\section{Introduction}

Although global stabilization of dynamical systems is of importance in system theory and engineering \cite{Kh:02,An:02},
it is sometimes difficult or impossible to synthesize a global stabilizing controller for certain linear and nonlinear systems \cite{Va:02}.
The reasons could be the poor controllability of system, e.g., systems that have uncontrollable linearizations \cite{Di:11}
and systems that have fewer degrees of control freedom than the degrees of freedom to be controlled \cite{Gu:13,Gu:14},
the input/output constraints in practice, e.g., an unstable linear time-invariant system cannot be globally stabilized in the presence of
input saturations \cite{Bl:99}, time delay \cite{Sun:11,Sun:13}, and/or the involved large disturbances \cite{Ki:06}, etc.
Moreover, in many applications the full stabilization, while possible, carries high penalty due to the cost of the control, thus is also not desirable.

Instead, minimizing a long-time average of the cost functional might be more realistic.
For instance, long-time-average cost analysis and control is often considered in irrigation, flood control, navigation, water supply, hydroelectric power, computer communication networks, and other applications \cite{Du:10,Bo:92}.
In addition, systems that include stochastic factors are often controlled in the sense of long-time average.
In \cite{Ro:83}, a summary of long-time-average cost problems for continuous-time Markov processes is given.
In \cite{Me:00}, the long-time-average control of a class of problems that arise in the modeling of semi-active suspension systems was considered,
where the cost includes a term based on the local time process diffusion.
Notice that the controller design methods proposed in \cite{Ro:83,Me:00} are highly dependent on the stochastic property of dynamical systems.

In certain cases, as, for example, turbulent flows of fluid, calculating the time averages is a big challenge even in the uncontrolled case.  As a result, developing the control aimed at reducing the time-averaged cost for turbulent flows, for example by using the receding horizon technique, leads to controllers too complicated for practical implementation  \cite{Bewley:01}.
To overcome this complexity, it was proposed \cite{Ph:14} to use an upper bound for the long-time average cost instead of the long-time average cost itself in cases when such an upper bound is easier to calculate. The idea is based on the hope that the control reducing an upper bound for a quantity will also reduce the quantity itself. Meanwhile, \cite{Ph:14} uses the sum of squares (SOS) decomposition of polynomials and semidefinite programming (SDP) and allows a trade-off between the quality of bound and the complexity of its calculation.

The SOS methods apply to systems defined by a polynomial vector field. Such systems may describe a wide variety of dynamics~\cite{Va:01} or approximate a system defined by an analytical vector field~\cite{Va:02}. A polynomial system can therefore yield a reliable model of a dynamical system globally or in larger regions than the linear approximation in the state-space~\cite{Va:03}. Recent results on SOS decomposition have transformed the verification of non-negativity of polynomials into SDP, hence providing promising algorithmic procedures for stability analysis of polynomial systems. However, using SOS techniques for optimal control, as for example in~\cite{Pr:02,Zh:07,Ma:10}, is subject to a generic difficulty: while the problem of optimizing the candidate Lyapunov function certifying the stability for a closed-loop system for a given controller and the problem of optimizing the controller for a given candidate Lyapunov function are reducible to an SDP and thus, are tractable, the problem of simultaneously  optimizing both the control and the Lyapuniov function is non-convex. Iterative procedures were proposed for overcoming this difficulty~\cite{Zh:07,Zh:09,Ng:11}.

While optimization of an upper bound with control proposed in~\cite{Ph:14} does not involve a Lyapunov function, it does involve a similar tunable function, and it shares the same difficulty of non-convexity. In the present work we propose a polynomial type state feedback controller design scheme for the long-time average upper-bound control, where the controller takes the structure of an asymptotic series in a small-amplitude perturbation parameter.
By fully utilizing the smallness of the perturbation parameter, the resultant SOS optimization problems are solved in sequence,
thus avoiding the non-convexity in optimization. We apply it to an illustrative example and demonstrate that it does allow to reduce the long-time average cost even without fully stabilizing the system. Notice the significant conceptual difference between our approach and the studies of control by small perturbations, often referred to as tiny feedback, see for example~\cite{tc:93}.

The paper is organized as follows. Section~\ref{seq:Background}
  presents some preliminary introduction on SOS and its application in bound estimation of long-time average cost for uncontrolled systems. Section~\ref{seq:Problem Formulation}
gives the problem formulation. Bound optimization of the long-time average cost for controlled polynomial systems is considered in Section~\ref{seq:Bound}.
 An illustrative example of a cylinder wake flow is addressed in Section~\ref{seq:Example}.
Section~\ref{seq:Conclusion}
 concludes the work.

\section{Background}\label{seq:Background}

In this section SOS of polynomials and a recently-proposed method of obtaining rigorous bounds of long-time average cost via SOS for uncontrolled polynomial systems are introduced.

\subsection{SOS of polynomials}

SOS techniques have been frequently used in the stability analysis and controller design for all kinds of systems, e.g., constrained ordinary differential equation systems \cite{An:02}, hybrid systems \cite{An:05}, time-delay systems \cite{An:04}, and partial differential equation systems \cite{Pa:06,Yu:08,GC:11}. These techniques help to overcome the common drawback of approaches based on Lyapunov functions: before \cite{Pr:02}, there were no coherent and tractable computational methods for constructing Lyapunov functions.

A multivariate polynomial $f({\bf x})$ is a SOS, if there exist polynomials $f_1({\bf x}), \cdots, f_m({\bf x})$ such that
\[
f({\bf x})=\sum_{i=1}^mf_i^2({\bf x}).
\]
If $f({\bf x})$ is a SOS then $f({\bf x})\ge 0, \forall{\bf x} $. In the general multivariate case, however,
$f({\bf x})\ge 0~  \forall \bf x$  does not necessarily imply that $f({\bf x})$ is SOS. While being stricter, the condition that $f({\bf x})$ is SOS is much more computationally tractable than non-negativity \cite{Par:00}. At the same time, practical
experience indicates that in many cases replacing non-negativity with the SOS property leads to satisfactory results.

In the present paper we will utilize the existence of efficient numerical methods and software \cite{Pra:04,Lo:09} for solving the optimization problems of the following type: minimize the linear objective function
\begin{equation}
{\bf w}^T{\bf c}
\label{linear}
\end{equation}
where ${\bf w}$ is the vector of weighting coefficients for the linear objective function, and
 ${\bf c}$ is a vector formed from the (unknown) coefficients of the
polynomials $p_i({\bf x})$ for $i=1,2,\cdots, \hat{N}$ and SOS $p_i({\bf x})$ for $i=(\hat{N}+1),\cdots, {N}$,
such that
\begin{eqnarray}
&a_{0,j}({\bf x})+\sum_{i=1}^Np_i({\bf x})a_{i,j}({\bf x})=0, ~ j=1,2,\cdots, \hat{J}, \label{c5} \\
[1ex]
&a_{0,j}({\bf x})+\sum_{i=1}^Np_i({\bf x})a_{i,j}({\bf x})  \mbox{~are SOS,~} j=(\hat{J}+1),\cdots, {J}.~
\label{c6}
\end{eqnarray}
In (\ref{c5}) and (\ref{c6}), the $a_{i,j}({\bf x})$ are given scalar constant coefficient polynomials.

The lemma below that provides a sufficient condition to test inclusions of sets defined by polynomials is frequently used for feedback controller design
in Section~~\ref{seq:Bound}. It is a particular case of the
Positivstellensatz
 Theorem \cite{Po:99} and is a generalized ${\mathcal S}$-procedure \cite{Ta:06}.

\begin{lemma}
Consider two sets of ${\bf x}$,
\begin{eqnarray*}
{\mathcal S}_1&\eqdef& \left\{{\bf x}\in {\mathbb R}^n~|~h({\bf x})=0, f_1({\bf x})\ge 0, \cdots, f_r({\bf x})\ge 0\right\}, \\
[1ex]
 {\mathcal S}_2&\eqdef& \left\{{\bf x}\in {\mathbb R}^n~|~f_0({\bf x})\ge 0\right\},
\end{eqnarray*}
where $f_i({\bf x}), i=0,\cdots, r$ and $h({\bf x})$ are scalar polynomial functions. The set containment ${\mathcal S}_1\subseteq {\mathcal S}_2$ holds if there exist a polynomial function $m({\bf x})$ and SOS polynomial functions $S_i({\bf x}), i=1,\cdots, r$ such that
\begin{eqnarray*}
f_0({\bf x})-\sum_{i=1}^rS_i({\bf x})f_i({\bf x})+m({\bf x})h({\bf x}) ~~\mbox{is SOS}.
\end{eqnarray*}
\end{lemma}

\subsection{Bound estimation of long-time average cost for uncontrolled systems}

For the convenience of the reader we outline here the method of obtaining bounds for long-time averages proposed in~\cite{Ph:14} and make some remarks on it.
Consider a system
\begin{eqnarray}
\dot{{\bf x}}={\bf f}({\bf x}),
\label{sys1}
\end{eqnarray}
where $\dot{{\bf x}}\eqdef d{\bf x}/dt$ and ${\bf f}({\bf x})$ is a vector of multivariate polynomials of the components of the state vector ${\bf x}\in{\mathbb R}^n$.  The long-time average of a function of the state $\Phi({\bf x})$ is defined as
\
\bar{\Phi}=\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T\Phi({{\bf x}}(t))\,dt,
\
where ${\bf x}(t)$ is the solution of (\ref{sys1}).

Define a polynomial function of the system state, $V({\bf x})$, of degree $d_V,$ and containing unknown decision variables as its coefficients.
The time derivative of $V$ along the trajectories of system (\ref{sys1}) is

\[
    \dot{V}({\bf x}) = \dot{{\bf x}} \cdot \nabla_{{\bf x}} V({\bf x})={\bf f}({\bf x})\cdot \nabla_{{\bf x}} V({\bf x}).
 \]
Consider the following quantity:
\[
    H({\bf x}) \eqdef \dot{V}({\bf x}) + \Phi({\bf x})= {\bf f}({\bf x}) \cdot\nabla_{{\bf x}} V({\bf x}) + \Phi({\bf x}).
\]

The following result is from \cite{Ph:14}:

\begin{lemma}
For the system (\ref{sys1}), assume that the state ${\bf x}$ is bounded in $\mathcal{D}\subseteq {\mathbb R}^n$. Then, $ H({\bf x})\le C, \forall {\bf x}\in \mathcal{D}$ implies $\bar{\Phi}\le C$.
\label{lemma1}
\end{lemma}

Hence, an upper bound of $\bar{\Phi}$ can be obtained by minimizing $C$ over $V$ under the constraint $H({\bf x})\le C$,
which can be formulated as a SOS optimization problem in the form:
\begin{eqnarray}
&\displaystyle\min_{V}~C \label{SOS_1}\\
[1ex]
&\mbox{s.t.} ~~-\left( {\bf f}({\bf x}) \cdot\nabla_{{\bf x}} V({\bf x})+\Phi({\bf x})-C\right) \mbox{~is~SOS},
\label{SOS}
\end{eqnarray}
which is a special case of (\ref{linear}). A better bound might be obtained by removing the requirement for $V({\bf x})$ to be a  polynomial and replacing (\ref{SOS}) with the requirement of non-negativeness.
However, the resulting problem would be too difficult, since the classical algebraic-geometry problem
of verifying positive-definiteness of a general multi-variate polynomial is NP-hard \cite{An:02,An:05}.

Notice that while  $V$ is similar to a Lyapunov function in a stability analysis, it is not required to be positive-definite. Notice also that a lower bound of any long-time average cost of the system (\ref{sys1}) can be analyzed in a similar way.

\begin{remark}\label{RemarkOnBoundedness}
For many systems the boundedness of system state immediately follows from energy consideration. In general, if the system state is bounded, this can often be proven using the SOS approach.
It suffices to check whether there exists a large but bounded global attractor, denoted by $\mathcal{D}_1.$
As an example, let $\mathcal{D}_1=\{{\bf x}~|~0.5{\bf x}^T{\bf x}\le \beta\}$, where the constant $\beta$ is sufficiently large. Then, the global attraction property of system in $\mathcal{D}_1$ may be expressed as
\begin{eqnarray}
{\bf x}^T\dot{{\bf x}}={\bf x}^T{\bf f}({\bf x})\le -(0.5{\bf x}^T{\bf x}-\beta).
\label{SOS1}
\end{eqnarray}
Introducing a tunable polynomial $S({\bf x})$ satisfying $S({\bf x})\ge 0 ~\forall {\bf x}\in{\mathbb R}^n$, by Lemma~1, (\ref{SOS1}) can be relaxed to
\begin{eqnarray}
\left\{
\begin{array}{c}
-\left({\bf x}^T{\bf f}({\bf x})-S({\bf x})(0.5{\bf x}^T{\bf x}-\beta)\right)\mbox{~is ~SOS},  \\
S({\bf x})\mbox{~is ~SOS}.
\end{array}
\right.
\label{SOS2}
\end{eqnarray}
Minimization of upper bound of long-time average cost for systems that have unbounded global attractor is usually meaningless, since the cost itself could be infinitely large.
\end{remark}

\section{Problem Formulation}\label{seq:Problem Formulation}

Consider a polynomial system with single input
\begin{eqnarray}
\dot{{\bf x}}={\bf f}({\bf x})+{\bf g}({\bf x}){\bf u}
\label{sys}
\end{eqnarray}
where ${\bf f}({\bf x}): {\mathbb R}^n\rightarrow {\mathbb R}^n$ and ${\bf g}({\bf x}): {\mathbb R}^n\rightarrow {\mathbb R}^{n\times m}$ are
polynomial functions of system state ${\bf x}$. The approach of this paper can easily be extended to multiple input systems.
The control ${\bf u}\in {\mathbb R}^m$, which is assumed to be a polynomial vector of the system state ${\bf x}$ with maximum degree $d_{{\bf u}}$,
is designed to minimize the upper bound of an average cost of the form:
\begin{eqnarray}
\bar{\Phi}=\lim_{T\rightarrow \infty}\frac{1}{T}\int_0^T\Phi({{\bf x}}(t),{\bf u}(t))\,dt,
\label{cost}
\end{eqnarray}
where ${{\bf x}}$ is the closed-loop solution of the system (\ref{sys}) with the control ${\bf u}$.
The continuous function $\Phi$ is a given non-negative polynomial cost in ${{\bf x}}$ and ${\bf u}$.

