of Hyderabad, Hyderabad, India
29: Also at University of Visva-Bharati, Santiniketan, India
30: Now at INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari
c, Bari, Italy
31: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
32: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
33: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
34: Also at Riga Technical University, Riga, Latvia, Riga, Latvia
35: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
36: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
37: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
38: Also at Institute for Nuclear Research, Moscow, Russia
39: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
40: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
41: Also at University of Florida, Gainesville, USA
42: Also at Imperial College, London, United Kingdom
43: Also at P.N. Lebedev Physical Institute, Moscow, Russia
44: Also at California Institute of Technology, Pasadena, USA
45: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
46: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
47: Also at Università degli Studi di Siena, Siena, Italy
48: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy,
Pavia, Italy
49: Also at National and Kapodistrian University of Athens, Athens, Greece
50: Also at Universität Zürich, Zurich, Switzerland
51: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria,
Vienna, Austria
52: Also at Burdur Mehmet Akif Ersoy University, BURDUR, Turkey
53: Also at Şırnak University, Sirnak, Turkey
54: Also at Department of Physics, Tsinghua University, Beijing, China,
Beijing, China
55: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey
56: Also at Istanbul Aydin University, Application and Research Center for
Advanced Studies (App. & Res. Cent. for Advanced Studies), Istanbul, Turkey
57: Also at Mersin University, Mersin, Turkey
58: Also at Piri Reis University, Istanbul, Turkey
59: Also at Gaziosmanpasa University, Tokat, Turkey
60: Also at Ozyegin University, Istanbul, Turkey
61: Also at Izmir Institute of Technology, Izmir, Turkey
62: Also at Marmara University, Istanbul, Turkey
63: Also at Kafkas University, Kars, Turkey
64: Also at Istanbul Bilgi University, Istanbul, Turkey
65: Also at Near East University, Research Center of Experimental Health
Science, Nicosia, Turkey
66: Also at Hacettepe University, Ankara, Turkey
67: Also at Adiyaman University, Adiyaman, Turkey
68: Also at Vrije Universiteit Brussel, Brussel, Belgium
69: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
70: Also at IPPP Durham University, Durham, United Kingdom
71: Also at Monash University, Faculty of Science, Clayton, Australia
72: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA
73: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
74: Also at Bingol University, Bingol, Turkey
75: Also at Georgian Technical University, Tbilisi, Georgia
76: Also at Sinop University, Sinop, Turkey
77: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
78: Also at Texas A&M University at Qatar, Doha, Qatar
79: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea


#  Sampled-data Model Predictive Control with zero-order hold for arbitrary
output constraints

Dario Dennstädt∗† Lukas Lanza† Karl Worthmann† We gratefully acknowledge
funding by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-IDs 471539468 and 507037103; Lukas Lanza is also
grateful for the support from the Carl Zeiss Foundation (VerneDCt – Project
No. 2011640173).† Technische Universität Ilmenau, Institute of Mathematics,
Optimization-based Control Group, Weimarer Straße 25, 98693 Ilmenau, Germany
{lukas.lanza<EMAIL_ADDRESS>Universität Paderborn, Institut
für Mathematik, Warburger Straße 100, 33098 Paderborn, Germany
<EMAIL_ADDRESS>
###### Abstract

We propose a sampled-data Model Predictive Control (MPC) scheme with zero-
order hold, which ensures output-reference tracking within predefined error
bounds for relative-degree-one systems. Hereby, we explicitly deduce bounds on
the required maximal control input and sampling frequency such that the MPC
scheme is both initially and recursively feasible despite using zero-order
hold sampled-data control. A key feature of the proposed approach is that
neither terminal conditions nor a sufficiently-large prediction horizon are
imposed, which renders the MPC scheme computationally very efficient.

##  I Introduction

Model Predictive Control (MPC) has gained widespread recognition as a control
technique suitable for both linear and nonlinear systems, owing to its
capacity to effectively manage multi-input multi-output systems while adhering
to control and state constraints, see the textbooks [1, 2] and the references
therein. Since MPC depends on the iterative solution of Optimal Control
Problems (OCPs) with finite prediction horizons, it is essential for the
successful application of MPC to ensure initial and recursive feasibility,
which ensures that the OCPs are solvable at any particular time instance.
Often used methods to ensure recursive feasibility are suitably designed
terminal conditions (costs and constraints), cf. [3, 4] or the more recent
textbook [2], or a combination of so-called cost controllability [5], with a
sufficiently long prediction horizon, see [6, 7]. However, introducing such
techniques increase the computational effort for solving the OPC and
complicate the problem of finding initially feasible control signals
significantly. This can result in a substantial reduction of the domain of
admissible controls for MPC, see e.g. [8, 9]. These methods are considerably
more involved in the presence of time-varying state or output constraints, see
e.g. [10]. In [11] a MPC scheme, called funnel MPC, was proposed, which
relaxes the aforementioned restrictions. Incorporating output constraints in
the OPC it was proven that this control algorithm is initially and recursively
feasible and ensures output reference tracking within (time-varying)
predefined error margins, i.e., satisfaction of hard output constraints. It
was shown in the successor work [12] that for systems with relative degree one
and, in a certain sense, input-to-state stable internal dynamics, these
constraints are dispensable. Utilizing a special kind of “funnel-like” stage
cost already ensures constraint satisfaction and feasibility – without
resorting to the usage of terminal conditions or requirements on the length of
the prediction horizon. These results were further generalized to systems with
higher relative degree in [13, 14]. To allow for the application of funnel MPC
in the presence of disturbances and even a structural plant-model mismatch a
two component control structure consisting of funnel MPC and a high-gain
adaptive controller, namely funnel control, was recently proposed in [15]. The
combined controller (funnel MPC + funnel control) was further extended by a
learning component in [16]. Funnel control, the original inspiration for
funnel MPC, is a robust adaptive feedback control technique of high-gain type
first proposed in [17], see also the recent work [18] and the survey paper
[19] for a comprehensive literature overview. Invoking merely structural
assumptions, it allows for output tracking with prescribed performance
guarantees for a fairly large class of systems by continuously adapting the
applied control signal based on the continuously measured output signal.

However, both in practical applications as well as in simulations, system
measurements and control input signals are typically only given at discrete
time instances. Consequently, the assumption underlying funnel control, namely
that the signal is continuously available, is not met. In the recent work [20]
this shortcoming was addressed. It was shown that the control objective of
ensuring output tracking with predefined error boundaries can be achieved by a
sampled-data controller which receives system measurements only at uniformly
sampled discrete time instances and emits a piecewise constant control signal;
the latter is usually called _sampled-data control with zero-order hold_
(ZoH). Although the system state usually is measured in MPC only at discrete
time instances and while simulations suggest that funnel MPC can be
implemented using piecewise constant step functions theoretical results, so
far, require application of control signals belonging to the set of bounded
measurable functions.

In this article, we address the question, if it is possible to achieve the
same tracking guarantees as funnel MPC [12] using piecewise constant input
signals. Invoking ideas from [20], we show that for systems with relative
degree one with sign-definite high-gain matrix and stable internal dynamics
this is possible. We show this without using a “funnel-like” stage cost as in
[12], but we allow a large class of cost functions. Moreover, explicit bounds
on the maximal control input as well as on the sufficient step length of the
applied control signals will be derived. We rigorously prove that the proposed
MPC algorithm is initially and recursively feasible.

The remainder of this article is organized as follows. In Section II we
introduce the system class under consideration, and define the tracking
control objective. In Section III we propose a MPC algorithm, which achieves
the control objective by using piecewise constant inputs only. As the main
result, we formulate a feasibility statement in Theorem III.1, the proof of
which is relegated to Section VI. A numerical simulation of the proposed MPC
algorithm is shown in Section IV. We conclude the article with a summary and
an outlook in Section V.

Nomenclature. $\mathds{N}$ and $\mathds{R}$ denote natural and real numbers,
respectively. $\mathds{N}_{0}:=\mathds{N}\cup\\{0\\}$ and $\mathds{R}_{\geq
0}:=[0,\infty)$. $\left\|x\right\|:=\sqrt{\left\langle x,x\right\rangle}$
denotes the Euclidean norm of $x\in\mathds{R}^{n}$, and
$\mathcal{B}_{v}:=\left\\{\ x\in\mathds{R}^{n}\ \left|\
\vphantom{x\in\mathds{R}^{n}}\|x\|\leq v\ \right.\right\\}$.
$\left\|A\right\|$ denotes the induced operator norm
$\left\|A\right\|:=\sup_{\left\|x\right\|=1}\left\|Ax\right\|$ for
$A\in\mathds{R}^{n\times n}$. $\text{GL}_{n}(\mathds{R})$ is the group of
invertible $\mathds{R}^{n\times n}$ matrices.
$\mathcal{C}^{p}(V,\mathds{R}^{n})$ is the linear space of $p$-times
continuously differentiable functions $f:V\to\mathds{R}^{n}$, where
$V\subset\mathds{R}^{m}$ and $p\in\mathds{N}_{0}\cup\\{\infty\\}$.
$\mathcal{C}(V,\mathds{R}^{n}):=\mathcal{C}^{0}(V,\mathds{R}^{n})$. On an
interval $I\subset\mathds{R}$, $L^{\infty}(I,\mathds{R}^{n})$ denotes the
space of measurable and essentially bounded functions $f:I\to\mathds{R}^{n}$
with norm $\left\|f\right\|_{\infty}:=\operatorname*{ess\,sup}_{t\in
I}\left\|f(t)\right\|$, $L^{\infty}_{\text{loc}}(I,\mathds{R}^{n})$ the set of
measurable and locally essentially bounded functions, and
$L^{p}(I,\mathds{R}^{n})$ the space of measurable and $p$-integrable functions
with norm $\left\|\cdot\right\|_{L^{p}}$ and with $p\geq 1$. Furthermore,
$W^{k,\infty}(I,\mathds{R}^{n})$ is the Sobolev space of all $k$-times weakly
differentiable functions $f:I\to\mathds{R}^{n}$ such that $f,\dots,f^{(k)}\in
L^{\infty}(I,\mathds{R}^{n})$.

##  II System class and control objective

In this section, we introduce the class of systems to be controlled, and
define the control objective precisely.

###  II-A System class

We consider nonlinear multi-input multi-output systems

$\displaystyle\dot{y}(t)$
$\displaystyle=f\big{(}\mathbf{T}(y)(t)\big{)}+g(\mathbf{T}(y)(t))u(t)$ (1)
$\displaystyle y|_{[t_{0}-\sigma,t_{0}]}$
$\displaystyle=y^{0}\in\mathcal{C}([t_{0}-\sigma,t_{0}],\mathds{R}^{m})$

with $t_{0}\geq 0$, “memory” $\sigma\geq 0$, control input $u\in
L^{\infty}_{\operatorname*{loc}}(\mathds{R}_{\geq 0},\mathds{R}^{m})$, and
output $y(t)\in\mathds{R}^{m}$ at time $t\geq 0$. Note that $u$ and $y$ have
the same dimension ${m\in\mathds{N}}$. The system consists of the nonlinear
functions $f\in\mathcal{C}(\mathds{R}^{q},\mathds{R}^{m})$,
$g\in\mathcal{C}(\mathds{R}^{q},\mathds{R}^{m\times m})$, and the nonlinear
operator $\mathbf{T}:\mathcal{C}([-\sigma,\infty),\mathds{R}^{m})\to
L^{\infty}_{\operatorname*{loc}}(\mathds{R}_{\geq 0},\mathds{R}^{q})$. The
operator $\mathbf{T}$ is causal, locally Lipschitz and satisfies a bounded-
input bounded-output property. It is characterised in detail in the following
definition.

