text:
\section{Introduction}

Control systems have been experiencing dramatic changes where advanced communication, computation and sensing technologies are integrated to improve system performance and produce new functions. This class of new-generation control systems is referred to as cyber-physical systems, e.g., mobile robotic networks, smart grids, smart buildings and intelligent transportation systems. The entities of cyber-physical systems are spatially distributed and thus distributed coordination becomes necessary to achieve network-wide goals. Substantial efforts have been spent on the synthesis, analysis and implementation of distributed algorithms in the last decades \cite{WR-YC:2011,fb-jc-sm:09,MZ-SM-distributed-book:2015,MM-ME:10}.

(Static) game theory provides a holistic and rigorous mathematical framework to reason about the behaviors of multiple self-interested agents; see \cite{Basar.Olsder:82,YS-KLB:2008} for instance. This paper investigates a class of convex games, which were first introduced in \cite{Arrow.Debreu:54}. There have been lots of efforts on studying the existence and structural properties of Nash equilibria in, e.g., \cite{Rosen:65} and the recent survey paper \cite{Facchinei.Kanzow:07}. A number of algorithms have been proposed to compute Nash equilibria of convex games, including, to name a few, ODE-based methods \cite{Rosen:65}, nonlinear Gauss-Seidel-type approaches \cite{Pang.Scutari.Facchinei.Wang:08}, iterative primal-dual Tikhonov schemes \cite{Yin.Shanbhag.Mehta:11}, consensus-based methods \cite{JK-AN-UVS:2016}, best-response dynamics \cite{PE:10} and finite difference approximation \cite{MZ-EF:Auto14}.

In many cyber-physical systems, the influence of agents' decisions on objective functions is restricted by inherent physical dynamics. However, none of the papers aforementioned explicitly considers physical dynamics. Recent papers \cite{Stankovic.Johansson.Stipanovic:11,PF-MK-TB:11} leverage extremum seeking to solve a class of convex games where agents' physical dynamics are non-trivial, and find applications in markets and mobile sensors, respectively. In paper \cite{Stankovic.Johansson.Stipanovic:11}, the agents' actions are filtered through stable linear time-invariant filters, e.g., single integrators and double integrators, before affecting objective functions. In paper \cite{PF-MK-TB:11}, the agents' actions act as inputs to a stable nonlinear dynamic system whose outputs are the agents' costs. The dynamic mapping is motivated by oligopoly games in microeconomics. In \cite{Stankovic.Johansson.Stipanovic:11,PF-MK-TB:11}, no constraint is included in the games. In the current paper, we are interested in the scenarios where physical systems consist of dynamically coupled, unstable and potentially nonlinear subsystems and underlying games include constraint sets as well as inequality and equality constraints. The scenarios are motivated by an optimal power flow problem and a multizone building temperature regulation problem in Section \ref{simulation section} and many others. The algorithms in \cite{Stankovic.Johansson.Stipanovic:11,PF-MK-TB:11} are inapplicable to this class of problems.

\emph{Contribution.} In this paper, we consider the synthesis of distributed algorithms under which dynamically coupled physical systems are regulated to a Nash equilibrium (NE) of a constrained convex game. We investigate two complementary cases where the physical dynamics are linear and nonlinear, respectively.

For linear dynamics with controllable subsystems, we propose an indirect approach for the distributed algorithm design. We synthesize distributed update rules for auxiliary variables such that they asymptotically converge to an NE of the game of interest, but their evolution is not restricted by the inherent physical dynamics. The distributed decision-making update rule extends the projected discontinuous dynamics in \cite{AC-EM-JC:2016} from convex optimization to convex games. We then design a no-communication distributed control law such that the states and inputs of the physical dynamics asymptotically keep track of the auxiliary variables and thus asymptotically converge to the same NE of the game. The distributed control law is new. The designed algorithm is verified by an optimal power flow problem.

For nonlinear dynamics in the strict-feedback form,
we develop a direct approach for the distributed algorithm design without introducing auxiliary variables. In particular, we perform a sequence of coordinate transformations such that each subsystem is converted into the one where its first state is driven by a primal-dual dynamics and it is followed by a chain of integrators. We then design a distributed control law such that the integrators are stabilized and the primal-dual dynamics asymptotically converges to an NE of the game of interest. The algorithm and its analysis developed for this case are novel. The proposed algorithm is verified by a multi-zone building temperature regulation problem.

Preliminary results of this paper were published in \cite{YL-MZ:NECSYS2015Privacy}, where the case of nonlinear dynamics was not investigated, no case study was included, and the proofs were omitted.

\emph{Notations and notions:} In this paper, the following notations and notions are used. Let $(x_i)_{i=1}^N$ denote $[x_1^T,\cdots,x_N^T]^T$ and $x_{-i}$ denote $(x_j)_{j\neq i}=[x_1^T,\cdots,x_{i-1}^T,x_{i+1}^T,\cdots,x_N^T]^T$. Let $\|  \cdot  \|_1$ and $\|  \cdot  \|$ denote the 1-norm and 2-norm of a vector or a matrix, respectively. Denote the zero column vector of size $n$ by $0_n$ and the identity matrix of size $n\times n$ by $I_n$. Let $\mathbb{R}_+^n$ denote the set of nonnegative real column vectors of size $n$.
Denote by $\mathrm{diag}(A_i)_{i=1}^M$ the block-diagonal matrix composed by sub-matrices $A_1,\cdots,A_M$, such that the $j$-th diagonal block is $A_j$ for each $j=1,\cdots,M$, and all the off-diagonal blocks are zero matrices. Given a function $f:\mathbb{R}^n\to\mathbb{R}$ of $x=(x_i)_{i=1}^n\in\mathbb{R}^n$, for a nonnegative integer $\ell$, we say that $f$ is of class $C^\ell$ if for all possible combinations of $\ell_1,\ell_2,\cdots,\ell_n$, where each of $\ell_1,\ell_2,\cdots,\ell_n$ is an integer between $0$ and $\ell$ such that $\ell_1+\ell_2+\cdots+\ell_n=\ell$, the partial derivative $\frac{\partial^\ell f}{\partial x_{1}^{\ell_1}\partial x_{2}^{\ell_2}\cdots\partial x_{n}^{\ell_n}}$ exists and is continuous. Given a function $V:\mathbb{R}^n\to\mathbb{R}$ and $\delta>0$, denote the sublevel set of $V$ by $V^{-1}(\leq\delta)=\{x\in\mathbb{R}^n:V(x)\leq\delta\}$.

Let $K\subseteq\mathbb{R}^n$ be a closed and convex set. Denote the boundary, interior and closure of $K$ by ${\rm bd}(K)$, ${\rm int}(K)$ and ${\rm cl}(K)$, respectively. Given $x\in{\rm bd}(K)$, define the normal cone of $K$ at $x$
by $N_K(x)=\{\gamma:\|\gamma\|=1,\,{\rm and}\,\gamma^T(x-y)\leq0,\,\forall y\in K\}$. Given $x\in\mathbb{R}^n$, the point projection of $x$ onto $K$ is ${\rm Proj}_K(x)={\rm argmin}_{z\in K}\|z-x\|$. Given $x\in K$ and $v\in\mathbb{R}^n$, the vector projection of $v$ at $x$ with respect to $K$ is $\Pi_K(x,v)=\lim_{\delta\to0^+}\frac{{\rm Proj}_K(x+\delta v)-x}{\delta}$.
The following lemma is adopted from Lemma 2.1 of \cite{AN-DZ:1996}.

\begin{lemma} \begin{rm}
\label{lemma vector projection}
If $x\in {\rm int}(K)$, then $\Pi_K(x,v)=v$; if $x\in{\rm bd}(K)$, then $\Pi_K(x,v)=v+\beta(x)\gamma^*(x)$, where $\gamma^*(x)={\rm argmin}_{\gamma\in N_K(x)}v^T\gamma$ and $\beta(x)=\max\{0,-v^T\gamma^*(x)\}$; and $\|\Pi_K(x,v)\|\leq\|v\|$ for any $x\in K$ and any $v\in\mathbb{R}^n$.
 \end{rm} \hfill\rule{2mm}{2mm

We next review some necessary concepts and results on projected discontinuous dynamical systems. Given a set $K\subseteq\mathbb{R}^n$ and a map $F:K\to\mathbb{R}^n$, consider the projected differential equation
\begin{align}
\label{differential inclusion}
\dot x=\Pi_K(x,F(x)).
\end{align}

Given $x^0\in K$, a solution of \eqref{differential inclusion} on the interval $[0,T)\subseteq\mathbb{R}$ is a map $\Gamma:[0,T)\to K$ with $\Gamma(0)=x^0$ that is absolutely continuous on $[0,T)$ and satisfies $\dot \Gamma(t)=\Pi_K(\Gamma(t),F(\Gamma(t)))$ almost everywhere on $[0,T)$.
A set $S\subseteq\mathbb{R}^n$ is invariant under \eqref{differential inclusion} if every solution starting from any point in $S$ remains in $S$. For a solution $\Gamma$ of \eqref{differential inclusion} defined on $[0,\infty)$, the omega-limit set $\Omega(\Gamma)$ is defined by $\Omega(\Gamma)=\{y\in\mathbb{R}^n:\exists\{t_k\}_{k=1}^\infty\subseteq[0,\infty)\;{\rm with}\;\lim_{k\to\infty}t_k=\infty\;{\rm and}\;\lim_{k\to\infty}\Gamma(t_k)=y\}$. Given a continuously differentiable function $V:K\to\mathbb{R}$, the Lie derivative of $V$ along \eqref{differential inclusion} at $x\in K$ is $\mathcal{L}_{K,F}V(x)=\nabla V(x)^T\Pi_K(x,F(x))$. The next result adopted from \cite{AB-FC:2006} provides an invariance principle for \eqref{differential inclusion}.

\begin{lemma} \begin{rm}
\label{lemma invariance}
Let $S\subseteq\mathbb{R}^n$ be compact and invariant under \eqref{differential inclusion}. Assume that for any $x^0\in S$ there exists a unique solution of \eqref{differential inclusion} starting at $x^0$ and that its omega-limit set is invariant under \eqref{differential inclusion}. Suppose that there exists a continuously differentiable function $V:K\to\mathbb{R}$ such that $V$ is positive definite and $\mathcal{L}_{K,F}V(x)\leq0$ for all $x\in S$. Then any solution of \eqref{differential inclusion} starting from $S$ converges to the largest invariant set contained in ${\rm cl}(\{x\in S:\mathcal{L}_{K,F}V(x)=0\})$.
 \end{rm} \hfill\rule{2mm}{2mm

We adopt the following notions of strictly and strongly monotone maps from \cite{FF-JSP:03}.

\begin{definition}\begin{rm}
A map $\Phi:\mathbb{R}^r\to\mathbb{R}^r$ is strictly monotone on $W\subseteq\mathbb{R}^r$ if $(w-w')^T(\Phi(w)-\Phi(w'))>0$ for all $w,w'\in W$ with $w\neq w'$. The map $\Phi$ is strongly monotone on $W$ if there exists $M>0$ such that $(w-w')^T(\Phi(w)-\Phi(w'))\geq M\|w-w'\|^2$ for all $w, w'\in W$.
\label{def: strict monotone}
 \end{rm} \hfill\rule{2mm}{2mm

\section{Problem statement\label{section problem statement}}

This section provides the general formulation of the distributed control problem for dynamically coupled networks concerned in this paper.

\subsection{Physical dynamics\label{problem formulation}}

Consider a set of agents $\mathcal{V} = \{ 1, \cdots ,N\}$.
The inherent physical dynamics is given by
\begin{align}
\label{physical dynamic}
\dot x_i (t) = \Xi_i(x(t),u_i(t))
\end{align}
where $x_i(t)\in \mathbb{R}^{n_i}$ and $u_i(t)\in\mathbb{R}$ are the state and control of agent $i$ at time $t$, respectively, $\Xi_i:\mathbb{R}^{n+1}\to\mathbb{R}^{n_i}$ with $n=\sum\nolimits_{i=1}^N{n_i}$. Let $x=(x_i)_{i=1}^N$ and $u=(u_i)_{i=1}^N$.

\subsection{Game formulation}

The control objective is formulated by a generalized Nash equilibrium problem. For each $i\in\mathcal{V}$, denote $w_i=[x_i^T,u_i]^T$ and $r_i=n_i+1$. Let $w=(w_i)_{i=1}^N$ and $r=\sum_{i=1}^Nr_i$. For each $i\in\mathcal{V}$, $w_i$ is agent $i$'s decision variable and is subject to a local constraint $W_i\subseteq\mathbb{R}^{r_i}$.
Let $W_{-i}=\prod_{j\neq i}W_j$ for each $i\in\mathcal{V}$ and let $W=\prod_{i=1}^NW_i$. Given any $w_{-i}\in W_{-i}$, the feasible set of $w_i$ is depicted by a set-valued map $\mathcal{D}_i: W_{-i}\to 2^{W_i}$ defined as ${\mathcal{D}_i}({w_{ - i}}) = \{ w_i\in W_i:H(w)=0_m,\,G(w)\leq0_l\}$, where $H=(H_j)_{j=1}^m:W\to\mathbb{R}^m$ and $G=(G_j)_{j=1}^l:W\to\mathbb{R}^l$ are coupled equality and inequality constraints, respectively, that are shared by all the agents. The abstracted constraints $H(w)=0_m$ and $G(w)\leq0_l$ are application-specific and may include constraints induced by economic concerns and physical dynamics. For each $i\in\mathcal{V}$, given any $w_{-i}\in W_{-i}$, agent $i$ aims to solve the optimization problem
\begin{align}
\label{general OP}
\mathop {\min }\nolimits_{w_i \in {\mathcal{D}_i}({w_{ - i}})} {f_i}(w_i,w_{ - i})
\end{align}
where $f_i:W\to\mathbb{R}$ is the objective function of agent $i$.