Similarly to (\ref{SOS_1})-(\ref{SOS}), we consider the following optimization problem:
\begin{eqnarray}
&\displaystyle \min_{{\bf u}, V} C \label{objective}\\
&{s.t.} -\left(({\bf f}({\bf x})+{\bf g}({\bf x}){\bf u})\cdot\nabla_{{\bf x}} V+\Phi({\bf x}, {\bf u})-C\right) \mbox{is SOS}.\quad
\label{optimization}
\end{eqnarray}
When it cannot be guaranteed that the closed-loop system state is bounded, SOS constraints (\ref{SOS2}) must be added to (\ref{optimization}) to make
our analysis rigorous.

Under the framework of SOS optimization, the main problem in solving (\ref{objective})-(\ref{optimization}) is due to the non-convexity of (\ref{optimization}) caused by the control input $u$ and the decision function $V,$ both of which are tunable, entering~(\ref{optimization}) nonlinearly.
Iterative methods \cite{Zh:07,Zh:09,Ng:11} may help to overcome this issue indirectly in the following way: first fix one subset of bilinear decision
variables and solve the resulting linear inequalities in the other decision variables; in the next step, the other bilinear decision variables are fixed and the procedure is repeated. For the particular long-time average cost control problem (\ref{objective})-(\ref{optimization}),
the non-convexity will be resolved in the following by considering a type of so-called small-feedback controller.
In such a new way, iterative updating of decision variables is exempted, and replaced by solving a sequence of SOS optimization problems.

\section{Bound optimization of long-time average cost for controlled polynomial systems}\label{seq:Bound}

In this section a small-feedback controller is designed to reduce the upper bound of the long-time average cost (\ref{cost}) for the controlled polynomial system (\ref{sys}).
It is reasonable to hope that a controller reducing the upper bound for the time-averaged cost will also reduce the time-averaged cost itself \cite{Ph:14}.

\subsection{Basic formalism of the controller design}

We will look for a controller in the form
\begin{eqnarray}
{\bf u}({\bf x},\epsilon)=\sum_{i=1}^{\infty}\epsilon^i {\bf u}_i({\bf x}),
\label{cc}
\end{eqnarray}
where $\epsilon>0$ is a parameter, and ${\bf u}_i({\bf x}), i= 1,2,\cdots$ are polynomial vector functions of system state ${\bf x}.$
In other words, we seek a family of controllers parameterised by $\epsilon$ in the form of a Taylor series in $\epsilon$. Notice that the expansion starts at the first-order term, so that $\epsilon=0$ gives the uncontrolled system.
To resolve the non-convexity problem of SOS optimization, we expand $V$  and $C$ in $\epsilon$:
\begin{eqnarray}
      V({\bf x},\epsilon)&=&\sum_{i=0}^{\infty} \epsilon^i V_i({\bf x}), \label{LF0} \\
      C(\epsilon)&=&\sum_{i=0}^{\infty}\epsilon^i C_i,
\label{LF1}
\end{eqnarray}
where $V_i$ and $C_i$ are the Taylor series coefficients for the tunable function and the bound, respectively, in the $i$th-order term of $\epsilon$.
Define
\begin{eqnarray}
F(V,u,C)\eqdef ({\bf f}({\bf x})+{\bf g}({\bf x}){\bf u})\cdot \nabla_{{\bf x}} V+\Phi({\bf x},{\bf u})-C.
\label{cc1}
\end{eqnarray}
Substituting (\ref{cc}), (\ref{LF0}), and (\ref{LF1}) into (\ref{cc1}), we have
\begin{eqnarray*}
F(V,u,C)&=& \left({\bf f}+{\bf g}\sum_{i=1}^{\infty}\epsilon^i{\bf u}_i\right)\cdot \sum_{i=0}^{\infty}\epsilon^i\nabla_{{\bf x}} V_i+\Phi\left({\bf x},\sum_{i=1}^{\infty}\epsilon^i{\bf u}_i\right)
-\sum_{i=0}^{\infty}\epsilon^i C_i.
\end{eqnarray*}
Noticing
\begin{eqnarray*}
\Phi\left({\bf x},\sum_{i=1}^{\infty}\epsilon^i{\bf u}_i\right
=
\sum_{i=0}^{\infty}\epsilon^i\left(\sum_{k=0}^i\frac{1}{k!}\frac{\partial^k\Phi}{\partial {\bf u}^k}({\bf x},0)\frac{1}{i!}\frac{\partial^i\left({\bf u}^k\right)}{\partial \epsilon^i}({\bf x},0)\right),
\end{eqnarray*}
it follows that
\begin{eqnarray}
F(V,u,C)=\sum_{i=0}^{\infty}\epsilon^i F_i(V_0,\cdots,V_i, {\bf u}_{1},\cdots,{\bf u}_i, C_i),
\label{inq2}
\end{eqnarray}
where
\begin{eqnarray}
F_i={\bf f}\cdot \nabla_{{\bf x}}V_i+\sum_{j+l=i}{\bf g}{\bf u}_j\cdot\nabla_{{\bf x}}V_l
+\sum_{k=0}^i\frac{1}{k!}\frac{\partial^k\Phi}{\partial {\bf u}^k}({\bf x},0)\frac{1}{i!}\frac{\partial^i\left({\bf u}^k\right)}{\partial \epsilon^i}({\bf x},0)-C_i.
\label{new1}
\end{eqnarray}
In (\ref{new1}), $({\partial^k\Phi}/{\partial {\bf u}^k})({\bf x},0)$ denotes the $k$th partial derivative of $\Phi$ with respect to ${\bf u}$ at ${\bf u}=0$, and
$({\partial^i({\bf u}^k)}/{\partial \epsilon^i})({\bf x},0)$ denotes the $i$th partial derivative of ${\bf u}^k({\bf x},\epsilon)\eqdef[u_1^k({\bf x},\epsilon),\cdots, u_m^k({\bf x},\epsilon)]^T$ with respect to $\epsilon$
at $\epsilon=0$.

Expression (\ref{inq2}) becomes more clear when a specific cost function $\Phi$ is considered.
For instance, let $\Phi=\Phi_0({\bf x})+{\bf u}^T{\bf u}$.
Then,
\begin{eqnarray*}
F(V,{\bf u},C)
=F_0(V_0,C_0)+\epsilon F_1(V_0,V_1,{\bf u}_1,C_1)+\epsilon^2F_2(V_0,V_1,V_2, {\bf u}_1,{\bf u}_2,C_2)+O(\epsilon^3),
\end{eqnarray*}
where
\begin{eqnarray*}
F_0&=&{\bf f}\cdot \nabla_{{\bf x}} V_0+\Phi_0-C_0, \\
[1ex]
F_1&=& {\bf f}\cdot \nabla_{{\bf x}} V_1+{\bf g} {\bf u}_1\cdot \nabla_{{\bf x}} V_0-C_1,\\
[1ex]
F_2&=&{\bf f}\cdot \nabla_{{\bf x}} V_2+{\bf g} {\bf u}_1\cdot \nabla_{{\bf x}} V_1+{\bf g} {\bf u}_2\cdot \nabla_{{\bf x}} V_0+{\bf u}_1^T{\bf u}_1-C_2,
\end{eqnarray*}
and $O(\epsilon^3)$ denotes all the terms with order of $\epsilon$ being equal or greater than 3.

It is clear that $F(V,{\bf u},C)\le 0$ holds if $F_i\le 0, i=0,1,2,\cdots$, simultaneously, and the series (\ref{cc})-(\ref{LF1}) converge.
Notice that $F_i$ includes tunable functions $V_j, j\le i$, and ${\bf u}_k, k\le i-1$. For any non-negative integers $i_1, i_2$ satisfying $i_1<i_2$, the tunable variables in $F_{i_1}$ are always a subset of the tunable variables in $F_{i_2}$. Hence (\ref{objective})-(\ref{optimization}) can be solved as a sequence of convex optimization problems. When the inequality constraints $F_i\le 0$ are relaxed to SOS conditions, our idea can be summarized as follows.

\noindent\rule{16.5cm}{0.1pt}
{\bf\it The sequential steps to solve (\ref{objective})-(\ref{optimization}): {\bf A-I}} \\
\noindent\rule{16.5cm}{0.1pt}

\begin{itemize}

\item[(s0)] First minimize $C_0$ over $V_0$ under the constraint $F_0(V_0,C_0)\le 0$, or more conservatively,
\begin{eqnarray*}
O_0: ~~\min_{V_0} C_0, {~~s.t.~~} -F_0(V_0,C_0)\mbox{~~is SOS}.
\end{eqnarray*}
Denote the optimal $C_0$ by $C_{0,SOS}$ and the associated $V_0$ by $V_{0,SOS}$.

\item[(s1)] Now, let $V_0=V_{0,SOS}$ in $F_1$, and then minimize $C_1$ over $V_1$ and ${\bf u}_1$ under the constraint $F_1(V_{0,SOS},V_1,{\bf u}_1,C_1)\le 0$, or under the framework of SOS optimization,
\begin{eqnarray*}
O_1: ~~\min_{V_1,{\bf u}_1} C_1, {~s.t.~} -F_1(V_{0,SOS},V_1,{\bf u}_1,C_1)\mbox{~is~ SOS}.
\end{eqnarray*}
Using the generalized $\mathcal{S}$-procedure given in Lemma~1 and the fact that
\begin{eqnarray}
-F_0(V_{0,SOS},C_{0,SOS})\ge 0,
\label{condition1}
\end{eqnarray}
$O_1$ can be revised by incorporating one more tunable function $S_0({\bf x})$:
\begin{eqnarray*}
O_1': \begin{array}{c}
\min_{V_1,{\bf u}_1,S_0} C_1, \\
[1ex]
{~~s.t.~~} \left\{
\begin{array}{c}
-F_1(V_{0,SOS},V_1,{\bf u}_1,C_1)
+S_0({\bf x})F_0(V_{0,SOS},C_{0,SOS}) \mbox{~~is SOS}, \\
[1ex]
S_0({\bf x})\mbox{~~is~ SOS}.
\end{array}
\right.
\end{array}
\end{eqnarray*}
Denote the optimal $C_1$ by $C_{1,SOS}$ and the associated $V_1$ and ${\bf u}_1$ by $V_{1,SOS}$ and ${\bf u}_{1,SOS}$, respectively.

\item[(s2)] Further let $V_0=V_{0,SOS}$, $V_1=V_{1,SOS}$, and ${\bf u}_1={\bf u}_{1,SOS}$ in $F_2$, and then minimize $C_2$ over $V_2$ and ${\bf u}_2$ under the constraint $F_2(V_{0,SOS},V_{1,SOS},V_2, {\bf u}_{1,SOS}, {\bf u}_2,C_2)\le 0$. In a more tractable way, consider
\begin{eqnarray*}
O_2:
\begin{array}{c}
 \displaystyle \min_{V_2,~ {\bf u}_2} C_2,~ {s.t.}\\
-F_2(V_{0,SOS},V_{1,SOS},V_2, {\bf u}_{1,SOS}, {\bf u}_2,C_2)\mbox{~is SOS}.
\end{array}
\end{eqnarray*}
Similarly as in (s1), noticing (\ref{condition1}) and
\begin{eqnarray*}
-F_1(V_{0,SOS},V_{1,SOS},{\bf u}_{1,SOS},C_{1,SOS})\ge 0,
\end{eqnarray*}
the SDP problem $O_2$ can be revised by the generalized $\mathcal{S}$-procedure to the following form:
\begin{eqnarray*}
O_2': \begin{array}{c}
 \min_{V_2,{\bf u}_2,S_0,S_1} C_2,  ~~~{s.t.} \\
 [1ex]
\left\{
\begin{array}{c}
 -F_2(V_{0,SOS},V_{1,SOS},V_2, {\bf u}_{1,SOS}, {\bf u}_2,C_2)
 +S_0({\bf x})F_0(V_{0,SOS},C_{0,SOS}) \\
~ +S_1({\bf x})F_1(V_{0,SOS},V_{1,SOS}, {\bf u}_{1,SOS},C_{1,SOS})\mbox{~is SOS},\\
 [1ex]
 S_0({\bf x})\mbox{~~is SOS}, \\
 [1ex]
 S_1({\bf x})\mbox{~~is SOS}.
\end{array}
\right.
\end{array}
\end{eqnarray*}
Denote the optimal $C_2$ by $C_{2,SOS}$ and the associated $V_2$ and ${\bf u}_2$ by $V_{2,SOS}$ and ${\bf u}_{2,SOS}$, respectively.

Notice that $S_0({\bf x})$ here might differ from the tunable function $S_0({\bf x})$ in $O_1'$. Throughout this paper we will use the same notations for the tunable functions like $S_0$ and $S_1$ in various instances of the  $\mathcal{S}$-procedure, to keep the notation simple.

\item[(s3)] The SOS-based controller design procedure is continued for higher-order terms.