###### Definition II.1

For $m,q\in\mathds{N}$ and $\sigma\geq 0$, the set
$\mathcal{T}_{\sigma}^{m,q}$ denotes the class of operators
$\textbf{T}:\mathcal{C}([t_{0}-\sigma,\infty),\mathds{R}^{m})\to
L^{\infty}_{\operatorname*{loc}}(\mathds{R}_{\geq 0},\mathds{R}^{q})$ for
which the following properties hold:

  * •

Causality: $\
\forall\,y_{1},y_{2}\in\mathcal{C}([t_{0}-\sigma,\infty),\mathds{R}^{m})$ $\
\forall\,t\geq t_{0}$:

$y_{1}|_{[t_{0}-\sigma,t]}=y_{2}|_{[t_{0}-\sigma,t]}\ \Longrightarrow\
\textbf{T}(y_{1})|_{[t_{0},t]}=\textbf{T}(y_{2})|_{[t_{0},t]}.$

  * •

Local Lipschitz: $\ \forall\,t\geq t_{0}$ $\
\forall\,y\in\mathcal{C}([t_{0}-\sigma,t];\mathds{R}^{m})$ $\
\exists\,\Delta,\delta,c>0$ $\
\forall\,y_{1},y_{2}\in\mathcal{C}([t_{0}-\sigma,\infty);\mathds{R}^{m})$ with
$y_{1}|_{[t_{0}-\sigma,t]}=y_{2}|_{[t_{0}-\sigma,t]}=y$ and
$\left\|y_{1}(s)-y(t)\right\|<\delta$, $\left\|y_{2}(s)-y(t)\right\|<\delta$
for all $s\in[t,t+\Delta]$:

$\operatorname*{ess\,sup}_{\mathclap{s\in[t,t+\Delta]}}\left\|\textbf{T}(y_{1})(s)\\!-\\!\textbf{T}(y_{2})(s)\right\|\\!\leq\\!c\
\sup_{\mathclap{s\in[t,t+\Delta]}}\
\left\|y_{1}(s)\\!-\\!y_{2}(s)\right\|\\!.$

  * •

Bounded-input bounded-output (BIBO): $\ \forall\,c_{0}>0$ $\ \exists\,c_{1}>0$
$\ \forall\,y\in\mathcal{C}([t_{0}-\sigma,\infty),\mathds{R}^{m})$:

$\sup_{t\in[t_{0}-\sigma,\infty)}\left\|y(t)\right\|\leq c_{0}\
\Longrightarrow\ \sup_{t\in[t_{0},\infty)}\left\|\textbf{T}(y)(t)\right\|\leq
c_{1}.$

Note that using the operator $\mathbf{T}$ many physical phenomena such as
_backlash_ , and _relay hysteresis_ , and _nonlinear time delays_ can be
modelled, where $\sigma\geq 0$ corresponds to the initial delay, cf. [18, Sec.
1.2]. In addition, systems with infinite-dimensional internal dynamics can be
represented by (1), see [21]. A practically relevant system with infinite-
dimensional internal dynamics modelled by an operator of this form is, for
example, the moving water tank system considered in [22].

###### Remark II.1

Consider a nonlinear control affine system

$\displaystyle\dot{x}(t)$ $\displaystyle=F(x(t))+G(x(t))u(t),\quad
x(t_{0})=x^{0}\in\mathds{R}^{n},$ (2) $\displaystyle y(t)$
$\displaystyle=H(x(t)),$

with $t_{0}\in\mathds{R}_{\geq 0}$, $x^{0}\in\mathds{R}^{n}$ and nonlinear
functions $F\in\mathcal{C}^{1}(\mathds{R}^{n},\mathds{R}^{n})$,
$G\in\mathcal{C}^{1}(\mathds{R}^{n},\mathds{R}^{n\times m})$, and
$H\in\mathcal{C}^{2}(\mathds{R}^{n},\mathds{R}^{m})$. Under assumptions
provided in [23, Cor. 5.6], there exists a diffeomorphism
$\Phi:\mathds{R}^{n}\to\mathds{R}^{n}$ which induces a coordinate
transformation putting the system (2) into the form (1) with new coordinates
$(y,\eta)\in\mathds{R}^{m}\times\mathds{R}^{n-m}$ (output and internal state)
for appropriate functions $f$, $g$, operator $\mathbf{T}$, and $\sigma=0$. In
this case $\mathbf{T}$ is the solution operator of the internal dynamics of
the transformed system. As in [12], exact knowledge about the coordinate
transformation and computation of the diffeomorphism $\Phi$ is not required to
apply Algorithm 1 to the system (2) – merely the existence of $\Phi$ has to be
assumed as a mean for the proofs.

In addition to the requirements for the operator detailed in Definition II.1,
we presume that the function $g$ satisfies the following assumption.

###### Assumption 1

The matrix function $g\in\mathcal{C}(\mathds{R}^{q},\mathds{R}^{m\times m})$
is strictly positive (negative) definite, that is for all $x\in\mathds{R}^{q}$
and for all $z\in\mathds{R}^{m}\setminus\\{0\\}$

$\ \left\langle z,g({x})z\right\rangle>0\quad(<0).$

With Assumption 1 and Definition II.1 we formally introduce the system class
under consideration.

###### Definition II.2

For $m\in\mathds{N}$ a system (1) belongs to the system class
$\mathcal{N}^{m}$, written ${(f,g,\mathbf{T})\in\mathcal{N}^{m}}$, if, for
some $p,q\in\mathds{N}$ and $\sigma\geq 0$, the following holds:
${f\in\mathcal{C}(\mathds{R}^{q},\mathds{R}^{m}})$, the function $g$ satisfies
Assumption 1, and ${\mathbf{T}\in\mathcal{T}^{m,q}_{\sigma}}$.

For $t\geq 0$ and a control function $u\in
L_{\operatorname*{loc}}^{\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$, the
system (1) has a solution in the sense of Carathéodory, meaning a function
$y:[t_{0}-\sigma,\omega)\to\mathds{R}^{m}$, $\omega>t_{0}$, with
$y|_{[t_{0}-\sigma,t_{0}]}=y^{0}\in\mathcal{C}([t_{0}-\sigma,t_{0}],\mathds{R}^{m})$
and $y|_{[t_{0},\omega)}$ is absolutely continuous and satisfies the ODE in
(1) for almost all $t\in[t^{0},\omega)$. A solution $y$ is called maximal, if
it has no right extension that is also a solution. A maximal solution is
called response associated with $u$ and denoted by $y(\cdot;t_{0},y^{0},u)$.
Note that in the case $\sigma=0$, we mean by $y|_{[t_{0}-\sigma,t_{0}]}$ the
evaluation of the function at $t_{0}$ , i.e.
$y|_{[t_{0}-\sigma,t_{0}]}=y(t_{0})$, and refer to the vector space
$\mathds{R}^{m}$ when using the notation
$\mathcal{C}([t_{0}-\sigma,t_{0}],\mathds{R}^{m})$.

###  II-B Control objective

The control objective is that the output $y$ of system (1) follows a given
reference $y_{\rm ref}$ with predefined accuracy. To be more precise, the
tracking error $t\mapsto e(t):=y(t)-y_{\operatorname*{ref}}(t)$ shall evolve
within the prescribed performance funnel

$\displaystyle\mathcal{F}_{\psi}=\left\\{\ (t,e)\in\mathds{R}_{\geq
0}\times\mathds{R}^{m}\ \left|\ \vphantom{(t,e)\in\mathds{R}_{\geq
0}\times\mathds{R}^{m}}\left\|e\right\|\leq\psi(t)\ \right.\right\\}.$

This funnel is determined by the choice of the function $\psi$ belonging to

$\displaystyle\mathcal{G}:=\left\\{\ \psi\in W^{1,\infty}(\mathds{R}_{\geq
0},\mathds{R})\ \left|\ \vphantom{\psi\in W^{1,\infty}(\mathds{R}_{\geq
0},\mathds{R})}\inf_{s\geq 0}\psi(s)>0\ \right.\right\\},$

see also Figure 1.

$t$•$\inf\limits_{t\geq 0}\psi(t)$$(0,e(0))$$\psi(t)$ Figure 1: Error
evolution in a funnel $\mathcal{F}_{\psi}$ with boundary $\psi(t)$; the figure
is based on [24, Fig. 1], edited for present purpose.

Note that keeping the tracking error in $\mathcal{F}_{\psi}$ does not mean
asymptotic convergence to zero. Moreover, the funnel boundary is not
necessarily monotonically decreasing. In some situations, like in the presence
of periodic disturbances, widening the funnel over some later time interval
might be advantageous in applications. The specific application usually
dictates the constraints on the tracking error and thus indicates suitable
choices for $\psi$.

We aim to develop a MPC algorithm, which achieves the aforementioned control
objective. In contrast to the funnel MPC scheme proposed in [12], the space of
admissible controls is restricted to step functions, i.e. the control signal
can only change finitely often between two sampling instances. To introduce
the control scheme properly, we formally define step functions in the
following definition.

###### Definition II.3

Let $I\subset\mathds{R}_{\geq 0}$ be an interval of the form $I=[a,b]$ with
$b>a$ or $I=[a,\infty)$. We call a strictly increasing sequence
$\mathcal{P}=(t_{k})_{k\in\mathds{N}_{0}}$ with
$\lim_{k\to\infty}t_{k}=\infty$ and ${t_{0}=a}$ a partition of $I$. The norm
of $\mathcal{P}$ is defined as $\left|\mathcal{P}\right|:=\sup\left\\{\
t_{i+1}-t_{i}\ \left|\ \vphantom{t_{i+1}-t_{i}}i\in\mathds{N}_{0}\
\right.\right\\}$. A function $f:I\to\mathds{R}^{m}$ is called step function
with partition $\mathcal{P}$ if $f$ is constant on every interval
$[t_{i},t_{i+1})\cap I$ for all $i\in\mathds{N}_{0}$. We denote the space of
all step functions on $I$ with partition $\mathcal{P}$ by
$\mathcal{T}_{\mathcal{P}}(I,\mathds{R}^{m})$.

Note that in the case of finite intervals of the form $I=[a,b]$ with $b>a$,
Definition II.3 can also be formulated using finite sequences
$\mathcal{P}=(t_{k})_{k=0}^{N}$ with $N\in\mathds{N}$ and $t_{N}=b$. However,
using infinite sequences every partition $\mathcal{P}$ of $[a,\infty)$ is also
a partition of $[a,b]$ for all $b>a$. Using this fact will simplify
formulating our results. Further note that Definition II.3 allows for non-
uniform step length, i.e., for $\tau_{i}=|t_{i-1}-t_{i}|$ we allow
$\tau_{k}\neq\tau_{j}$ for $k\neq j$, where $i,j,k\in\mathds{N}$. However, in
practice, a uniform step length often will be used.

##  III Sampled-data MPC with zero-order hold

Before formulating a MPC algorithm, which achieves the control objective
described in Section II-B, we introduce the class of admissible stage-costs.
Let $\mathcal{D}^{\psi}:=\left\\{\ \\!\\!(t,y)\in\mathds{R}_{\geq
0}\times\mathds{R}^{m}\\!\ \left|\ \vphantom{\\!\\!(t,y)\in\mathds{R}_{\geq
0}\times\mathds{R}^{m}\\!}\\!\|y-y_{\rm ref}(t)\|<\psi(t)\\!\\!\
\right.\right\\}$, and consider functions $\tilde{\ell}\in
L^{1}(\mathcal{D}^{\psi}\times\mathcal{B}_{\bar{u}};\mathds{R}_{\geq 0})$ for
$\bar{u}>0$. Then, we define the stage-cost $\ell$ piecewise by

$\displaystyle\ell(t,y,u)\\!=\\!\begin{dcases}\\!\tilde{\ell}(t,y,u)&\left\|e(y,t)\right\|\\!\leq\\!\psi(t)\\\
\\!\infty,&\text{else},\end{dcases}$ (3)

where $e(y,t):=y-y_{\operatorname*{ref}}(t)$. A suitable choice is

$\displaystyle\ell(t,y,u)=\begin{cases}\|e(y,t)\|^{2}+\lambda_{u}\|u\|^{2},&\left\|e(y,t)\right\|\leq\psi(t)\\\
\infty,&\text{else},\end{cases}$ (4)

where $\lambda_{u}\geq 0$ is a design parameter. Note that in contrast to
funnel MPC investigated in [11, 12], the stage-cost (3) allows the error to be
“on the funnel boundary”.

Invoking stage-costs like in (3), we propose the sampled-data MPC with zero-
order hold Algorithm 1, where the input is restricted to piecewise constant
step functions with given step length.