Let $F=(f_i)_{i=1}^N$, $\mathcal{D}(w)=\prod_{i=1}^N\mathcal{D}_i(w_{-i})$ and $\mathcal{\hat D}=\{w\in W:H(w)=0_m,\,G(w)\leq0_l\}$. Notice that $\mathcal{D}(\cdot)$ is a set-valued map while $\mathcal{\hat D}$ is a constant set that represents the overall feasible set of $w$. The collection of \eqref{general OP} for all $i\in\mathcal{V}$ is a generalized Nash equilibrium problem with shared constraints \cite{Facchinei.Kanzow:07}, denoted by GNEP$(\mathcal{D},F)$. Its solution concept is given by the notion of Nash equilibrium (NE), defined next.

\begin{definition}\begin{rm}
A joint state $\tilde w=(\tilde w_i)_{i=1}^N\in W$ is an NE of the GNEP$(\mathcal{D},F)$ if, for any $i\in\mathcal{V}$, $\tilde w_i \in \mathcal{D}_i(\tilde w_{-i})$ and ${f_i}(\tilde w) \leq {f_i}({w_i},\tilde w_{ - i})$ for all ${w_i}\in \mathcal{D}_i(\tilde w_{-i})$.
\label{def: constrained game with dynamics NE definition}
 \end{rm} \hfill\rule{2mm}{2mm

\begin{remark} \begin{rm}
The following separable optimization problem is a special case of the above GNEP: $\min_{x\in X}\sum\nolimits_{i =1}^N {{f_i}({x_i})}$, s.t. $H(x) = {0_m}$ and $G(x) \le {0_l}$,
Since the objective function is separable, this problem is equivalent to the collection of optimization problems: given $x_{-i}\in X_{-i}$, agent $i$ minimizes $f_i(x_i)$ over $x_i\in X_i$ with shared constraints $H(x)= 0_m$ and $G(x)\le0_l$.
\label{applicability to separable distributed optimization}
 \end{rm} \hfill\rule{2mm}{2mm

Let $\nabla F(w) = (\nabla_{w_i}f_i(w))_{i=1}^N$, $\nabla H(w)=(\nabla_{w}H_j(w))_{j=1}^m$ and $\nabla G(w)=(\nabla_{w}G_j(w))_{j=1}^l$. The following assumptions on the GNEP hold throughout the paper.

\begin{assumption} \begin{rm}
$\forall j=1,\cdots,m$, $H_{j}(\cdot)$ is affine in $w\in W$; $\forall j=1,\cdots,l$, $G_{j}(\cdot)$ is of class $C^1$ and convex in $w\in W$; $\forall i\in\mathcal{V}$, $\forall w_{-i}\in W_{-i}$, $f_i(\cdot,w_{-i})$ is of class $C^1$ and convex in $w_i\in W_i$; $\nabla G$ and $\nabla F$ are locally Lipschitz on $W$.
\label{asm: convexity}
\end{rm} \hfill\rule{2mm}{2mm

\begin{assumption} \begin{rm}
$\forall i\in\mathcal{V}$, $W_i$ is closed and convex.
\label{asm: convex feasible set}
\end{rm} \hfill\rule{2mm}{2mm

\begin{assumption} \begin{rm}
The Slater condition holds for $\mathcal{\hat D}$, i.e., $\exists\bar w\in W$ such that $H(\bar w)=0_m$ and $G(\bar w)<0_l$.
\label{asm: Slater condition}
\end{rm} \hfill\rule{2mm}{2mm

\begin{assumption} \begin{rm}
$\nabla F$ is strictly monotone over $W$.
\label{asm: strict monotone}
\end{rm} \hfill\rule{2mm}{2mm

Assumptions \ref{asm: convexity}--\ref{asm: strict monotone} are standard in the literature on NE convergence analysis, e.g., \cite{MZ-EF:Auto14,UR-UVS:2011,SL-AN:2013,GS-FF-JSP:2014}.

\subsection{Communication graph}

Denote by $\mathcal{G}=(\mathcal{V},\mathcal{E})$ the communication graph of the agents, where $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ is the set of communication links such that $(i,j)\in\mathcal{E}$ if and only if agent $i$ can receive messages from agent $j$. Throughout the paper, we assume that $\mathcal{G}$ is fixed, undirected and connected. We also assume that $\mathcal{G}$ includes the dependency graph defined by problem \eqref{general OP} as a subgraph, i.e., for each $i\in\mathcal{V}$, if agent $i$'s optimization problem \eqref{general OP} depends on an entry of $w_j$, then $(i,j)\in\mathcal{E}$. Denote by $\mathcal{N}_i\subseteq\mathcal{V}$ the set of neighbors of agent $i$ in $\mathcal{G}$, i.e., $\mathcal{N}_i=\{j\in\mathcal{V}\setminus\{i\}:(i,j)\in\mathcal{E}\}$. Denote by ${\rm dm}(\mathcal{G})$ the diameter of $\mathcal{G}$.

\subsection{From GNEP to variational inequality}

GNEPs with shared constraints are in general difficult to solve, but certain types of solutions can be calculated relatively easily by using the variational inequality (VI) approach \cite{GS-ea:10}. This subsection reviews relevant results of VI in the literature. We refer to \cite{FF-JSP:03,PE:10} for a comprehensive discussion on VI.

Given a set $\mathcal{Q}\subseteq \mathbb{R}^r$ and a map $\Phi:\mathcal{Q}\to\mathbb{R}^r$, the generalized variational inequality, denoted by GVI$(\mathcal{Q},\Phi)$, is the problem of finding $\tilde w\in\mathcal{Q}$ such that $(w-\tilde w)^T\Phi(\tilde w)\geq0$ for all $w\in\mathcal{Q}$.

We recall that if Assumptions \ref{asm: convexity} and \ref{asm: convex feasible set} hold, then any solution of the GVI$(\mathcal{\hat D},\nabla F)$, if exists, is an NE of the GNEP$(\mathcal{D},F)$, but not vice versa (\cite{PE:10}, Proposition 12.4). An NE of the GNEP$(\mathcal{D},F)$ that is also a solution of the GVI$(\mathcal{\hat D},\nabla F)$ is termed as a variational equilibrium (VE). The following assumption has been used in many references that studied convergence algorithms for convex games \cite{AK-UVS:2012,JK-AN-UVS:2013}.

\begin{assumption} \begin{rm}
The GNEP$(\mathcal{D},F)$ has a VE.
\label{asm: VE nonempty}
\end{rm} \hfill\rule{2mm}{2mm

We refer to \cite{PE:10} for a thorough review of VE existence. With Assumptions \ref{asm: convexity}, \ref{asm: convex feasible set} and \ref{asm: strict monotone}, the GNEP$(\mathcal{D},F)$ has at most one VE (\cite{PE:10}, Proposition 12.9). Together with Assumption \ref{asm: VE nonempty}, the GNEP$(\mathcal{D},F)$ has a unique VE.

\subsection{Objective\label{section objective}}

This paper aims to design distributed algorithms such that $(x(t),u(t))$ of system \eqref{physical dynamic} globally asymptotically converge to the unique VE of the GNEP$(\mathcal{D},F)$.

\section{Case I: Linear dynamics\label{section linear dynamics}}

In this section, we consider the case where system \eqref{physical dynamic} is in the following linear form
\begin{align}
\label{linear physical dynamic}
\dot x_i (t) = A_{ii} x_i (t) + \sum\nolimits_{j \ne i} {{A_{ij}}{x_j}(t)} + B_i u_i (t)
\end{align}
where $A_{ij} \in \mathbb{R}^{n_i \times n_j}$ for each $j \in \mathcal{V}$ and $B_i\in \mathbb{R}^{n_i}$. If \eqref{linear physical dynamic} does not depend on some $x_j$, then $A_{ij}$ is a zero matrix. Denote $A_i=[A_{i1},\cdots,A_{iN}]$, $A=(A_i)_{i=1}^N$ and $B=\mathrm{diag}(B_i)_{i=1}^N$. The following assumption is adopted in this section.

\begin{assumption} \begin{rm}
$\forall i\in\mathcal{V}$, $(A_{ii}, B_i)$ is controllable.
\label{asm: CCF}
\end{rm} \hfill\rule{2mm}{2mm

\subsection{Algorithm design\label{linear dynamic properties}}

In this section, we achieve the objective claimed in Section \ref{section objective} by an indirect approach. Roughly speaking: 1) we design an update rule for auxiliary variables such that they globally asymptotically converge to the VE, however, their evolution is not restricted by \eqref{linear physical dynamic}; 2) for the physical variables $(x,u)$, we design a distributed control law for $u$ such that, governed by \eqref{linear physical dynamic}, $(x,u)$ globally asymptotically keep track of the auxiliary variables and thus also asymptotically converge to the VE. The algorithm is presented by Algorithm \ref{Algorithm 1} and detailed next.

\begin{algorithm2e}[htbp]
\caption{Distributed update rule and control law for linear system dynamics}
\label{Algorithm 1}
Each agent $i\in\mathcal{V}$ picks any $(\bar w_i(0),u_i(0))\in W_i\times\mathbb{R}$;

All the agents agree on any $(\lambda(0),\mu(0))\in\mathbb{R}^m\times\mathbb{R}_+^l$;

\While{$t\geq0$}{
Each agent $i\in\mathcal{V}$ updates its auxiliary variables by
\begin{align}
\label{decision update rule}
&\dot{\bar w}_i(t)=\Pi_{W_i}(\bar w_i(t),-\nabla_{\bar w_i}L_i(\bar w(t),\lambda(t),\mu(t))),\nonumber\\
&\dot\lambda(t)=H(\bar w(t)),\;\;\dot\mu(t)=\Pi_{\mathbb{R}_+^l}(\mu(t),G(\bar w(t))),
\end{align}
and executes the control law
\begin{align}
\label{control law for linear system}
u_i (t) = (K_i(\varepsilon)+a_i)\mathcal{T}_i(x_i(t)-\bar x_i(t))+\bar u_i(t).
\end{align}
}
\end{algorithm2e}

First we illustrate the design of the update rule for auxiliary variables. For each $i\in\mathcal{V}$, define auxiliary variables $\bar x_i\in\mathbb{R}^{n_i}$ and $\bar u_i\in\mathbb{R}$, and let $\bar w_i=[\bar x_i^T,\bar u_i]^T$. Denote $\bar x=(\bar x_i)_{i=1}^N$, $\bar u=(\bar u_i)_{i=1}^N$ and $\bar w=(w_i)_{i=1}^N$. To handle constraints, for each $i\in\mathcal{V}$, define Lagrangian $L_i:W\times\mathbb{R}^m\times\mathbb{R}_+^l\to\mathbb{R}$ as $L_i({\bar w},\lambda,\mu) = {f_i}(\bar w)+\lambda^{T}H(\bar w)+\mu^TG(\bar w)$, where $\lambda=(\lambda_{j})_{j=1}^{m}\in\mathbb{R}^m$ and $\mu=(\mu_{j})_{j=1}^{l}\in\mathbb{R}_+^l$ are Lagrange multipliers associated with $H$ and $G$, respectively.

The update rule for the auxiliary variables is given by \eqref{decision update rule}. For each $i\in\mathcal{V}$, the primal state $\bar w_i$ moves along the direction of $-\nabla_{\bar w_i}L_i(\bar w,\lambda,\mu)$ so as to minimize $L_i$, and the projection operation $\Pi_{W_i}$ guarantees that $\bar w_i(t)\in W_i$ for all $t$. The dynamics of the dual states $(\lambda,\mu)$ are defined by the positive gradients of $L_i$ so as to maximize $L_i$. Again, the projection operation $\Pi_{\mathbb{R}_+^l}$ on $\mu$ is to guarantee that $\mu(t)\in\mathbb{R}_+^l$ for all $t$.

We next illustrate the design of the control law for the physical variables. By Theorem 8.2 of \cite{chen}, for each $i\in\mathcal{V}$, since $(A_{ii},B_i)$ is controllable, there exists a nonsingular matrix $\mathcal{T}_i\in \mathbb{R}^{n_i\times n_i}$ such that $(\bar A_{ii},\hat B_i)$ with $\bar A_{ii}=\mathcal{T}_iA_{ii}\mathcal{T}_i^{-1}$ and $\hat B_i=\mathcal{T}_iB_i$ is in the controllable canonical form, i.e., $\bar A_{ii}=[[0_{n_i-1},I_{n_i-1}]^T,-a_i^T]^T$ with $a_i=[a_{i1},\cdots,a_{in_i}]$ a constant row vector, and $\hat B_i=[0_{n_i-1}^T,1]^T$.
Let $\hat x_i=\mathcal{T}_ix_i$, $\bar{\bar x}_i=\mathcal{T}_i\bar x_i$, $v_i=u_i-a_i\hat x_i$, $\bar v_i=\bar u_i-a_i\bar{\bar x}_i$, $\hat A_{ij}=\mathcal{T}_iA_{ij}\mathcal{T}_j^{-1}$ for all $j\neq i$, and $\hat A_{ii}=[[0_{n_i-1},I_{n_i-1}]^T,0_{n_i}]^T$. We then have
\begin{align}
\label{transformed linear dynamics}
&\dot{\hat x}_i(t)=\mathcal{T}_i\dot x_i(t)=\mathcal{T}_i(A_{ii}x_i(t)+\sum\nolimits_{j\neq i}A_{ij}x_j(t)+B_iu_i(t))\nonumber\\
&=\bar A_{ii}\hat x_i(t)+\sum\nolimits_{j\neq i}\hat A_{ij}\hat x_j(t)+\hat B_iu_i(t)\nonumber\\
&=\hat A_{ii}\hat x_i(t)+\sum\nolimits_{j\neq i}\hat A_{ij}\hat x_j(t)+\hat B_iv_i(t).
\end{align}

For each $i\in\mathcal{V}$, notice that $(\hat A_{ii},\hat B_i)$ is controllable. Then there exists $K_{0i}\in \mathbb{R}^{1\times n_i}$ such that $\hat A_{ii}+\hat B_iK_{0i}$ is Hurwitz (\cite{chen}, Theorem 8.3). We further have that there exists a symmetric positive definite matrix $P_{0i}\in \mathbb{R}^{n_i\times n_i}$ such that $P_{0i}(\hat A_{ii}+\hat B_iK_{0i}) + (\hat A_{ii}+\hat B_iK_{0i})^TP_{0i} = -I_{n_i}$ (\cite{chen}, Theorem 5.5). For each $i\in\mathcal{V}$, let ${S_i}(\varepsilon ) \buildrel \Delta \over = \mathrm{diag}(\varepsilon^{j-1})_{j=1}^{n_i}$ and ${{\bar S}_i}(\varepsilon )\buildrel \Delta \over = \mathrm{diag}(\varepsilon^{j-n_i})_{j=1}^{n_i}$, where $\varepsilon\in \mathbb{R}$ is a constant scalar such that $0<\varepsilon<1$. Let $S(\varepsilon)=\mathrm{diag}(S_i(\varepsilon))_{i=1}^N$, $K_i(\varepsilon) = \frac{1}{\varepsilon} K_{0i} \bar S_i(\varepsilon)$, $P_0=\mathrm{diag}(P_{0i})_{i=1}^N$ and $K(\varepsilon)=\mathrm{diag}(K_i(\varepsilon))_{i=1}^N$.