\end{itemize}
\noindent\rule{16.5cm}{0.1pt}

Now, define three series
\begin{eqnarray}
C_{SOS}=\sum_{i=0}^{\infty}\epsilon^iC_{i,SOS}, ~~{\bf u}_{SOS}=\sum_{i=1}^{\infty}\epsilon^i {\bf u}_{i,SOS}, ~~V_{SOS}=\sum_{i=0}^{\infty}\epsilon^iV_{i,SOS}.
\label{series}
\end{eqnarray}
When all of them converge, the following statement will be true.

\begin{theorem}
By applying the state-feedback controller ${\bf u}={\bf u}_{SOS}$ for the system (\ref{sys}), if the trajectories of the closed-loop system are bounded
\footnote{In the context of long-time average cost controller design and analysis, it is actually enough to assume the boundedness of the global attractor of the system to ensure the existence of $C_{SOS}$.
}, then $C_{SOS}$ is an upper bound of the long-time average cost $\bar{\Phi}$.
\end{theorem}

{\it Proof}.  Using the algorithm {\bf A-I}, we obtain
\begin{eqnarray*}
F_i(V_{0,SOS},\cdots,V_{i,SOS}, {\bf u}_{1,SOS},\cdots, {\bf u}_{i,SOS},C_{i,SOS})\le 0, \forall ~i.
\label{new2}
\end{eqnarray*}
Then, it follows that
\begin{eqnarray*}
\sum_{i=0}^{\infty}F_i(V_{0,SOS},\cdots,V_{i,SOS}, {\bf u}_{1,SOS},\cdots, {\bf u}_{i,SOS},C_{i,SOS})
=F(V_{SOS}, {\bf u}_{SOS}, C_{SOS})\le 0,
\label{new3}
\end{eqnarray*}
where $C_{SOS}, {\bf u}_{SOS}, V_{SOS}$ are given in (\ref{series}).
By virtue of a same analysis as in proving Lemma~2 (see \cite{Ph:14}), we can conclude that $\bar{\Phi}\le C_{SOS}$.
\rule{0.09in}{0.09in}

\begin{remark}
After specifying the structure of controller to be of the form (\ref{cc}), the non-convexity in solving the optimization problem (\ref{objective})-(\ref{optimization}) has been avoided by solving the linear SDPs $O_0, O_1', O_2', \cdots$ in sequence.
During the process, all the involved decision variables are optimized sequentially, but not iteratively as in other methods \cite{Zh:07,Zh:09,Ng:11}.
\end{remark}

\begin{remark}
The smallness of $\epsilon$ can be used to relax $O_1', O_2', \cdots$ further.
For instance, in $O_1'$, in order to prove
$
F_0+\epsilon F_1\le 0,
$
we prove $F_1(V_{0,SOS},V_1,{\bf u}_1,C_1)\le 0$ with the aid of the known constraint $F_0(V_{0,SOS},C_{0,SOS})\le 0$,
thus not using that $\epsilon$ is small.
In fact, when $\epsilon$ is small, for $F_0+\epsilon F_1$ to be negative $F_1$ has to be negative only for those ${\bf x}$ where $F_0({\bf x})$ is small,
and not for all ${\bf x}$ as required in $O_1'$.
Meanwhile, checking the convergence of the series (\ref{series}) would be challenging or even impractical.
These points will be addressed in what follows.
\end{remark}

\subsection{Design of small-feedback controller}

Next, the sequential design method {\bf A-I} is revised to utilize that $\epsilon\ll 1$.

\noindent\rule{16.5cm}{0.1pt}
{\bf\it The revised sequential steps to solve (\ref{objective})-(\ref{optimization}): {\bf A-II}}

\noindent\rule{16.5cm}{0.1pt}

\begin{itemize}

\item[(s0)] Same as in {\bf A-I}, first solve the SOS optimization problem $O_0$.
Denote the optimal $C_0$ by $C_{0,SOS}$ and the associated $V_0$ by $V_{0,SOS}$.

\item[(s1)] Let $V_0=V_{0,SOS}$ in $F_1$, and then consider the following SDP problem:
\begin{eqnarray*}
&O_1'': \begin{array}{c}
\displaystyle \min_{V_1,{\bf u}_1,S_0} C_1, \\
[1ex]
{~~s.t.~~}
\begin{array}{c}
-F_1(V_{0,SOS},V_1,{\bf u}_1,C_1)
+S_0({\bf x})F_0(V_{0,SOS},C_{0,SOS}) \mbox{~is SOS},
\end{array}
\end{array}
\end{eqnarray*}
where $S_0$ is any tunable polynomial function of ${\bf x}$ of fixed degree.
Denote the optimal $C_1$ by $C_{1,SOS}$ and the associated $V_1$ and ${\bf u}_1$ by $V_{1,SOS}$ and ${\bf u}_{1,SOS}$, respectively.
Unlike $O_1'$, here the non-negativity requirement of $S_0$ is not imposed.
This can be understood as that the non-negativity constraint  is imposed only for ${\bf x}$ such that $F_0(V_{0,SOS},C_{0,SOS})=0$.

\item[(s2)] Further let $V_0=V_{0,SOS}$, $V_1=V_{1,SOS}$, and ${\bf u}_1={\bf u}_{1,SOS}$ in $F_2$, and then consider
\begin{eqnarray*}
O_2'': \begin{array}{c}
\displaystyle  \min_{V_2,{\bf u}_2,S_0,S_1} C_2, ~~~{s.t.} \\
 [1ex]
\left\{
\begin{array}{c}
-F_2(V_{0,SOS},V_{1,SOS},V_2, {\bf u}_{1,SOS}, {\bf u}_2,C_2)
 +S_0({\bf x})F_0(V_{0,SOS},C_{0,SOS}) \\
~+S_1({\bf x}) F_1(V_{0,SOS},V_{1,SOS},{\bf u}_{1,SOS},C_{1,SOS})
\mbox{~~is~ SOS},
\end{array}
\right.
\end{array}
\end{eqnarray*}
where $S_0$ and $S_1$ are any tunable polynomial functions of fixed degrees.
$S_0$ here does not need to be the same as in $O_1''$.
Denote the optimal $C_2$ by $C_{2,SOS}$ and the associated $V_2$ and ${\bf u}_2$ by $V_{2,SOS}$ and ${\bf u}_{2,SOS}$, respectively.
Similarly as in $O_1''$, here the non-negativity constraint is in effect imposed only where $F_0(V_{0,SOS},C_{0,SOS})=F_1(V_{0,SOS},V_{1,SOS}, {\bf u}_{1,SOS},C_{1,SOS})=0$.

\item[(s3)] The revised SOS-based controller design procedure is continued for higher-order terms.
\end{itemize}
\noindent\rule{16.5cm}{0.1pt}

Since the constraints of $S_i$ being SOS imposed in {\bf A-I} are removed in {\bf A-II}, the coefficients $C_{i,SOS}$ obtained in {\bf A-II} can be smaller than the coefficients $C_{i,SOS}$ obtained in {\bf A-I}. This advantage comes at a price: even if all the relevant series converge for a particular value of $\epsilon$, the procedure {\bf A-II} does not guarantee that the value $C_{SOS}$ given in (\ref{series}) is an upper bound for the time-averaged cost of the closed-loop system with the controller ${\bf u}_{SOS}$. We have now to consider (\ref{series}) as asymptotic expansions rather than Taylor series. Accordingly, we have to truncate the series and hope that the resulting controller will work for (sufficiently) small $\epsilon$
\footnote{It is worthy of noticing that the series truncation here does not mean that our controller design and analysis are conducted in a non-rigorous way.
The truncated controller would be effective if it leads to a better (lower) bound of the long-time average cost.}. It is possible to prove that this is, indeed, the case.

For illustration, the first-order truncation is considered only.

\begin{theorem}
Consider the first-order small-feedback controller for the system (\ref{sys}),
\begin{eqnarray}
{\bf u}_{SOS}=\epsilon {\bf u}_{1,SOS}
\label{new3}
\end{eqnarray}
where $\epsilon>0$ is sufficiently small. Assume that the trajectories of the closed-loop system are bounded, and that $C_{1,SOS}<0$.
Then, $C_{\kappa,SOS}\eqdef C_{0,SOS}+\epsilon\kappa C_{1,SOS}, \kappa\in(0,1)$ is an upper bound of the long-time average cost $\bar{\Phi}$.
Clearly, $C_{\kappa,SOS}<C_{0,SOS}$.
\end{theorem}

{\it Proof}. Let $V_{SOS}=V_{0,SOS}+\epsilon V_{1,SOS}$.
By substituting $V=V_{SOS}, C=C_{\kappa, SOS}, {\bf u}={\bf u}_{SOS}$ in the constraint function $F(V,{\bf u},C)$ that is defined in (\ref{cc1}), the remaining task
is to seek small $\epsilon>0$ such that
\begin{eqnarray}
F(V_{SOS},{\bf u}_{SOS},C_{\kappa, SOS}
\le
0.
\label{cc1_1}
\end{eqnarray}
Notice that
\begin{eqnarray}
F(V_{SOS},{\bf u}_{SOS},C_{\kappa,SOS}
=
F_0(V_{0,SOS},C_{0,SOS})
+\epsilon F_1(V_{0,SOS},V_{1,SOS},{\bf u}_{1,SOS},C_{1,SOS})
+\epsilon (1-\kappa)C_{1,SOS}+\epsilon^2 w({\bf x},\epsilon),
\label{boundx}
\end{eqnarray}
where
\begin{eqnarray*}
w({\bf x},\epsilon)={\bf g}{\bf u}_1\cdot  \nabla_{{\bf x}}V_{1,SOS}
+\frac{1}{\epsilon^2}\left(\Phi({\bf x},\epsilon {\bf u}_{1,SOS})-\Phi({\bf x},0)-\epsilon\frac{\partial \Phi}{\partial {\bf u}}({\bf x},0){\bf u}_{1,SOS}\right),
\end{eqnarray*}
and $F_0, F_1$, being polynomial in ${\bf x}$, possess all the continuity properties implied by the proof.
Let $\mathcal{D}\in{\mathbb R}^n$ be the phase domain that interests us, where the closed-loop trajectories are all bounded.
Then,
\begin{eqnarray}
F_{1,max}\eqdef \max_{{\bf x}\in\mathcal{D}}F_1(V_{0,SOS},V_{1,SOS},{\bf u}_{1,SOS},C_{1,SOS})<\infty,
\label{bound1}
\end{eqnarray}
and $w({\bf x},\epsilon)$ is bounded for any ${\bf x}\in\mathcal{D}$ and any finite $\epsilon$ (the latter following from the standard mean-value-theorem-based formula for the Lagrange remainder). By (\ref{boundx}) and (\ref{bound1}),
\begin{eqnarray}
F(V_{SOS},{\bf u}_{SOS},C_{\kappa,SOS})\le
F_0(V_{0,SOS},C_{0,SOS})+\epsilon F_{1,max}+\epsilon (1-\kappa)C_{1,SOS}+O(\epsilon^2).
\label{bound1_ex}
\end{eqnarray}
Meanwhile, consider the two inequality constraints obtained by solving $O_0$ and $O_1''$:
\begin{eqnarray}
\left\{
\begin{array}{c}
 F_0(V_{0,SOS},C_{0,SOS})\le 0, \\
[1ex]
 F_1(V_{0,SOS},V_{1,SOS}, {\bf u}_{1,SOS},C_{1,SOS})\le 0
~~\forall {\bf x} ~~\mbox{such that}~ F_0(V_{0,SOS},C_{0,SOS})= 0.
\end{array}
\right.
\label{cons}
\end{eqnarray}

Define $\mathcal{D}_{\delta}\eqdef \left\{{\bf x}\in \mathcal{D} ~|~ \delta\le F_0(V_{0,SOS},C_{0,SOS})\le 0\right\}$ for a given constant $\delta\le 0$.
Clearly, $\mathcal{D}_{\delta}\rightarrow \mathcal{D}_{0}$ as $\delta\rightarrow 0$.
Further define
\begin{eqnarray}
F_{1,\delta}(\delta)\eqdef \max_{{\bf x}\in \mathcal{D}_{\delta}} F_1(V_{0,SOS},V_{1,SOS},{\bf u}_{1,SOS},C_{1,SOS}).
\label{bound2}
\end{eqnarray}
By the second constraint in (\ref{cons}), $\lim_{\delta\rightarrow 0}F_{1,\delta}(\delta)\le 0$.
Therefore, by continuity and the fact $C_{1,SOS}<0$, for any $0<\kappa<1$ there exists a constant $\delta_{\kappa}<0$ such that
\begin{eqnarray}
F_1(V_{0,SOS},V_{1,SOS}, {\bf u}_{1,SOS},C_{1,SOS})\le F_{1,\delta_{\kappa}}<-\frac{1}{2}(1-\kappa)C_{1,SOS}, ~~\forall {\bf x}\in \mathcal{D}_{\delta_{\kappa}}.
\label{bound3}
\end{eqnarray}
In consequence, (\ref{boundx}), the first constraint in (\ref{cons}), and (\ref{bound3}) render to
\begin{eqnarray}
F(V_{SOS},{\bf u}_{SOS},C_{\kappa,SOS}) &\le&
F_0(V_{0,SOS},C_{0,SOS})+\epsilon F_{1,\delta_{\kappa}}+\epsilon (1-\kappa)C_{1,SOS}+O(\epsilon^2) \nonumber \\
&\le& \frac{\epsilon}{2} (1-\kappa)C_{1,SOS}+O(\epsilon^2)\le 0, ~~\forall {\bf x}\in \mathcal{D}_{\delta_{\kappa}},
\label{bound4}
\end{eqnarray}
for sufficiently small $\epsilon$.

