text:
\section{Introduction}
\label{sec:introduction}

A basic problem of control theory is to achieve a global stabilization of a nonlinear system. However, in many cases it is impossible (as in quantized control) or too costly to construct a feedback which lets all states of a system uniformly converge to the unique equilibrium point of a closed loop system.
In such cases, it is natural to let the trajectories converge to a certain ball of a positive radius.

A basic concept which formalizes such a behavior is practical stability \cite{LLM90}.
This concept was extremely useful for stabilization of stochastic control systems \cite{ZhX13}, for control under quantization errors \cite{ShL12} and under communication bandwidth constraints \cite{WoB99}, etc. In \cite{JTP94,JMW96} conditions of small-gain type have been developed for stability of interconnections of input-to-state practically stable nonlinear systems.
In view of the importance of practical stability, it is essential to understand a relationship of practical stability to other stability notions. In this paper \textit{we prove a general criterion for practical uniform global asymptotic stability (pUGAS) for a wide class of control systems with disturbances} including evolution equations in Banach spaces, many classes of partial differential equations, switched systems and ordinary differential equations (ODEs). A key concept helping to achieve this aim is a \textit{uniform weak attractivity} \cite{MiW17a,MiW17b}.

A subset $\A$ of the state space $X$ is called \textit{globally weakly attractive} if for any neighborhood $S$ of $\A$ and any $x\in X$
the trajectory of a system, corresponding to initial condition $x$ intersects $S$.
The concept of weak attractivity has been introduced in \cite{Bha66} and played a prominent role in stability theory. In particular, this notion was one of the cornerstones for the development of stability theory in locally compact metric spaces in \cite{BhS02} and it was central for the proof of characterizations of input-to-state stability (ISS) of ODE systems in \cite{SoW95, SoW96}.

In \cite{KaK15} conditions in the spirit of LaSalle's theorem which
guarantee that a set is a weak attractor are provided for finite-dimensional systems, without assuming forward completeness and boundedness of the set.

As useful the concept of weak attractivity is for ODEs and control systems over locally compact metric spaces, it is far too weak for general infinite-dimensional systems to imply any kind of uniform stability.
There are fundamental reasons for that: closed bounded balls are never compact in the norm topology
of infinite-dimensional normed linear spaces, nonuniformly globally
asymptotically stable nonlinear systems do not need to have bounded
finite-time reachability sets and even if they do, this still does not
guarantee uniform global stability \cite{MiW17b} (all such unpleasant things cannot occur in ODE setting).

In order to overcome these obstacles, in \cite{MiW17a,MiW17b} a notion of uniform weak attractivity has been introduced.
A subset $\A$ of the state space is called \textit{uniformly globally weakly attractive} if for any neighborhood $S$ of $\A$ and any bounded subset $B$ of $X$ there is a \textit{uniform} finite time $\tau$ so that any trajectory starting in $B$ intersects $S$ within time not larger than $\tau$.
It is crucial in this definition that $\tau$ depends on the set $B$ and does not depend on a particular $x\in B$.

Uniform limit property (which is closely related to uniform weak attractivity) has already played a central role in the recent proof of characterizations of ISS for general infinite-dimensional control systems \cite{MiW17b} and in the theory of non-coercive Lyapunov functions for such systems \cite{MiW17a}. On the other hand, uniform weak attractivity was implicitly used in \cite[Proposition 2.2]{Tee13} and in \cite[Proposition 3]{SuT16} for characterizations of uniform global asymptotic stability of \textsl{finite-dimensional} hybrid and stochastic systems, and it is closely connected to the notion of recurrent sets \cite{BhS02,Kha11,TSS14}.

In this paper we use the concept of uniform weak attractivity to derive criteria for pUGAS of wide classes of distributed parameter systems. Our main technical result is that uniform global weak attractivity of a bounded set $\A$ together with a robust forward completeness implies the existence of a superset of $\A$ which is bounded, invariant and uniformly globally attractive. Using this fact \textit{we prove that existence of a bounded globally uniformly weakly attractive set is equivalent to practical UGAS of a system}.