The distributed control law is given by \eqref{control law for linear system}. Each agent $i$ takes $(\bar x_i,\bar u_i)$ as an intermediate estimate of the VE and aims to drive $x_i$ by $u_i$ to asymptotically keep track of $\bar x_i$ and $u_i$ itself asymptotically keep track of $\bar u_i$. The logic of \eqref{control law for linear system} lies in that we estimate the equilibrium of $u_i$ by $\bar u_i$ and correct this estimation by the feedback of the tracking error from $x_i$ to $\bar x_i$ gained by $(K_i(\varepsilon)+a_i)\mathcal{T}_i$. The introduction of $\varepsilon$ is to enable arbitrarily fast tracking (by choosing $\varepsilon$ arbitrarily small). Notice that \eqref{control law for linear system} does not require inter-agent communications.

\subsection{Convergence results}

The following assumptions are needed for convergence.

\begin{assumption} \begin{rm}
$Ax+Bu=0_n$ is included in $\hat{\mathcal{D}}$.
\label{asm: physical feasibility}
\end{rm} \hfill\rule{2mm}{2mm

\begin{remark} \begin{rm}
Assumption \ref{asm: physical feasibility} guarantees that the VE of the GNEP satisfies the steady state condition of \eqref{linear physical dynamic} so that the VE tracking problem is feasible.
If $Ax+Bu=0_n$ is not originally included in $\hat{\mathcal{D}}$, one can deliberately add it into $\hat{\mathcal{D}}$ and impose Assumption \ref{asm: VE nonempty} on the resulted GNEP.
 \end{rm} \hfill\rule{2mm}{2mm

\begin{assumption} \begin{rm}
For each $i\in\mathcal{V}$, there exists a constant scalar $\sigma_i>0$ such that $\| S_i (\varepsilon )\hat A_{ij} S_j^{ - 1} (\varepsilon )\|_1 < \sigma_i$ for any $ j\in \mathcal{V}$ with $j\ne i$ and any $0<\varepsilon<1$.
\label{asm: S constraint}
\end{rm} \hfill\rule{2mm}{2mm

\begin{remark} \begin{rm}
For any $i,j\in \mathcal{V}$, the entry of $S_i (\varepsilon )\hat A_{ij} S_j^{ - 1} (\varepsilon )$ at the intersection of the $s$-th row and the $c$-th column, denoted by $\bar a_{sc}$, is $\varepsilon^{s-c}a_{sc}$, where $a_{sc}$ is the corresponding entry of $\hat A_{ij}$. A sufficient condition for Assumption \ref{asm: S constraint} is that $a_{sc}=0$ for all $s<c$. Under this condition, $\bar a_{sc}=0$ for all $s<c$. For the entries with $s\geq c$, since $0<\varepsilon<1$, $|\bar a_{sc}|\leq|a_{sc}|$. Thus, $\|S_i (\varepsilon )\hat A_{ij} S_j^{ - 1} (\varepsilon )\|_1\leq \|\hat A_{ij}\|_1$ for any $0<\varepsilon<1$. If $n_i=n_j$, this condition means that $\hat A_{ij}$ is a lower triangular matrix. If $n_i>n_j$, $\hat A_{ij}$ is denser than lower triangular; if $n_i<n_j$, it is sparser. We provide a concrete optimal power flow example that satisfies Assumption \ref{asm: S constraint} in Section \ref{OPF simulation}.
 \end{rm} \hfill\rule{2mm}{2mm

The following theorem summarizes the convergence results of Algorithm \ref{Algorithm 1}.

\begin{theorem} \begin{rm}
Suppose Assumptions \ref{asm: convexity}--\ref{asm: VE nonempty} and \ref{asm: CCF}--\ref{asm: S constraint} hold. Choose $\varepsilon>0$ such that $1/\varepsilon> 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i \| {{P_{0i}}} \|_1\}+2$. By Algorithm \ref{Algorithm 1}, $(x(t),u(t))$ of linear system \eqref{linear physical dynamic} globally asymptotically converge to the unique VE of the GNEP$(\mathcal{D},F)$.
\label{theorem: Algorithm 1}
 \end{rm} \hfill\rule{2mm}{2mm

The proof of Theorem \ref{theorem: Algorithm 1} is given in Section \ref{appendix 2}.

\begin{remark} \begin{rm}
\label{remark max-consensus}
In Algorithm \ref{Algorithm 1}, all the agents have to choose the same (fixed) $\varepsilon$ such that $1/\varepsilon> 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i \| {{P_{0i}}} \|_1\}+2$. This can be realized distributively by the max-consensus scheme \cite{ROS-JAF-RMM:07}. Each agent $i$ initializes the scheme by picking any $\eta_i^0$ such that $\eta_i^0>\sigma_i \| {{P_{0i}}} \|_1$. Notice that this initialization only requires an agent's local information. At each step $\ell$ from $\ell=0$ to $\ell={\rm dm}(\mathcal{G})-1$, each agent $i$ communicates with its neighbors and computes $\eta_i^{\ell+1}=\max_{j\in\mathcal{N}_i\cup\{i\}}\eta_j^\ell$. Given that $\mathcal{G}$ is undirected and connected, by the analysis in \cite{ROS-JAF-RMM:07}, after ${\rm dm}(\mathcal{G})$ steps, max-consensus is reached, i.e., $\eta_i^{{\rm dm}(\mathcal{G})}=\max_{j\in\mathcal{V}}\eta_j^0$ for all $i\in\mathcal{V}$. Each agent $i$ sets $\varepsilon_i=\frac{1}{2\sqrt n (N - 1)\eta_i^{{\rm dm}(\mathcal{G})}+2}$. We then have $\varepsilon_1=\cdots=\varepsilon_N=\varepsilon$ with $1/\varepsilon=2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\eta_i^0+2>2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i \| {{P_{0i}}} \|_1\}+2$.
 \end{rm} \hfill\rule{2mm}{2mm

\section{Case II: Nonlinear dynamics\label{nonlinear section}}

In this section, we assume $n_1=\cdots=n_N=\bar n$ and consider the case where system \eqref{physical dynamic} is nonlinear and in the following strict-feedback form
\begin{align}
\label{nonlinear state dynamics}
&\dot x_{i\ell}=\Gamma_{i\ell}(x^{[1\sim\ell]})+\Theta_{i\ell}(x^{[1\sim\ell]})x_{i(\ell+1)},\;\;\ell=1,\cdots,\bar n-1,\nonumber\\
&\dot x_{i\bar n}=\Gamma_{i\bar n}(x)+\Theta_{i\bar n}(x)u_i
\end{align}
where $x^{[1\sim\ell]}=[x^{[1]T},\cdots,x^{[\ell]T}]^T$ with $x^{[\ell]}=(x_{i\ell})_{i=1}^N$, $\Gamma_{i\ell}:\mathbb{R}^{\ell N}\to\mathbb{R}$ and $\Theta_{i\ell}:\mathbb{R}^{\ell N}\to\mathbb{R}$ are nonlinear in $x^{[1\sim\ell]}$. The following assumption is standard for systems in the strict-feedback form (e.g., Section 14.3 of \cite{Khalil:02}).

\begin{assumption} \begin{rm}
$\forall i\in\mathcal{V}$, $f_i$ is of class $C^{\bar n}$; $\forall j\in\{1,\cdots,l\}$, $G_j$ is of class $C^{\bar n}$; $\forall i\in\mathcal{V}$, $\forall \ell=1,\cdots,\bar n$, $\Theta_{i\ell}$ and $\Gamma_{i\ell}$ are of class $C^\ell$, and $\Theta_{i\ell}(y)\neq0$ for any $y\in\mathbb{R}^{\ell N}$.
\label{asm: smoothness}
\end{rm} \hfill\rule{2mm}{2mm

We note that Algorithm \ref{Algorithm 1} is inapplicable to the nonlinear system \eqref{nonlinear state dynamics}. The reasons are twofold. First, the equality constraints in the GNEP can only include affine functions. It is the case for the steady states of the linear system \eqref{linear physical dynamic}. However, the steady states of the nonlinear system \eqref{nonlinear state dynamics} introduce nonlinear functions, making problem \eqref{general OP} non-convex. Second, in centralized control, stabilization of nonlinear systems is completely different from and much more challenging than that of linear systems. So the nonlinear system \eqref{nonlinear state dynamics} requires a completely different distributed stabilizing controller from \eqref{control law for linear system} for the linear system \eqref{linear physical dynamic}.

If a linear system is stabilizable, there are many systematic ways to find its stabilizing controllers, e.g., cyclic design, Lyapunov-equation method and canonical-form method \cite{chen}. However, such general methods do not exist for nonlinear systems. In the literature of nonlinear control, researchers have studied stabilization of several classes of nonlinear systems with certain special structures. Notable examples include strict-feedback systems (or systems in lower triangular form) which can be stabilized by the technique of backstepping, and strict-feedforward systems (or systems in upper triangular form) which can be stabilized by the technique of nested saturation or forwarding \cite{AI:99,Khalil:02}. Many coordinate transformations have been proposed to cast nonlinear systems in one of the two forms \cite{JLM-JKH,WZ-LYW:1994,WR-IAT:2004}. In this section, we focus on strict-feedback systems. Such systems have broad real-world applications, e.g., HVAC systems \cite{ESK-MJY-CL:2008}, unmanned autonomous vehicles \cite{JC-WY-SS-IC-RL:2016,BA-HRP-MG:2010} and power systems \cite{ADM-JMC-JFAS-FRV:2015,SYL-SSL-YTY-JKP:2009}. Strict-feedforward systems are left as future works.

In this section, we assume that the GNEP only depends on $x^{[1]}$, i.e., the first entry of each agent's state, and each agent $i$'s decision variable in the GNEP is $x_{i1}$. Moreover, for each $i\in\mathcal{V}$, assume $W_i=\mathbb{R}$. The optimization problem for agent $i$ is then $\mathop {\min }\nolimits_{x_{i1} \in {\mathcal{D}_i}({x_{ - i}^{[1]}})} {f_i}(x_{i1},x_{-i}^{[1]})$, where $f_i:\mathbb{R}^N\to\mathbb{R}$ and ${\mathcal{D}_i}({x_{ - i}^{[1]}})=\{x_{i1}\in\mathbb{R}:H(x_{i1},x_{-i}^{[1]})=0_m,G(x_{i1},x_{-i}^{[1]})\leq0_l\}$ with $H:\mathbb{R}^N\to\mathbb{R}^m$ and $G:\mathbb{R}^N\to\mathbb{R}^l$.

\begin{remark} \begin{rm}
With the strict-feedback form of \eqref{nonlinear state dynamics} and the restriction that the GNEP only depends on $x^{[1]}$, given the VE of the GNEP, denoted by $\tilde x^{[1]}$, it is guaranteed that there exists a uniquely determined pair $(\tilde x,\tilde u)$ which is a steady state of system \eqref{nonlinear state dynamics}. This is observed as follows. For each $i\in\mathcal{V}$ and each $\ell=1,\cdots,\bar n$, since $\Gamma_{i\ell}$ and $\Theta_{i\ell}$ only depend on $x^{[1\sim\ell]}$, we can sequentially uniquely determine $\tilde x_{i(\ell+1)}=-\Gamma_{i\ell}(\tilde x^{[1\sim\ell]})/\Theta_{i\ell}(\tilde x^{[1\sim\ell]})$ for all $\ell=1,\cdots,\bar n-1$ and $\tilde u_i=-\Gamma_{i\bar n}(\tilde x)/\Theta_{i\bar n}(\tilde x)$ (by Assumption \ref{asm: smoothness}, $\Theta_{i\ell}(\tilde x^{[1\sim\ell]})\neq0$ for all $i\in\mathcal{V}$ and all $\ell=1,\cdots,\bar n$). Due to this fact, we do not need to include the equality constraints induced by the steady sate condition of \eqref{nonlinear state dynamics} into the constraint set of the GNEP.
 \end{rm} \hfill\rule{2mm}{2mm

In the rest of this section, to save space, we omit the time index $t$ and the arguments of the component functions.

\subsection{Algorithm design\label{section nonlinear algorithm design}}

Different from the indirect method of Algorithm \ref{Algorithm 1}, in this section, we directly work on $(x,u)$ and design a control law for $u$ such that $(x,u)$ of system \eqref{nonlinear state dynamics} globally asymptotically converge to the steady state of \eqref{nonlinear state dynamics}, $(\tilde x,\tilde u)$, with $\tilde x^{[1]}$ being the VE of the GNEP. The algorithm design is inspired by the technique of backstepping \cite{KKK:95}. A major step is to convert system \eqref{nonlinear state dynamics} into a chain of stabilizable integrators by a coordinate transformation, detailed next.