Next, we prove (\ref{cc1_1}) for any ${\bf x}\in \mathcal{D}\setminus\mathcal{D}_{\delta_{\kappa}}$. By the definition of the set $\mathcal{D}_{\delta_{\kappa}}$, we have
\begin{eqnarray}
F_0(V_{0,SOS},C_{0,SOS})<\delta_{\kappa}<0,  ~~\forall {\bf x}\in \mathcal{D}\setminus\mathcal{D}_{\delta_{\kappa}}.
\label{bound5}
\end{eqnarray}
Then, (\ref{bound1_ex}) and (\ref{bound5}) yield
\begin{eqnarray}
F(V_{SOS},{\bf u}_{SOS},C_{\kappa,SOS})\le
\delta_{\kappa}+\epsilon F_{1,max}+\epsilon (1-\kappa)C_{1,SOS}+O(\epsilon^2)\le \delta_{\kappa}+O(\epsilon)\le 0, ~~\forall {\bf x}\in \mathcal{D}\setminus\mathcal{D}_{\delta_{\kappa}},
\label{bound1_ex1}
\end{eqnarray}
if $\epsilon$ is sufficiently small.

(\ref{bound4}) and (\ref{bound1_ex1}) imply that (\ref{cc1_1}) holds $\forall {\bf x}\in \mathcal{D}$. The proof is complete.
\rule{0.09in}{0.09in}

In practice, once the form of the controller has been specified in (\ref{new3}), the upper bound $C$ and the corresponding $V$ actually can be obtained by solving the following optimization problem directly:
\begin{eqnarray*}
O_{\epsilon}: \begin{array}{c}
\displaystyle  \min_{V,~\epsilon} C, ~~~ \\
 [1ex]
{s.t.}~~-F(V,\epsilon {\bf u}_{1,SOS},C)\mbox{~is~ SOS}.
\end{array}
\end{eqnarray*}
This problem can be further relaxed by incorporating the known constraints (\ref{cons}).
In $O_{\epsilon}$, if $\epsilon$ is set as one of the tunable variables, the SOS optimization problem will become non-convex again, thus causing additional trouble in solving it. Alternatively, one can fix $\epsilon$ here, and investigate its effect on the upper bound of $\bar{\Phi}$ by trial and error. We will follow this route in Section~\ref{seq:Example}.

\section{Illustrative example}\label{seq:Example}

As an illustrative example we consider a system proposed in \cite{Ki:05} as a model for studying control of oscillatory vortex shedding behind a cylinder. The actuation was assumed to be achieved by a volume force applied in a compact support region downstream of the cylinder. The Karhunen-Lo\`{e}ve (KL) decomposition \cite{No:03} was used and the first two KL modes and an additional shift mode were selected. For the Reynolds number equal to 100 the resulting low-order Galerkin model of the cylinder flow with control was given as follows
\begin{eqnarray}
\left[
\begin{array}{c}
\dot{a}_1 \\
\dot{a}_2 \\
\dot{a}_3
\end{array}
\right]&=&
\left[
\begin{array}{ccc}
\sigma_r & -\omega-\gamma a_3 & -\beta a_1 \\
\omega+\gamma a_3 & \sigma_r & -\beta a_2\\
\alpha a_1 & \alpha a_2 & -\sigma_3
\end{array}
\right]
\left[
\begin{array}{c}
{a}_1 \\
{a}_2 \\
{a}_3
\end{array}
\right]
+
\left[
\begin{array}{c}
g_1 \\
g_2 \\
0
\end{array}
\right]u,
\label{cf}
\end{eqnarray}
where $\sigma_r=0.05439, \sigma_3=0.05347, \alpha=0.02095, \beta=0.02116,$
$\gamma=-0.03504, \omega=0.9232, g_1=-0.15402$, and $g_2=0.046387$.
More details on deriving the reduced-order model (\ref{cf}) are given in~\cite{Ro:14}.

The system (\ref{cf}) possesses a unique equilibrium when $u=0$, which is at the origin.
Let $\Phi=1/2{\bf a}^T{\bf a}+u^2$, where ${\bf a}=[a_1 ~a_2 ~a_3]^T$.
The proposed algorithms {\bf A-I} and {\bf A-II} were applied to (\ref{cf}), with
the system state assumed to be available. In experiment, it could be estimated by designing a state observer with some sensed output measurement at a typical position~\cite{Ro:14}.

\subsection{Performance of algorithm {\bf A-I}}

The SDP problem $O_0$ is solved first. It corresponds to the uncontrolled sysytem.
The minimal upper bound we could achieve was $C_{0,SOS}=6.59.$ It was obtained with
\begin{eqnarray*}
V_{0,SOS}=-96.63 a_3+14.01a_1^2+14.01a_2^2+14.15 a_3^2.
\end{eqnarray*}
Increasing the degree of $V_0$ cannot give a better bound because there exists a stable limit cycle in the phase space of (\ref{cf}), on which $a_1^2+a_2^2=6.560,$ and $ a_3=2.570$.
Since $\bar{\Phi}=1/2{\bf a}^T{\bf a}=6.584$ on the limit cycle, the minimal upper bound achieved by SOS optimization is tight in the sense
that the difference between $C_{0,SOS}$ and $\bar{\Phi}$ is less than the prescribed precision for $C$, $0.01$.

Solving the SDP problem $O_1$, where $V_1$ and $u_1$ are tunable functions, gave $C_{1,SOS}=0$.
Solving $O_1',$ with $V_1, u_1, S_0$ being tuning functions, gave the same result: $C_{1,SOS}=0$.
In both cases, increasing the degrees of the tuning functions did not reduce the upper bound.
The consequent SOS optimization problems, $O_i',$ with $i=2,3$ also gave $C_{i,sos}=0, i=2,3$.
Therefore, by (\ref{series}),
\begin{eqnarray*}
C_{SOS}=C_{0,SOS}+\epsilon C_{1,SOS}+\epsilon^2 C_{2,SOS}+O(\epsilon^3)
       \approx C_{0,SOS}=6.59,
\end{eqnarray*}
implying that {\bf A-I} does not generate a control ensuring a better upper bound of $\bar{\Phi}$ than the bound obtained in the uncontrolled case.

\subsection{Performance of algorithm {\bf A-II}}

Without any control, it has been obtained in {\bf A-I} that $C_{0,SOS}=6.59$.

We first solve $O_1''$. Given the vectors of monomials in ${\bf x}$ without repeated elements \cite{Ch:09}, $Z_i, i=1,\cdots, 3$, define $V_1=P_1^TZ_1$, $u_1=P_2^TZ_2$, and $S_0=P_3^TZ_3$, where the parametric vectors $P_i$, $i=1,\cdots, 3$ consist of tuning vector variables. The degrees of $V_1, u_1$ and $S_0$ are specified by the maximum degrees of monomials in $Z_i, i=1,\cdots, 3$, and denoted by $ d_{V_1}, d_{u_1}$, and $d_{S_0}$, respectively. Consider two subcases: $d_{V_1}=d_{u_1}=d_{S_0}=2$ and $d_{V_1}=d_{u_1}=d_{S_0}=4$.
For the former case, we have $C_{1,SOS}=-354$, induced by
\begin{eqnarray*}
u_{1,SOS,2}=45.37 a_1-28.47 a_2-142.76 a_2a_3+399.49 a_1a_3.
\end{eqnarray*}
For the latter case, we have $C_{1,SOS}=-1965$, induced by
\begin{eqnarray*}
u_{1,SOS,4}&=&233.08a_1-54.73a_2-67.61a_2a_3+218.56a_1a_3+717.28a_1^3+13.16a_1^2a_2 \\
&&+571.67a_1a_2^2-277.73a_2^3+466.61a_1a_3^2-141.41a_2a_3^2+230.53a_1^3a_3\\
&&+106.32a_1^2a_2a_3+220.19a_1a_2^2a_3-161.44a_2^3a_3+628.40a_1a_3^3-173.78a_2a_3^3.
\end{eqnarray*}

We then solve $O_{\epsilon}$ with a fixed $\epsilon$.
For simplicity we considered $u_{1,SOS}=u_{1,SOS,2}$ and $d_V\le 10$ only.
The upper-bound results for different $\epsilon$ are summarized in Fig. \ref{inf0}.
The long-time average cost $\bar{\Phi}$, which is obtained by direct numerical experiment, and the linear truncated bound $C_{0,SOS}+\epsilon C_{1,SOS}$ are also presented for comparison.
From Fig. \ref{inf0}, we can see the following.

Let $\epsilon_1= 1.267\times 10^{-2}$ and $\epsilon_2=7.416\times 10^{-2}$.
The small-feedback controller
\begin{eqnarray}
u=\epsilon u_{1,SOS,2}
 \label{test}
\end{eqnarray}
 reduces $\bar{\Phi},$ and the reduction in $\bar{\Phi}$ increases monotonically with $\epsilon$ when $0<\epsilon<\epsilon_2$. In particular, $\bar{\Phi}=0$ for $\epsilon_1 \le \epsilon< \epsilon_2,$ that is in this range of $\epsilon$ the controller fully stabilizes the system. When $\epsilon\ge \epsilon_2$, the controller makes the long-time average cost worse than in the uncontrolled case. The effect of $\epsilon$ on $\bar{\Phi}$ can be seen more clearly by investigating the qualitative properties of the closed-loop system. A simple check gives that when $0\le \epsilon< \epsilon_1$, the closed-loop system has a unique unstable equilibrium at the origin and a stable limit cycle, thus yielding a non-zero but finite $\bar{\Phi}$;
when $\epsilon_1 \le \epsilon< \epsilon_2$, the limit cycle disappears and the unique equilibrium becomes globally stable, thus implying
the vanishness of $\bar{\Phi}$;
when $\epsilon\ge \epsilon_2$ but is close to $\epsilon_2$, besides the equilibrium at the origin, there exist four additional non-zero equilibria, and as a result $\bar{\Phi}$ becomes large immediately. For instance, at the bifurcation point $\epsilon=\epsilon_2$, the non-zero equilibria of the closed-loop system are $(\pm 0.6988, \pm 2.362, 2.377)$ and $(\pm 0.7000, \pm 2.364,  2.382)$, resulting in $\bar{\Phi}=171.55$.

Solving $O_{\epsilon}, 0<\epsilon\le 8.7\times 10^{-4}$ yields a tight upper bound $C_{\epsilon, SOS}$ for $\bar{\Phi}$.
However, the obtained upper bound becomes non-tight when $\epsilon> 8.7\times 10^{-4}$.
The conservativeness of $C_{\epsilon, SOS}$ can be fully overcome by considering additional relaxation constraint (\ref{cons})
for $ 8.7\times 10^{-4}<\epsilon\le 4\times 10^{-3}$, but only mitigated to certain extend for larger $\epsilon$.

The two-term expansion $C_{0,SOS}+\epsilon C_{1,SOS}$ is only a linear approximation of $C_{SOS}$ in (\ref{series}).
Thus, as an upper bound of $\bar{\Phi}$, it behaves well when $\epsilon$ is very small, but it becomes conservative when $\epsilon$ is further increased, and meaningless as $\epsilon>-C_{0,SOS}/C_{1,SOS}=0.0186$.

In summary, for small $\epsilon$, the proposed small-feedback controller
yields a better bound of the long-time average cost than in the uncontrolled case. Further, the controller indeed reduces the long-time average cost itself.

\begin{figure}
  \centerline{\includegraphics[trim=0mm 5mm 0mm 0mm,clip,width=10cm]{result1.eps}}
\centerline{$\epsilon$}
  \caption{The long-time average cost $\bar{\Phi}$ and its upper bounds for different $\epsilon$.
   $C_{0,SOS}, C_{1,SOS}, C_{\epsilon,SOS}, C_{\epsilon,SOS}'$ are obtained by solving $O_0, O_1'', O_{\epsilon}$, and $O_{\epsilon}$ with the relaxation (\ref{cons}), respectively.
}
\label{inf0}
\end{figure}

\begin{figure}

  \centerline {\includegraphics[trim=0mm 10mm 0mm 0mm,clip,width=10cm]{ep20.eps}}

\centerline{$t$}
  \caption{Control input profile.
}
\label{inf2}
\end{figure}

\begin{figure}
  \centerline{\includegraphics[height=5cm,width=9cm]{ep30.eps}}
  \caption{Closed-loop trajectory starting at ${\bf a}=[-0.3~-0.3~0.3]^T$.
  Owing to the small-feedback control, the magnitude of the periodic oscillation has been reduced.
}
\label{inf3}
\end{figure}

Figs. \ref{inf2}-\ref{inf3} show more details of the control performance of the proposed controller (\ref{test}) with $\epsilon=8.7\times 10^{-4}$ and
the initial state ${\bf a}=[-0.3 ~-0.3 ~0.3]^T$.

\section{Conclusion}\label{seq:Conclusion}

Based on sum-of-squares decomposition of polynomials and semidefinite programming, a numerically tractable approach is presented for long-time average cost control of polynomial dynamical systems.
The obtained controller possesses a structure of small feedback, which is an asymptotic expansion in a small parameter, with all the coefficients being polynomials of the system state.
The derivation of the small-feedback controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities.
The non-convexity in SOS optimization can be resolved by making full use of the smallness of the perturbation parameter while not using any iterative
algorithms.
The efficiency of the control scheme has been tested on a low-order model of  cylinder wake flow stabilization problem.
In the next research phase, we will consider SOS-based long-time average cost control under modelling uncertainties and in the presence of noise, as well as direct numerical simulations of small-feedback control for actual fluid flows.

The proof of concept of the idea of using the upper bound of long-time average cost control as the objective of the control design, and the method of overcoming the non-convexity of simultaneous optimization of the control law and the tunable function are the two main contributions of the present paper.