Algorithm 1 Sampled-data MPC with zero-order hold

Given: System (1), reference $y_{\operatorname*{ref}}\in
W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$, funnel function
$\psi\in\mathcal{G}$, control bound $u_{\max}>0$, maximal step length $\tau>0$
of the control signal, and initial data
$y^{0}\in\mathcal{C}([t_{0}-\sigma,t_{0}],\mathds{R}^{m})$.
Set the time shift $\delta>0$, the pediction horizon $T\geq\delta$, initialize
the current time $\hat{t}=t_{0}$, and choose a partition
$\mathcal{P}=(t_{k})_{k\in\mathds{N}_{0}}$ of the interval $[t_{0},\infty)$
with $\left|\mathcal{P}\right|\leq\tau$ and which contains
$(t_{0}+k\delta)_{k\in\mathds{N}_{0}}$ as a subsequence.
Steps:

  1. 1.

Obtain a measurement of the output $y$ of (1) on the interval
$[t_{0}-\sigma,\hat{t}]$ and set $\hat{y}:=y|_{[t_{0}-\sigma,\hat{t}]}$.

  2. 2.

Compute a solution
$u^{\star}\in\mathcal{T}_{\mathcal{P}}([t_{k},t_{k}+T],\mathds{R}^{m})$ of

$\mathop{\operatorname{minimize}}_{\begin{subarray}{c}u\in\mathcal{T}_{\mathcal{P}}([t_{k},t_{k}+T],\mathds{R}^{m}),\\\
\left\|u\right\|_{\infty}\leq
u_{\max}\end{subarray}}\quad\int_{t_{k}}^{t_{k}+T}\ell(t,y(t;\hat{t},\hat{y},u),u)\
\text{d}t.$ (5)

  3. 3.

Apply the piecewise constant control signal
$\mu:[\hat{t},\hat{t}+\delta)\times\mathcal{C}([t_{0}-\sigma,\hat{t}],\mathds{R}^{m})\to\mathds{R}^{m}$,
defined by

$\mu(t,\hat{y})=u^{\star}(t)$ (6)

to system (1). Increase $\hat{t}$ by $\delta$ and go to Step (1).

###### Remark III.1

If the system is given as nonlinear control affine system (2), availability of
the output signal $y$ is not required on the whole interval $[t_{0},\hat{t}]$
during Step 1 of Algorithm 1, but measurement of the state
$\hat{x}=x(\hat{t};t_{0},x^{0},u_{\rm MPC})$ is sufficient.

###### Remark III.2

Note that while the time shift $\delta>0$ is an upper bound for the step
length $\tau>0$ of the control signals, $\delta$ is allowed to be larger than
$\tau$. In the latter case, several control signals are applied to the system
between two steps of the MPC Algorithm 1. This can also be interpreted as a
multistep MPC scheme, cf. [1].

In the following main result we show that for a funnel function
$\psi\in\mathcal{G}$, and a reference signal $y_{\operatorname*{ref}}$, there
exists a sufficiently large upper bound $u_{\max}>0$ on the control input, and
sufficiently small step length $\tau>0$ of the piecewise constant control such
that the MPC Algorithm 1 is initially and recursively feasible for every
prediction horizon ${T>0}$, and that it guarantees the evolution of the
tracking error within the predefined performance funnel $\mathcal{F}_{\psi}$.

###### Theorem III.1

Consider system (1) with $(f,g,\mathbf{T})\in\mathcal{N}^{m}$ and initial data
$y^{0}\in\mathcal{C}([t_{0}-\sigma,t_{0}],\mathds{R}^{m})$. Let
$\psi\in\mathcal{G}$ and $\ y_{\operatorname*{ref}}\in
W^{1,\infty}(\mathds{R}_{\geq 0},\mathds{R}^{m})$. Then, there exists
$u_{\max}>0$ and $\tau>0$ such Algorithm 1 with $T>0$ and $\delta>0$ is
initially and recursively feasible, i.e., at time $\widehat{t}=t^{0}$ and at
each successor time $\widehat{t}\in t^{0}+\delta\mathds{N}$ the OCP (5) has a
solution. In particular, the closed-loop system consisting of (1) and feedback
(6) has a (not necessarily unique) global solution
$y:[t^{0},\infty)\to\mathds{R}^{m}$ and the corresponding input is given by

$u_{\rm MPC}(t)=\mu(t,y|_{[t_{0}-\sigma,\hat{t}]}).$

Furthermore, each global solution $x$ with corresponding input $u_{\rm MPC}$
satisfies:

  1. (i)

$\ \forall\,t\geq t^{0}:\quad\left\|u_{\rm MPC}(t)\right\|\leq u_{\rm max}$.

  2. (ii)

The error $e=y-y_{\operatorname*{ref}}$ evolves within the error boundaries
$\mathcal{F}_{\psi}$, i.e., $\left\|e(t)\right\|\leq\psi(t)$ for all $t\geq
t^{0}$.

The proof is relegated to Section VI. Here we present the main ideas. First,
using the construction of the sampled-data ZoH controller [20], we show that
there exist piecewise constant controls satisfying the input constraints, and
achieving the output constraints. Invoking the definition of admissible cost
functions (3) we establish that the cost function is finite if and only if the
corresponding input belongs to the set of admissible inputs. In the next step
we show that minimal costs can be achieved by piecewise constant controls, and
in fact, that these piecewise constant controls are contained in the set of
admissible controls.

Although Theorem III.1 is an existence results, we emphasize that feasible
choices for the step length $\tau>0$ and the maximal control $u_{\rm max}$ are
given explicitly in (11) and (12). Since the derivation of both quantities
requires some additional notation, we relegated the explicit expression into
Section VI.

##  IV Simulation

The theoretical results are illustrated by a numerical example. We consider a
torsional oscillator with two flywheels, which are connected by a rod, see
Figure 2. Such a system can be interpreted as a simple model of a driving
train, cf. [25, 26].

$z_{1}$
---
$z_{2}$
---
$u$
---
$k,d$
---
Figure 2: Torsional oscillator. The figure is based on [26, Fig. 2.7], edited
to the case of two flywheels for the present purpose.

The equations of motion for the torsional oscillator are given by

$\displaystyle\begin{bmatrix}I_{1}&0\\\
0&I_{2}\end{bmatrix}\begin{pmatrix}\ddot{z}_{1}(t)\\\
\ddot{z}_{2}(t)\end{pmatrix}=$ $\displaystyle\begin{bmatrix}-d&d\\\
d&-d\end{bmatrix}\begin{pmatrix}\dot{z}_{1}(t)\\\ \dot{z}_{2}(t)\end{pmatrix}$
$\displaystyle+$ $\displaystyle\begin{bmatrix}-k&k\\\
k&-k\end{bmatrix}\begin{pmatrix}z_{1}(t)\\\
z_{2}(t)\end{pmatrix}+\begin{bmatrix}1\\\ 0\end{bmatrix}u(t),$

where for $i=1,2$ (the index $1$ refers to the lower flywheel) $z_{i}$ is the
rotational position of the flywheel, $I_{i}>0$ is the inertia, $d,k>0$ are
damping and torsional-spring constant, respectively. We aim to control the
oscillator such that the lower flywheel follows a given velocity profile.
Hence, we choose $y(t)=\dot{z}_{1}(t)$ as output. To remove the rigid body
motion from the dynamics, we introduce the quantity $\hat{z}:=z_{1}-z_{2}$.
With this new variable, setting $x:=(\hat{z},\dot{z}_{1},\dot{z}_{2})$ the
dynamics can be written as

$\dot{x}(t)=Ax(t)+Bu(t),\quad y(t)=Cx(t)=\dot{z}_{1}(t),$

where

$\displaystyle M:=\begin{bmatrix}1&0&0\\\ 0&I_{1}&0\\\
0&0&I_{2}\end{bmatrix},\ \tilde{A}:=\begin{bmatrix}0&1&-1\\\ -k&-d&d\\\
k&d&-d\end{bmatrix},\ \tilde{B}:=\begin{bmatrix}0\\\ 1\\\ 0\end{bmatrix},$
$\displaystyle A:=M^{-1}\tilde{A},\ B:=M^{-1}\tilde{B},\
C=\begin{bmatrix}0&1&0\end{bmatrix}.$

Using standard techniques, see e.g. [27] and invoking Remark II.1, the reduced
dynamics of the torsional oscillator can then be written in input/output form

$\displaystyle\dot{y}(t)$ $\displaystyle=Ry(t)+S\eta(t)+\Gamma u(t),$ (7)
$\displaystyle\dot{\eta}(t)$ $\displaystyle=Q\eta(t)+Py(t),$

where $\eta$ is the internal state, and $R=\frac{-d}{I_{1}},\
S=\frac{1}{I_{1}}\left[\begin{smallmatrix}k&d\end{smallmatrix}\right]$,
$Q=\frac{1}{I_{2}}\left[\begin{smallmatrix}0&I_{2}\\\
-k&-d\end{smallmatrix}\right]$,
$P=\frac{1}{I_{2}}\left[\begin{smallmatrix}-I_{2}\\\
d\end{smallmatrix}\right]$. Note that $Q$ is a stable matrix, i.e., the
internal dynamics are BIBS. The high-gain matrix is given by
$\Gamma:=CB=1/I_{1}>0$. For purpose of simulation, we choose the reference

$y_{\rm
ref}(t)=\frac{250}{2}\left(1+\frac{2}{\pi}\int_{0}^{t}e^{-\frac{1}{2}(t-3)^{2}}\right),$

which is a modified version of the error function (erf), and represents a
smooth transition from zero rotation to a (approximately) constant angular
velocity of $250$ rotations per unit time; we have $\|y_{\rm
ref}\|_{\infty}\leq 250$, $\|\dot{y}_{\rm ref}\|_{\infty}=250/\pi$. Inserting
the dimensionless parameters $I_{1}=0.136$, $I_{2}=0.12$, $k=10$, and $d=16$,
and invoking the reference $y_{\rm ref}$ and the constant error tolerance
$\psi=25$ (we allow $10\%$ deviation), we may derive worst case bounds on the
system dynamics by estimating the explicit solution of the linear equations
(7). For the sake of simplicity, we will assume $\eta(0)=0$, which does not
cause loss of generality. For $\|y\|_{\infty}\leq\|y_{\rm
ref}\|_{\infty}+\psi$ we estimate

$\displaystyle\forall\,t\geq 0\,:\ \ \|\eta(t)\|$
$\displaystyle\leq\frac{M}{\mu}\|P\|(\|y_{\rm ref}\|_{\infty}+\psi);$

where $M:=\sqrt{\|K^{-1}\|\|K\|}$ and $\mu:=1/(2\|K\|)$, and
$K\in\mathds{R}^{2\times 2}$ solves the Lyapunov equation
$KQ+Q^{\top}K+I(2)=0$, where $I(2)$ is the two dimensinal identity matrix.
Inserting the values, we find by simple calculations that the estimates for
step lenght of the control signal $\tau$ and maximal control provided in (11),
(12) are satisfied with $\tau=0.0048$, and $u_{\rm max}=357$. We choose the
time shift $\delta=\tau$, i.e., a constant control is applied to the system
between two iterations in Algorithm 1. Further, the prediction horizon is set
as $T=10\delta$. For the purpose of simulation, we use the cost function (4)
with $\lambda_{u}=10^{-1}.$ The results are depicted in Figures 3 and 4.

Figure 3: Outputs and reference, with error boundary.

We stress that the estimate in (12) is very conservative. To demonstrate this
aspect, we run a second simulation, where we chose $\tau=0.2$, and $T=1$. The
results of this simulation are labeled as $\hat{y},\hat{u}$, respectively.
With this much larger uniform step length, the tracking objective can be
satisfied as well, cf. Figures 3 and 4.

Figure 4: Control inputs.

Note that the maximal applied control value is much smaller than the
(conservative) estimate $u_{\rm max}$ which satisfies (11). The simulations
have been performed with Matlab using the CasADi framework.