For each $i\in\mathcal{V}$, define Lagrangian $L_i(x^{[1]},\lambda,\mu)=f_i(x^{[1]})+\lambda^TH(x^{[1]})+\mu^TG(x^{[1]})$, where $\lambda\in\mathbb{R}^m$, $\mu\in\mathbb{R}_+^l$ are shared dual states. To achieve VE convergence, we aim that each $\dot x_{i1}$ converges to $-k_1\nabla_{x_{i1}}L_i$ for some $k_1>0$. However, $\dot x_{i1}$ is governed by \eqref{nonlinear state dynamics}. We then aim that the error between $\dot x_{i1}$ and $-k_1\nabla_{x_{i1}}L_i$ asymptotically diminishes.
For each $i\in\mathcal{V}$, let $z_{i1}=x_{i1}$. We have $\dot z_{i1}=\Gamma_{i1}+\Theta_{i1}x_{i2}=-k_{1}\nabla_{z_{i1}}L_i+z_{i2}$, where $z_{i2}=k_{1}\nabla_{z_{i1}}L_i+\Gamma_{i1}+\Theta_{i1}x_{i2}$. We then aim that $z_{i2}$ asymptotically diminishes. We have $\dot z_{i2}=\frac{d}{dt}(k_{1}\nabla_{z_{i1}}L_i+\Gamma_{i1}+\Theta_{i1}x_{i2})=\frac{d}{dt}(k_{1}\nabla_{z_{i1}}L_i+\Gamma_{i1})+x_{i2}\frac{d}{dt}\Theta_{i1}+\Theta_{i1}(\Gamma_{i2}+\Theta_{i2}x_{i3})
=-k_{i2}z_{i2}+z_{i3}$, where $z_{i3}=k_{i2}z_{i2}+\frac{d}{dt}(k_{1}\nabla_{z_{i1}}L_i+\Gamma_{i1})+x_{i2}\frac{d}{dt}\Theta_{i1}+\Theta_{i1}(\Gamma_{i2}+\Theta_{i2}x_{i3})$ with constant $k_{i2}>0$. Intuitively, if $z_{i3}$ asymptotically diminishes, then $\dot z_{i2}$ asymptotically converges to $-k_{i2}z_{i2}$, which implies that $z_{i2}$ asymptotically diminishes. Hence, we further aim that $z_{i3}$ asymptotically diminishes. With this logic, for each $i\in\mathcal{V}$, we aim to sequentially define a coordinate transformation for $z_{i2},\cdots,z_{i\bar n}$ and design a control law for $u_i$ such that the dynamics of $z_{i\ell}$'s are as follows
\begin{align}
\label{transformed states dynamics}
\dot z_{i1}&=-k_{1}\nabla_{z_{i1}}L_i+z_{i2},\nonumber\\
\dot z_{i\ell}&=-k_{i\ell}z_{i\ell}+z_{i(\ell+1)},\;\;\forall\ell=2,\cdots,\bar n-1,\nonumber\\
\dot z_{i\bar n}&=-k_{i\bar n}z_{i\bar n}
\end{align}
where $k_1,k_{i\ell},\cdots,k_{i\bar n}>0$. Intuitively, $z_{i\bar n}$ asymptotically diminishes, which implies that $z_{i(\bar n-1)}$ asymptotically diminishes, which further implies that $z_{i(\bar n-2)}$ asymptotically diminishes. With this logic of backstepping from $z_{i\bar n}$ back to $z_{i2}$, we have that $z_{i\bar n},z_{i(\bar n-1)},\cdots,z_{i2}$ all asymptotically diminish for all $i\in\mathcal{V}$. We next provide the general form of the coordinate transformation and the control law.
For any $\alpha,\beta\in\{1,\cdots,\bar n-1\}$, let $T_{\alpha,\beta}^i$ be the sum of all the $\alpha$-th order products of the first $\beta$ elements of $\{k_{i2},\cdots,k_{i\bar n}\}$ with repeat. For example, $T_{1,2}^i=k_{i2}+k_{i3}$ and $T_{2,3}=k_{i2}^2+k_{i3}^2+k_{i4}^2+k_{i2}k_{i3}+k_{i2}k_{i4}+k_{i3}k_{i4}$. Let $T_{\alpha,\beta}^i=0$ for any $\alpha\leq0$. For each $i\in\mathcal{V}$, for each $\ell\in\{1,\cdots,\bar n\}$, let $\Theta_{i,1\sim\ell} =\Theta_{i1}\Theta_{i2}\cdots\Theta_{i\ell}$. For each $i\in\mathcal{V}$, agent $i$ performs the following coordinate transformation
\begin{align}
\label{coordinate transformation}
z_{i1}=&x_{i1},\;\;z_{i2}=k_{1}\nabla_{z_{i1}}L_i+\Theta_{i1}x_{i2}+\Gamma_{i1},\nonumber\\
z_{i\ell}=&(-1)^{\ell-1}T_{\ell-2,1}^iz_{i2}+(-1)^{\ell-2}T_{\ell-3,2}^iz_{i3}+\cdots\nonumber\\
&+(-1)^2T_{1,\ell-2}^iz_{i(\ell-1)}+\frac{d^{\ell-2}}{dt^{\ell-2}}(k_1\nabla_{z_{i1}}L_i+\Gamma_{i1})\nonumber\\
&+\frac{d^{\ell-3}}{dt^{\ell-3}}(x_{i2}\frac{d}{dt}\Theta_{i1}+\Theta_{i1}\Gamma_{i2})+\cdots+\nonumber\\
&\frac{d}{dt}(x_{i(\ell-2)}\frac{d}{dt}\Theta_{i,1\sim(\ell-3)}+\Theta_{i,1\sim(\ell-3)}\Gamma_{i(\ell-2)})\nonumber\\
&+x_{i(\ell-1)}\frac{d}{dt}\Theta_{i,1\sim(\ell-2)}+\Theta_{i,1\sim(\ell-2)}\Gamma_{i(\ell-1)}\nonumber\\
&+\Theta_{i,1\sim(\ell-1)}x_{i\ell},\quad\forall\ell=3,4,\cdots,\bar n.
\end{align}

For each $i\in\mathcal{V}$, agent $i$'s control $u_i$ is designed as follows
\begin{align}
\label{control law for nonlinear dynamics}
u_i=&-\frac{1}{\Theta_{i,1\sim\bar n}}[(-1)^{\bar n}T_{\bar n-1,1}^iz_{i2}+(-1)^{\bar n-1}T_{\bar n-2,2}^iz_{i3}\nonumber\\
&+\cdots+(-1)^2T_{1,\bar n-1}^iz_{i\bar n}+\frac{d^{\bar n-1}}{dt^{\bar n-1}}(k_1\nabla_{z_{i1}}L_i+\Gamma_{i1})\nonumber\\
&+\frac{d^{\bar n-2}}{dt^{\bar n-2}}(x_{i2}\frac{d}{dt}\Theta_{i1}+\Theta_{i1}\Gamma_{i2})+\cdots+\nonumber\\
&\frac{d}{dt}(x_{i(\bar n-1)}\frac{d}{dt}\Theta_{i,1\sim(\bar n-2)}+\Theta_{i,1\sim(\bar n-2)}\Gamma_{i(\bar n-1)})\nonumber\\
&+x_{i\bar n}\frac{d}{dt}\Theta_{i,1\sim(\bar n-1)}+\Theta_{i,1\sim(\bar n-1)}\Gamma_{i\bar n}].
\end{align}

\begin{remark} \begin{rm}
By Assumption \ref{asm: smoothness}, all the derivatives in \eqref{coordinate transformation} and \eqref{control law for nonlinear dynamics} exist and are continuous. With the strict-feedback form of \eqref{nonlinear state dynamics}, for each $i\in\mathcal{V}$ and each $\ell=1,\cdots,\bar n$, $z_{i\ell}$ only depends on $z^{[1\sim(\ell-1)]}$ but does not depend on $z^{[\ell']}$ for any $\ell'\geq\ell$, and $u_i$ only depends on $z$ but does not depend on any other $u_j$. Hence, \eqref{coordinate transformation} and \eqref{control law for nonlinear dynamics} can be sequentially computed in the order of $z^{[1]}\to z^{[2]}\to\cdots\to z^{[\bar n]}\to u$.

For each $i\in\mathcal{V}$, for each $\ell=3,\cdots,\bar n$, $z_{i\ell}$ is actually defined from the dynamics of $z_{i(\ell-1)}$ by letting $\dot z_{i(\ell-1)}=-k_{i(\ell-1)}z_{i(\ell-1)}+z_{i\ell}$, which leads to the simple expression $z_{i\ell}=k_{i(\ell-1)}z_{i(\ell-1)}+\dot z_{i(\ell-1)}$. The more complicated form \eqref{coordinate transformation} is obtained by just expanding the term $\dot z_{i(\ell-1)}$. This expansion helps to obtain the explicit expression of $u_i$. Recall that we aim to achieve $\dot z_{i\bar n}=-k_{i\bar n}z_{i\bar n}$. Thus, $u_i$ is designed such that $\dot z_{i\bar n}+k_{i\bar n}z_{i\bar n}=0$, which leads to \eqref{control law for nonlinear dynamics}.
 \end{rm} \hfill\rule{2mm}{2mm

The following lemma summarizes the dynamics of the transformed system derived by \eqref{coordinate transformation} and \eqref{control law for nonlinear dynamics}.

\begin{lemma} \begin{rm}
Under Assumption \ref{asm: smoothness}, by \eqref{coordinate transformation} and \eqref{control law for nonlinear dynamics}, the dynamics of the transformed system follows \eqref{transformed states dynamics}.
\label{lemma: coordinate transformation dynamics}
 \end{rm} \hfill\rule{2mm}{2mm

The algorithm is summarized by Algorithm \ref{Algorithm 2}. At time $t$, each agent $i$ executes the control law \eqref{control law for nonlinear dynamics} and updates the transformed state $z_i$ by \eqref{transformed states dynamics}. The update rules for the dual variables $(\lambda,\mu)$ are similar to those in Algorithm \ref{Algorithm 1}.

\begin{algorithm2e}[htbp]
\caption{Distributed control law for nonlinear system dynamics}
\label{Algorithm 2}
Each agent $i\in\mathcal{V}$ picks any $(x_i(0),u_i(0))\in \mathbb{R}^{\bar n}\times\mathbb{R}$ and computes $z_i(0)$ by \eqref{coordinate transformation};

All the agents agree on any $(\lambda(0),\mu(0))\in\mathbb{R}^m\times\mathbb{R}_+^l$;

\While{$t\geq0$}{
Each agent $i\in\mathcal{V}$ executes the control law \eqref{control law for nonlinear dynamics}; updates $z_i$ by \eqref{transformed states dynamics}; and updates $(\lambda,\mu)$ by
\begin{align}
\label{nonlinear dual update}
&\dot\lambda=k_1H(z^{[1]}),\;\;\dot\mu=\Pi_{\mathbb{R}_+^l}(\mu,k_1G(z^{[1]}));
\end{align}
}
\end{algorithm2e}

\normalsize

\subsection{Convergence results}

The following assumption is needed for convergence.

\begin{assumption} \begin{rm}
$\nabla F$ is strongly monotone on $\mathbb{R}^N$ with constant $M>0$.
\label{asm: strong monotone}
\end{rm} \hfill\rule{2mm}{2mm

Notice that the GNEP in this section only involves $x^{[1]}$ and the primal-dual dynamics is perturbed by a chain of integrators. Strict monotonicity of $\nabla F$ is not enough for convergence guarantee because it provides zero stability margin for the primal-dual dynamics. Instead, strong monotonicity is needed.

The following theorem summarizes the convergence results of Algorithm \ref{Algorithm 2}.

\begin{theorem} \begin{rm}
Suppose Assumptions \ref{asm: convexity}, \ref{asm: Slater condition}, \ref{asm: VE nonempty}, \ref{asm: smoothness} and \ref{asm: strong monotone} hold. Choose $k_1,k_{i2},\cdots,k_{i\bar n}$ such that $k_1M>1$ and $k_{i\ell}>1$ for each $i\in\mathcal{V}$ and each $\ell=2,\cdots,\bar n$. By Algorithm \ref{Algorithm 2}, $(x(t),u(t))$ globally asymptotically converge to the steady state of system \eqref{nonlinear state dynamics}, denoted by $(\tilde x,\tilde u)$, with $\tilde x^{[1]}$ being the unique VE of the GNEP$(\mathcal{D},F)$.
\label{theorem: Algorithm 2}
 \end{rm} \hfill\rule{2mm}{2mm

The proof of Theorem \ref{theorem: Algorithm 2} is given in Section \ref{proof of theorem 2}.

\begin{remark} \begin{rm}
Similar to Remark \ref{remark max-consensus}, the agents can distributively choose the same $k_1$ by the max-consensus scheme \cite{ROS-JAF-RMM:07}.
 \end{rm} \hfill\rule{2mm}{2mm

\subsection{Extension}

In this subsection, we discuss how to extend the results derived in this section to the case where the GNEP depends on some $x_{j\ell}$'s with $\ell>1$ and/or some $u_j$'s. The overall idea is to derive the steady state of each $x_{j\ell}$ with $\ell>1$ and $u_j$ as a function of $x^{[1]}$ and substitute $x_{j\ell}$'s and $u_j$'s with their steady state forms in the GNEP. Then the resulted GNEP only depends on $x^{[1]}$. The change of variables is feasible due to the strict-feedback form of \eqref{nonlinear state dynamics}, and is detailed next.

For each $i\in\mathcal{V}$, the steady state condition of $x_{i1}$ is $\Gamma_{i1}(x^{[1]})+\Theta_{i1}(x^{[1]})x_{i2}=0$, by which we obtain the steady state of $x_{i2}$ as a function of $x^{[1]}$ as $\bar x_{i2}=-\Gamma_{i1}(x^{[1]})/\Theta_{i1}(x^{[1]})$. For convenience of notation, let $\bar x^{[2]}=\Upsilon_2(x^{[1]})$. The steady state condition of $x_{i2}$ is $\Gamma_{i2}(x^{[1\sim2]})+\Theta_{i2}(x^{[1\sim2]})x_{i3}=0$, by which we can obtain the steady state of $x_{i3}$ as a function of $x^{[1]}$ as $\bar x_{i3}=-\Gamma_{i2}(x^{[1]},x^{[2]})/\Theta_{i2}(x^{[1]},x^{[2]})=-\Gamma_{i2}(x^{[1]},\Upsilon_2(x^{[1]}))/\Theta_{i2}(x^{[1]},\Upsilon_2(x^{[1]}))$. Let $\bar x^{[3]}=\Upsilon_3(x^{[1]})$. Go on with this procedure of change of variables in the order of $x^{[2]}\to\cdots\to x^{\bar n}\to u$ to obtain the steady state of each $x^{[\ell]}$ and $u$ as a function of only $x^{[1]}$, denoted by $\bar x^{[\ell]}=\Upsilon_\ell(x^{[1]})$ and $\bar u=\Upsilon_u(x^{[1]})$, respectively. We then substitute $(x^{[2]},\cdots,x^{[\bar n]},u)$ with $(\bar x^{[2]},\cdots,\bar x^{[\bar n]},\bar u)$ in the GNEP. The resulted GNEP then only depends on $x^{[1]}$ and Algorithm \ref{Algorithm 2} can be applied. Notice that, to have the convergence result of Theorem \ref{theorem: Algorithm 2}, all the assumptions required by Theorem \ref{theorem: Algorithm 2} should be imposed on the GNEP after the change of variables, rather than the original one.

\section{Proofs}

This section proves the results of Sections \ref{section linear dynamics} and \ref{nonlinear section}.