\section{Introduction}

The discovery of the Higgs boson in 2012 at the LHC has attested the success
of the standard model (SM) in describing the observed fermions and their
interactions. However, there exist many theoretical issues or open questions
that have no satisfactory answer. In particular, the observed flavour
pattern lacks of a definitive explanation, i.e., the quark Yukawa coupling
matrices $Y_u$ and $Y_d$, which in the SM reproduce the six quark masses,
three mixings angles and a complex phase to account for CP violation
phenomena, are general complex matrices, not constrained by any gauge
symmetry.

Experimentally the flavour puzzle is very intricate. First, there is the
quark mass hierarchy in both sectors. Secondly, the mixings in the SM,
encoded in the Cabibbo-Kobayashi-Maskawa (CKM) unitary matrix, turns out to
be close to the identity matrix. If one takes also the lepton sector into
account, the hierarchy there is even more puzzling~\cite%
{Emmanuel-Costa:2015tca}. On the other hand, in the SM there is in general
no connection between the quark masses hierarchy and the CKM mixing pattern.
In fact, if one considers the Extreme Chiral Limit, where the quark masses
of the first two generations are set to zero, the mixing does not
necessarily vanish~\cite{Botella:2016krk}, and one concludes that the CKM
matrix~$V$ being close to the identity matrix has nothing to do with the
fact that the quark masses are hierarchical. Indeed, in order to have $%
V\approx \mathbf{1}$, one must have a definite alignment of the quark mass
matrices in the flavour space, and to explain this alignment, a flavour
symmetry or some other mechanism is required~\cite{Botella:2016krk}.

Among many attempts made in the literature to address the flavour puzzle,
extensions of the SM with new Higgs doublet are particularly motivating.
This is due to fact that the number of Higgs doublets is not constrained by
the SM symmetry. Moreover, the addition of scalar doublets gives rise to new
Yukawa interactions and as a result it provides a richer framework in
approaching the theory of flavour. On the other hand, any new extension of
the Higgs sector must be very much constrained, since it naturally leads to
flavour changing neutral currents. At tree level, in the SM, all the flavour
changing transitions are mediated through charged weak currents and the
flavour mixing is controlled by the CKM matrix~\cite%
{Cabibbo:1963yz,Kobayashi:1973fv}. If new Higgs doublets are added, one
expects large FCNC effects already present at tree level. Such effects have
not been experimentally observed and they constrain severely any model with
extra Higgs doublets, unless a flavour symmetry suppresses or avoids large
FCNC~\cite{Branco:2011iw}.

Minimal flavour violating models~\cite%
{Joshipura:1990pi,Antaramian:1992ya,Hall:1993ca,Mantilla2017, Buras:2001,
Dambrosio:2002} are examples of a multiHiggs extension where FCNC are
present at tree-level but their contributions to FCNC phenomena involve only
off-diagonal elements of the CKM matrix or their products. The first
consistent models of this kind were proposed by Branco, Grimus and Lavoura
(BGL)~\cite{Branco:1996bq}, and consisted of the SM with two Higgs doublets
together with the requirement of an additional discrete symmetry. BGL models
are compatible with lower neutral Higgs masses and FCNC's occur at tree
level, with the new interactions entirely determined in terms of the CKM
matrix elements.

The goal of this paper is to generalize the previous BGL models and to,
systematically, search for patterns where a discrete flavour symmetry
naturally leads to the alignment of the flavour space of both the quark
sectors. Although the quark mass hierarchy does not arise from the symmetry,
the effect of both is such that the CKM matrix is near to the identity and
has the correct overall phenomenological features, determined by the quark
mass hierarchy, \cite{Branco:2011aa}. To do this we extend the SM with two
extra Higgs doublets to a total of three Higgs $\phi _{a}$. The choice for
discrete symmetries is to avoid the presence of Goldstone bosons that appear
in the context of any global continuous symmetry, when the spontaneous
electroweak symmetry breaking occurs. For the sake of simplicity, we
restrict our search to the family group $Z_{N}$, and demand that the
resulting up-quark mass matrix $M_{u}$ is diagonal. This is to say that, due
to the expected strong up-quark mass hierarchy, we only consider those cases
where the contribution of the up-quark mass matrix to quark mixing is
negligible.

If one assumes that all Higgs doublets acquire vacuum expectation values
with the same order of magnitude, then each Higgs doublet must couple to the
fermions with different strengths. Possibly one could obtain similar results
assuming that the vacuum expectation values (VEVs) of the Higgs have a
definite hierarchy instead of the couplings, but this is not considered
here. Combining this assumption with the symmetry, we obtain the correct
ordered hierarchical pattern if the coupling with $\phi _{3}$ gives the
strength of the third generation, the coupling with $\phi _{2}$ gives the
strength of the second generation and the coupling with $\phi _{1}$ gives
the strength of the first generation. Therefore, from our point of view, the
three Higgs doublets are necessary to ensure that there exists three
different coupling strengths, one for each generation, to guarantee
simultaneously an hierarchical mass spectrum and a CKM matrix that has the
correct overall phenomenological features e.g. $\left\vert V_{cb}\right\vert
^{2}+\left\vert V_{ub}\right\vert ^{2}=O(m_{s}/m_{b})^{2}$, \ and denoted
here by $V\approx \mathbf{1.}$

Indeed, our approach is within the BGL models, and such that the FCNC
flavour structure is entirely determined by CKM. Through the symmetry, the
suppression of the most dangerous FCNC's, by combinations of the CKM matrix
elements and light quark masses, is entirely natural.

The paper is organised as follows. In the next section, we present our model
and classify the patterns allowed by the discrete symmetry in combination with our assumptions.
In Sec. \ref{sec:num}, we give a brief numerical analysis of
the phenomenological output of our solutions. In Sec. \ref{sec:fcnc}, we
examine the suppression of scalar mediated FCNC in our framework for each
pattern. Finally, in Sec. \ref{sec:conc}, we present our conclusions.

\section{The Model}

\label{sec:model}

We extend the Higgs sector of the SM with two extra new scalar doublets,
yielding a total of three scalar doublets, as $\phi _{1}$, $\phi _{2}$, $%
\phi _{3}$. As it was mentioned in the introduction, the main idea for
having three Higgs doublets is to implement a discrete flavour symmetry,
that leads to the alignment of the flavour space of the quark sectors. The
quark mass hierarchy does not arise from the symmetry, but together with the
symmetry the effect of both is such that the CKM matrix is near to the
identity and has the correct overall phenomenological features, determined
by the quark mass hierarchy.

Let us start by considering the most general quark Yukawa coupling
Lagrangian invariant in our setup
\begin{equation}
-\mathcal{L}_{\text{Y}}=(\Omega _{a})_{ij}\,\overline{Q}_{Li}\ \widetilde{%
\phi }_{a}\ u_{R_{j}}+(\Gamma _{a})_{ij}\,\overline{Q}_{Li}\ \phi _{a}\
d_{R_{j}}+h.c.,  \label{eq:lag}
\end{equation}%
with the Higgs labeling $a=1,2,3$ and $i,j$ are just the usual flavour
indexes identifying the generations of fermions. In the above Lagrangian,
one has three Yukawa coupling matrices $\Omega _{1}$, $\Omega _{2}$, $\Omega
_{3}$ for the up-quark sector and three Yukawa coupling matrices $\Gamma
_{1} $, $\Gamma _{2}$, $\Gamma _{3}$ for the down sector, corresponding to
each of the Higgs doublets $\phi _{1}$, $\phi _{2}$, $\phi _{3}$. Assuming
that only the neutral components of the three Higgs doublets acquire vacuum
expectation value (VEV), the quark mass $M_{u}$ and $M_{d}$ are then easily
generated as
\begin{subequations}
\label{eq:mass}
\begin{align}
M_{u}& =\Omega _{1}\left\langle \phi _{1}\right\rangle \,^{\ast }+\,\Omega
_{2}\left\langle \phi _{2}\right\rangle \,^{\ast }+\,\Omega
_{3}\,\left\langle \phi _{3}\right\rangle ^{\ast },  \label{eq:massup} \\
M_{d}& =\Gamma _{1}\left\langle \phi _{1}\right\rangle \,+\,\Gamma
_{2}\left\langle \phi _{2}\right\rangle \,+\,\Gamma _{3}\left\langle \phi
_{3}\right\rangle ,
\end{align}%
where VEVs $\langle \phi _{i}\rangle $ are parametrised as
\end{subequations}
\begin{equation}
\left\langle \phi _{1}\right\rangle =\frac{v_{1}}{\sqrt{2}},\quad
\left\langle \phi _{2}\right\rangle =\frac{v_{2}e^{i\alpha _{2}}}{\sqrt{2}}%
,\quad \left\langle \phi _{3}\right\rangle =\frac{v_{3}e^{i\alpha _{3}}}{%
\sqrt{2}},
\end{equation}%
with $v_{1}$, $v_{2}$ and $v_{3}$ being the VEV moduli and $\alpha _{2}$, $%
\alpha _{3}$ just complex phases. We have chosen the VEV of $\phi _{1}$ to
be real and positive, since this is always possible through a proper gauge
transformation. As stated, we assume that the moduli of VEVs $v_{i}$ are of
the same order of magnitude, i.e.,
\begin{equation}
v_{1}\sim v_{2}\sim v_{3}.  \label{vs}
\end{equation}

Each of the $\phi _{a}$ couples to the quarks with a coupling $(\Omega
_{a})_{ij},(\Gamma _{a})_{ij}$ which we take be of the same order of
magnitude, unless some element vanishes by imposition of the flavour
symmmetry. In this sence, each $\phi _{a}$ and $(\Omega _{a},\Gamma _{a})$
will generate it's own respective generation: i.e., our model is such that
by imposition of the flavour symmmetry, $\phi _{3}$, $\Omega _{3}$, $\Gamma
_{3}$ will generate $m_{t}$ respectivelly $m_{b}$, that $\phi _{2}$, $\Omega
_{2}$, $\Gamma _{2}$ will generate $m_{c}$ respectivelly $m_{s}$, and that $%
\phi _{1}$, $\Omega _{1}$, $\Gamma _{1}$ will generate $m_{u}$ respectivelly
$m_{d}$. Generically, we have
\begin{subequations}
\label{eq:hierarchy}
\begin{align}
v_{1}\left\vert (\Omega _{1})_{ij}\right\vert & \sim m_{u},\;v_{2}\left\vert
(\Omega _{2})_{ij}\right\vert \sim m_{c},\;v_{3}\left\vert (\Omega
_{3})_{ij}\right\vert \sim m_{t}, \\
v_{1}\left\vert (\Gamma _{1})_{ij}\right\vert & \sim m_{d},\;v_{2}\left\vert
(\Gamma _{2})_{ij}\right\vert \sim m_{s},\;v_{3}\left\vert (\Gamma
_{3})_{ij}\right\vert \sim m_{b},
\end{align}%
which together with Eq. \ref{vs} implies a definite hierarchy amongst the
non-vanishing Yukawa coupling matrix elements:
\end{subequations}
\begin{subequations}
\label{eq:hier}
\begin{align}
\left\vert (\Omega _{1})_{ij}\right\vert & \ll \left\vert (\Omega
_{2})_{ij}\right\vert \ll \left\vert (\Omega _{3})_{ij}\right\vert ,
\label{eq:hierup} \\[2mm]
\left\vert (\Gamma _{1})_{ij}\right\vert & <\left\vert (\Gamma
_{2})_{ij}\right\vert \ll \left\vert (\Gamma _{3})_{ij}\right\vert .
\end{align}

Next, we focus on the required textures for the Yukawa coupling matrices $%
\Omega _{a}$ and $\Gamma _{a}$ that naturally lead to an hierarchical mass
quark spectrum and at the same time to a realistic CKM mixing matrix. These
textures must be reproduced by our choice of the flavour symmetry. As
referred in the introduction, we search for quark mass patterns where the
mass matrix $M_{u}$ is diagonal. Therefore, one derives from Eqs.~%
\eqref{eq:massup}, \eqref{eq:hierup} the following textures for $\Omega _{a}$
\end{subequations}
\begin{equation}
\Omega _{1}=%
\begin{pmatrix}
\mathsf{x} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0%
\end{pmatrix}%
,\,\Omega _{2}=%
\begin{pmatrix}
0 & 0 & 0 \\
0 & \mathsf{x} & 0 \\
0 & 0 & 0%
\end{pmatrix}%
,\,\Omega _{3}=%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & \mathsf{x}%
\end{pmatrix}%
.  \label{eq:textureOs}
\end{equation}%
The entry $\mathsf{x}$ means a non zero element. In this case, the up-quark
masses are given by $m_{u}=v_{1}\left\vert (\Omega _{1})_{11}\right\vert $, $%
m_{c}=v_{2}\left\vert (\Omega _{2})_{22}\right\vert $ and $%
m_{t}=v_{3}\left\vert (\Omega _{3})_{33}\right\vert $.

Generically, the down-quark Yukawa coupling matrices must have the following
indicative textures
\begin{equation}
\Gamma _{1}=%
\begin{pmatrix}
\boldsymbol{\mathsf{x}} & \boldsymbol{\mathsf{x}} & \boldsymbol{\mathsf{x}}
\\
\mathsf{x} & \mathsf{x} & \mathsf{x} \\
\mathsf{x} & \mathsf{x} & \mathsf{x}%
\end{pmatrix}%
,\,\Gamma _{2}=%
\begin{pmatrix}
0 & 0 & 0 \\
\boldsymbol{\mathsf{x}} & \boldsymbol{\mathsf{x}} & \boldsymbol{\mathsf{x}}
\\
\mathsf{x} & \mathsf{x} & \mathsf{x}%
\end{pmatrix}%
,\,\Gamma _{3}=%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
\boldsymbol{\mathsf{x}} & \boldsymbol{\mathsf{x}} & \boldsymbol{\mathsf{x}}%
\end{pmatrix}%
.  \label{eq:textureGs}
\end{equation}%
We distinguish rows with bold $\boldsymbol{\mathsf{x}}$ in order to indicate
that it is mandatory that at least one of matrix elements within that row
must be nonvanishing. Rows denoted with $\mathsf{x}$ may be set to zero,
without modifying the mass matrix hierarchy. These textures ensure that not only is
the mass spectrum hierarchy respected but it also leads to the alignment of the
flavor space of both the quark sectors \cite{Branco:2011aa}
and to a CKM matrix $V\approx \mathbf{1}$. For instance, if one would not have
a vanishing, or comparatively very small, $(1,3)$ entry in the $\Gamma _{2}$%
, this would not necessarily spoil the scale of $m_{s}$, but it would
dramatically change the predictions for the CKM mixing matrix.