##  V Conclusion

We proposed a MPC algorithm, which achieves for nonlinear control systems that
the output tracks a given reference with predefined accuracy. This control
objective was also under consideration in [12] (funnel MPC). While the MPC
scheme in [12] uses $L^{\infty}$–functions in the optimal control problem, we
showed in this article that the control signal can be restricted to piecewise
constant functions with given step length while still guaranteeing the
tracking of a reference signal within predefined boundaries. By giving bounds
on the sufficient step length for the applied control function, this solves an
open problem regarding the applicability of MPC using zero-order hold inputs
while maintaining satisfaction of hard output constraints. It is a subject of
future research to show that the funnel MPC schemes from [13, 14] for systems
with higher relative degree can also be restricted to piecewise constant step
functions while still achieving the control objective. Another topic for
future research is to investigate whether the sampled-data controller from
[20] can be combined with the presented MPC algorithm similar as in [15] to
robustly allow for the application of zero-order hold MPC to systems with
disturbances and a plant-model mismatch while receiving system data only at
sampling times, and restricting the control signal to piecewise constant
functions. In the case of a positive outcome, it might be possible to further
combine the resulting control scheme with learning techniques like done in
[16].

##  VI Proof of the main result

Throughout this section, let the assumptions of Theorem III.1 hold. In
preparation of proving Theorem III.1, we note some observations for later use
and recall some results from [20] adapted to the current setting. To achieve
that the tracking error $e:=y-y_{\operatorname*{ref}}$ evolves within
$\mathcal{F}_{\psi}$ with $\psi\in\mathcal{G}$, it is necessary that the
output $y(t)$ of the system (1) is at every time $t\geq t_{0}$ an element of
the set

$\displaystyle\mathcal{D}_{t}:=\left\\{\ y\in\mathds{R}^{m}\ \left|\
\vphantom{y\in\mathds{R}^{m}}\left\|y-y_{\operatorname*{ref}}(t)\right\|\leq\psi(t)\
\right.\right\\}.$ (8)

For a $L^{\infty}$–control function $u$ bounded by $u_{\max}>0$ to achieve the
control objective on an interval $I\subset\mathds{R}_{\geq 0}$ with
$\hat{t}:=\min I$, i.e. ensuring that the tracking error $e$ evolves within
$\mathcal{F}_{\psi}$, it is necessary for $u$ to be an element of the set

$\mathcal{U}_{I}(u_{\max},\hat{y})\\!:=\\!\left\\{\ \\!\\!\\!u\in
L^{\infty}(I,\mathds{R}^{m})\\!\\!\\!\ \left|\ \vphantom{\\!\\!\\!u\in
L^{\infty}(I,\mathds{R}^{m})\\!\\!\\!}\\!\\!\\!\\!\\!\begin{array}[]{l}y(t;\hat{t},\hat{y},u)\in\mathcal{D}_{t}\\!\\!\
\forall\,t\in I,\\\ \left\|u\right\|_{\infty}\leq
u_{\max}\end{array}\\!\\!\\!\\!\\!\\!\\!\ \right.\right\\}\\!,$

where $\hat{y}\in\mathcal{C}([t_{0}-\sigma,\hat{t}],\mathds{R}^{m})$ with
$\hat{y}(t)\in\mathcal{D}_{t}$ for all $t\in[t_{0},\hat{t}]$. Consequently,
for a step function with partition $\mathcal{P}$ to achieve the control
objective it has to be an element of

$\mathcal{U}_{I}^{\mathcal{P}}(u_{\max},\hat{y}):=\mathcal{T}_{\mathcal{P}}(I,\mathds{R}^{m})\cap\mathcal{U}_{I}(u_{\max},\hat{y}).$

A solution of the system (1) which fulfills the control objective up to a time
$\nu>t_{0}$ is an element of the set

$\mathcal{Y}_{\nu}\\!:=\\!\left\\{\
\\!\\!\zeta\\!\in\\!\mathcal{C}([t_{0}-\sigma,\infty),\mathds{R}^{m})\\!\\!\\!\
\left|\
\vphantom{\\!\\!\zeta\\!\in\\!\mathcal{C}([t_{0}-\sigma,\infty),\mathds{R}^{m})\\!\\!\\!}\\!\\!\\!\\!\begin{array}[]{l}\zeta=y^{0},\\!\
\forall\,t\\!\in\\![t_{0},\nu]\\!:\\\
\zeta(t)\\!-\\!y_{\operatorname*{ref}}(t)\\!\in\\!\mathcal{D}_{t}\end{array}\\!\\!\\!\\!\\!\
\right.\right\\}\\!.$

This is the set of all functions
$\zeta\in\mathcal{C}([t_{0}-\sigma,\infty),\mathds{R}^{m})$ which coincide
with $y^{0}$ on $[t_{0}-\sigma,t_{0}]$ and evolve within the funnel
$\mathcal{D}_{t}$ on the interval $[t_{0},\nu]$. With this notation, we may
state the following existence result, which is a particular version of [20,
Lemma 1.2].

###### Lemma VI.1

Under the assumptions of Theorem III.1, there exist constants $f_{\max}$,
$g_{\max}$, $g_{\min}>0$ such that for every $\nu>t_{0}$,
$\zeta\in\mathcal{Y}_{\nu}$, $z\in\mathds{R}^{n}\backslash\left\\{0\right\\}$
and $t\in[t_{0},\nu)$

$\displaystyle f_{\max}$
$\displaystyle\geq\left\|f((\mathbf{T}(\zeta))|_{[t_{0},\nu)})\right\|_{\infty},$
(9) $\displaystyle g_{\max}$
$\displaystyle\geq\left\|g((\mathbf{T}(\zeta))|_{[t_{0},\nu)})\right\|_{\infty},$
$\displaystyle g_{\min}$ $\displaystyle\leq\frac{\left\langle
z,g((\mathbf{T}(\zeta))|_{[t_{0},\delta)}(t))z\right\rangle}{\left\|z\right\|^{2}}.$

In virtue of Lemma VI.1 let

$\displaystyle\kappa_{0}$
$\displaystyle:=\left\|\psi\tfrac{\text{d}}{\text{d}t}1/\psi\right\|_{\infty}+\left\|1/\psi\right\|_{\infty}(f_{\max}+\|\dot{y}_{\operatorname*{ref}}\|_{\infty}),$
$\displaystyle\beta$
$\displaystyle>\tfrac{2\kappa_{0}}{g_{\min}\left\|\psi\right\|_{\infty}},$
$\displaystyle\kappa_{1}$
$\displaystyle:=\kappa_{0}+\left\|1/\psi\right\|_{\infty}g_{\max}\beta,$

and $\tau\in(0,\kappa_{0}/\kappa_{1}^{2}]$. It has been proven in [20, Theorem
2.8] that the application of the ZoH control

$\,u_{\rm
ZoH}(t)=\begin{cases}0,\\!\\!&\|e(t_{i})\|<\psi(t_{i})\left(1-\tfrac{\kappa_{0}^{2}}{\kappa_{1}^{2}}\right),\\\
-\beta\tfrac{e(t_{i})}{\|e(t_{i})\|^{2}},\\!\\!&\|e(t_{i})\|\geq\psi(t_{i})\left(1-\tfrac{\kappa_{0}^{2}}{\kappa_{1}^{2}}\right),\end{cases}$
(10)

to the system (1), for $t\in[t_{i},t_{i}+\tau)$ with $t_{i}=i\tau$,
$i\in\mathds{N}_{0}$, yields $y(t;t_{0},y^{0},u_{\rm ZoH})\in\mathcal{D}_{t}$
for all $t\in[t_{0},\infty)$. Note that $\left\|u_{\rm
ZoH}\right\|_{\infty}\leq\tfrac{\beta}{1-\kappa_{0}^{2}/\kappa_{1}^{2}}$.
Consequently
$\mathcal{U}_{[t_{0},\infty)}^{\mathcal{P}}(u_{\max},y_{0})\neq\emptyset$ for
$u_{\max}\geq\tfrac{\beta}{1-\kappa_{0}^{2}/\kappa_{1}^{2}}$ and all
partitions $\mathcal{P}$ of $[t_{0},\infty)$ with
$\left|P\right|\leq\kappa_{0}/\kappa_{1}^{2}$. If, for
$\hat{t}\in\tau\mathds{N}$, a on the interval $[t_{0},\hat{t}]$ piece-wise
constant control
$u\in\mathcal{U}_{[t_{0},\hat{t}]}^{\mathcal{P}}(u_{\max},y^{0})$ is applied
to the system (1) on the interval $[t_{0},\hat{t}]$, then the tracking error
$e$ evolves within the funnel $\mathcal{F}_{\psi}$, i.e.
$y(t;t_{0},y^{0},u)\in\mathcal{D}_{t}$ for all for all $t\in[t_{0},\hat{t}]$.
In particular, $y(\hat{t};t_{0},y^{0},u)\in\mathcal{D}_{\hat{t}}$. Then,
$y(\cdot;t_{0},y^{0},u)$ can be extended to a function
$\zeta\in\mathcal{C}([t_{0}-\sigma,\infty),\mathds{R}^{m})$, i.e.
$\zeta|_{[t_{0},\hat{t}]}\equiv y(\cdot;t_{0},y^{0},u)$. The function $\zeta$
is an element of $\mathcal{Y}_{\hat{t}}$. Therefore, the prerequisites of [20,
Theorem 2.8] are fulfilled with the same constants as in (10). This means that
the application of $u_{\rm ZoH}$ to the system (1) on the interval
$[\hat{t},\infty)$ ensures that the tracking error $e$ evolves within the
funnel $\mathcal{F}_{\psi}$ on the interval $[\hat{t},\infty)$. Thus, we have
$u_{\rm
ZoH}\in\mathcal{U}_{[t_{0},\hat{t}]}^{\mathcal{P}}(u_{\max},y(\hat{t};t_{0},y^{0},u))$.

Summing up our observations, we state the following direct consequence of [20,
Lemma 1.2, Theorem 2.8].

###### Corollary VI.1

Under the assumptions of Theorem III.1, let

$u_{\max}\geq\tfrac{\beta}{1-\kappa_{0}^{2}/\kappa_{1}^{2}}$ (11)

and $\mathcal{P}=(t_{k})_{k\in\mathds{N}_{0}}$ be a partition of the interval
$[t_{0},\infty)$ with

$\left|\mathcal{P}\right|\leq\kappa_{0}/\kappa_{1}^{2}.$ (12)

Then,
$\mathcal{U}_{[t_{0},\infty)}^{\mathcal{P}}(u_{\max},y_{0})\neq\emptyset$.
Furthermore, for all $k,j\in\mathds{N}_{0},k>j$, we have

$\displaystyle\
\forall\,u\in\mathcal{U}_{[t_{0},t_{j}]}^{\mathcal{P}}(u_{\max},y^{0})\\!:\mathcal{U}_{[t_{0},t_{k}]}^{\mathcal{P}}(u_{\max},y(t_{j};t_{0},y^{0},u))\neq\emptyset.$
(13)

With these preliminaries at hand, we may now prove Theorem III.1.

_Proof of Theorem III.1_: According to Corollary VI.1, there exist
$u_{\max}>0$ and $\tau>0$ such that
$\mathcal{U}_{[t_{0},\infty)}^{\mathcal{P}}(u_{\max},y^{0})\neq\emptyset$ for
a partition $\mathcal{P}$ with $\left|\mathcal{P}\right|\leq\tau$. Let
$T\geq\delta$ be arbitrary but fixed.
$\mathcal{U}_{[t_{0},\infty)}^{\mathcal{P}}(u_{\max},y^{0})\neq\emptyset$
implies that $\mathcal{U}_{[t_{0},t_{0}+T]}^{\mathcal{P}}(u_{\max},y^{0})$ is
non-empty as well. Fact (13) implies that if, at every time instance
$\hat{t}\in t_{0}+\delta\mathds{N}_{0}$ in Step 3 of Algorithm 1, a control
$u\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})$ is
applied to the system (1), where $\hat{y}=y(\hat{t};t_{0},y^{0},u_{\rm MPC})$
is the output of the system at time $\hat{t}$, then
$\left\|e(t)\right\|=\left\|y(t)-y_{\operatorname*{ref}}(t)\right\|\leq\psi(t)$
for all $t\in[\hat{t},\hat{t}+\delta]$ and
$\mathcal{U}_{[\hat{t}+\delta,\hat{t}+\delta+T]}^{\mathcal{P}}(u_{\max},y(\hat{t}+\delta;\hat{t},\hat{y},u))\neq\emptyset$.
In particular, this inductively implies that (i) and (ii) are fulfilled.