Making a bridge to the framework of non-coercive Lyapunov functions, initiated in \cite{MiW17a}, \textit{we show that for a robustly forward complete system existence of a non-coercive Lyapunov function implies pUGAS.}

Although our main intention is to give novel criteria for practical UGAS, we show how one can modify the arguments to obtain the characterizations of UGAS.
{\it We specialize our results to linear infinite-dimensional systems and to ODEs with uniformly bounded disturbances. In the latter case, pUGAS is equivalent to the mere existence of a bounded globally weakly attractive set}.

\subsection{Notation}

The following notation will be used throughout these notes. Denote $\R_+:=[0,+\infty)$.
For an arbitrary set $S$ and any $n\in\N$ the $n$-fold Cartesian product is $S^n:=S\times\ldots\times S$.
The norm in a normed linear space $X$ is denoted by $\|\cdot\|$. The closure of a set $S \subset X$ w.r.t. norm $\|\cdot\|$ is denoted by $\overline{S}$.

For each $\A \subset X$ and any $x \in X$ we define the distance between $x$ and $\A$ by $\|x\|_{\A}:=\inf_{y\in \A}\|x-y\|$.
Define also $\|\A\|:=\sup_{x\in\A} \|x\|$.
The open ball in a normed linear space $X$ with radius $r$ around $\A \subset X$ is denoted by $B_r(\A):=\{x\in X \ :  \|x\|_{\A}<r\}$.
For $x\in X$ we denote $B_r(x) := B_r(\{x\})$ and $B_r:=B_r(0)$.

We use the comparison functions formalism:
\begin{equation*}
\begin{array}{ll}
{\K} &:= \left\{\gamma:\R_+\rightarrow\R_+:\ \gamma\mbox{ is continuous, strictly} \right.  \\
&\phantom{aaaaaaaaaaaaaaaaaaaa}\left. \mbox{ increasing and } \gamma(0)=0 \right\} \\
{\K_{\infty}}&:=\left\{\gamma\in\K:\ \gamma\mbox{ is unbounded}\right\}\\
{\LL}&:=\left\{\gamma:\R_+\rightarrow\R_+:\ \gamma\mbox{ is continuous and strictly}\right.\\
&\phantom{aaaaaaaaaaaaaaaa} \text{decreasing with } \lim\limits_{t\rightarrow\infty}\gamma(t)=0\}\\
{\KL} &:= \left\{\beta:\R_+\times\R_+\rightarrow\R_+:\ \beta \mbox{ is continuous,}\right.\\
&\phantom{aaaaaaaa}\left.\beta(\cdot,t)\in{\K},\ \beta(r,\cdot)\in {\LL},\ \forall t\geq 0,\ \forall r >0\right\} \\
\end{array}
\end{equation*}

\section{Preliminaries}
\label{sec:problem-statement}

The concept of a (time-invariant) system we understand in the following way:
\begin{definition}
\label{Steurungssystem}
Consider the triple $\Sigma=(X,\Dc,\phi)$, consisting of
\begin{enumerate}[label=(\roman*)]
	\item A normed linear space $(X,\|\cdot\|)$, called the {state space}, endowed with the norm $\|\cdot\|$.
	\item A set of  disturbance values $D$, which is a nonempty subset of a certain normed linear space
	\item A {space of disturbances} $\Dc \subset \{d:\R_+ \to D\}$
          satisfying the following two axioms.
\begin{itemize}
  \item \textit{The axiom of shift invariance} states that for all $d \in \Dc$ and all $\tau\geq0$ the time
shift $d(\cdot + \tau)$ is in $\Dc$.
  \item \textit{The axiom of concatenation} is defined by the requirement that for all $d_1,d_2 \in \Dc$ and for all $t>0$ the concatenation of $d_1$ and $d_2$ at time $t$
\begin{equation}
d(\tau) :=
\begin{cases}
d_1(\tau), & \text{ if } \tau \in [0,t], \\
d_2(\tau-t),  & \text{ otherwise},
\end{cases}
\label{eq:Composed_Input}
\end{equation}
belongs to $\Dc$.
\end{itemize}