\subsection{Proof of Theorem \ref{theorem: Algorithm 1}\label{appendix 2}}

This subsection is devoted to proving Theorem \ref{theorem: Algorithm 1}.

\begin{IEEEproof} Let $\nabla_{\bar w}L(\bar w,\lambda,\mu)=(\nabla_{\bar w_i}L_i(\bar w,\lambda,\mu))_{i=1}^N$. Under Assumptions \ref{asm: convexity}, \ref{asm: convex feasible set} and \ref{asm: Slater condition}, by Section 12.2.3 of \cite{PE:10}, $\tilde w\in W$ is the VE of the GNEP$(\mathcal{D},F)$ if and only if there exist $(\tilde \lambda,\tilde\mu)\in\mathbb{R}^m\times\mathbb{R}_+^l$ such that the KKT conditions hold
\begin{align}
\label{game KKT condition}
&\nabla_{\bar w}L(\tilde w,\tilde\lambda,\tilde\mu)=0_r,\,\,H(\tilde w)=0_m,\nonumber\\
&G(\tilde w)\leq0_l\,\,{\rm and}\,\,\tilde \mu^{T}G(\tilde w)=0.
\end{align}
A triple $(\tilde w,\tilde\lambda,\tilde\mu)$ satisfying \eqref{game KKT condition} is called a primal-dual optimizer of \eqref{general OP}. Denote by $Q$ the set of primal-dual optimizers of \eqref{general OP}. We first prove the convergence of the auxiliary variables $(\bar w,\lambda,\mu)$. This is formally stated by the following theorem.

\begin{theorem} \begin{rm}
\label{theorem auxiliary variable convergence}
Any solution of the primal-dual dynamics \eqref{decision update rule} starting from any point in $W\times\mathbb{R}^m\times\mathbb{R}_+^l$ asymptotically converges to a point of $Q$.
 \end{rm} \hfill\rule{2mm}{2mm

\begin{IEEEproof}
The update rule \eqref{decision update rule} can be written in the standard form of a projected dynamical system (see \eqref{differential inclusion}) as
\begin{align}
\label{update rule standard form}
&(\dot{\bar w},\dot\lambda,\dot\mu)=\Pi_{W\times\mathbb{R}^m\times\mathbb{R}_+^l}((\bar w,\lambda,\mu),\nonumber\\
&(-\nabla_{\bar w}L(\bar w,\lambda,\mu),H(\bar w),G(\bar w))).
\end{align}

Define the Lyapunov function as $V(\bar w,\lambda,\mu)=\frac{1}{2}(\|\bar w-\tilde w\|^2+\|\lambda-\tilde \lambda\|^2+\|\mu-\tilde \mu\|^2)$, where $(\tilde w,\tilde\lambda,\tilde\mu)$ is any point of $Q$. For convenience of notation, denote by $\mathcal{L}V(\bar w,\lambda,\mu)$ the Lie derivative of $V$ along \eqref{update rule standard form} at $(\bar w,\lambda,\mu)\in W\times\mathbb{R}^m\times\mathbb{R}_+^l$. We next show that system \eqref{update rule standard form} satisfies all the hypotheses of Lemma \ref{lemma invariance} by proving the following series of claims.

\begin{claim} \begin{rm}
\label{middle lemma monotonicity of Lyapunov function}
$\mathcal{L}V(\bar w,\lambda,\mu)\leq0$ for all $(\bar w,\lambda,\mu)\in W\times\mathbb{R}^m\times\mathbb{R}_+^l$.
 \end{rm} \hfill\rule{2mm}{2mm

\begin{IEEEproof}
We first show for all $(\bar w,\lambda,\mu)\in W\times\mathbb{R}^m\times\mathbb{R}_+^l$, $(\bar w-\tilde w)^T\Pi_W(\bar w,-\nabla_{\bar w}L(\bar w,\lambda,\mu))\leq-(\bar w-\tilde w)^T\nabla_{\bar w}L(\bar w,\lambda,\mu)$. For convenience of notation, let $y=-\nabla_{\bar w}L(\bar w,\lambda,\mu)$. By Lemma \ref{lemma vector projection}, we only have to consider the case where $\bar w\in{\rm bd}(W)$ and $y^T\gamma^*(\bar w)<0$ with $\gamma^*(\bar w)={\rm argmin}_{\gamma\in N_W(\bar w)}y^T\gamma$. By Lemma \ref{lemma vector projection}, we then have $\Pi_W(\bar w,y)=y-(y^T\gamma^*(\bar w))\gamma^*(\bar w)$. Then we have $(\bar w-\tilde w)^T\Pi_W(\bar w,y)=(\bar w-\tilde w)^T(y-(y^T\gamma^*(\bar w))\gamma^*(\bar w))=(\bar w-\tilde w)^Ty-(y^T\gamma^*(\bar w))(\bar w-\tilde w)^T\gamma^*(\bar w)$. Since $\bar w,\tilde w\in W$ and $\gamma^*(\bar w)\in N_W(\bar w)$, by the definition of normal cone, we have $(\bar w-\tilde w)^T\gamma^*(\bar w)\leq0$. Since $y^T\gamma^*(\bar w)<0$, we then have $(y^T\gamma^*(\bar w))(\bar w-\tilde w)^T\gamma^*(\bar w)\geq0$ and thus $(\bar w-\tilde w)^T\Pi_W(\bar w,y)\leq(\bar w-\tilde w)^Ty$. Following the similar argument, one can obtain $(\mu-\tilde \mu)^T\Pi_{\mathbb{R}_+^l}(\mu,G(\bar w))\leq(\mu-\tilde\mu)^TG(\bar w)$.
By the affinity of $H$ and the convexity of $G$ (see Assumption \ref{asm: convexity}) and the first equation of \eqref{game KKT condition}, we have
\begin{align}
\label{middle 1}
&-(\bar w-\tilde w)^T\nabla_{\bar w}L(\bar w,\lambda,\mu)=-(\bar w-\tilde w)^T(\nabla F(\bar w)\nonumber\\
&+\sum\nolimits_{j=1}^m\lambda_j\nabla H_j(\bar w)+\sum\nolimits_{j=1}^l\mu_j\nabla G_j(\bar w)-\nabla F(\tilde w)\nonumber\\
&-\sum\nolimits_{j=1}^m\tilde\lambda_j\nabla H_j(\tilde w)-\sum\nolimits_{j=1}^l\tilde\mu_j\nabla G_j(\tilde w))\nonumber\\
&\leq-(\bar w-\tilde w)^T(\nabla F(\bar w)-\nabla F(\tilde w))-(\lambda-\tilde\lambda)^T\nonumber\\
&\quad(H(\bar w)-H(\tilde w))-(\mu-\tilde\mu)^T(G(\bar w)-G(\tilde w)).
\end{align}

By \eqref{game KKT condition}, we have $H(\tilde w)=0_m$, $G(\tilde w)\leq0_l$ and $\tilde\mu^TG(\tilde w)=0$. Since $\mu\in\mathbb{R}_+^l$, we have $\mu^TG(\tilde w)\leq0$. By \eqref{middle 1}, we have $-(\bar w-\tilde w)^T\nabla_{\bar w}L(\bar w,\lambda,\mu)\leq-(\bar w-\tilde w)^T(\nabla F(\bar w)-\nabla F(\tilde w))-(\lambda-\tilde\lambda)^TH(\bar w)-(\mu-\tilde\mu)^TG(\bar w)$.
Then, by strict monotonicity of $\nabla F$, we have
\begin{align}
\label{middle 2}
&\mathcal{L}V(\bar w,\lambda,\mu)=(\bar w-\tilde w)^T\Pi_W(\bar w,-\nabla_{\bar w}L(\bar w,\lambda,\mu))\nonumber\\
&\quad+(\lambda-\tilde\lambda)^TH(\bar w)+(\mu-\tilde\mu)^T\Pi_{\mathbb{R}_+^l}(\mu,G(\bar w))\nonumber\\
&\leq-(\bar w-\tilde w)^T\nabla_{\bar w}L(\bar w,\lambda,\mu)+(\lambda-\tilde\lambda)^TH(\bar w)+(\mu-\tilde\mu)^T\nonumber\\
&\quad G(\bar w)\leq-(\bar w-\tilde w)^T(\nabla F(\bar w)-\nabla F(\tilde w))\leq0.
\end{align}

By strict monotonicity of $\nabla F$, the equality sign of the last equation of \eqref{middle 2} holds if and only if $\bar w=\tilde w$. Since $(\bar w,\lambda,\mu)$ is an arbitrary triple in $W\times\mathbb{R}^m\times\mathbb{R}_+^l$, we have $\mathcal{L}V(\bar w,\lambda,\mu)\leq0$ for all $(\bar w,\lambda,\mu)\in W\times\mathbb{R}^m\times\mathbb{R}_+^l$.
\end{IEEEproof}

\begin{claim} \begin{rm}
\label{middle lemma existence and uniqueness}
For any starting point $(\bar w(0),\lambda(0),\mu(0))\in W\times\mathbb{R}^m\times\mathbb{R}_+^l$, a unique solution $\Gamma:[0,\infty)\to W\times\mathbb{R}^m\times\mathbb{R}_+^l$ of \eqref{decision update rule} exists and $\Gamma(t)\in (W\times\mathbb{R}^m\times\mathbb{R}_+^l)\cap V^{-1}(\leq V(\bar w(0),\lambda(0),\mu(0)))$ for any $t\geq0$.
 \end{rm} \hfill\rule{2mm}{2mm

\begin{IEEEproof}
By Assumption \ref{asm: convexity}, $\nabla F$ and $\nabla G$ are all locally Lipschitz on $W$. Since $H$ is affine in $w$, $\nabla H$ is Lipschitz on $W$. The proof of Claim \ref{middle lemma existence and uniqueness} then follows the same argument of the proof of Lemma 4.3 of \cite{AC-EM-JC:2016}.
\end{IEEEproof}

\begin{claim} \begin{rm}
\label{lemma invariance omega limit set}
The omega-limit set of any solution of \eqref{update rule standard form} is invariant under \eqref{update rule standard form}.
 \end{rm} \hfill\rule{2mm}{2mm

\begin{IEEEproof}
The proof of Claim \ref{lemma invariance omega limit set} follows the same argument of the proof of Lemma 4.1 of \cite{Khalil:02}.
\end{IEEEproof}

For any $\delta>0$, consider the compact set $S=V^{-1}(\leq \delta)\cap(W\times\mathbb{R}^m\times\mathbb{R}_+^l)$. By Claim \ref{middle lemma existence and uniqueness}, starting from any point in $S$, there exists a unique solution of \eqref{update rule standard form} and $S$ is invariant under \eqref{update rule standard form}. Furthermore, by Claim \ref{lemma invariance omega limit set}, the omega-limit set of each solution starting from any point in $S$ is invariant. Finally, by Claim \ref{middle lemma monotonicity of Lyapunov function}, $\mathcal{L}V(\bar w,\lambda,\mu)\leq0$ for all $(\bar w,\lambda,\mu)\in S$. By Lemma \ref{lemma invariance}, starting from any point in $S$, the solution of \eqref{update rule standard form} converges to the largest invariant set $M$ contained in ${\rm cl}(Z)$, where $Z=\{(\bar w,\lambda,\mu)\in S:\mathcal{L}V(\bar w,\lambda,\mu)=0\}$. By the proof of Claim \ref{middle lemma monotonicity of Lyapunov function}, $\mathcal{L}V(\bar w,\lambda,\mu)=0$ if and only if $\bar w=\tilde w$ and $\mu^TG(\tilde w)=0$. Hence, we have $Z=\{(\bar w,\lambda,\mu)\in S:\bar w=\tilde w,\;\mu^TG(\tilde w)=0\}$. It is clear that $Z$ is closed. Let $(\tilde w,\lambda,\mu)\in M\subseteq Z$. The solution of \eqref{update rule standard form} starting from $(\tilde w,\lambda,\mu)$ remains in $M$ only if $\nabla_{\bar w}L(\tilde w,\lambda,\mu)=0_r$. Together with $H(\tilde w)=0_m$, $G(\tilde w)\leq 0_l$ and $\mu^TG(\tilde w)=0$, we have that the triple $(\tilde w,\lambda,\mu)$ satisfies the KKT conditions \eqref{game KKT condition}. Hence, we have $M\subseteq Q$. Since $\delta$ is arbitrary, we have that $Q$ is globally asymptotically stable on $W\times\mathbb{R}^m\times\mathbb{R}_+^l$. Finally, by the definition of omega-limit set and Claim \ref{middle lemma monotonicity of Lyapunov function}, the omega-limit set of any solution of \eqref{update rule standard form} is a singleton. This implies that any solution of \eqref{update rule standard form} converges to a point in $Q$. This completes the proof of Theorem \ref{theorem auxiliary variable convergence}.
\end{IEEEproof}

Next we prove the convergence of the physical variables $(x,u)$. For each $i\in\mathcal{V}$, let $e_i=\hat x_i-\bar{\bar x}_i=\mathcal{T}_i(x_i-\bar x_i)$, $\hat e_i=S_i(\varepsilon)e_i$ and $e_i^v=v_i-\bar v_i=K_i(\varepsilon)e_i$. Let $(e,\hat e,e^v)=(e_i,\hat e_i,e_i^v)_{i=1}^N$. By the definition of $S_i(\varepsilon)$ and $\bar S_i(\varepsilon)$, we can obtain $\varepsilon {S_i(\varepsilon)}{\hat A_{ii}} = {\hat A_{ii}}{S_i(\varepsilon)}$, ${S_i(\varepsilon)}{\hat B_i} = {\varepsilon ^{{n_i} - 1}}{\hat B_i}$ and ${\varepsilon ^{{n_i} - 1}}{{\bar S}_i(\varepsilon)} = {S_i(\varepsilon)}$. Choose the Lyapunov function as ${V}(\hat e) = {\hat e^T}P_0\hat e$. We then have
$\dot V(\hat e) = 2\hat e^T P_0\dot {\hat {e}}=2\hat e^TP_0S(\varepsilon)\dot e= 2\hat e^TP_0S(\varepsilon)(\hat Ae+\hat Be^v+\hat A{\bar {\bar x}}+\hat B\bar v-\dot {\bar{\bar x}})$. Let $c=\hat A{\bar {\bar x}}+\hat B\bar v-\dot {\bar{\bar x}}$. Then we have ${\dot {V}}(\hat e)=2\hat e^TP_0S(\varepsilon)(\hat Ae+\hat Be^v)+2\hat e^TP_0S(\varepsilon)c$. First consider $2\hat e^TP_0S(\varepsilon)(\hat Ae+\hat Be^v)$. We have
\begin{align}
\label{Theorem 3 step 1}
&2\hat e^TP_0S(\varepsilon)(\hat Ae+\hat Be^v)=\sum\nolimits_{i = 1}^N [2\hat e_i^T{P_{0i}}{S_i}(\varepsilon )({\hat A_{ii}} \nonumber\\
&+ \frac{1}{\varepsilon }{\hat B_i}{K_{0i}}{{\bar S}_i}(\varepsilon )){e_i}]+ \sum\nolimits_{i = 1}^N {[2\hat e_i^T{P_{0i}}{S_i}(\varepsilon )\sum\nolimits_{j \ne i} {{\hat A_{ij}}{e_j}} ]} \nonumber\\
&=\frac{2}{\varepsilon }\sum\nolimits_{i = 1}^N {[\hat e_i^T{P_{0i}}({\hat A_{ii}} + {\hat B_i}{K_{0i}}){{\hat e}_i}]}+ \sum\nolimits_{i = 1}^N [2\hat e_i^T{P_{0i}}{S_i}(\varepsilon )\nonumber\\
&\sum\nolimits_{j \ne i} {{\hat A_{ij}}{e_j}} ]=2\hat e^TP_0S(\varepsilon)\hat A_{-ii}S^{-1}(\varepsilon)\hat e-\frac{1}{\varepsilon}{\| {\hat e} \|^2}
\end{align}
where $\hat A_{-ii}=\hat A-\mathrm{diag}(\hat A_{ii})_{i=1}^N$. For $P_0S(\varepsilon )\hat A_{-ii}S^{ - 1} (\varepsilon )$, the blocks at the diagonal positions are zero matrices and the block at the $i$-th row and $j$-th column position with $i\ne j$ is $P_{0i}S_i(\varepsilon)\hat A_{ij}S_j^{-1}(\varepsilon)$. By Assumption \ref{asm: S constraint}, for each $i,j\in \mathcal{V}$, $i\ne j$, $\|P_{0i}S_i(\varepsilon)\hat A_{ij}S_j^{-1}(\varepsilon)\|_1<\sigma_i\|P_{0i}\|_1$, which implies that the maximum absolute column sum for $P_{0i}S_i(\varepsilon)\hat A_{ij}S_j^{-1}(\varepsilon)$ is upper bounded by $\sigma_i\|P_{0i}\|_1$. Each column of $P_0S(\varepsilon )\hat A_{-ii}S^{ - 1} (\varepsilon )$ is composed of columns of $N-1$ blocks $P_{0i}S_i(\varepsilon)\hat A_{ij}S_j^{-1}(\varepsilon)$ at the off-diagonal position and one column of the zero matrix at the diagonal position.
Thus, we have ${\| P_0S(\varepsilon )\hat A_{-ii}S^{ - 1} (\varepsilon ) \|_1} \le (N-1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\}$. By 5.6.P5 of \cite{Horn}, the 2-norm of a matrix with $n$ columns is upper bounded by $\sqrt{n}$ times of its 1-norm. Hence, $\| P_0S(\varepsilon )\hat A_{-ii}S^{ - 1} (\varepsilon ) \| \le \sqrt{n}{\| P_0S(\varepsilon )\hat A_{-ii}S^{ - 1} (\varepsilon ) \|_1} \le \sqrt n (N-1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\}$. By \eqref{Theorem 3 step 1}, we have $2\hat e^TP_0S(\varepsilon)(\hat Ae+\hat Be^v)\le -\frac{1}{\varepsilon}{\| {\hat e} \|^2}+2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\}{\| {\hat e} \|^2}=- (\frac{1}{\varepsilon } - 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\}){\| {\hat e} \|^2}$. For any $\|\hat e\|\geq \|P_0\|\|c\|$, given $\|S(\varepsilon)\|=1$ for any $0<\varepsilon<1$, we have
\begin{align}
\label{Theorem 3 step 3}
& 2\hat e^TP_0S(\varepsilon)(\hat Ae+\hat Be^v)+2\hat e^TP_0S(\varepsilon)c\nonumber\\
&\!\leq 2\|\hat e\|\|P_0\|\|c\| - (\frac{1}{\varepsilon } - 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\}){\| {\hat e} \|^2}\nonumber\\
&\!\leq - (\frac{1}{\varepsilon } - 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\} ){\| {\hat e} \|^2}+2\|\hat e\|^2\nonumber\\
&\!\leq - (\frac{1}{\varepsilon } - 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\} -2){\| {\hat e} \|^2}.
\end{align}