Therefore, it only remains to show that if
$\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})$ is non-
empty for some $\hat{t}\in t_{0}+\delta\mathds{N}_{0}$ and
$\hat{y}\in\mathcal{C}([t_{0}-\sigma,\hat{t}],\mathds{R}^{m})$ with
$\hat{y}(t)\in\mathcal{D}_{t}$ for all $t\in[t_{0},\hat{t}]$, then the OCP (5)
has a solution
$u^{\star}\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})$.
To prove this, assume
$\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})\neq\emptyset$
for $\hat{t}\in t_{0}+\delta\mathds{N}_{0}$ and
$\hat{y}\in\mathcal{C}([t_{0}-\sigma,\hat{t}],\mathds{R}^{m})$ with
$\hat{y}(t)\in\mathcal{D}_{t}$ for all $t\in[t_{0},\hat{t}]$. Define the
function
$J:L^{\infty}([\hat{t},\hat{t}+T],\mathds{R}^{m})\to\mathds{R}\cup\left\\{\infty\right\\}$
by

$J(u)=\int_{\hat{t}}^{\hat{t}+T}\ell(t,y(t;\hat{t},\hat{y},u),u)\ \text{d}t.$

_Step 1_ : Adapting [12, Theorem 4.3] to the current setting we show that
$J(u)<\infty$ for $u\in L^{\infty}([\hat{t},\hat{t}+T],\mathds{R}^{m})$ with
$\left\|u\right\|\leq u_{\max}$ if and only if
$u\in\mathcal{U}_{[\hat{t},\hat{t}+T]}(u_{\max},\hat{y})$. Given
$u\in\mathcal{U}_{T}(u_{\max},\hat{y})$, it follows from the definition of
$\mathcal{U}_{T}(u_{\max},\hat{y})$ that
$e(t):=y(t;\hat{t},\hat{y},u)-y_{\operatorname*{ref}}(t)\in\mathcal{D}_{t}$
for all $t\in[\hat{t},\hat{t}+T]$. Thus,

$\forall\,t\in[\hat{t},\hat{t}+T]:\ \left\|e(t)\right\|\leq\psi(t).$

Therefore,
$\ell(t,y(t;\hat{t},\hat{y},u),u)=\tilde{\ell}(t,y(t;\hat{t},\hat{y},u),u)$
for all $t\in[\hat{t},\hat{t}+T]$. Hence,

$\displaystyle
J(u)=\\!\int_{\hat{t}}^{\hat{t}+T}\\!\\!\\!\tilde{\ell}(t,y(t;\hat{t},\hat{y},u),u)\
\text{d}{t}\leq\|{\tilde{\ell}}\|_{{L^{1}{(\mathcal{D}^{\varphi}}\times\mathcal{B}_{u_{\max}})}}\\!\\!<\\!\infty.$

To show the opposite direction, let $u\in
L^{\infty}([\hat{t},\hat{t}+T],\mathds{R}^{m})$ with $J(u)<\infty$. Assume
there exists $\tilde{t}\in(\hat{t},\hat{t}+T]$ with
$\left\|e(\tilde{t})\right\|>\psi(\tilde{t})$. By continuity of the involved
functions, there exists $\varepsilon\in(0,\tilde{t}-\hat{t})$ with
$\left\|e(t)\right\|>\psi(t)$ for all $t\in(\tilde{t}-\varepsilon,\tilde{t})$.
Thus,

$\displaystyle J(u)$
$\displaystyle=\int_{\hat{t}}^{\hat{t}+T}\ell(t,y(t;\hat{t},\hat{y},u),u)\
\text{d}t$
$\displaystyle\geq\int_{\tilde{t}-\varepsilon}^{\tilde{t}}\ell(t,y(t;\hat{t},\hat{y},u),u)\
\text{d}t=\varepsilon\cdot\infty=\infty.$

_Step 2_ : We prove that
$\min_{u\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})}J(u)$
exists. Since
$\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})\subset\mathcal{U}_{[\hat{t},\hat{t}+T]}(u_{\max},\hat{y})$,
the set $\mathcal{U}_{[\hat{t},\hat{t}+T]}(u_{\max},\hat{y})$ is non-empty by
assumption. Since ${J(u)\geq 0}$ for all
$u\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}$, the infimum
$J^{\star}:=\inf_{u\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})}J(u)$
exists. Let
$(u_{k})_{k\in\mathds{N}_{0}}\in\left(\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})\right)^{\mathds{N}_{0}}$
be a minimizing sequence, meaning $J(u_{k})\to J^{\star}$. By choice in the
Algorithm 1 we known that $(t_{0}+\delta k)_{k\in\mathds{N}_{0}}$ is a
subsequence of the partition $\mathcal{P}=(t_{k})_{k\in\mathds{N}_{0}}$.
Therefore, there exists $M,N\in\mathds{N}_{0}$, $N\geq M$ with $t_{M}=\hat{t}$
and $t_{n}>\hat{t}+T$ for all $n>N$. Define $u_{i,k}:=u_{k}(t_{i})$ for
$i=M,\ldots,N$. For every $i=M,\ldots,N$, $(u_{i,k})_{k\in\mathds{N}_{0}}$ is
a sequence in $\mathds{R}^{m}$ with $\left\|u_{i,k}\right\|\leq u_{\max}$ for
all $k\in\mathds{N}$. Thus, it has a limit point
$u_{i}^{\star}\in\mathds{R}^{m}$. The function $u^{\star}$ defined by
$u^{\star}|_{[t_{i},t_{i+1})\cap[\hat{t},\hat{t}+T]}:=u^{\star}_{i}$ is an
element of $\mathcal{T}_{\mathcal{P}}([\hat{t},\hat{t}+T],\mathds{R}^{m})$
with $\left\|u^{\star}\right\|\leq u_{\max}$. Up to subsequence, $u_{k}$
converges uniformly to $u^{\star}$. Let $y_{k}:=y(\cdot;t_{0},y^{0},u_{k})$ be
the sequence of associated responses. By an adaption of the Steps 2, 3 of the
proof of [12, Theorem 4.6] to the current setting, we may infer that $(y_{k})$
has a subsequence (which we do not relabel) that converges uniformly to
$y^{\star}(\cdot;\hat{t},\hat{y},u^{\star})$. It remains to show
$u^{\star}\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})$.
This means to show $y^{\star}(t)\in\mathcal{D}_{t}$ for all
$t\in[\hat{t},\hat{t}+T]$. Assume there exists $\tau\in(\hat{t},\hat{t}+T]$
with $y^{\star}(\tau)\notin\mathcal{D}_{\tau}$, i.e.
$\left\|y^{\star}(\tau)-y_{\operatorname*{ref}}(\tau)\right\|>\psi(\tau)$.
There exists $\varepsilon>0$ with
$\left\|y^{\star}(\tau)-y_{\operatorname*{ref}}(\tau)\right\|>\psi(\tau)+\varepsilon$.
Since the uniform convergence of $(y_{k})$ towards $y^{\star}$ implies
pointwise convergence of $(y_{k})$, there exists $K>0$ such that
$\left\|y^{\star}(\tau)-y_{k}(\tau)\right\|<\varepsilon$ for all $k\geq K$.
Furthermore,
$\left\|y_{k}(\tau)-y_{\operatorname*{ref}}(\tau)\right\|\leq\psi(\tau)$ since
$u_{k}\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})$
for all $k\in\mathds{N}_{0}$. This raises the following contradiction for
$k\geq K$.

$\displaystyle\psi(\tau)+\varepsilon$
$\displaystyle<\left\|y^{\star}(\tau)-y_{\operatorname*{ref}}(\tau)\right\|$
$\displaystyle\leq\left\|y^{\star}(\tau)-y_{k}(\tau)\right\|+\left\|y_{k}(\tau)-y_{\operatorname*{ref}}(\tau)\right\|$
$\displaystyle\leq\varepsilon+\psi(\tau).$

Thus,
$u^{\star}\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})$.
It remains to show that $J^{\star}=J(u^{\star})$ and
$J(u^{\star})=\min_{u\in\mathcal{U}_{[\hat{t},\hat{t}+T]}^{\mathcal{P}}(u_{\max},\hat{y})}J(u)$,
which follows along the lines of Steps 6, 7 of the proof of [12, Theorem 4.6].
$\Box$

## Acknowledgements

We are deeply indebted to Thomas Berger (U Paderborn) for fruitful
discussions, several helpful comments, and remarks.

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$p_{u+1}<p_{u}$ is satisfied by some, but not all, pairs of primes
$(p_{u},p_{u+1})$ from the intervals
$(B_{u}^{\prime},B_{u}^{\prime}+B_{u}^{\prime}/L]$ and
$(B_{u+1}^{\prime},B_{u+1}^{\prime}+B_{u+1}^{\prime}/L]$. Clearly this problem
cannot arise when $L\geq 2P_{u}$ since then the intervals
$(B_{u}^{\prime},B_{u}^{\prime}+B_{u}^{\prime}/L]$ and
$(B_{u+1}^{\prime},B_{u+1}^{\prime}+B_{u+1}^{\prime}/L]$ contain at most one
prime each. It follows that any such $n$ to be counted in this case must have
two prime factors $p^{\prime}>p\geq P_{u}\geq L/2$ with
$p^{\prime}=(1+O(L^{-1})p$. Hence the corresponding contribution is

$O\left(\sum_{\begin{subarray}{c}p^{\prime}>p\geq L/2\\\
p^{\prime}=(1+O(L^{-1})p\end{subarray}}\sum_{\begin{subarray}{c}n\ll N\\\
p^{\prime}p|n\end{subarray}}\tau(n)\right)=O_{\varepsilon}\left(\sum_{\begin{subarray}{c}p^{\prime}>p\geq
L/2\\\
p^{\prime}=(1+O(L^{-1}))p\end{subarray}}\frac{N^{1+\varepsilon}}{p^{\prime}p}\right)=O_{\varepsilon}\left(N^{1+\varepsilon}L^{-1}\right).$

We therefore find that our sum is bounded by

$O_{\varepsilon}\left(L^{5}N^{1-1/1200+\varepsilon}+N^{1+\varepsilon}L^{-1}\right),$

whenever $L\leq Y$. We may then choose $L=N^{10^{-5}}$ say, to achieve the
claimed power saving in the case of large $d$.

###  10.2. Small $d$

To handle small $d$ it suffices to show that for any
$\mathfrak{k}\leq{\mathcal{C}},d\leq D_{1}$, and any non-principal
$\chi\pmod{d}$ that

$\sideset{}{{}^{\ast}}{\sum}_{c\pmod{d\mathfrak{k}}}\sideset{}{{}^{\flat}}{\sum}_{\begin{subarray}{c}(z,y)\in{\mathcal{U}}_{1}\times{\mathcal{U}}_{2}\\\
cz\equiv
y\pmod{d\mathfrak{k}}\end{subarray}}\beta_{z}\beta_{y}\chi(c)\ll_{C}\frac{N^{2}}{(\log
X)^{C}}$

for every $C>0$. Since

$\sideset{}{{}^{\ast}}{\sum}_{c\pmod{d\mathfrak{k}}}\chi(c)=0,$

it suffices to prove that if ${\mathcal{U}}={\mathcal{U}}_{1}$ or
${\mathcal{U}}_{2}$ then there is a number
$\mathfrak{M}=\mathfrak{M}({\mathcal{U}},d\mathfrak{k})$ such that

$\sum_{\begin{subarray}{c}z\in{\mathcal{U}}\\\
z\equiv\alpha\pmod{2d\mathfrak{k}}\end{subarray}}\beta_{z}=\mathfrak{M}+O_{C}\left(\frac{N}{(\log
X)^{C}}\right)$

for any $\gcd(\alpha,2d\mathfrak{k})=1$ and $C>0$, since $\beta_{z}$ is
supported on those $z$ free of small prime factors, and $2d\mathfrak{k}$ is
small. As before we may drop the summation condition $\flat$. For notational
convenience, we set $q=2d\mathfrak{k}$ and note that $q\leq(\log X)^{C_{0}}$
for some $C_{0}>0$.