	\item A transition map $\phi:\R_+ \times X \times \Dc \to X$.
\end{enumerate}
The triple $\Sigma$ is called a system, if the following properties hold:
\begin{sysnum}
	\item Forward completeness: for every $(x,d) \in X \times \Dc$ and
          for all $t \geq 0$ the value $\phi(t,x,d) \in X$ is well-defined.
	\item\label{axiom:Identity} The identity property: for every $(x,d) \in X \times \Dc$
          it holds that $\phi(0, x,d)=x$.
	\item Causality: for every $(t,x,d) \in \R_+ \times X \times
          \Dc$, for every $\tilde{d} \in \Dc$, such that $d(s) =
          \tilde{d}(s)$, $s \in [0,t]$ it holds that $\phi(t,x,d) = \phi(t,x,\tilde{d})$.
	\item \label{axiom:Continuity} Continuity: for each $(x,d) \in X \times \Dc$ the map $t \mapsto \phi(t,x,d)$ is continuous  on $\R_+$.
		\item \label{axiom:Cocycle} The cocycle property: for all $t,h \geq 0$, for all
                  $x \in X$, $d \in \Dc$ we have
$\phi(h,\phi(t,x,d),d(t+\cdot))=\phi(t+h,x,d)$.
\end{sysnum}
\end{definition}
Here  $\phi(t,x,d)$ denotes the state of a system $\Sigma=(X,\Dc,\phi)$ at the moment $t \in
\R_+$ corresponding to the initial condition $x \in X$ and the disturbance $d \in \Dc$.

Definition~\ref{Steurungssystem} covers a very broad class of systems, including differential equations in Banach spaces, time-delay systems, ODEs, many classes of partial differential equations etc.

\begin{remark}
\label{rem:axiom_of_concatenation}
The axiom of concatenation restricts the choice of a space of admissible inputs. Possible candidates are e.g. $L_p(\R_+,D)$, for $p\in[0,+\infty]$ or $PC(\R_+,D)$ (the class of piecewise-continuous functions which are right-continuous). At the same time, the space of continuous functions $C(\R_+,D)$ does not satisfy the axiom of concatenation (but is used in control theory much less frequently than $L_p$ and $PC$ spaces).
This axiom has been used only in Remark~\ref{rem:Invariance_Aeps} in order to prove invariance of reachability sets.
Hence, the results developed in this paper remain valid if instead of the axiom of concatenation we assume in advance that reachability sets are invariant.
\end{remark}

For each $T\geq 0$, and each subset $S\subseteq X$, we define the sets of states which can be reached from $S$ under acting disturbances from $\Dc$ at time not exceeding $T$:
\[
\Rc^T(S):=\lbrace \phi(t,x,d):\ 0\leq t\leq T,\ d\in\Dc,\ x\in S\rbrace,
\]
\vspace{-7mm}
\begin{align*}
\Rc(S) :=\bigcup_{T\geq 0}\Rc^T(S) = \lbrace \phi(t,x,d):\ t\geq 0,\ d\in\Dc,\ x\in S\rbrace.
\end{align*}

Many classes of systems exhibit a stronger version of forward completeness, see \cite[p. 31]{KaJ11b}
\begin{definition}
\label{Def_RFC}
A system $\Sigma=(X,\Dc,\phi)$ is called robustly forward complete (RFC)
if for any $C>0$ and any $\tau>0$ it holds that $\|\Rc^\tau(\overline{B_C})\| <\infty$.
\end{definition}

The next technical lemma will be useful in the sequel.
\begin{lemma}
\label{lem:Norms_Changing}
For each $\A\subset X$ and $x\in X$ it holds that
\begin{eqnarray}
\|x\| - \|\A\| \leq \|x\|_{\A} \leq \|x\| + \|\A\|.
\label{eq:Relations_between_norms_1}
\end{eqnarray}
\vspace{-8mm}
\begin{eqnarray}
\|x\|_{\A} - \|\A\| \leq \|x\| \leq \|x\|_{\A} + \|\A\|.
\label{eq:Relations_between_norms_2}
\end{eqnarray}
\end{lemma}