By \eqref{Theorem 3 step 3}, by choosing $\varepsilon$ such that $\frac{1}{\varepsilon } > 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\} +2$, we have that, for any $\|\hat e\|\geq \|P_0\|\|c\|$, ${\dot {V}}(\hat e)\leq - (\frac{1}{\varepsilon } - 2\sqrt n (N - 1)\max_{i\in\mathcal{V}}\{\sigma_i\|P_{0i}\|_1\}-2){\| {\hat e} \|^2}<0$ for any $\| {\hat e} \|\neq0$. Since $P_0$ is symmetric, we have $\lambda_{\min}(P_0)\| {\hat e} \|^2\leq V(\hat e)\leq\lambda_{\max}(P_0)\| {\hat e} \|^2$, where $\lambda_{\min}(P_0)$ and $\lambda_{\max}(P_0)$ are the smallest and largest eigenvalues of $P_0$, respectively. Because $P_0$ is positive definite, $\lambda_{\min}(P_0)$ and $\lambda_{\max}(P_0)$ are both positive. View $\hat e$ as state and $c$ as input. Then, by Theorem 10.4.1 of \cite{AI:99}, the system is input-to-state stable. By Theorem 10.4.5 of \cite{AI:99}, we then have $\mathop {\lim \sup}\nolimits_{t \to \infty }\|\hat e(t)\|\leq \gamma (\mathop {\lim \sup}\nolimits_{t \to \infty }\|c(t)\|)$, where $\gamma(\cdot)$ is a class $\mathcal{K}$ function (\cite{AI:99}, Definition 10.1.1). In the convergence analysis of the auxiliary decision-making variables $(\bar x,\bar u)$, we have shown that $(\bar x(t),\bar u(t))$ asymptotically converge to the VE $(\tilde x,\tilde u$) which satisfies $A\tilde x+B\tilde u=0_n$. Thus, as $t\to \infty$, we have $A\bar x(t)+B\bar u(t)\to 0_n$ and $\dot {\bar x}(t)\to 0_n$, by which we can obtain $\hat A\bar{\bar x}+\hat B\bar v\to0_n$ and $\dot{\bar{\bar x}}(t)\to0_n$. Therefore, we have $\mathop {\lim \sup}\nolimits_{t \to \infty }\|c(t)\|=0$, which implies that $\mathop {\lim \sup}\nolimits_{t \to \infty }\|\hat e(t)\|=0$. Hence, $\hat e(t)$ asymptotically converges to zero which implies that $x(t)$ keeps track of $\bar x(t)$ asymptotically. Thus, $e^v(t)$ diminishes asymptotically which implies that $v(t)$ asymptotically keeps track of $\bar v(t)$, which further implies that $u(t)$ asymptotically keeps track of $\bar u(t)$. Hence, $(x(t),u(t))$ also asymptotically converge to the VE $(\tilde x,\tilde u)$. This completes the proof of Theorem \ref{theorem: Algorithm 1}.
\end{IEEEproof}

\subsection{Proof of Lemma \ref{lemma: coordinate transformation dynamics}\label{proof of lemma}}

In this subsection, we provide the proof of Lemma \ref{lemma: coordinate transformation dynamics}.

\begin{IEEEproof}
By the construction of $z_{i1},z_{i2},z_{i3}$ in Section \ref{section nonlinear algorithm design}, we see that the dynamics of $z_{i1}$ and $z_{i2}$ satisfy \eqref{transformed states dynamics}. Assume that the dynamics of $z_{i1},\cdots,z_{i(\ell-1)}$ satisfy \eqref{transformed states dynamics} for some $\ell\in\{3,\cdots,\bar n-1\}$. By \eqref{coordinate transformation} and the inductive assumption, we have
\begin{align*}
&\dot z_{i\ell}=(-1)^{\ell-1}T_{\ell-2,1}^i\dot z_{i2}+(-1)^{\ell-2}T_{\ell-3,2}^i\dot z_{i3}+\cdots+(-1)^2\nonumber\\
&T_{1,\ell-2}^i\dot z_{i(\ell-1)}+\frac{d^{\ell-1}}{dt^{\ell-1}}(k_1\nabla_{z_{i1}}L_i+\Gamma_{i1})+\frac{d^{\ell-2}}{dt^{\ell-2}}(x_{i2}\frac{d}{dt}\Theta_{i1}\nonumber\\
&+\Theta_{i1}\Gamma_{i2})+\cdots+\frac{d^2}{dt^2}(x_{i(\ell-2)}\frac{d}{dt}\Theta_{i,1\sim(\ell-3)}+\Theta_{i,1\sim(\ell-3)}\nonumber\\
&\Gamma_{i(\ell-2)})+\frac{d}{dt}(x_{i(\ell-1)}\frac{d}{dt}\Theta_{i,1\sim(\ell-2)}+\Theta_{i,1\sim(\ell-2)}\Gamma_{i(\ell-1)})\nonumber\\
&+x_{i\ell}\frac{d}{dt}\Theta_{i,1\sim(\ell-1)}+\Theta_{i,1\sim(\ell-1)}(\Gamma_{i\ell}+\Theta_{i\ell}x_{i(\ell+1)})\nonumber\\
&=(-1)^{\ell-1}T_{\ell-2,1}^i(z_{i3}-k_{i2}z_{i2})+(-1)^{\ell-2}T_{\ell-3,2}^i(z_{i4}-k_{i3}\nonumber\\
&z_{i3})+\cdots+(-1)^2T_{1,\ell-2}^i(z_{i\ell}-k_{i(\ell-1)}z_{i(\ell-1)})+\frac{d^{\ell-1}}{dt^{\ell-1}}\nonumber\\
&(k_1\nabla_{z_{i1}}L_i+\Gamma_{i1})+\frac{d^{\ell-2}}{dt^{\ell-2}}(x_{i2}\frac{d}{dt}\Theta_{i1}+\Theta_{i1}\Gamma_{i2})+\cdots+\nonumber\\
&\frac{d^2}{dt^2}(x_{i(\ell-2)}\frac{d}{dt}\Theta_{i,1\sim(\ell-3)}+\Theta_{i,1\sim(\ell-3)}\Gamma_{i(\ell-2)})+\frac{d}{dt}(x_{i(\ell-1)}\nonumber\\
&\frac{d}{dt}\Theta_{i,1\sim(\ell-2)}+\Theta_{i,1\sim(\ell-2)}\Gamma_{i(\ell-1)})+x_{i\ell}\frac{d}{dt}\Theta_{i,1\sim(\ell-1)}\nonumber\\
&+\Theta_{i,1\sim(\ell-1)}(\Gamma_{i\ell}+\Theta_{i\ell}x_{i(\ell+1)})=-k_{i\ell}z_{i\ell}+z_{i(\ell+1)}.
\end{align*}

Thus, the dynamics of $z_{i\ell}$ also satisfies \eqref{transformed states dynamics}. By induction, the dynamics of $z_{i\ell}$ satisfies \eqref{transformed states dynamics} for all $\ell=1,\cdots,\bar n-1$. Finally, one can expand the dynamics of $z_{i\bar n}$ as above and, by \eqref{control law for nonlinear dynamics}, obtain $\dot z_{i\bar n}=-k_{i\bar n}z_{i\bar n}$.
\end{IEEEproof}

\subsection{Proof of Theorem \ref{theorem: Algorithm 2}\label{proof of theorem 2}}

This subsection proves Theorem \ref{theorem: Algorithm 2}.

\begin{IEEEproof} The dynamics of $(z,\lambda,\mu)$ consist of \eqref{transformed states dynamics} and \eqref{nonlinear dual update}. Let $\nabla_{z^{[1]}}L(z^{[1]},\lambda,\mu)=(\nabla_{z_{i1}}L_i(z^{[1]},\lambda,\mu))_{i=1}^N$. Under Assumptions \ref{asm: convexity}--\ref{asm: Slater condition}, by Section 12.2.3 of \cite{PE:10}, $\tilde x^{[1]}\in \mathbb{R}^N$ is the VE of the GNEP$(\mathcal{D},F)$ if and only if there exist $(\tilde \lambda,\tilde\mu)\in\mathbb{R}^m\times\mathbb{R}_+^l$ such that the KKT conditions hold: $\nabla_{z^{[1]}}L(\tilde x^{[1]},\tilde\lambda,\tilde\mu)=0_N$, $H(\tilde x^{[1]})=0_m$, $G(\tilde x^{[1]})\leq0_l$, and $\tilde \mu^{T}G(\tilde x^{[1]})=0$.
Denote by $Q$ the set of $(\tilde x^{[1]},\tilde\lambda,\tilde\mu)$ satisfying the KKT conditions. Let $\tilde z_{i1}=\tilde x_{i1}$, $\tilde z_{i\ell}=0$ for all $\ell=2,\cdots,\bar n$, $y=[z^T,\lambda^T,\mu^T]^T$ and $\tilde y=[\tilde z^T,\tilde\lambda^T,\tilde\mu^T]^T$ with $(\tilde x^{[1]},\tilde\lambda,\tilde\mu)$ being any point of $Q$.
Choose the Lyapunov function as $V(y)=\frac{1}{2}\|y-\tilde y\|^2$. Following the same arguments, one can show that counterparts of Claim \ref{middle lemma existence and uniqueness} and Claim \ref{lemma invariance omega limit set} hold for \eqref{transformed states dynamics} and \eqref{nonlinear dual update}. To apply Lemma \ref{lemma invariance}, we are left to prove a counterpart of Claim \ref{middle lemma monotonicity of Lyapunov function}. By the proof of Claim \ref{middle lemma monotonicity of Lyapunov function}, we can obtain $(\mu-\tilde \mu)^T\Pi_{\mathbb{R}_+^l}(\mu,k_1G(z^{[1]}))\leq(\mu-\tilde\mu)^Tk_1G(z^{[1]})$. By Lemma \ref{lemma: coordinate transformation dynamics}, we have
\begin{align}
\label{dot V}
&\mathcal{L}V(y)=\sum\nolimits_{i\in\mathcal{V}}[(z_{i1}-\tilde z_{i1})\dot z_{i1}+z_{i2}\dot z_{i2}+\cdots+z_{i\bar n}\dot z_{i\bar n}]\nonumber\\
&+(\mu-\tilde\mu)^T\dot\mu+(\lambda-\tilde\lambda)^T\dot\lambda\nonumber\\
&\leq\sum\nolimits_{i\in\mathcal{V}}[(z_{i1}-\tilde z_{i1})(z_{i2}-k_{1}\nabla_{z_{i1}}L_i(y))+z_{i2}(z_{i3}-k_{i2}\nonumber\\
&z_{i2})+\cdots+z_{i(\bar n-1)}(z_{i\bar n}-k_{i(\bar n-1)}z_{i(\bar n-1)})-z_{i\bar n}k_{i\bar n}z_{i\bar n}]\nonumber\\
&+(\mu-\tilde\mu)^Tk_1G(z^{[1]})+(\lambda-\tilde\lambda)^Tk_1H(z^{[1]})\nonumber\\
&=-\sum\nolimits_{i\in\mathcal{V}}k_1(z_{i1}-\tilde z_{i1})\nabla_{z_{i1}}L_i(y)+k_1(\mu-\tilde\mu)^TG(z^{[1]})\nonumber\\
&+k_1(\lambda-\tilde\lambda)^TH(z^{[1]})+\sum\nolimits_{i\in\mathcal{V}}[-k_{i2}z_{i2}^2-\cdots-k_{i\bar n}z_{i\bar n}^2\nonumber\\
&+(z_{i1}-\tilde z_{i1})z_{i2}+z_{i2}z_{i3}+\cdots+z_{i(\bar n-1)}z_{i\bar n}].
\end{align}
By the affinity of $H$ and the convexity of $G$ and the KKT condition $\nabla_{z^{[1]}}L(\tilde x^{[1]},\tilde\lambda,\tilde\mu)=0_N$, we can derive
\begin{align*}
&-\sum\nolimits_{i\in\mathcal{V}}(z_{i1}-\tilde z_{i1})\nabla_{z_{i1}}L_i(y)\nonumber\\
&\leq-(z^{[1]}-\tilde z^{[1]})^T(\nabla F(z^{[1]})-\nabla F(\tilde z^{[1]}))-(\lambda-\tilde\lambda)^T\nonumber\\
&\quad(H(z^{[1]})-H(\tilde z^{[1]}))-(\mu-\tilde\mu)^T(G(z^{[1]})-G(\tilde z^{[1]})).
\end{align*}
Then, by strong monotonicity of $\nabla F$, we have
\begin{align}
\label{dot V z1}
&-\sum\nolimits_{i\in\mathcal{V}}k_1(z_{i1}-\tilde z_{i1})\nabla_{z_{i1}}L_i(y)+k_1(\mu-\tilde\mu)^TG(z^{[1]})\nonumber\\
&+k_1(\lambda-\tilde\lambda)^TH(z^{[1]})\nonumber\\
&\leq-k_1(z^{[1]}-\tilde z^{[1]})^T(\nabla F(z^{[1]})-\nabla F(\tilde z^{[1]}))\nonumber\\
&\leq-k_1M\sum\nolimits_{i\in\mathcal{V}}(z_{i1}-\tilde z_{i1})^2.
\end{align}