As in the previous subsection we may assume that $\beta_{z}=\beta_{N(z)}$,
where $\beta_{n}$ is the indicator function of either $Q_{j}$ or $R$. We
describe the procedure for $Q_{j}$, the method for $R$ being similar. We
decompose $z$ as $s_{1}s_{2}$ with $N(s_{1})$ being the largest prime factor
of $N(s_{1}s_{2})$. The requirement that $n\in Q_{j}$ is then equivalent to a
condition of the form $N(s_{2})\in Q_{j}^{\prime}$ together with a restriction
of the type $N(s_{1})\in I(s_{2})$ for some real interval $I(s_{2})$.
Specifically, we have

$Q_{j+1}^{\prime}=\\{p_{2}\cdots p_{j+1}:p_{j=1}\in J,p_{j+1}<\cdots<p_{2}\\}$

and

$I(s_{2})=(p_{2},\infty)\cap\left(\frac{N^{\prime}}{N(s_{2})},\frac{N^{\prime}(1+\varpi)}{N(s_{2})}\right]\cap\left[\frac{Y}{N(s_{2})},\frac{X^{1/2-\delta}}{N(s_{2})}\right),$

where $p_{2}$ is the largest prime factor of $N(s_{2})$. When ${\mathcal{U}}$
is given by (7.10) the condition on the size of $N(s_{1}s_{2})$ is exactly the
condition

$N(s_{1})\in\left(\frac{N^{\prime}}{N(s_{2})},\frac{N^{\prime}(1+\omega)}{N(s_{2})}\right],$

and we have $\theta_{0}<\arg z\leq\theta_{0}+\omega_{2}$ exactly when $\arg
s_{1}$ is constrained in a small interval of length $O(\varpi_{2})$ dependent
on $\arg s_{2}$. It follows that

(10.4) $\sum_{\begin{subarray}{c}z\in{\mathcal{U}}\\\
z\equiv\alpha\pmod{q}\end{subarray}}\beta_{z}=\sum_{\begin{subarray}{c}N(s_{2})\in
Q_{j}^{\prime}\\\ \gcd(s_{2},q)=1\end{subarray}}{\mathcal{N}}(s_{2},\alpha),$

where ${\mathcal{N}}(s_{2},\alpha)$ is the number of ideal numbers $s_{1}$
satisfying

$s_{1}s_{2}\equiv\alpha\pmod{q},N(s_{1})\in I(s_{2}),\text{ and }\arg
s_{1},\arg s_{2}$

and for which $N(s_{1})$ is prime. We can estimate
${\mathcal{N}}(s_{2},\alpha)$ using a form of the Prime Number Theorem for
arithmetic progressions over number fields, due to Mitsui. We note that, as we
remarked earlier, we can easily re-divide our sectors in accordance with the
condition $N(z)\sim N$ as opposed to $\lVert{\mathbf{z}}\rVert_{2}\sim N$, so
we may apply Mitsui’s theorem without worry in each of our sectors. If we put
$\pi(X;q,\alpha,\theta)$ for the number of prime ideal numbers $\mathfrak{p}$
in a fixed ideal class satisfying $\mathfrak{p}\equiv\alpha\pmod{q}$ and
having norm at most $X$ with $0\leq\arg(\mathfrak{p})\leq\theta$, then
Mitsui’s theorem gives the estimate

(10.5) $\pi(X;q,\alpha,\theta)=\frac{w\theta
R_{K}}{2^{r_{1}}h_{K}\varphi_{K}(\mathfrak{a})}\operatorname{Li}(X)+O_{K}\left(X\exp\left(-c\sqrt{\log
X}\right)\right)$

where $r_{1}$ is the number of real embeddings of $K$, $w$ the number of roots
of unity in $K$, $R_{K}$ the regulator of $K$, and $h_{K}$ the class number o
$K$. Here $c$ is an absolute constant. Since we do not care about dependence
on $K$, we may take the implied constant in (10.5) as an absolute constant. We
emphasize that (10.5) holds uniformly for $\theta\in[0,2\pi]$ and for all
$q\leq(\log X)^{A}$.

Applying (10.5) with $q=2d\mathfrak{k}$ to estimate
${\mathcal{N}}(s_{2},\alpha)$, we have $I(s_{2})\subseteq(0,2N/N(s_{2})]$ and
so we will need to know that $q=2d\mathfrak{k}\leq(\log 2N/N(s_{2}))^{A}$ for
some constant $A$. This holds whenever $p$ divides an element of $Q_{j}$ then
one has $p\geq X^{\delta_{1}}$ with $\delta=(A\log\log X)/\log X$. Thus we
will have $2N/N(s_{2})\geq X^{\delta_{1}}$ and so

$\delta_{1}\log X\leq\log\left(\frac{N}{N(s_{2})}\right),$

which implies that

$\log
X\leq\left(\log\left(\frac{N}{N(s_{2})}\right)\right)^{\frac{1}{\varpi}}.$

Therefore whenever $2d\mathfrak{k}\leq(\log X)^{C_{0}}$ we have

$2d\mathfrak{k}\leq(\log
X)^{C_{0}}\leq\left(\log\left(\frac{2N}{N(s_{2})}\right)\right)^{\frac{C_{0}}{\varpi}}$

The required condition therefore holds when $\mathfrak{k}\leq\mathfrak{K}$ and
$d\leq D_{1}$.

We may then conclude, as in [9], that

${\mathcal{N}}(s_{2},\alpha)=\mathfrak{M}(s_{2},d\mathfrak{k},j,{\mathcal{U}})+O\left(\frac{N}{N(s_{2})}\exp\left(-c(\log
X)^{\varpi/2}\right)\right)$

where the main term crucially is independent of $\alpha$. Feeding this into
(10.4) then completes our treatment of small $d$, and hence the proof of
Proposition 3.7.

##  11\. Character sums

In this section our goal is to introduce and prove analogues of Proposition
23.1 and Theorem $\psi$ in [3]. To wit, we introduce, for an ideal number
$\alpha$ in a fixed class $A$, the vector

$\widehat{\alpha}=(a_{1},a_{2})\in{\mathbb{Z}}^{2}$

corresponding to the class $A$ with basis produced as in Section 4. We then
introduce the symbol

$[\alpha]=i^{\frac{a_{1}-1}{2}}\left(\frac{a_{2}}{|a_{1}|}\right)$

where $\left(\frac{\cdot}{\cdot}\right)$ is the Jacobi symbol. Note that the
symbol $[\cdot]$ depends on the class $A$ _and the choice of basis_ , which we
have suppressed.

Our goal is to obtain an analogue of Lemma 20.1 in [3], which shows that while
$[\cdot]$ is not multiplicative, a suitable result exists to separate $[zw]$
into $[z][w]\kappa(zw)$, where $|\kappa(zw)|=1$ and $\kappa$ can be described
explicitly. To do so we need to introduce an analogue of the so-called Jacobi-
Kubota symbol $\xi_{w}(z)$ in [3]. Defining the analogue of $\xi_{w}(z)$ in
the present setting is tricky, due to the fact that in general
${\mathcal{O}}_{K}$ need not be a unique factorization domain and could have
an infinite unit group.

To prepare for our definition, we first gather several of the key properties
satisfied by Friedlander and Iwaneic’s $\xi_{w}(z)$ in [3]. In particular, it
satisfies the following:

  1. (1)

It satisfies an equation of the form

$[z][w]=\varepsilon[zw]\xi_{w}(z)$

where $\varepsilon=\pm 1$ depending only on the quadrants containing $z,w$
respectively;

  2. (2)

It is multiplicative for each $w\in{\mathbb{Z}}[i]$: one has
$\xi_{w}(z_{1})\xi_{w}(z_{2})=\xi_{w}(z_{1}z_{2})$;

  3. (3)

It is symmetric: $\xi_{w}(z)=\xi_{z}(w)$ for $w,z\in{\mathbb{Z}}[i]$;

  4. (4)

(Lemma 21.1 in [5]) For $q=|w_{1}w_{2}|^{2}$ and
$d=|\gcd(w_{1},\overline{w_{2}})|^{2}$ one has

$\sum_{\zeta\pmod{q}}\xi_{w_{1}}(\zeta)\xi_{w_{2}}(\zeta)=\begin{cases}q\varphi(d)\varphi(q/d)&\text{if
}q,d\text{ are squares}\\\ 0&\text{otherwise.}\end{cases}.$

  5. (5)

For $w=u+iv$ and $\omega\equiv-v\overline{u}\pmod{q}$ with $q=|w|^{2}$, one
has

$\xi_{w}(z)=\left(\frac{ur-vs}{q}\right)\text{ and
}\xi_{w}(z)=\left(\frac{r+\omega s}{q}\right),$

where $z=r+is$.

We would like to define our function $\xi_{\alpha}(z)$ to have the same
properties. Unfortunately, it seems that at least some of these properties
require special structures of the Gaussian integers ${\mathbb{Z}}[i]$. Thus,
some more preparatory work is needed before we can define our stand-in for the
Jacobi-Kubota symbol. We then check that our analogous symbol has the
necessary properties to carry out the proofs of analogous statements in [3].

First we note that our symbol $\xi_{\alpha}(z)$ depends on $\alpha$, and in
particular, depends on the class $A$ of $\alpha$. This of course is a trivial
point when $K={\mathbb{Q}}(i)$, since ${\mathbb{Z}}[i]$ has unique
factorization. Next we will also need to restrict the class of the _inputs_
$z$, in order for our symbol to be well-behaved. This is far from ideal and is
likely too restrictive, but it suffices for our purposes in this paper.
Indeed, later we will see that it is necessary to define a separate symbol
$\xi$ for each class of ideal numbers _along with a basis_ of said ideal
numbers.

The most important property turns out to be (1), so we define our symbol with
this in mind. To simplify matters we will assume that in our composition law
the bilinear form $Q_{A,B}(w,z)$ is given by $w_{1}z_{1}+w_{2}z_{2}$. In
particular, we fix bases $\\{\alpha_{1},\alpha_{2}\\}\subset
A,\\{\beta_{1},\beta_{2}\\}\subset B,\\{\gamma_{1},\gamma_{2}\\}\subset
C=\cdot B$ so that

$(\alpha_{1}x_{1}+\alpha_{2}x_{2})(\beta_{1}y_{1}+\beta_{2}y_{2})=(x_{1}y_{1}+x_{2}y_{2})\gamma_{1}+(x_{1}\ell_{1}(y_{1},y_{2})+x_{2}\ell_{2}(y_{1},y_{2}))\gamma_{2}.$

Observe that the roles of $R_{A,B},Q_{A,B}$ are switched from the previous
sections, but this is due the freedom to choose our bases.

We begin with the Jacobi symbol

$\left(\frac{w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2})}{|w_{1}z_{1}+w_{2}z_{2}|}\right)$

where $R_{A,B}(w,z)=w_{1}z_{1}+w_{2}z_{2}$. Note that we can extend the
definition of the Jacobi symbol by setting

$\left(\frac{a}{b}\right)=\left(\frac{a}{|b|}\right)(a,b)_{\infty},$

where

$(a,b)_{\infty}=\begin{cases}-1&\text{if }a,b<0\\\
1&\text{otherwise}\end{cases}$

is the Hilbert symbol. Next we note quadratic reciprocity, which states for
$a,b$ odd and coprime that

(11.1)
$\left(\frac{a}{|b|}\right)\left(\frac{b}{|a|}\right)=(-1)^{\frac{a-1}{2}\cdot\frac{b-1}{2}}(a,b)_{\infty}.$

Clearly, not both $Q_{A,B},R_{A,B}$ can be even otherwise the corresponding
ideal number is not primitive. Without loss of generality, let us suppose that
$w_{1}z_{1}+w_{2}z_{2}$ is odd. Let $2^{k}$ be the highest power of $2$
dividing $w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2})$. Then

$\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{|w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2})|}\right)=\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{2^{-k}|w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2})|}\right).$

We put

$u=w_{1}z_{1}+w_{2}z_{2},v=w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2})$

for simplicity. Applying quadratic reciprocity (11.1) then gives

$\displaystyle\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{2^{-k}(w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2}))}\right)(u,v)_{\infty}$
$\displaystyle=\left(\frac{2^{-k}v}{u}\right)(-1)^{\frac{u-1}{2}\cdot\frac{2^{-k}v-1}{2}}$
$\displaystyle=\left(\frac{2^{k}}{u}\right)(-1)^{\frac{u-1}{2}\cdot\frac{2^{-k}v-1}{2}}\left(\frac{v}{u}\right).$