\begin{proof}
For any $x,a\in X $ it holds that
\[
\|x\| \leq \|x-a\| + \|a\| \leq \|x-a\| + \|\A\|.
\]
Taking $\inf_{a\in \A}$, we obtain $\|x\| {-} \|\A\| \leq \inf_{a\in \A} \|x-a\| {=} \|x\|_{\A}.$
Computing $\inf_{a\in \A}$ in the inequality $\|x-a\| \leq \|x\| + \|a\|_\A$ we show \eqref{eq:Relations_between_norms_1}. Inequality \eqref{eq:Relations_between_norms_2} follows from \eqref{eq:Relations_between_norms_1}.
\end{proof}

\begin{proposition}
\label{prop:RFC_equivalence_wrt_A}
Let $\Sigma$ be a system. $\Sigma$ is RFC if and only if there is a bounded set $\A \subset X$ so that
\begin{eqnarray*}
\hspace{-4mm} C>0,\ \tau>0 \srs \sup_{\|x\|_{\A}\leq C,\ d \in \Dc,\ t \in [0,\tau]}\hspace{-7mm}\|\phi(t,x,d)\|_{\A} < \infty.
\end{eqnarray*}
\end{proposition}

\begin{proof}
Let $\Sigma$ be an RFC system. By means of Lemma~\ref{lem:Norms_Changing} we obtain
\begin{multline*}
\sup_{\|x\|_{\A}\leq C,\ d \in \Dc,\ t \in [0,\tau]}\hspace{-7mm}\|\phi(t,x,d)\|_{\A} \\
\leq \sup_{\|x\|\leq C + \|\A\|,\ d \in \Dc,\ t \in [0,\tau]}\hspace{-7mm}\big(\|\phi(t,x,d)\| + \|\A\|\big) < \infty.
\end{multline*}
The proof of a converse implication is analogous.
\end{proof}

\begin{definition}
\label{def:0-invariance}
Let $\Sigma=(X,\Dc,\phi)$ be a system.
\begin{enumerate}[label=(\roman*)]
	\item A set $\A \subset X$ is called invariant if $\Rc(\A) \subset \A$.
	\item An invariant set $\A \subset X$ is called robustly invariant if $\forall \eps >0$, $\forall h>0$  $\exists \delta = \delta (\eps,h)>0$, so that:
\begin{eqnarray}
\hspace{-15mm} t\in[0,h],\ \|x\|_{\A} \leq \delta,\ d\in\Dc \; \Rightarrow \;  \|\phi(t,x,d)\|_{\A} \leq \eps.
\label{eq:RobEqPoint}
\end{eqnarray}

	\item A robust invariant point $x_0 \in X$ is called a robust equilibrium.
\end{enumerate}
\end{definition}

\begin{remark}
\label{rem:Robustness_of_Equilibirum}
  Robustness of a zero equilibrium is equivalent to continuity of a function $x\mapsto \sup_{t\in[0,h],\ d\in \Dc}\|\phi(t,x,d)\|$ at $x=0$, for all $h>0$. An example of a system with a non-robust equilibrium is given in Example~\ref{exam:UGATT_Robustness_Invariant_Sets}.
\end{remark}

We will investigate the following stability properties:

\begin{definition}{}
    \label{d:stability_new}
Consider a system $\Sigma=(X,\Dc,\phi)$. A set $\A \subset X$ is called
\begin{enumerate}[label=(\roman*)]
  \item  Lagrange stable, if there are $\sigma \in \Kinf$ and $c>0$:
\[
\hspace{-5mm} x\in X,\ t \geq 0,\ d\in\Dc \srs  \|\phi(t,x,d)\|_{\A} \leq \sigma(\|x\|_{\A}) + c.
\]
  \item uniformly (locally) stable (ULS), if $\forall \eps >0$ \ $\exists \delta>0$:
\begin{eqnarray}
\hspace{-10mm} \|x\|_\A \leq \delta,\ d\in\Dc,\ t \geq 0 \srs\|\phi(t,x,d)\|_\A \leq \eps.
\label{eq:Uniform_Stability}
\end{eqnarray}

\item uniformly globally stable (UGS), if it is Lagrange stable with $c=0$.