For each $i\in\mathcal{V}$, let $\hat z_{i1}=z_{i1}-\tilde z_{i1}$. Given that $k_1,k_{i2},\cdots,k_{i\bar n}$ are chosen such that $k_1M>1,k_{i2}>1,\cdots,k_{i\bar n}>1$, plugging \eqref{dot V z1} into \eqref{dot V} yields
\begin{align}
\label{dot V overall}
&\mathcal{L}V(y)\leq-k_1M\sum\nolimits_{i\in\mathcal{V}}\hat z_{i1}^2+\sum\nolimits_{i\in\mathcal{V}}[-k_{i2}z_{i2}^2-\cdots-k_{i\bar n}z_{i\bar n}^2\nonumber\\
&+\hat z_{i1}z_{i2}+z_{i2}z_{i3}+\cdots+z_{i(\bar n-1)}z_{i\bar n}]\nonumber\\
&\leq\sum\nolimits_{i\in\mathcal{V}}[-k_1M\hat z_{i1}^2-k_{i2}z_{i2}^2-\cdots-k_{i\bar n}z_{i\bar n}^2+\frac{\hat z_{i1}^2+z_{i2}^2}{2}\nonumber\\
&+\frac{z_{i2}^2+z_{i3}^2}{2}+\cdots+\frac{z_{i(\bar n-1)}^2+z_{i\bar n}^2}{2}]\nonumber\\
&\leq-\sum\nolimits_{i\in\mathcal{V}}[(k_1M-1)\hat z_{i1}^2+(k_{i2}-1)z_{i2}^2+\cdots\nonumber\\
&+(k_{i\bar n}-1)z_{i\bar n}^2]\leq0.
\end{align}

This proves a counterpart result of Claim \ref{middle lemma monotonicity of Lyapunov function} and we are ready to apply Lemma \ref{lemma invariance}. For any $\delta>0$, consider the compact set $S=V^{-1}(\leq \delta)\cap(\mathbb{R}^n\times\mathbb{R}^m\times\mathbb{R}_+^l)$. By Lemma \ref{lemma invariance}, starting from any point in $S$, the solution of \eqref{transformed states dynamics} and \eqref{nonlinear dual update} converges to the largest invariant set $M$ contained in ${\rm cl}(Z)$, where $Z=\{(z,\lambda,\mu)\in S:\mathcal{L}V(z,\lambda,\mu)=0\}$. By the above proof, we have $\mathcal{L}V(z,\lambda,\mu)=0$ if and only if $z=\tilde z$ and $\mu^TG(\tilde z^{[1]})=0$. Hence, $Z=\{(z,\lambda,\mu)\in S:z=\tilde z,\;\mu^TG(\tilde z^{[1]})=0\}$. It is clear that $Z$ is closed. Let $(\tilde z,\lambda,\mu)\in M\subseteq Z$. The solution of \eqref{transformed states dynamics} and \eqref{nonlinear dual update} starting from $(\tilde z,\lambda,\mu)$ remains in $M$ only if $\nabla_{z^{[1]}}L(\tilde z^{[1]},\lambda,\mu)=0_N$. Together with $H(\tilde z^{[1]})=0_m$, $G(\tilde z^{[1]})\leq 0_l$ and $\mu^TG(\tilde z^{[1]})=0$, we have that the triple $(\tilde z^{[1]},\lambda,\mu)$ satisfies the KKT conditions. Hence, we have $M\subseteq Q$. By the definition of omega-limit set and \eqref{dot V overall}, the omega-limit set of any solution of \eqref{transformed states dynamics} and \eqref{nonlinear dual update} is a singleton. Since $\delta$ is arbitrary, we have that, starting from any point in $\mathbb{R}^n\times\mathbb{R}^m\times\mathbb{R}_+^l$, the solution of \eqref{transformed states dynamics} and \eqref{nonlinear dual update} converges to a point $(\tilde z,\tilde\lambda,\tilde\mu)$ with $(\tilde z^{[1]},\tilde\lambda,\tilde\mu)$ being a point in $Q$ and $\tilde z^{[\ell]}=0_N$ for all $\ell=2,\cdots,\bar n$.

We now check the coordinate transformation \eqref{coordinate transformation}. Since $z_{i1}=x_{i1}$ for each $i\in\mathcal{V}$, we have that $x^{[1]}$ asymptotically converges to the VE $\tilde x^{[1]}$. Next, since $\tilde z_{i2}=0$ and $\nabla_{z_{i1}}L_i(\tilde y)=0$, by the coordinate transformation of $z_{i2}$, we have $\Theta_{i1}(\tilde x^{[1]})\tilde x_{i2}+\Gamma_{i1}(\tilde x^{[1]})=0$. Notice that this is just the steady state condition for the dynamics of $x_{i1}$. We next show by induction that, for each $\ell=2,\cdots,\bar n$, $x_{i\ell}$ asymptotically converges to $\tilde x_{i\ell}$ which is the unique steady state of $x_{i\ell}$ given $\tilde x^{[1]}$. We have proved the case of $\ell=2$. Assume that this argument holds for all $\ell$ from $2$ up to $k$ for some $k\in\{2,\cdots,\bar n-1\}$. For $\ell=k+1$, we check the coordinate transformation of $z_{i\ell}$ given by \eqref{coordinate transformation}. Notice that $\tilde z_{i2}=\cdots=\tilde z_{i(k+1)}=0$. Also notice that on the right-hand-side of the equation, all the terms with the derivative operation $\frac{d}{dt}$ only depend on $x^{[1]},\cdots,x^{[k]}$, which, by assumption, asymptotically converge to fixed values $\tilde x^{[1]},\cdots,\tilde x^{[k]}$. Thus, the derivatives asymptotically converge to zero. Therefore, as $t\to\infty$, we have $\Theta_{i,1\sim(\ell-2)}\Gamma_{i(\ell-1)}+\Theta_{i,1\sim(\ell-1)}x_{i\ell}\to0$, which further implies $\Gamma_{i(\ell-1)}+\Theta_{i(\ell-1)}x_{i\ell}\to0$. This is just the steady state condition for the dynamics of $x_{i(\ell-1)}$. Thus, the argument holds for $\ell=k+1$. By induction, the argument holds for all $\ell=2,\cdots,\bar n$. Thus, $x^{[2]},\cdots,x^{[\bar n]}$ asymptotically converge to $\tilde x^{[2]},\cdots,\tilde x^{\bar n}$ such that, together with $\tilde x^{[1]}$, $\tilde x$ is a steady state of system \eqref{nonlinear state dynamics}.
\end{IEEEproof}

\section{Conclusion}

We study two complementary cases of distributed optimization-based control problems: linear systems with controllable subsystems and nonlinear systems in the strict-feedback form. The proposed algorithms are verified by two case studies on an optimal power flow problem and a multizone building temperature regulation problem, respectively.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Q: Save and Load Data from Amazon Cloud (AWS) I have following data model, and i am saving and loading data with userdefaults in Swift (Local) successfully. However, instead of locally, i want to save and load data directly from AWS Cloud. Which service i have to use; Cognito User Sync data or DynamoDB or S3?? I have only checked Cognito user sync, i think that's only for strings.
While considering the method, the following requirements must be met:
a) all data i want to collect in one table to summarise values
b) each user will retrieve their own data from there.
c) i need that each user has to share his own data with another user, is it possible? i will be glad if you let me know about which method i have to follow.
    static var networks = [Network]()

    struct Network: Codable {

            var label: Label
            let topic: String
            var rssi: String
            var ssid: String
            var IP: String
            var MAC: String
            var result: String

            init(label lbl: Label, topic: String, rssi: String, ssid: String, IP: String, MAC: String, result: String) {
                self.label = lbl
                self.topic = topic
                self.rssi = rssi
                self.ssid = ssid
                self.IP = IP
                self.MAC = MAC
                self.result = result
            }
    }
Local save & load functions;

    static func saveDefaults() {
          let encodedData = try? JSONEncoder().encode(networks)
          UserDefaults.standard.set(encodedData, forKey: "network")
    }

    static func loadDefaults() {
          if let decoded = UserDefaults.standard.object(forKey: "network") {
                networks = try! JSONDecoder().decode([NETWORK].self, from: decoded as! Data)
          }
    }

redpajama_set_name: RedPajamaStackExchange


text:
Home News Covid-19: 520 new cases, four deaths, 961 recoveries Friday in QatarJuly 10...
Covid-19: 520 new cases, four deaths, 961 recoveries Friday in QatarJuly 10 2020 03:10 PM
By BenArmani -
The Ministry of Public Health (MoPH) Friday announced 520 new confirmed cases of the coronavirus disease (Covid-19), 4 deaths and 961 recoveries from the infection during the past 24 hours in Qatar, bringing the total number of deaths to 146 and recoveries to 98232.
The Ministry said the deceased patients, who were aged 30, 57, 70 and 91 respectively, were all receiving the necessary medical care.
All new cases have been introduced to isolation and are receiving necessary healthcare according to their health status, MoPH added.
The Ministry also pointed put that the measures to tackle Covid-19 in Qatar have succeeded as is evident from the flattening of the graph of the daily count of infections and the reduced number of new daily cases and hospital admissions.
It is the proactive and extensive testing of suspected cases that enabled Qatar
to identify a high number of positive cases in the community, the Ministry added. Qatar has one of the lowest Covid-19 death rates in the world, thanks to the factors like country's young population, proactive testing that leads to early detection of the cases, expanded hospital capacity, especially the intensive care facilities.
MoPH also cautioned that, the lifting of restrictions and the reduction in the number of infections do not mean that the pandemic is ended in Qatar. It should be kept in mind that each day between 50 and 100 people are being admitted to hospitals with moderate to severe Covid-19 symptoms.
Unless all precautionary measures are strictly followed a second wave of the virus and a surge in the number of cases can not be ruled out, the Ministry said. Exactly the same scenario can be seen in other countries around the world, MoPH reminded.
The Ministry also said that, now more than ever, all must remain careful and protect the most vulnerable in the society.
Even as the Covid-19 restrictions are being gradually lifted in Qatar, it is important for everyone to play their role in controlling the virus by following precautionary measures: ?Adhering to physical distancing, wearing face masks and washing hands regularly. Most importantly, it is vital that the elderly and those with chronic medical conditions are continue to be protected.
The Ministry also said anyone suffering from Covid-19 symptoms should either contact 16000 helpline or go directly to one of the designated health centers to undergo the necessary checks: Muaither, Rawdat Al Khail, Umm Slal, and Al Gharafa. This is important as the earlier the disease is detected the easier it will be to receive the right treatment and recover from it.
The MoPH also recommended visiting this section of its website for regular updates and new information on Covid-19.?
SOURCEGulf Times
BenArmani
Is it possible expired QID holders return back to Qatar?(latest update)
HOW TO CONVERT QATAR FAMILY VISIT VISA TO FAMILY RESIDENCE VISA
getting a work visa in Qatar
Airport Infrastructure – Everything You Need To Know
Mermaid-pop musician Gedina talks new single and creating while in quarantine
Ten famous people from Qatar
Editor of WQ - 5 October, 2015
QCB witnesses robust growth in official reserves, assets in Feb
The Ministry of Labor issues a guide to amend the profession(latest updates)
The Supreme Committee for Delivery and Legacy sheds light on the sustainability of the Qatar 2022 World Cup
The digital transformation reinforces the drivers of the real estate sector
Travel to Qatar is subject to entry restrictions???(LATEST UPDATE VISAS & IMMIGRATION)
Request Assistance from the Ministry of Foreign Affairs
The conclusion of the eighth annual international conference of the Center for the Study of Islamic Legislation and Ethics

redpajama_set_name: RedPajamaCommonCrawl


Scooter is a Siamese cat that was recently named by the Guinness Book of World Records as the oldest living cat in the world. Scooter was 30 years old when named by Guinness as the oldest living cat. Scooter died a couple weeks after his 30th birthday.
Guinness World Records' report says Scooter was owned by Gail Floyd. He was with her since the day he was born as a kitten.
We know Gail is really going to miss her friend Scooter. He lived a very long and eventually life. He traveled to 45 of the 50 states. 30 years is a very long life for a cat. Guinness says the longest living cat ever was Creme Puff of Austin, TX who lived to be 38.