Now we use the fact that

$w_{1}z_{1}+w_{2}z_{2}\equiv 0\pmod{u}$

implies

$w_{2}\equiv-z_{2}^{-1}w_{1}z_{1}\pmod{u}.$

Substituting this into $R_{A,B}(w,z)$ gives

$\displaystyle w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2})$
$\displaystyle\equiv
w_{1}\ell_{1}(z_{1},z_{2})-z_{2}^{-1}w_{1}z_{1}\ell_{2}(z_{1},z_{2})\pmod{u}$
$\displaystyle\equiv
z_{2}^{-1}w_{1}\left(z_{2}\ell_{1}(z_{1},z_{2})-z_{1}\ell_{2}(z_{1},z_{2})\right)\pmod{u}.$

Here we require an interpretation of the quadratic form

$g(z_{1},z_{2})=z_{2}\ell_{1}(z_{1},z_{2})-z_{1}\ell_{2}(z_{1},z_{2}).$

By definition, our composition law gives the relation

(11.2)
$\displaystyle(z_{2}\alpha_{1}-z_{1}\alpha_{2})(z_{1}\beta_{1}+z_{2}\beta_{2})$
$\displaystyle=R_{A,B}(z_{2},-z_{1};z_{1},z_{2})\gamma_{1}+Q_{A,B}(z_{2},-z_{1};z_{1},z_{2})\gamma_{2}$
$\displaystyle=(z_{2}\ell_{1}(z_{1},z_{2}))-z_{1}\ell_{2}(z_{1},z_{2})\gamma_{1}+(z_{2}z_{1}-z_{1}z_{2})\gamma_{2}$
$\displaystyle=g(z_{1},z_{2})\gamma_{1}.$

Dividing both sides by $\gamma_{1}$ we then see that $g(z_{1},z_{2})$ must be
equivalent to the norm form of ${\mathcal{O}}_{K}$.

We must now relate $g(z_{1},z_{2})$ to
$N(z)=N(J(z_{1}\beta_{1}+z_{2}\beta_{2}))$. Note that

$g(z_{1},z_{2})=\gamma_{1}^{-1}(\alpha_{1}z_{2}-\alpha_{2}z_{1})(\beta_{1}z_{1}+\beta_{2}z_{2})$

is divisible by $z=\beta_{1}z_{1}+\beta_{2}z_{2}$, which implies that
$g(z_{1},z_{2})$ is a rational integer divisible by $N(z)$. By primitivity we
then see that $g(z_{1},z_{2})$ must be a constant multiple of $N(z)$, the
constant depending only on the classes $A,B$. We summarize this as a lemma:

######  Lemma 11.1.

Let $g(x,y)$ be the integral binary quadratic form which arises from the
composition law (11.2). Then $g(z_{1},z_{2})$ is a constant multiple of
$N(J(\beta_{1}z_{1}+\beta_{2}z_{2}))$, with the constant depending only on the
classes $A,B$ and choices of bases of $A,B,A\cdot B$.

Similarly, since $v=w_{1}\ell_{1}+w_{2}\ell_{2}$ is divisible by $2^{k}$, we
may assume without loss of generality that $\ell_{1}$ is odd to obtain

$w_{1}\equiv-\ell_{1}^{-1}w_{2}\ell_{1}\pmod{2^{k}}$

and this implies that

$\displaystyle w_{1}z_{1}+w_{2}z_{2}$
$\displaystyle\equiv-\ell_{1}^{-1}w_{2}z_{1}+w_{2}z_{2}\pmod{2^{k}}$
$\displaystyle\equiv-\ell_{1}^{-1}w_{2}(z_{1}\ell_{1}-z_{2}\ell_{2})\pmod{2^{k}}$
$\displaystyle\equiv-\ell_{1}^{-1}w_{2}g_{C}(z_{1},z_{2})\pmod{2^{k}}.$

Since $u=w_{1}z_{1}+w_{2}z_{2}$ is odd by assumption, it follows that
$g_{C}(z_{1},z_{2})$ must be odd as well.

Continuing on, we then have

$\displaystyle
z_{2}^{-1}w_{1}\left(z_{2}\ell_{1}(z_{1},z_{2})-z_{1}\ell_{2}(z_{1},z_{2})\right)$
$\displaystyle\equiv z_{2}^{-1}w_{1}g_{C}(z_{1},z_{2})\pmod{u},$

which implies that

$\left(\frac{v}{u}\right)=\left(\frac{z_{2}^{-1}w_{1}g_{C}(z_{1},z_{2})}{u}\right)=\left(\frac{z_{2}w_{1}}{u}\right)\left(\frac{g_{C}(z_{1},z_{2})}{u}\right).$

Applying quadratic reciprocity again we obtain

$\left(\frac{z_{2}w_{1}}{u}\right)\left(\frac{g_{C}(z_{1},z_{2})}{u}\right)=\left(\frac{2^{k_{1}+k_{2}}}{u}\right)\left(\frac{u}{w_{1}z_{2}}\right)\left(\frac{u}{g_{C}(z_{1},z_{2})}\right).$

Here $2^{k_{1}}$ is the highest power of $2$ dividing $w_{1}$ and $2^{k_{2}}$
the highest power of $2$ dividing $z_{2}$. Note that

$\displaystyle\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{w_{1}z_{2}}\right)$
$\displaystyle=\left(\frac{w_{2}z_{2}}{w_{1}}\right)\left(\frac{w_{1}z_{1}}{z_{2}}\right)$
$\displaystyle=\left(\frac{w_{2}}{w_{1}}\right)\left(\frac{z_{2}}{w_{1}}\right)\left(\frac{w_{1}}{z_{2}}\right)\left(\frac{z_{1}}{z_{2}}\right)$
$\displaystyle=\left(\frac{w_{2}}{|w_{1}|}\right)\left(\frac{z_{1}}{|z_{2}|}\right)\left(\frac{2^{k_{2}}}{w_{1}}\right)\left(\frac{2^{k_{1}}}{z_{2}}\right)(-1)^{\frac{2^{-k_{1}}w_{1}-1}{2}\frac{2^{-k_{2}}z_{2}-1}{2}}(z_{1},z_{2})_{\infty}(w_{1},w_{2})_{\infty}$
$\displaystyle=\left(\frac{w_{2}}{|w_{1}|}\right)\left(\frac{z_{2}}{|z_{1}|}\right)\left(\frac{2^{k_{2}}}{z_{1}}\right)\left(\frac{2^{k_{2}}}{w_{1}}\right)\left(\frac{2^{k_{1}}}{z_{2}}\right)(-1)^{\frac{2^{-k_{1}}w_{1}-1}{2}\frac{2^{-k_{2}}z_{2}-1}{2}}(z_{1},z_{2})_{\infty}(w_{1},w_{2})_{\infty}.$

Collecting these calculations we conclude that

(11.3)
$\left(\frac{w_{1}\ell_{1}(z_{1},z_{2})+w_{2}\ell_{2}(z_{1},z_{2})}{|w_{1}z_{1}+w_{2}z_{2}|}\right)=\left(\frac{w_{2}}{|w_{1}|}\right)\left(\frac{z_{2}}{|z_{1}|}\right)\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{g(z_{1},z_{2})}\right)\varepsilon(w,z)\theta(w,z),$

where $\varepsilon(w,z)$ is the product of all of the Hilbert symbols and the
terms of the shape $(-1)^{x}$ for some $x\in{\mathbb{Z}}$ which appear. It is
clear that $\varepsilon(w,z)$ depends on the congruence class of $w,z$ with a
bounded conductor, and thus is of little consequence. Here $\theta(w,z)$ is
given by

$\theta(w,z)=\left(\frac{2^{k_{2}}}{z_{1}}\right)\left(\frac{2^{k_{2}}}{w_{1}}\right)\left(\frac{2^{k_{1}}}{z_{2}}\right)\left(\frac{2^{k_{1}+k_{2}}}{u}\right)\left(\frac{2^{k}}{u}\right).$

Since we have insisted that $w,z$ belong to fixed congruence classes modulo
$8etn$ as in (10.2) it follows that $\theta(w,z)$ can be determined as a
function of the congruence class alone, and is therefore a constant for our
purposes.

These calculations compels us to define our analogue of the Jacobi-Kubota
symbol as

(11.4) $\xi_{w}(z)=\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{g(w_{1},w_{2})}\right).$

Note that $\xi_{w}(z)$ depends on the ideal classes of $w,z$ and a choice of
basis for the ideal classes.

Next we observe for $w,z$ satisfying (10.2), $w,z$ are in the same class and
therefore $R_{A,B}(w,z)=R_{A,A}(w,z)$ must be symmetric in $w,z$. From here it
follows that

$\displaystyle z_{2}^{-1}w_{1}g(z_{1},z_{2})$ $\displaystyle\equiv
R_{A,A}(w,z)\pmod{u}$ $\displaystyle\equiv R_{A,A}(z,w)\pmod{u}$
$\displaystyle\equiv w_{1}^{-1}z_{2}g(w_{1},w_{2})\pmod{u}.$

This implies that

$\left(\frac{g(z_{1},z_{2})}{u}\right)\left(\frac{g(w_{1},w_{2})}{u}\right)=1.$

Thus, up to a factor $\varepsilon$ depending at most on congruence classes and
signs of $w,z$, we have

(11.5) $\xi_{w}(z)=\varepsilon\xi_{z}(w).$

Summarizing, we obtain the following analogue of Lemma 20.1 in [5]:

######  Lemma 11.2.

Let $w,z$ satisfy the hypothesis given in (10.2). Then there exist numbers
$\varepsilon(w,z)\in\\{-1,1\\}$ depending only on the signs and congruence
classes of $w,z$ modulo $8etn$ such that

(11.6)
$\left(\frac{Q_{A,B}(w,z)}{|R_{A,B}(w,z)|}\right)=\varepsilon(w,z)\left(\frac{w_{2}}{|w_{1}|}\right)\left(\frac{z_{2}}{|z_{1}|}\right)\xi_{z}(w).$

Next we show that the analogue of Lemma 21.1 in [3] holds:

######  Lemma 11.3.

For fixed elements $w,v$ in the class $A$ and $q=g(w_{1},w_{2})g(v_{1},v_{2})$
and $d=\gcd(g(w_{1},w_{2}),g(v_{1},v_{2}))$, we have

$\sum_{z\pmod{q}}\xi_{w_{1}}(z)\xi_{w_{2}}(z)=\begin{cases}q\varphi(d)\varphi(q/d)&\text{if
}q,d\text{ are squares}\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

We have

$\displaystyle\sum_{z\pmod{q}}\xi_{w}(z)\xi_{v}(z)$
$\displaystyle=\sum_{z\pmod{q}}\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{g(w_{1},w_{2})}\right)\left(\frac{v_{1}z_{1}+v_{2}z_{2}}{g(v_{1},v_{2})}\right)$
$\displaystyle=\sum_{z\pmod{q}}\left(\frac{(w_{1}z_{1}+w_{2}z_{2})(v_{1}z_{1}+v_{2}z_{2})}{d}\right)\left(\frac{w_{1}z_{1}+w_{2}z_{2}}{g(w_{1},w_{2})/d}\right)\left(\frac{v_{1}z_{1}+v_{2}z_{2}}{g(v_{1},v_{2})/d}\right).$

From here we see that the final sum is zero unless each of the summands is
equal to $1$ or $0$ identically. This is only the case when
$d,g(w_{1},w_{2})/d,g(w_{1},w_{2})/d$ are all squares. Since
$d|\gcd(g(w_{1},w_{2}),g(v_{1},v_{2}))$ and $d\nmid\Delta(f)$ it follows that
$w_{1}x+w_{2}y,v_{1}x+v_{2}y$ are not proportional modulo $d$. From here we
see that, modulo $d$, the number of solutions to
$\gcd(w_{1}x+w_{2}y,d)=\gcd(v_{1}x+v_{2}y,d)=1$ is equal to $\varphi(d)^{2}$.
Similarly, modulo $g(w_{1},w_{2})/d$ and $g(v_{1},v_{2})/d$ there are
$\frac{g(w_{1},w_{2})\varphi(g(w_{1},w_{2})/d)}{d}$ solutions to
$\gcd(w_{1}x+w_{2}y,g(w_{1},w_{2})/d)=1$ and
$\gcd(v_{1}x+v_{2}y,g(v_{1},v_{2})/d)=1$ respectively. Lifting to the modulus
$q$ yields

$\frac{q^{2}}{d^{2}}\cdot\varphi(d)^{2}\cdot\frac{g(w_{1},w_{2})g(v_{1},v_{2})}{d^{2}}\varphi(g(w_{1},w_{2})/d)\varphi(g(v_{1},v_{2})/d)=q\varphi(d)\varphi(q/d),$

since $\gcd(q/d^{2},d)=1$. This completes the proof. ∎

Lemma 11.3 is analogous to Lemma 21.1 in [3].