\item uniformly globally asymptotically stable (UGAS)
if there is a $\beta \in \KL$ s.t. for all $x \in X$, $d\in \Dc$ and $t \geq 0$
\begin{equation}
\label{eq:UGAS_wrt_A_estimate}
\hspace{-1mm}\|\phi(t,x,d)\|_\A \leq \beta(\|x\|_\A,t).
\end{equation}

\item practically uniformly globally asymptotically stable (pUGAS)
if there are $\beta \in \KL$ and $c>0$ such that for all $x \in X$, $d\in \Dc$ and $t \geq 0$
\begin{equation}
\label{eq:practical_UGAS_wrt_A_estimate}
\hspace{-1mm} \|\phi(t,x,d)\|_\A \leq \beta(\|x\|_\A,t) +c.
\end{equation}

\item globally weakly attractive, if
\begin{eqnarray}
\hspace{-12mm} x\in X,\ d \in\Dc,\ \eps>0 \srs \exists t: \ \|\phi(t,x,d)\|_{\A} \leq \eps.
\label{eq:LIM_inf_form}
\end{eqnarray}

\item \label{def:UniformGlobalWeakAttractivity} uniformly globally weakly attractive, if for every $\eps>0$ and
for every $r>0$ there is a $\tau = \tau(\eps,r)$ such that
\begin{eqnarray}
\hspace{-13mm} \|x\|_{\A}\leq r,\ d\in\Dc \ \Rightarrow \ \exists t  \leq \tau:\ \|\phi(t,x,d)\|_{\A} \leq \eps.
\label{eq:Uniform_weak2}
\end{eqnarray}
\item \label{def:UniformGlobalAttractivity} uniformly globally attractive (UGATT), if for any $r,\eps >0$ there exists $\tau=\tau(r,\eps)$ so that
\begin{eqnarray}
\hspace{-16mm} \|x\|_{\A} \leq r,\ d\in\Dc,\ t \geq \tau(r,\eps) \ \Rightarrow \ \|\phi(t,x,d)\|_{\A} \leq \eps.
\label{eq:UAG_with_zero_gain}
\end{eqnarray}
\end{enumerate}
\end{definition}

\begin{remark}
\label{rem:UGS_equals_LagrStab_and_ULS}
For any system and any $\A\subset X$ the following holds:
\begin{center}
$\A$ is UGS \quad$\Iff$\quad $\A$ is ULS $\ \wedge\ $ $\A$ is Lagrange stable,
\end{center}
see \cite[Lemma 1.2]{SoW96} for a similar result.
\end{remark}

\begin{remark}
\label{rem:Invariance_of_pUGAS_Lagrange_Stability_wrt_choice_of_aboundedset}
  Note that a bounded set $\A\subset X$ is Lagrange stable if and only if $\{0\}$ is Lagrange stable (which easily follows from Lemma~\ref{lem:Norms_Changing}).
 Similarly $\A\subset X$ is pUGAS iff $\{0\}$ is pUGAS. In view of these facts \textit{we will call a system $\Sigma$ Lagrange stable or pUGAS respectively if there is a bounded subset of $X$ which possesses such properties}.
\end{remark}

\begin{definition}
\label{def:Unif_Ultimate_Boundedness}
A system $\Sigma=(X,\Uc,\phi)$ is called uniformly ultimately bounded, if there is $K>0$ so that for each $r>0$ there is $T=T(r)$:
\begin{eqnarray}
\hspace{-5mm} \|x\| \leq r,\ d \in\Dc,\ t\geq T \srs  \|\phi(t,x,d)\|\leq K.
\label{eq:Unif_Ult_Boundedness}
\end{eqnarray}
\end{definition}

\begin{remark}
\label{rem:UGATT_implies_Unif_Boundedness}
Let $\Sigma$ be any system. Existence of a bounded UGATT set implies uniform ultimate boundedness of $\Sigma$.
Indeed, let $\A$ be a bounded invariant UGATT set. Pick $\eps:=1$
and set $T(r):=\tau(r,1)$. Then
\begin{eqnarray*}
\hspace{-8mm} \|x\|_\A \leq r,\ d \in\Dc,\ t\geq T(r) \srs  \|\phi(t,x,d)\|\leq \|\A\| + 1,
\end{eqnarray*}
and Lemma~\ref{lem:Norms_Changing} ensures uniform ultimate boundedness \text{of $\Sigma$} as:
\begin{eqnarray*}
\hspace{-9mm} \|x\| \leq r,\ d \in\Dc,\ t\geq T(r{+}\|\A\|) \srs  \|\phi(t,x,d)\|\leq \|\A\| {+} 1.
\end{eqnarray*}
\end{remark}