text:
Is ASCD Embracing Market-Driven Education Reform?
Federal Opinion
By Anthony Cody — December 05, 2013 6 min read
Anthony Cody
Another day, another headline on ASCD's Smartbrief extolling some new Common Core strategy. ASCD has signed on as an endorser of Common Core, and in 2011, the Gates Foundation awarded ASCD a grant of $3 million "to provide teachers and school leaders with supports to implement the Common Core State Standards at the district, school, and classroom levels." In July of this year, ASCD won an additional $244,000, "to support implementation of the Common Core State Standards."
I have generally positive feelings towards ASCD. I recall in 2010, when ASCD's Gene Carter wrote a public letter in response to Oprah's show on Waiting for Superman. He wrote:
As a career educator and the executive director of ASCD, an education association of 160,000 educators worldwide, I was dismayed that your show on education reform excluded a key demographic from the dialogue: teachers. Yet the research---and your high-profile guests---say a child's teacher is the most important factor to determining his or her success. Moreover, simplistically dividing a profession of 5 million people into "good teachers" and "bad teachers" misses an important opportunity to show how all educators must continue to learn, develop, and grow throughout their careers.
In the more distant past, Educational Leadership, ASCD's journal, carried articles like this one, detailing the problems with the high stakes testing paradigm. And Educational Leadership continues to feature strong explorations of issues like teacher evaluation and professional development, including this article I contributed recently on the subject of teacher leadership.
Their coverage of the controversies regarding Common Core have been a bit less balanced. Their "Common Core State Standards Myths and Facts" perpetuates the idea that state adoption of the standards was "voluntary" when we all know adoption has been the result of a combination of RttT carrots and NCLB waiver sticks. The article presents a host of other glowing promises about Common Core that neither ASCD nor anyone else can deliver upon.
Educational Leadership has treated the controversy over Common Core literacy instruction in a one-sided fashion, by presenting this article by Timothy Shanahan, derisively titled The Common Core Ate My Baby and Other Urban Legends. There are respected literacy experts who would disagree with his stance, but their views were not shared, except to be dismissed by him as nonsense. The journal has also published articles expressing skepticism such as this one by Tom Loveless, The Common Core Initiative: What are the Chances of Success. But the thrust of the journal and work of ASCD has become dominated by the push to implement Common Core.
I was also a bit stunned last week when I saw the program for ASCD's Leadership Institute for Legislative Action, and their choice for this year's keynote speaker; Rick Hess of the American Enterprise Institute.
This raises some questions for me as to the direction the ASCD is taking in its legislative advocacy. This is not to slight Rick Hess. He writes a thought-provoking blog here at Education Week, and often has interesting things to say. But he represents the American Enterprise Institute, and is a strong advocate of market-based reform. Earlier this year, Hess wrote about the conservative school board in Douglas County, Colorado, describing the place as "the most interesting school district in America." He wrote:
The district's distinctive aim of going from good to great, rather than from poor to passable, is remarkable in the annals of contemporary school reform. For Douglas County, school choice is not seen not as a "ticket out" of failing schools, but a way to encourage customization and to offer more paths for students to choose.
It came out (and Hess subsequently acknowledged in his blog) that Hess had been paid a substantial fee for consulting with the district he was praising. This article from Our Castle Rock News reveals how much the District got for their money:
AEI's Rick Hess and Max Eden wrote a paper, "The Most Interesting School District in America," that the district emailed to parents Sept. 18 as a "just-released white paper" and did not identify as a district-paid product. The Feb. 6, 2013, contract between DCSD and AEI outlines a $30,000 payment and scope of services, asking the organization to "research, create, publish and publicize" a 25-30 page white paper with three to five sidebars. DCSD's requirements for the paper included a description of the district, the problems its reforms are meant to address, how the reforms are "new and different," district challenges in the face of its reforms and lessons learned. The district's lead spokeswoman and its foundation director, Cinamon Watson, signed as DCSD's representative. Hess and Watson also exchanged emails about the paper in which the writer asked for further guidance. "Ideally, we would love for you all to help us help you," Hess wrote in a March 22 email to Watson. "Rather, we would prefer it if you would tell us what you want us to focus on, what is most worthy of attention, what you'd like to see written about, and what your general angle on it and the paper is."
Remember, this paper was commissioned and released in the context of a contentious election in which conservative "reformers" were challenged by candidates less supportive of vouchers and other market-based reforms. Apparently this is the way the free market works in education policy writing.
When I tweeted a question, asking if the choice of Hess as keynote reflected an embrace of free market education reform, ASCD responded: "Hosting speakers isn't endorsement of their positions. ASCD strives to provide members with diverse viewpoints & opinions." This response is supported by the fact that the speaker at last year's event was Diane Ravitch, who no doubt delivered a very different message to the group.
I spoke with David Griffith, ASCD's Director of Public Policy, and he said:
We are supporters of public education. Just because we have someone speak does not mean we agree with them. Hess is a compelling speaker, and has provocative things to say. There is a privatization movement, so it behooves us to hear from them. Conference participants will be visiting Capital Hill, and the House is controlled by Republicans. It is not that we're supporting one view or another, we want them be prepared and understand what the arguments are around this.
ASCD is free to invite whomever they wish to speak to their organization, but at this moment in history, when the future of public education hangs in the balance, the choice of an outspoken advocate of privatization is a bit disturbing.
Taking a look at the actual 2013 policy agenda of the ASCD, there is likewise reason for concern. The document says that the organization will be advocating:
Support meaningful accountability systems - Any comprehensive determination of student proficiency, school quality, or educator effectiveness must take into account student growth, use multiple measures of evaluation beyond standardized test scores, and differentiate among levels of performance.
We are seeing how this "multiple measures" language translates into practice. As much as 50% of a teacher's evaluation may be based on highly unstable "value added" scores drawn from standardized tests. Evaluations go "beyond" this, by including administrator observations. And did we mention - the administrator's evaluations are likewise required to include test scores.
Everyone - even Michelle Rhee - will use this "multiple measures" language, to distance themselves from a straw man they are implicitly creating - an evaluation system based ONLY on test scores. Such systems have never existed, so far as I know. But you do not have to make test scores 100% of an evaluation scheme to make them hugely consequential.
Common Core is illuminating the way our democratic processes have been systematically bypassed by the combined efforts of big philanthropies and the federal gevernment. Race to the Top was used to get state officials to sign on to Common Core with little discussion, and apparently professional organizations such as ASCD have simply accepted this as a fait accompli, forgoing any responsibility for real debate.
Organizations like ASCD are being directly paid to "support implementation" of Common Core, which in effect converts them into advocates for the controversial standards. These organizations are also directly benefitting from the bonanza associated with professional development and curriculum made necessary by the shift to the Common Core standards. Is the organization capable of taking an independent stance, once it has accepted grants such as these?
The ASCD's Legislative agenda is silent on the issue of privatization of public education, or on the expansion of vouchers. It is silent on the spread of low-quality virtual charter schools. It is silent about the expansion of testing that is coming with the new Common Core tests, or the diversion of billions of education dollars into technology these tests will require.
I have perhaps a fanciful vision of what professional organizations like ASCD should be. In the absence of genuine public debate about the adoption of the Common Core, I wish that our professional organizations would create space for that debate to occur. Of course that is hard to do once you have accepted millions of dollars to promote the project!
As the very institution of public education is threatened by the push to privatize, I wish that ASCD would provide some real debate over this trend, and the effects of market-driven systems on public schools. I would like to see organizations like ASCD step forward as advocates for policies that serve all children, and against the transformation of schools into profit centers. But choosing a keynote speaker from the American Enterprise Institute will not deliver this debate.
As we step forward into implementation of Common Core tests, I think it is the absolute responsibility of leaders of organizations like ASCD, who have the expertise to understand the terrible effects that high stakes tests have on children, to take a clear and public stand against the consequences attached to them. That includes their use as one of "multiple measures" of teacher performance, and as justification for the closing of low-scoring schools. Otherwise the transition to Common Core will have accomplished one of its tacit goals - the conversion of critics of NCLB into backhanded promoters of the next generation of high stakes tests.
I realize this is opening a tough conversation, but I think it is an important one for our profession at this juncture. I extend an invitation to anyone from ASCD to respond, and will be happy to post it.
What do you think? Was ASCD wise to endorse Common Core and engage in promoting the project? Is there reason to be concerned about their choice of keynote speaker this year?
Continue the dialogue with Anthony on Twitter.
Common Core Standardized Testing Race to the Top Gates Foundation Department of Education No Child Left Behind Act
The opinions expressed in Living in Dialogue are strictly those of the author(s) and do not reflect the opinions or endorsement of Editorial Projects in Education, or any of its publications.


The New England Fishery Management Council has POSTPONED two groundfish meetings.
Groundfish Committee, Tuesday, March 27, 2018: This meeting also is being RESCHEDULED to a date in early May.
Both meetings were intended to focus on Groundfish Monitoring Amendment 23, which currently is under development. However, the technical analyses related to this amendment are not ready for discussion at either meeting. New meeting dates will be announced soon on the Council's Northeast Multispecies Webpage.
WHAT HAPPENED: Staff from both the New England Council and the National Marine Fisheries Service (NMFS – NOAA Fisheries) have been working at full strength to complete two critical actions that need to be implemented in time for the May 1 start of the 2018 groundfish fishing year.
Framework Adjustment 57 to the Northeast Multispecies Fishery Management contains catch limits and other fishery specifications and measures for 2018. NMFS published the proposed rule for this action on March 22. It's available at comment now on Framework 57.
Fishing Year 2018 Recreational Management Measures for Gulf of Maine cod and haddock and Georges Bank cod also were published on March 22 in separate proposed rule. It's available at comment now on 2018 recreational cod/haddock measures.
Completion of these two actions took top priority, and now work will resume on Groundfish Monitoring Amendment 23.
ADDITIONAL GROUNDFISH ACTIONS: Here are several other important groundfish-related actions that stakeholders should be aware of.
Groundfish Sectors: On March 23, NMFS published a proposed rule containing: (a) 2018 annual catch entitlements (ACE) to groundfish sectors; and (b) a new sector exemption pertaining to day gillnet vessels fishing in the Gulf of Maine. Details are available at comment now on proposed sector ACE and day gillnet exemption.
Yellowtail Flounder: On March 21, NMFS transferred 30 metric tons of unused quota of Southern New England/Mid-Atlantic yellowtail flounder from the Atlantic sea scallop fishery to the commercial groundfish fishery. Learn more at yellowtail flounder transfer.
Groundfish Charter/Party Control Date: As a reminder, NMFS published a new control date for the Northeast multispecies charter/party fishery. The new control date is March 19, 2018. Specifics can be found at comment now on charter/party control date.
Juvenile cod: NMFS's Greater Atlantic Regional Fisheries Office has written a feature story on the Council's Habitat Area of Particular Concern (HAPC) for inshore juvenile Atlantic cod. The HAPC went into effect January 3, 2018. Learn more at juvenile cod HAPC.
MORE INFORMATION: Visit "Related News," "Upcoming Meetings," and "Past Meetings" on the Council's groundfish webpage for press releases and documents associated with upcoming and past meetings. Go to stay in touch.


Table topper Cape Town Blitz will be hosting bottom-placed Durban Heat in their last group encounter at Cape Town on Sunday night.
With 6 wins in 9 encounters, Cape Town Blitz has advanced into the next round and would be eyeing to book a place directly into the finals by remaining at the top position through another win.
Durban Heat, on the opposite, has nothing left but to play for pride. in eight encounters, they have secured a single win coming off under DLS methods against Tshwane Spartans in the second encounter.
Cape Town Blitz and Heat met at Durban for the first time where Blitz turned out to be the winner by 3 wickets. Put into bat, Heat posted 157-5 with the help of Khaya Zondo's unbeaten 49 and Albie Morkel's 45 of 25 deliveries.
In response, Blitz chased down the target with three wickets in hands and 5 balls to spare with the help of Asif Ali's 33 balls unbeaten 88 runs knock.
The probable CTB vs DUR playing 11 is listed below.
Note- The CTB vs DUR dream11 match preview provided here is based on our research and analysis. We suggest you pick your CTB vs DUR playing 11 and make your own CTB vs DUR Dream11 team. Also, hit the bell icon to never miss an important update from us.


Interlibrary Loan is a privilege extended to the PSU borrowing community. It is designed to open up access to material that is not owned locally, and by nature, we do not ask to borrow material from outside sources if it is owned in the community. In these cases you should place a hold for the material at the Circulation desk. This includes material that is currently checked out. Faculty and graduate students have semester checkout, however the due date is shortened if a hold has been placed. Please see a representative at the Circulation desk for more information.
Unfortunately, we cannot tell you ahead of time when your material will be available. Items arrive at different times from different libraries, and in many cases more than one library must be tried before the request is filled.
Libraries are chosen, and your request is sent out typically within one business day. It could take a bit longer during peak times of the semester.
Once your request has been sent, it usually takes one or two days for photocopied articles to start arriving. Many are delivered online, and they are forwarded directly into your ILLiad account.
Books take at least one week before they start to come in. These arrive in the mail or other delivery system.
The majority of our material is obtained at no cost to you. We have a subsidized charge of $10.00 for items obtained from libraries that charge us for their service, and you have the option to approve those charges in advance. If your request states "Free Only", we'll only try free libraries, and return the request to you for approval if we are limited to libraries that charge for their Interlibrary Loan services.
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That is strictly up to the owning library. The average loan period is three weeks, and some libraries will renew one time. Renewals may be requested from within your account starting five days before your due date. You may want to find another source if you need a book for a semester long project.
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The first and second days after the due date are $.50 each. The third and following days are $1.00 per day.We do charge fines even when the Library is closed, and on holidays. Please return borrowed items directly to the Interlibrary Loan desk. If the Library is closed, you may place them in the outside book return.
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When items arrive, ILLiad automatically sends an e-mail to the contact address you list in your account. Check your spam filters, we can't get around those. Many people set up an ILLiad subdirectory, and have the e-mails filtered into that, so new messages are easy to spot. E-mails come from "ill@library.pittstate.edu", "ill_borrowing@library.pittstate.edu", and "rsamford@pittstate.edu".
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Bring materials back, or request renewals if they are allowed, on or before the due date. There is a paper band on the front of each item, please leave it there, as it identifies the material as an Interlibrary Loan, and not Axe material. We want to be able to properly credit you for the return of all material.
Be specific in what you request. The Reference department is great at helping narrow down your sources.
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Each lending library sets the conditions concerning the loan of its property. You must abide by those conditions, but options may be available. Books may be requested from a second library if necessary, and if requested early enough, a second copy might arrive before the first loan must be returned.
Lenders have the right to recall their material at any time if requested by their own patrons.
What is not available on ILL?
Textbooks: Textbooks and other material currently sold for courses offered by PSU or other institutions are not obtained on Interlibrary Loan. Those campus bookstores can supply the material needed.
Material owned locally: Books owned by Axe or other Pittsburg libraries should be requested through the Axe Circulation desk. Items that are checked out will be recalled for you. Material at the Pittsburg Public Library may be checked out directly. PSU students are not charged a fee to obtain a Public Library checkout card.
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Exceptions to these restrictions can be made by conferring with interlibrary loan and demonstrating a research or other academic need.
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Requests are canceled for a number of reasons. Many are requested, but unavailable, some are only held at libraries that do not loan, others are canceled because of Copyright issues. When requests are canceled, the status is shown in your account, and an e-mail is automatically sent stating the reason.
If there is a condition that you can correct, you may do so, then re-submit the item, using the original record in your ILLiad account.