In order to force the Yukawa coupling matrices $\Omega _{a}$ and $\Gamma
_{a} $ to have the indicative forms outlined in Eqs.~\eqref{eq:textureOs}
and~\eqref{eq:textureGs}, we introduce a global flavour symmetry. Since any
global continuous symmetry leads to the presence of massless Goldstone
bosons after the spontaneous electroweak breaking, one should instead
consider a discrete symmetry. Among many possible discrete symmetry
constructions, we restrict our searches to the case of cycle groups $Z_{N}$.
Thus, we demand that any quark or boson multiplet $\chi $ transforms
according to $Z_{N}$ as
\begin{equation}
\chi \rightarrow \chi ^{\prime }=e^{i\,\mathcal{Q}(\chi )\,\frac{2\pi }{N}%
}\chi ,
\end{equation}%
where $\mathcal{Q}(\chi )\in \{0,1,\dots ,N\}$ is the $Z_{N}$-charge
attributed for the multiplet $\chi $.

We have chosen the up-quark mass matrix $M_{u}$ to be diagonal. This
restricts the flavour symmetry $Z_{N}$. We have found that, in order to
ensure that all Higgs doublet charges are different, and to
have appropriate charges for fields ${Q_{L}}_{i}$ and ${u_{R}}_{i}$, we must
have $N\geq 7$. We simplify our analysis by fixing $N=7$ and choose:
\begin{subequations}
\label{eq:fix}
\begin{align}
\mathcal{Q}({Q_{L}}_{i})& =(0,1,-2), \\
\mathcal{Q}({u_{R}}_{i})& =(0,2,-4),
\end{align}%
In addition, we may also fix
\end{subequations}
\begin{equation}
\mathcal{Q}({Q_{L}}_{i})=\mathcal{Q}(\phi _{i})  \label{eq:fix1}
\end{equation}%
It turns out that these choices do not restrict the results, i.e. the
possible textures that one can have for the $\Gamma _{i}$ matrices. Other
choices would only imply that we reshuffle the charges of the multiplets.

With the purpose of enumerating the different possible textures for the $%
\Gamma _{i}$ matrices implementable in $Z_{7}$, we write down the charges of
the trilinears $\mathcal{Q}({\overline{Q}_{L}}_{i}\phi _{a}{d_{R}}_{j})$
corresponding to each $\phi _{a}$ as
\begin{subequations}
\begin{equation}
\mathcal{Q}({\overline{Q}_{L}}_{i}\phi _{1}{d_{R}}_{j})=%
\begin{pmatrix}
d_{1} & d_{2} & d_{3} \\
d_{1}-1 & d_{2}-1 & d_{3}-1 \\
d_{1}+2 & d_{2}+2 & d_{3}+2%
\end{pmatrix}%
,
\end{equation}%
\begin{equation}
\mathcal{Q}({\overline{Q}_{L}}_{i}\phi _{2}{d_{R}}_{j})=%
\begin{pmatrix}
d_{1}+1 & d_{2}+1 & d_{3}+1 \\
d_{1} & d_{2} & d_{3} \\
d_{1}+3 & d_{2}+3 & d_{3}+3%
\end{pmatrix}%
,
\end{equation}%
\begin{equation}
\mathcal{Q}({\overline{Q}_{L}}_{i}\phi _{3}{d_{R}}_{j})=%
\begin{pmatrix}
d_{1}-2 & d_{2}-2 & d_{3}-2 \\
d_{1}-3 & d_{2}-3 & d_{3}-3 \\
d_{1} & d_{2} & d_{3}%
\end{pmatrix}%
,
\end{equation}%
where $d_{i}\equiv \mathcal{Q}({d_{R}}_{i})$. One can check that, in order
to have viable solutions, one must vary the values of $d_{i}\in
\{0,1,-2,-3\} $.

We summarise in Table \ref{tab:downTextures} all the allowed textures for
the $\Gamma _{a}$ matrices and the resulting $M_{d}$ mass matrix texture,
excluding all cases which are irrelevant, e.g. matrices that have too much
texture zeros and are singular, or matrices that do not accommodate CP
violation. It must be stressed, that these are the textures obtained by the
different charge configurations that one can possibly choose. However, if
one assumes a definite charge configuration, then the entire texture, $M_{d}$
and $M_{u}$ and the respective phenomenology are fixed. As stated, the list
of textures in Table~\ref{tab:downTextures} remains unchanged even if one
chooses any other set than in Eqs.~\eqref{eq:fix}, \eqref{eq:fix1}. As
stated, that all patterns presented here are of the Minimal Flavour
Violation (MFV) type \cite%
{Joshipura:1990pi,Antaramian:1992ya,Hall:1993ca,Mantilla2017, Buras:2001,
Dambrosio:2002}.

Pattern~I in the table was already considered in Ref.~\cite{Botella:2009pq}
in the context of $Z_{8}$. We discard Patterns~IV, VII and X, because
contrary to our starting point, at least one of three non-zero couplings
with $\phi _{1}$ will turn out be of the same order as the larger coupling
with $\phi _{2}$ in order to meet the phenomenological requirements of the
CKM matrix.

Notice also, that the structure of other $M_{d}$'s cannot be trivially
obtained, e.g. from Pattern I, by a transformation of the right-handed down
quark fields.

Our symmetry model may be extended to the charged leptons and neutrinos,
e.g. in the context of type one see-saw. Choosing for the lepton doublets ${L%
}_{i}$ the charges $\mathcal{Q}({L}_{i})=(0,-1,2)$, opposite to the Higgs
doublets in Eq. \eqref{eq:fix1},~and e.g. for the charges $\mathcal{Q}({e_{R}%
}_{i})=(0,-2,4)$ of the right-handed fields ${e_{R}}_{i}$, we force the
charged~ lepton mass matrix to be diagonal. Then for the right-handed
neutrinos ${\nu _{R}}_{i}$, choosing $\mathcal{Q}({\nu _{R}}_{i})=(0,0,0)$,
we obtain for the neutrino Dirac mass matrix a pattern similar to pattern I.
Of course, for this case, the heavy right-handed neutrino Majorana mass
matrix is totally arbitrary. In other cases, i.e. for other patterns and
charges, in particular for the right-handed neutrinos, we could introduce
scalar singlets with suitable charges, which would then lead to certain
heavy right-handed neutrino Majorana mass matrices.

Next, we address an important issue of the model, namely, whether accidental
$U(1)$ symmetries may appear in the Yukawa sector or in the potential. One
may wonder whether a continuous accidental $U(1)$ symmetry could arise, once
the $Z_{7}$ is imposed at the Lagrangian level in Eq.~\eqref{eq:lag}. This
is indeed the case, i.e., for all realizations of $Z_{7}$, one has the
appearance of a global $U(1)_{X}$. However, any consistent global $U(1)_{X}$
must obey to the anomaly-free conditions of global symmetries~\cite%
{Babu:1989ex}, which read for the anomalies $SU(3)^{2}\times U(1)_{X}$, $%
SU(2)^{2}\times U(1)_{X}$ and $U(1)_{Y}^{2}\times U(1)_{X}$ as
\end{subequations}
\begin{subequations}
\begin{equation}
A_{3}\equiv \frac{1}{2}\sum_{i=1}^{3}\biggl(2X({Q_{L}}_{i})-X({u_{R}}_{i})-X(%
{d_{R}}_{i})\biggr)=0,  \label{eq:A3}
\end{equation}%
\begin{equation}
A_{2}\equiv \frac{1}{2}\sum_{i=1}^{3}\biggl(3X({Q_{L}}_{i})+X({\ell _{L}}%
_{i})\biggr)=0,  \label{eq:A2}
\end{equation}%
\begin{equation}
A_{1}\equiv \frac{1}{6}\sum_{i=1}^{3}\biggl(X({Q_{L}}_{i})+3X({\ell _{L}}%
_{i})-8X({u_{R}}_{i})-2X({d_{R}}_{i})-6X({e_{R}}_{i})\biggr)=0,
\end{equation}%
where $X(\chi )$ is the $U(1)_{X}$ charge of the fermion multiplet $\chi $.
We have properly shifted the $Z_{7}$-charges in Eq.\eqref{eq:fix} and in
Table \ref{tab:downTextures} so that $X(\chi )=\mathcal{Q}(\chi )$, apart of
an overall $U(1)_{X}$ convention. In general to test those conditions, one
needs to specify the transformation laws for all fermionic fields. Looking
at the Table~1, we derive that all the cases, except the first case
corresponding to $d_{i}=(0,0,0)$, violate the condition given in Eq.~%
\eqref{eq:A3} that depends only on coloured fermion multiplets. In the case $%
d_{i}=(0,0,0)$, if one assigns the charged lepton charges as $X({\ell _{L}}%
_{i})=X({Q_{L}}_{i})$ one concludes that the condition given in Eq.~%
\eqref{eq:A2} is violated. One then concludes that the global $U(1)_{X}$
symmetry is anomalous and therefore only the discrete symmetry $Z_{7}$
persists.

We also comment on the scalar potential of our model. The most general
scalar potential with three scalars invariant under $Z_{7}$ reads as
\end{subequations}
\begin{equation}
V(\phi )=\sum_{i}\left[ -\mu _{i}^{2}\phi _{i}^{\dagger }\phi _{i}+\lambda
_{i}(\phi _{i}^{\dagger }\phi _{i})^{2}\right] +\sum_{i<j}\left[ +C_{i}(\phi
_{i}^{\dagger }\phi _{i})(\phi _{j}^{\dagger }\phi _{j})+\,\bar{C}%
_{i}\left\vert \phi _{i}^{\dagger }\phi _{j}\right\vert ^{2}\right] ,
\label{eq:pot}
\end{equation}%
where the constants $\mu _{i}^{2}$, $\lambda _{i}$, $C_{i}$ and $\bar{C}_{i}$
are taken real for $i,j=1,2,3$. Analysing the potential above, one sees that
it gives rise to the accidental global continuous symmetry $\phi
_{i}\rightarrow e^{i\alpha _{i}}\phi _{i}$, for arbitrary $\alpha _{i}$,
which upon spontaneous symmetry breaking leads to a massless neutral scalar,
at tree level. Introducing soft-breaking terms like $m_{ij}^{2}\phi
_{i}^{\dagger }\,\phi _{j}\,+\text{H.c.}$ can erase the problem. Another
possibility without spoiling the $Z_{7}$ symmetry is to add new scalar
singlets, so that the coefficients $m_{ij}^{2}$ are effectively obtained
once the scalar singlets acquire VEVs.

\begin{table}[]
\caption{The table shows the viable configurations for the right-handed
down-quark ${d_{R}}_{i}$ and their corresponding $\Gamma _{1}$, $\Gamma _{2}$%
, $\Gamma_{3}$ and $M_{d}$ matrices. It is understood that, for each pattern
and coupling, the parameters expressed here by the same symbol, are in fact
different, but denoting he same order of magnitude, (or possibly smaller).
E.g. in pattern I, coupling $\Gamma_1$, the three $\protect\delta$, $\protect%
\delta$, $\protect\delta$, stand for $\protect\delta_1$, $\protect\delta_2$,
$\protect\delta_3$. The same applies to the $\protect\varepsilon$'s and $c$%
's. For patterns IV, VII, and X, which will be excluded, one of the
couplings in $\Gamma _{1}$ turns out to be much larger. }
\label{tab:downTextures}\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Pattern & $\mathcal{Q}({d_R}_i)$ & $\Gamma_1$ & $\Gamma_2$ & $\Gamma_3$ & $%
M_d$ \\ \hline
I & $(0, 0, 0)$ & $%
\begin{pmatrix}
\delta & \delta & \delta \\
0 & 0 & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
\varepsilon & \varepsilon & \varepsilon \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
c & c & c%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\delta & \delta & \delta \\
\varepsilon & \varepsilon & \varepsilon \\
c & c & c%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
II & $(0, 0, 1)$ & $%
\begin{pmatrix}
\delta & \delta & 0 \\
0 & 0 & \delta \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
\varepsilon & \varepsilon & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
c & c & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\delta & \delta & 0 \\
\varepsilon & \varepsilon & \delta \\
c & c & 0%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
III & $(0, 0, -3)$ & $%
\begin{pmatrix}
\delta & \delta & 0 \\
0 & 0 & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
\varepsilon & \varepsilon & 0 \\
0 & 0 & \varepsilon%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
c & c & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\delta & \delta & 0 \\
\varepsilon & \varepsilon & 0 \\
c & c & \varepsilon%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
IV & $(0, 0, -2)$ & $%
\begin{pmatrix}
\delta & \delta & 0 \\
0 & 0 & 0 \\
0 & 0 & \varepsilon%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
\varepsilon & \varepsilon & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
c & c & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\delta & \delta & 0 \\
\varepsilon & \varepsilon & 0 \\
c & c & \varepsilon%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
V & $(0, 1, 0)$ & $%
\begin{pmatrix}
\delta & 0 & \delta \\
0 & \delta & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
\varepsilon & 0 & \varepsilon \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
c & 0 & c%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\delta & 0 & \delta \\
\varepsilon & \delta & \varepsilon \\
c & 0 & c%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
VI & $(0, -3, 0)$ & $%
\begin{pmatrix}
\delta & 0 & \delta \\
0 & 0 & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
\varepsilon & 0 & \varepsilon \\
0 & \varepsilon & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
c & 0 & c%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\delta & 0 & \delta \\
\varepsilon & 0 & \varepsilon \\
c & \varepsilon & c%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
VII & $(0, -2, 0)$ & $%
\begin{pmatrix}
\delta & 0 & \delta \\
0 & 0 & 0 \\
0 & \varepsilon & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
\varepsilon & 0 & \varepsilon \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
c & 0 & c%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\delta & 0 & \delta \\
\varepsilon & 0 & \varepsilon \\
c & \varepsilon & c%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
VIII & $(1, 0, 0)$ & $%
\begin{pmatrix}
0 & \delta & \delta \\
\delta & 0 & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & \varepsilon & \varepsilon \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & c & c%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & \delta & \delta \\
\delta & \varepsilon & \varepsilon \\
0 & c & c%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
IX & $(-3, 0, 0)$ & $%
\begin{pmatrix}
0 & \delta & \delta \\
0 & 0 & 0 \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & \varepsilon & \varepsilon \\
\varepsilon & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & c & c%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & \delta & \delta \\
0 & \varepsilon & \varepsilon \\
\varepsilon & c & c%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
X & $(-2, 0, 0)$ & $%
\begin{pmatrix}
0 & \delta & \delta \\
0 & 0 & 0 \\
\varepsilon & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & \varepsilon & \varepsilon \\
0 & 0 & 0%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & c & c%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
0 & \delta & \delta \\
0 & \varepsilon & \varepsilon \\
\varepsilon & c & c%
\end{pmatrix}%
$\rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\ \hline
\end{tabular}%
\end{table}
\newpage