Let a system $\Sigma$ be fixed and let $V:X \to\R$ be continuous.
Given $x\in X$ and $d\in \mathcal{D}$, we consider the (right-hand lower) Dini
derivative of the continuous function $t \mapsto V(\phi(t,x,d))$ at $t=0$:
\begin{equation}
\label{UGAS_wrt_D_LyapAbleitung}
\dot{V}_d(x):=\mathop{\underline{\lim}} \limits_{t \rightarrow +0} {\frac{1}{t}\big(V(\phi(t,x,d))-V(x)\big) }.
\end{equation}
{\nolinebreak We call this the Dini derivative of $V$ along trajectories \text{of $\Sigma$}.}

Recently in \cite{MiW17a} a novel concept of a non-coercive Lyapunov function has been introduced.
\begin{definition}
\label{def:UGAS_LF_With_Disturbances}
Let $\A \subset X$ be a bounded set.
A continuous function $V:X \to \R_+$ is called a \textit{non-coercive Lyapunov function (or functional)} for a system $\Sigma=(X,\Dc,\phi)$ w.r.t. $\A$,  if there exist
$\psi_2 \in \Kinf$ and $\alpha \in \K$
such that $V(x)=0$ for all $x\in \A$, the inequalities
\begin{equation}
    \label{eq:1}
    0 < V(x) \leq \psi_2(\|x\|_{\A}) \quad \forall x \in X \backslash \A
\end{equation}
hold
and the Dini derivative of $V$  along trajectories of $\Sigma$ satisfies
\begin{equation}
\label{DissipationIneq_UGAS_With_Disturbances}
\dot{V}_d(x) \leq -\alpha(\|x\|_{\A})
\end{equation}
for all $x \in X\backslash\A$ and all $d \in \Dc$.
\end{definition}

The importance of non-coercive Lyapunov functions is due to the following result, see \cite{MiW17a}:

\begin{proposition}
    \label{t:noncoeLT}
    Consider an RFC system $\Sigma=(X,\Dc,\phi)$	and assume that $\{0\}$ is its robust equilibrium.
		If $V$ is a non-coercive Lyapunov function for $\Sigma$ w.r.t $\{0\}$, then $\{0\}$ is UGAS.
\end{proposition}

Non-coercive Lyapunov functions are easier to construct than coercive ones.
For example, construction of Lyapunov functions for linear systems over Banach spaces by means of the Lyapunov equation provides in general merely non-coercive Lyapunov functions, \cite[Theorem 5.1.3]{CuZ95}.

We will use non-coercive Lyapunov functions to derive a sufficient condition for pUGAS of systems with disturbances.

\section{Characterization of pUGAS}
\label{sec:pUGAS_equals_ULIM}

In this section we characterize pUGAS as follows:
\begin{theorem}
\label{thm:RFC_weak_attr_imply_pUGAS}
Let $\Sigma$ be a system. The following statements are equivalent:
\begin{enumerate}[label=(\roman*)]
	\item\label{enum:pUGAS_item1} $\Sigma$ is pUGAS
	\item\label{enum:pUGAS_item11} $\Sigma$ is Lagrange stable and there is a bounded invariant UGATT set.
	\item\label{enum:pUGAS_item2} $\Sigma$ is RFC and there is a bounded invariant UGATT set.
	\item\label{enum:pUGAS_item21} $\Sigma$ is RFC and uniformly ultimately bounded.
	\item\label{enum:pUGAS_item3} $\Sigma$ is RFC and there is a bounded uniformly globally weakly attractive set.
\end{enumerate}
\end{theorem}