We now prove the following analogue of Lemma 21.2 in [3]:

######  Proposition 11.4.

Let $A,B$ be classes of ideal numbers with $A\cdot B=\operatorname{Cl}[f]$.
Put

(11.7)
${\mathcal{Q}}(M,N)=\sideset{}{{}^{\ast}}{\sum}_{w}\sum_{z}\alpha_{w}\beta_{z}\xi_{w}(z),$

where $\alpha_{w},\beta_{z}$ are bounded real coefficients supported in
appropriate fundamental domains for $A,B$ having norm bounded by $M,N$
respectively. Then for all $\varepsilon>0$ we have

(11.8)
${\mathcal{Q}}(M,N)\ll_{\varepsilon}(M+N)^{\frac{1}{12}}(MN)^{\frac{11}{12}+\varepsilon}.$

###### Proof.

Applying Cauchy’s inequality we obtain

$\displaystyle\lvert{\mathcal{Q}}(M,N)\rvert^{2}$
$\displaystyle\leq\lVert\beta\rVert_{2}^{2}\sum_{z}\left\lvert\sideset{}{{}^{\ast}}{\sum}_{w}\alpha_{w}\xi_{w}(z)\right\rvert^{2}$
$\displaystyle=\lVert\beta\rVert_{2}^{2}\sideset{}{{}^{\ast}}{\sum}_{w_{1}}\sideset{}{{}^{\ast}}{\sum}_{w_{2}}\alpha_{w_{1}}\alpha_{w_{2}}\sum_{z}\xi_{w_{1}}(z)\xi_{w_{2}}(z).$

We then find that splitting $z$ into congruence classes modulo
$q=g(w_{1})g(w_{2})$ that

$\sum_{z}\xi_{w_{1}}(z)\xi_{w_{2}}(z)=\sum_{\zeta\pmod{q}}\xi_{w_{1}}(\zeta)\xi_{w_{2}}(\zeta)\cdot\left(\frac{c_{f}N}{q^{2}}+O_{f}\left(\frac{\sqrt{N}}{q}+1\right)\right)$

where

$c_{f}=\lim_{s\rightarrow 1}(s-1)\zeta_{K}(s).$

We obtain, by Lemma 11.1 and using (11.5) if necessary,

(11.9) ${\mathcal{Q}}(M,N)^{2}\ll
N^{2}\mathop{\sum\sum}_{\begin{subarray}{c}m_{1},m_{2}\leq M\\\
m_{1}m_{2}=\square\end{subarray}}\tau(m_{1}m_{2})+NM^{4}\left(\sqrt{N}+M^{2}\right),$

which gives the bound

${\mathcal{Q}}(M,N)\ll_{\varepsilon}\left(M^{3}N^{\frac{1}{2}}+M^{2}N^{\frac{3}{4}}+M^{\frac{1}{2}}N\right)(MN)^{\varepsilon}.$

In the next step we shall apply Hölder’s inequality to obtain

${\mathcal{Q}}(M,N)^{k}\ll
M^{k-1}\sideset{}{{}^{\ast}}{\sum}_{w}\left\lvert\sum_{z}\beta_{z}\xi_{w}(z)\right\rvert^{k}=M^{k-1}\widetilde{{\mathcal{Q}}}\left(M,N^{k}\right),$

say. In [3] the next step is to argue that
$\widetilde{{\mathcal{Q}}}(M,N^{k})$ can be written as a bilinear form of the
shape (11.7), using the fact that in the case $K={\mathbb{Q}}(i)$ that
$\xi_{w}(z)$ is multiplicative in $z$. In general this is not the case.
However, we are free to choose a basis for the class $B^{k}$ for each positive
integer $k$ one can write

(11.10) $\xi_{w}(z_{1})\cdots\xi_{w}(z_{k})=\xi_{w}^{(k)}(z_{1}\cdots z_{k})$

in a consistent way. Recall (11.4), we note that

$\xi_{w}(z_{1})\xi_{w}(z_{2})=\left(\frac{Q_{B,B}(z_{1})Q_{B,B}(z_{2})}{g(w_{1},w_{2})}\right).$

The numerator is a bilinear form in $z_{1},z_{2}$. Using composition laws to
write

$z_{1}z_{2}=R_{B^{2}}(z_{1},z_{2})\gamma_{1}^{(2)}+Q_{B^{2}}(z_{1},z_{2})\gamma_{2}^{(2)}$

as ideal numbers, we see that we can apply a change of variables, depending
only on $w$, the class $B$, and the choice of bases, so that the numerator
$Q_{B,B}(z_{1})Q_{B,B}(z_{2})$ as a linear form in
$R_{B^{2}}(z_{1},z_{2}),Q_{B^{2}}(z_{1},z_{2})$. Inductively, we then find
that

$\xi_{w}(z_{1})\cdots\xi_{w}(z_{k})=\left(\frac{L_{w}(z_{1}\cdots
z_{k})}{g(w_{1},w_{2})}\right)$

where $L_{w}$ is a linear form in two variables with coefficients depending at
most on $w$ and evaluates $z_{1}\cdots z_{k}$ in terms of its representation
as an element in the lattice of the corresponding ideal numbers. Defining the
right hand side as $\xi_{w}^{(k)}(z_{1}\cdots z_{k})$ we obtain (11.10).
Replacing $\xi_{w}(\cdot)$ with $\xi_{w}^{(k)}(\cdot)$ in (11.7) shows that
(11.9) holds, and therefore we may proceed as in [3] after applying Hölder’s
inequality to conclude

${\mathcal{Q}}(M,N)^{k}\ll_{\varepsilon}M^{k-1}\left\\{M^{3}N^{\frac{k}{2}}+M^{2}N^{\frac{3k}{4}}+M^{\frac{1}{2}}N^{k}\right\\}(MN)^{\varepsilon},$

which upon taking $k$-th roots gives us the bound

${\mathcal{Q}}(M,N)\ll_{\varepsilon}\left\\{M^{1+\frac{2}{k}}N^{\frac{1}{2}}+M^{1+\frac{1}{k}}N^{\frac{3}{4}}+M^{1-\frac{1}{2k}}N\right\\}(MN)^{\varepsilon}$

for all positive $k\in{\mathbb{N}}$. Switching the roles of $M,N$ and applying
Lemma 11.3, we obtain as in [3] that

${\mathcal{Q}}(M,N)\ll_{\varepsilon}(M+N)^{\frac{1}{12}}(MN)^{\frac{11}{12}+\varepsilon}$

upon setting $k=6$. ∎

Next we move on to proving the analogue of Proposition 22.1 in [3]. We define,
for any ideal number $z$, a rational integer $k$, and a character $\chi$
modulo $4d$ the Hecke character

(11.11) $\psi(z)=\chi(z)\left(\frac{z}{|z|}\right)^{k}.$

Consider the sum

${\mathcal{K}}(N)=\sideset{}{{}^{\wedge}}{\sum}_{z\in\mathfrak{B}}\psi(z)[wz]$

and

${\mathcal{K}}^{\ast}(N)=\sideset{}{{}^{\wedge}}{\sum}_{\begin{subarray}{c}z\in\mathfrak{B}\\\
\gcd(z,w)=1\end{subarray}}\psi(z)[wz],$

where $\mathfrak{B}$ is narrow sector contained in the intersection of a
fundamental domain for the ideal class numbers containing $z$ having norm
bounded $N$. We treat $w$ as a fixed primitive ideal number. Our analogue of
Proposition 22.1 in [3] is thus:

######  Proposition 11.5.

Given $\psi$ and $w$ as above we have

(11.12) ${\mathcal{K}}(N)\ll d(|k|+1)|w|N^{\frac{3}{4}}\log(|w|N)$

and

(11.13) ${\mathcal{K}}^{\ast}(N)\ll
d(|k|+1)|w|\tau(N(w))N^{\frac{3}{4}}\log(|w|N).$

###### Proof.

Just like the proof of Proposition 22.1 in [3], the key result needed to
obtain the necessary cancellation is the Polya-Vinogradov theorem, which
asserts that

$\sum_{n\leq N}\chi(n)\ll\sqrt{q}\log q$

for every non-trivial Dirichlet character $\chi\pmod{q}$ with an absolute
implied constant. To estimate ${\mathcal{K}}(N)$ we apply Lemma 11.6 to obtain

${\mathcal{K}}(N)=[w]\sideset{}{{}^{\wedge}}{\sum}_{z\in\mathfrak{B}}\varepsilon(w,z)\psi(z)[z]\xi_{w}(z),$

and by breaking the sum up to finitely many congruence classes if necessary,
we may factor the $\varepsilon$-factor out (because it will be constant) to
obtain

${\mathcal{K}}(N)=[w]\varepsilon\sideset{}{{}^{\wedge}}{\sum}_{z\in\mathfrak{B}}\psi(z)[z]\xi_{w}(z).$

Breaking the sum up into a double sum over rational integers forming vectors
running over $\mathfrak{B}$ as in [3] and applying Polya-Vinogradov we obtain
(11.12) and (11.13) as required. ∎

Put

$\lambda(n)=\sideset{}{{}^{\wedge}}{\sum}_{N(z)=n}\psi(z)[z],$

the sum restricted to a fundamental domain of ideal numbers so each ideal is
represented at most once. Consider the sum

(11.14) ${\mathcal{L}}(M,N)=\sum_{m}\sum_{n}\alpha(m)\beta(n)\lambda(cmn)$

where $\alpha,\beta$ are complex coefficients having norm at most $1$ and
supported on $1\leq m\leq M$ and $n\leq N$. Like wise, let
${\mathcal{L}}^{\ast}(M,N)$ be the subsum of (11.14) restricted to
$\gcd(m,n)=1$. Combining Proposition 11.4 and Lemma 11.2 then gives the
following analogue of Proposition 23.1 in [3]:

######  Proposition 11.6.

For any complex coefficients $\alpha(m),\beta(n)$ as above and for any
positive integer $c$ we have

(11.15)
${\mathcal{L}}(M,N)\ll\tau(c)(M+N)^{\frac{1}{12}}(MN)^{\frac{11}{12}+\varepsilon}.$

We also introduce the analogues of ${\mathcal{K}}(N),{\mathcal{K}}^{\ast}(N)$:

(11.16) ${\mathcal{L}}(N)=\sum_{n\leq
N}\lambda(mn),{\mathcal{L}}^{\ast}(N)=\sum_{\begin{subarray}{c}n\leq N\\\
\gcd(m,n)=1\end{subarray}}\lambda(mn)$

and obtain the following analogue of Proposition 23.2 in [3] by applying
Proposition 11.5:

######  Proposition 11.7.

For $\psi$ as defined by (11.11) and positive integer $m$ we have the bounds

(11.17) ${\mathcal{L}}(N)\ll
d(|k|+1)\tau(m)^{4}\sqrt{m}N^{\frac{3}{4}}\log(mN)$

and

(11.18) ${\mathcal{L}}^{\ast}(N)\ll
d(|k|+1)\tau(m)^{2}\sqrt{m}N^{\frac{3}{4}}\log(mN).$

These estimates then imply the following analogue of Theorem $\psi$ in [3]:

######  Proposition 11.8.

For any $c\geq 1$ we have

(11.19) $\sum_{n\leq X}\Lambda(n)\lambda(cn)\ll cd(|k|+1)X^{\frac{76}{77}}$

with the absolute constant dependent only on $f$.

###### Proof.

This is the same as the proof of Theorem $\psi$ in [3] with Propositions 23.1
and 23.2 replaced by Propositions 11.6 and 11.7 respectively. ∎

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