\section{Numerical analysis}

\label{sec:num}

In this section, we give the phenomenological predictions obtained by the
patterns listed in Table~\ref{tab:downTextures}. Note that, although these
patterns arrize directly from the chosen discrete charge configuration of
the quark fields, one may further preform a residual flavour transformation
of the right-handed down quark fields, resulting in an extra zero entry in $%
M_{d}$. Taking this into account, all the parameters in each pattern may be
uniquely expressed in terms of down quark masses and the CKM matrix elements
$V_{ij}$. This follows directly from the diagonalization equation of $M_{d}:$
\begin{equation}
V\ ^{\dagger }M_{d}\ W=diag(m_{d},m_{s},m_{b})\quad \Longrightarrow \quad
M_{d}=V\ diag(m_{d},m_{s},m_{b})\ W^{\dagger }  \label{diag}
\end{equation}%
with $V$ being the CKM mixing matrix, since $M_{u}$ is diagonal. Because of
the zero entries in $M_{d}$, it is easy to extract the right-handed
diagonalization matrix $W$, completely in terms of the down quark masses and
the $V_{ij}$. Thus, all parameters, modulo the residual transformation of
the right-handed down quark fields, are fixed, i.e., all parameters in each
pattern may be uniquely expressed in terms of down quark masses and the CKM
matrix elements $V_{ij}$, including the right-handed diagonalization matrix $%
W$ of $M_{d}$. More precisely, all matrix elements of $V$ are written in
terms of Wolfenstein real parameters $\lambda $, $A$, $\overline{\rho }$ and
$\overline{\eta }$, defined in terms of rephasing invariant quantities as
\begin{subequations}
\begin{equation}
\lambda \equiv \frac{|V_{us}|}{\sqrt{|V_{us}|^{2}+|V_{ud}|^{2}}},\qquad
A\equiv \frac{1}{\lambda }\left\vert \frac{V_{cb}}{V_{us}}\right\vert \,,
\end{equation}%
\begin{equation}
\overline{\rho }+i\,\overline{\eta }\equiv -\frac{V_{ud}^{\phantom{\ast}%
}V_{ub}^{\ast }}{V_{cd}^{\phantom{\ast}}V_{cb}^{\ast }}
\end{equation}%
and $diag(m_{d},m_{s},m_{b})$ in Eq.~\eqref{diag}
\end{subequations}
\begin{equation}
\begin{array}{l}
\sqrt{\frac{m_{d}}{m_{s}}}=\sqrt{{\frac{k_{d}}{k_s}}}\ \lambda \\
\\
\frac{m_{s}}{m_{b}}=k_{s}\ \lambda ^{2}%
\end{array}%
\quad \Longrightarrow \quad
\begin{array}{l}
m_{d}=k_{d}\ \lambda ^{4}m_{b} \\
\\
m_{s}=k_{s}\ \lambda ^{2}m_{b}%
\end{array}
\label{ks}
\end{equation}%
with phenomenologically, $k_{d}$ and $k_{s}$ being factores of order one.
Writing $W^{\dagger }$ in Eq.~\eqref{diag} as $W^{\dagger
}=(v_{1},v_{2},v_{3})$, with the $v_{i}$ vectors formed by the $i$-th column
of $W^{\dagger }$, we find e.g. for pattern II,
\begin{equation}
v_{3}=\frac{1}{n_{3}}\left(
\begin{array}{r}
\frac{m_{d}}{m_{b}}V_{11} \\
\frac{m_{s}}{m_{b}}V_{12} \\
V_{13}%
\end{array}%
\right) \times \left(
\begin{array}{r}
\frac{m_{d}}{m_{b}}V_{31} \\
\frac{m_{s}}{m_{b}}V_{32} \\
V_{33}%
\end{array}%
\right)  \label{v3}
\end{equation}%
where $n_{3}$ is the norm of the vector obtained from the external product
of the two vectors. Taking into account the extra freedom of transformation
of the right-handed fields, we may choose $M_{31}^{d}=0$, corresponding to $%
c_{1}=0$ in Table~\ref{tab:downTextures}, and we conclude that%
\begin{equation}
v_{1}=\frac{1}{n_{1}}\left(
\begin{array}{r}
\frac{m_{d}}{m_{b}}V_{31} \\
\frac{m_{s}}{m_{b}}V_{32} \\
V_{33}%
\end{array}%
\right) \times v_{3}^{\ast }  \label{v1}
\end{equation}%
Obviously, then $v_{2}=\frac{1}{n_{2}}v_{1}^{\ast }\times v_{3}^{\ast }$.
This process is replicated for all patterns. Thus, $V$ and $W$, are entirely
expressed in terms of Wolfenstein parameters and $k_{d}$ and $k_{s}$ of Eq.~%
\eqref{ks}. These two matrices will be later used to compute the patterns of
the FCNC's in Table \ref{tb:FCNCpatterns}. Indeed, in this way, we find e.g.
for pattern II, \ in leading order order,
\begin{equation}
M_{d}=m_{b}\ \left(
\begin{array}{ccc}
-k_{d}\ \lambda ^{3} & \left( \overline{\rho }-i\,\overline{\eta }\right) \
A\ \lambda ^{3} & 0 \\
-k_{d}\ \lambda ^{2} & A\ \lambda ^{2} & -k_{s}\ \lambda ^{3} \\
0 & 1 & 0%
\end{array}%
\right)  \label{mdl}
\end{equation}%
which corresponds to the expected power series where the couplings in $%
\Gamma _{1}$ to the first Higgs $\phi _{1}$ are comparatively smaller than
then couplings in $\Gamma _{2}$, and these smaller to\ the couplings in $%
\Gamma _{3}$. Similar results are obtained for all patterns in Table~\ref%
{tab:downTextures}, except for patterns IV, VII and X, where e.g. for
pattern IV, we find that the coupling in $(\Gamma _{1})_{33}$ is
proportional to$\ \lambda $, which is too large and contradicts our initial
assumption that all couplings in $\Gamma _{1}$ to the first Higgs $\phi _{1}$
must be smaller than the couplings in $\Gamma _{2}$ to the second Higgs $%
\phi _{2}$. Therefore, we exclude Patterns IV, VII and X.

We give in Table {\ref{Yukawa_example}} a numerical example of a Yukawa
coupling configuration for each pattern. We use the following quark running
masses at the electroweak scale $M_{Z}$:
\begin{subequations}
\begin{align}
m_{u}& =1.3_{-0.2}^{+0.4}\,\text{MeV},\quad m_{d}=2.7\pm 0.3\,\text{MeV}%
,\quad m_{s}=55_{-3}^{+5}\,\text{MeV}, \\
m_{c}& =0.63\pm 0.03\,\text{GeV},\quad m_{b}=2.86_{-0.04}^{+0.05}\,\text{GeV}%
,\quad m_{t}=172.6\pm 1.5\,\text{GeV}.
\end{align}%
which were obtained from a renormalisation group equation evolution at
four-loop level \cite{1674-1137-38-9-090001}, which, taking into account all
experimental constrains \cite{Charles:2015gya}, implies:
\end{subequations}
\begin{subequations}
\begin{align}
\lambda & =0.2255\pm 0.0006,\qquad A=0.818\pm 0.015, \\
\overline{\rho }& =0.124\pm 0.024,\qquad \overline{\eta }=0.354\pm 0.015.
\end{align}

\begin{table}[]
\caption{A numerical example of a Yukawa coupling configuration for each
pattern that gives the correct hierarchy among the quark masses and mixing.}{%
} {\label{Yukawa_example}}
\par
\begin{center}
\setlength{\tabcolsep}{0.5pc}
\resizebox{\textwidth}{!}{\begin{tabular}{|c|c|c|c|c|c|}
\hline
Pattern  & $v_1Y_1$ & $v_2Y_2$ & $v_3Y_3$ &  $M_d$ \\
\hline
I &
$\begin{pmatrix}
	0.00277 & 0.0124 & 0.0101\,e^{1.907 \,i} \\
	0 & 0 & 0 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0.0537 & 0.119 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0 & 0 \\
	0 & 0 & 2.86
\end{pmatrix}$
 &
$\begin{pmatrix}
	0.00277 & 0.00124 & 0.0101\,e^{1.907 \,i} \\
	0 & 0.0537 & 0.119 \\
	0 & 0 & 2.86
\end{pmatrix}$
\\
\hline
II  &
$\begin{pmatrix}
	0.0123 &  0.0101\,e^{-1.235 \,i} & 0 \\
	0 & 0 & 0.012 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0.0524 & 0.119 & 0 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0 & 0 \\
	0 & 2.86 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0.0123 &  0.0101\,e^{-1.235 \,i} & 0 \\
	0.0524 & 0.119 & 0.012 \\
	0 & 2.86 & 0
\end{pmatrix}$
\\
\hline
III  &
$\begin{pmatrix}
	0.0127 & 0.0102\, e^{-1.253 \,i}& 0 \\
	0 & 0 & 0 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0.0523 & 0.120 & 0 \\
	0 & 0 & 0.295
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0 & 0 \\
	0 & 2.844 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0.0127 & 0.0102\, e^{-1.253 \,i}  & 0 \\
	0.0523 & 0.120  & 0 \\
	0 & 2.844 & 0.295
\end{pmatrix}$
\\
\hline
V  &
$\begin{pmatrix}
	0.0127 & 0 & 0.0101\,e^{-1.234 \,i}\\
	0 & 0.0117  & 0 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0.0524 & 0 & 0.112 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0 & 0 \\
	0 & 0 & 2.86
\end{pmatrix}$
 &
$\begin{pmatrix}
	0.0127 & 0 &0.0101\,e^{-1.234 \,i} \\
	0.0524  & 0.0117& 0.112 \\
	0 & 0 & 2.86
\end{pmatrix}$
\\
\hline
VI &
$\begin{pmatrix}
	0.0127 & 0 & 0.0102\,e^{-1.253 \,i} \\
	0 & 0 & 0 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0.0523 & 0 & 0.120 \\
	0 & 0.295  & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0 & 0 \\
	0 & 0 & 2.844
\end{pmatrix}$
 &
$\begin{pmatrix}
	0.0127 & 0 & 0.0102\,e^{-1.253 \,i} \\
0.0523 & 0 & 0.120 \\
	0 & 0.295 & 2.844
\end{pmatrix}$
\\
\hline
VIII &
$\begin{pmatrix}
	0 &0.0127 & 0.0102\,e^{1.907 \,i} \\
	0.0117 & 0 & 0 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0.0524 & 0.119 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0 & 0 \\
	0 & 0 & 2.86
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 &0.0127 & 0.0102\,e^{1.907 \,i} \\
	0.0117 & 0.0524 & 0.119 \\
	0 & 0 & 2.86
\end{pmatrix}$
\\
\hline
IX &
$\begin{pmatrix}
	0 & 0.0127 & 0.0101\,e^{-1.253 \,i} \\
	0 & 0 & 0 \\
	0 & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0.0523& 0.120 \\
	0.295  & 0 & 0
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0 & 0 \\
	0 & 0 & 0 \\
	0 & 0 &2.844
\end{pmatrix}$
 &
$\begin{pmatrix}
	0 & 0.0127 & 0.0101\,e^{-1.253 \,i} \\
	0 & 0.0523& 0.120 \\
	0.295& 0 &2.844
\end{pmatrix}$
\\
\hline
\end{tabular}}
\end{center}
\end{table}

\section{Predictions of flavour changing neutral currents}\label{sec:fcnc}

In the SM, flavour changing neutral currents (FCNC) are forbidden at tree
level, both in the gauge and the Higgs sectors. However, by extending the SM
field content, one obtains Higgs Flavour Violating Neutral Couplings \cite%
{Branco:2011iw}. In terms of the quark mass eigenstates, the Yukawa couplings
to the Higgs neutral fields are:

\end{subequations}
\begin{equation}
\begin{aligned} -\mathcal{L}_{\text{Neutral Yukawa}}= &\frac{H_0}{v}\left(
\overline{d_L}\, D_d \, d_R + \overline{u_L}\, D_u \, u_R \right) +
\frac{1}{v'} \overline{d_L} \, N^d_{1}\, \left( R_1 + i\, I_1 \right) \, d_R
\\ & + \frac{1}{v'} \overline{u_L} \, N^u_{1} \, \left( R_1 - i\, I_1
\right) \, u_R + \frac{1}{v''} \overline{d_L} \, N^d_{2}\, \left( R_2 + i\,
I_2 \right) \, d_R \\ &+ \frac{1}{v''} \overline{u_L} \, N^u_{2} \, \left(
R_2 - i\, I_2 \right) \, u_R + h.c. \end{aligned}
\end{equation}%
where the $N_{i}^{u,d}$ are the matrices which give the strength and the
flavour structure of the FCNC,
\begin{subequations}
\begin{align}  \label{eq:FCNC}
& N_{1}^{d}=\frac{1}{\sqrt{2}}V^{\dagger }\,\left( v_{2}\Gamma
_{1}-v_{1}e^{i\,\alpha _{2}}\Gamma _{2}\right) \,W, \\
& N_{2}^{d}=\frac{1}{\sqrt{2}}V^{\dagger }\left( v_{1}\Gamma
_{1}+v_{2}e^{i\,\alpha _{2}}\Gamma _{2}-\frac{v_{1}^{2}+v_{2}^{2}}{v_{3}}%
e^{i\,\alpha _{3}}\Gamma _{3}\right) \,W, \\
& N_{1}^{u}=\frac{1}{\sqrt{2}}\left( v_{2}\Omega _{1}-v_{1}e^{-i\,\alpha
_{2}}\Omega _{2}\right) , \\
& N_{2}^{u}=\frac{1}{\sqrt{2}}\left( v_{1}\Omega _{1}+v_{2}e^{-i\,\alpha
_{2}}\Omega _{2}-\frac{v_{1}^{2}+v_{2}^{2}}{v_{3}}e^{-i\,\alpha _{3}}\Omega
_{3}\right) .
\end{align}%
Since in our case the $N_{i}^{u}$ are diagonal, there are no flavour
violating terms in the up-sector. Therefore, the analysis of the FCNC
resumes only to the down-quark sector. One can use the equations of the mass
matrices presented in Eq.~\eqref{eq:mass} to simplify the Higgs mediated
FCNC matrices for the down-sector:
\end{subequations}
\begin{subequations}
\label{eq:simplefcnc}
\begin{align}
N_{1}^{d}& =\frac{v_{2}}{v_{1}}D_{d}-\frac{v_{2}}{\sqrt{2}}\left( \frac{v_{2}%
}{v_{1}}+\frac{v_{1}}{v_{2}}\right) e^{i\alpha _{2}}\,V^{\dagger }\,\Gamma
_{2}\,W-\frac{v_{2}\,v_{3}}{v_{1}\sqrt{2}}e^{i\alpha _{3}}V^{\dagger
}\,\,\Gamma _{3}\,W \\[2mm]
N_{2}^{d}& =D_{d}-\frac{v^{2}}{v_{3}\sqrt{2}}e^{i\alpha _{3}}\,V^{\dagger
}\,\Gamma _{3}\,W
\end{align}

In order to satisfy experimental constraints arising from $K^{0}-\overline{%
K^{0}}$, $B^{0}-\overline{B^{0}}$ and $D^{0}-\overline{D^{0}}$, the
off-diagonal elements of the Yukawa interactions $N_{1}^{d}$ and $N_{2}^{d}$
must be highly suppressed \cite{Botella:2014ska} \cite{AndreasCrivellin2013}%
. For each of our 10 solutions in Table~\ref{tab:downTextures}, we summarize
in Table {\ref{tb:FCNCpatterns}} all FCNC patterns, for each solution, and
for $v_{1}=v_{2}=v_{3}$ and $\alpha _{2}=\alpha _{3}=0$. These patterns are
of the BGL type, since in Eq.~\eqref{eq:simplefcnc} all matrices can be
expressed in terms of the CKM mixing matrix elements and the down quark
masses. As explained, to obtain these patterns, we express the CKM matrix $V$
and the matrix $W$ in terms of Wolfenstein parameters.

\begin{table}[]
\caption{For all allowed patterns, we find that the matrices $N^d_1-D_d$ and
$N^d_2$ are proportional to the following paterns, where $\protect\lambda$
is the Cabibbo angle.}{\label{tb:FCNCpatterns}}
{\ } \setlength{\tabcolsep}{14pt} \centering
\begin{tabular}{|c|c|c|}
\hline
Pattern & $(N^d_1-D_d)\sim$ & $N^d_{2}\sim$ \\ \hline
I & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^5 & \lambda^2 & \lambda^2 \\
\lambda^7 & \lambda^4 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^7 & \lambda^3 \\
\lambda^9 & \lambda^2 & \lambda^2 \\
\lambda^7 & \lambda^4 & 1%
\end{pmatrix}
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
II & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda^5 & \lambda^4 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^7 & \lambda^3 \\
\lambda^9 & \lambda^2 & \lambda^2 \\
\lambda^7 & \lambda^4 & 1%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
III & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^5 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
IV & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^5 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$\rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
V & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda^5 & \lambda^4 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^7 & \lambda^3 \\
\lambda^7 & \lambda^2 & \lambda^2 \\
\lambda^5 & \lambda^4 & 1%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
VI & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^5 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
VII & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^5 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
VIII & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda^5 & \lambda^4 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^7 & \lambda^3 \\
\lambda^7 & \lambda^2 & \lambda^2 \\
\lambda^5 & \lambda^4 & 1%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
IX & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^5 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ \rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\
X & $%
\begin{pmatrix}
\lambda^4 & \lambda^3 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$ & $%
\begin{pmatrix}
\lambda^4 & \lambda^5 & \lambda^3 \\
\lambda^3 & \lambda^2 & \lambda^2 \\
\lambda & \lambda^2 & 1%
\end{pmatrix}%
$\rule{0pt}{0.9cm}\rule[-0.9cm]{0pt}{0pt} \\ \hline
\end{tabular}%
\end{table}

The tree level Higgs mediated $\Delta S=2$ amplitude must be suppressed.
This may allways be achieved if one chooses the masses of the flavour
violating neutral Higgs scalars sufficiently heavy. However, from the
experimental point of view, it would be interesting to have these masses as
low as possible. Therefore, we also estimate the lower bound of these
masses, by considering the contribution to $B^{0}-\overline{B^{0}}$ mixing.
We choose this mixing, since for our patterns, the $(3,1)$ entry of the
matrix $N_{1}^{d}$ is the less suppressed in certain cases and would require
very heavy flavour violating neutral Higgses. The relevant quantity is the
off-diagonal matrix element $M_{12}$, which connects the B meson with the
corresponding antimeson. This matrix element, $M_{12}^{NP}$, receives
contributions \cite{Botella:2014ska} both from a SM box diagram and a
tree-level diagram involving the FCNC:
\end{subequations}
\begin{equation}
M_{12}=M_{12}^{SM}+M_{12}^{NP},
\end{equation}%
where the New Physics (NP) short distance tree level contribution to the meson-antimeson
contribution is:
\begin{equation}
\begin{aligned} M_{12}^{NP}=& \sum_i^2 {\frac{f_B^2 \, m_B}{ 96\, v^2
m^2_{R_i}} } \left\{ \left[ \left(1+ \left( \frac{m_B}{m_d+m_b} \right)^2
\right)\left(a^R_i\right)_{12} \right] - \left[ \left(1+ 11 \left(
\frac{m_B}{m_d+m_b} \right)^2 \right] \right)\left(b^R_i\right)_{12}
\right\} \\ &+\sum_i^2 {\frac{f_B^2 \, m_B}{ 96\, v^2 m^2_{I_i}} } \left\{
\left[ \left( 1+ \left( \frac{m_B}{m_d+m_b}
\right)^2\right)\left(a^I_i\right)_{12}\right] - \left[ \left(1+ 11 \left(
\frac{m_B}{m_d+m_b} \right)^2\right]\right) \left(b^I_i\right)_{12} \right\}
\end{aligned}
\end{equation}%
with $v^{2}=v_{1}^{2}+v_{2}^{2}+v_{2}^{2}$ and
\begin{equation}
\begin{array}{l}
\left( a_{i}^{R}\right) _{12}=\left[ \left( N_{i}^{d}\right) _{31}^{\ast
}+\left( N_{i}^{d}\right) _{13}\right] ^{2} \\
\left( a_{i}^{I}\right) _{12}=-\left[ \left( N_{i}^{d}\right) _{31}^{\ast
}-\left( N_{i}^{d}\right) _{13}\right] ^{2}%
\end{array}%
~,\qquad
\begin{array}{l}
\left( b_{i}^{R}\right) _{12}=\left[ \left( N_{i}^{d}\right) _{31}^{\ast
}-\left( N_{i}^{d}\right) _{13}\right] ^{2} \\
\left( b_{i}^{I}\right) _{12}=-\left[ \left( N_{i}^{d}\right) _{31}^{\ast
}+\left( N_{i}^{d}\right) _{13}\right] ^{2}%
\end{array}%
~,\qquad i=1,2  \label{ab}
\end{equation}%
In order to obtain a conservative measure, we have tentatively expanded the
original expression in \cite{Botella:2014ska} and, for the three Higgs case,
included all neutral Higgs mass eigenstates.

Adopting as input values the PDG experimental determinations of $f_{B}$, $%
m_{B}$ and $\Delta \,m_{B}$ and considering a common VEV for all Higgs
doublets, we impose the inequality $M_{12}^{NP}<\Delta m_{B}$. The following
plots show an estimate of the lower bound for the flavour-violating Higgs
masses for two different patterns. We plot two masses chosen from the set $%
\left( m_{1}^{R},m_{2}^{R},m_{1}^{I},m_{2}^{I}\right) $, while the other two
are varied over a wide range. In Fig. 1, we illustrate these lower bounds
for Pattern III, which are restricted by the $(3,1)$ entry of $N_{1}^{d}$
matrix and suppressed by a factor of $\lambda $. For Pattern VIII, in Fig. 2
we find the flavour violating neutral Higgs to be much lighter and possibly
accessible at LHC.

\section{Conclusions}

\label{sec:conc}

We have presented a model based on the SM with 3 Higgs and an additional
flavour discrete symmetry. We have shown that there exist flavour discrete
symmetry configurations which lead to the alignment of the quark sectors.\
By allowing each scalar field to couple to each quark generation with a
distinctive scale, one obtains the quark mass hierarchy, and although this
hierarchy does not arise from the symmetry, the effect of both is such that
the CKM matrix is near to the identity and has the correct overall
phenomenological features. In this context, we have obtained 7 solutions
fulfilling these requirements, with the additional constraint of the up
quark mass matrix being diagonal and real.

We have also verified if accidental $U(1)$ symmetries may appear in the
Yukawa sector or in the potential, particularly the case where a continuous
accidental $U(1)$ symmetry could arise, once the $Z_{7}$ is imposed at the
Lagrangian level. This was indeed the case, however we shown that the
anomaly-free conditions of global symmetries are violated. Thus, the global $%
U(1)_{X}$ symmetry is anomalous and therefore only the discrete symmetry $%
Z_{7}$ persists.

As in this model new Higgs doublets are added, one expects large FCNC
effects, already present at tree level. However, such effects have not been
experimentally observed. We show that, for certain of our specific
implementations of the flavour symmetry, it is possible to suppress the FCNC
effects and to ensure that the flavour violating neutral Higgs are
light enough to be detectable at LHC. Indeed, in this respect, our model
is a generalization of the BGL models for 3HDM, since the FCNC flavour
structure is entirely determined by CKM.

\begin{figure}[h!]
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=1.\linewidth]{P3-a.png}
  \caption{Estimate of the lower bound for the flavour-violating Higgs masses for $R_1$ and $I_1$.}
  \label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=1.\linewidth]{P3-b.png}
  \caption{Estimate of the lower bound for the flavour-violating Higgs masses for $R_2$ and $I_2$.}
  \label{fig:sub2}
\end{subfigure}
\caption{Lower bound for the flavour-violating Higgs masses for case III. }
\label{fig:test}
\end{figure}

\begin{figure}[h!]
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=1.\linewidth]{P8-a-good.png}
  \caption{Estimate of the lower bound for the flavour-violating Higgs masses for $R_1$ and $I_1$.}
  \label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=1.\linewidth]{P8-b-good.png}
  \caption{Estimate of the lower bound for the flavour-violating Higgs masses for $R_2$ and $I_2$.}
  \label{fig:sub2}
\end{subfigure}
\caption{Lower bound for the flavour-violating Higgs masses for case VIII. }
\label{fig:test1}
\end{figure}
\acknowledgments

This work is partially supported by Funda\c{c}\~{a}o para a Ci\^{e}ncia e a
Tecnologia (FCT, Portugal) through the projects CERN/FP/123580/2011,
PTDC/FIS-NUC/0548/2012, EXPL/FIS-NUC/0460/2013, and CFTP-FCT Unit 777
(PEst-OE/FIS/UI0777/2013) which are partially funded through POCTI (FEDER),
COMPETE, QREN and EU. The work of D.E.C. is also supported by Associa\c c\~
ao do Instituto Superior T\'ecnico para a Investiga\c c\~ao e
Desenvolvimento (IST-ID). N.R.A is supported by European Union Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 674896. N.R.A is grateful to CFTP for the hospitality during
his stay in Lisbon.

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