We start with
\begin{definition}
\label{def:first_prolongation}
For each $\eps>0$ and any $\A\neq \emptyset$ define
\begin{eqnarray}
\A_\eps:=\Rc\big(B_\eps(\A)\big),\qquad P_+(\A):=\bigcap_{\eps>0} \A_\eps.
\label{eq:A_eps_definition}
\end{eqnarray}
\end{definition}

\begin{remark}
\label{rem:Invariance_Aeps}
If $\A\neq\emptyset$, then for any $\eps>0$ the set $\A_\eps$ is invariant. Indeed, pick any $x\in\A_\eps$, $t\geq0$ and $d\in\Dc$.
Then there are $t_1\geq0$, $x_1\in\A$ and $d_1\in\Dc$ so that $x=\phi(t_1,x_1,d_1)$.
Due to the cocycle axiom ($\Sigma$\ref{axiom:Cocycle}) we have that $\phi(t,x,d)=\phi(t+t_1,x_1,d_2)$, where $d_2$ is a concatenation of $d_1$ and $d$ (which belongs to $\Dc$ in view of the axiom of concatenation).
Due to the axiom of shift-invariance $d_2\in \Dc$, and thus $\phi(t,x,d) \in\A_\eps$.
\end{remark}

We start by showing invariance of $P_+(\A)$

\begin{lemma}
\label{lem:properties_A_epsilon_0}
 Let a system $\Sigma$ be given and let $\A \subset X$, $\A \neq \emptyset$. Then $P_+(\A)$ is a nonempty invariant set.
\end{lemma}

\begin{proof} Since $\A \subset \A_\eps$ for any $\eps>0$, we have $\A \subset P_+(\A)$, and hence $P_+(\A)$ is nonempty.
For any $\eps>0$ the set $\A_\eps$ is invariant by Remark~\ref{rem:Invariance_Aeps}.

Pick any $x\in \bigcap_{\eps>0} \A_\eps$. Since $\A_\eps$ is invariant for all $\eps>0$, for any $t\geq 0$ and any $d\in\Dc$ it holds that $\phi(t,x,d) \in \A_\eps$, $\eps>0$. Thus, $\phi(t,x,d) \in \bigcap_{\eps>0} \A_\eps=P_+(\A)$, which shows invariance of $P_+(\A)$.
\end{proof}

Next  we present our main technical result, showing that for any bounded uniformly globally weakly attractive set there is an  invariant superset with a much stronger UGATT property.
\begin{proposition}
\label{prop:from_ULIM_to_UGATT}
Assume that $\Sigma$ is an RFC system and that $\A$ is a bounded uniform global weak attractor
for $\Sigma$. Then for any $\eps>0$ the set $\A_\eps$ is bounded, invariant and UGATT.

\end{proposition}

\begin{proof}
Pick any $\eps>0$, any $R>0$ and fix them. Since $\A$ is a global uniform weak attractor of an RFC system $\Sigma$, there exists $\tilde\tau=\tilde\tau(\eps, R)>0$ so that
\begin{eqnarray}
\hspace{-8mm} \|x\|_{\A} \leq R,\ d\in\Dc \srs \exists \bar{t}\leq \tilde\tau:\ \|\phi(\bar{t},x,d)\|_{\A} \leq \eps.
\label{eq:ULIM_UGATT_Section0}
\end{eqnarray}
Without loss of generality assume that $\tilde\tau$ is decreasing w.r.t. $\eps$ and increasing w.r.t. $R$.
Then $\tilde\tau$ is integrable on compact sets.
Define
\begin{eqnarray*}
\tau(\eps,R):= \frac{2}{\eps R}\int_R^{2R}\int_{\eps/2}^{\eps} \tilde\tau(\eps_1,R_1)d\eps_1dR_1.
\end{eqnarray*}
Clearly, $\tau$ is increasing w.r.t. $R$, decreasing w.r.t. $\eps$, continuous and
$\tau(\eps,R)\geq\tilde\tau(\eps,R)$ for any $\eps,R>0$. Thus,
\begin{eqnarray}
\hspace{-8mm} \|x\|_{\A} \leq R,\ d\in\Dc \srs \exists \bar{t}\leq \tau:\ \|\phi(\bar{t},x,d)\|_{\A} \leq \eps.
\label{eq:ULIM_UGATT_Section1}
\end{eqnarray}

In particular, since $\tau$ is continuous, for any $\eps>0$ and $d\in\Dc$ it holds that
\begin{eqnarray}
\hspace{-8mm} \|x\|_{\A} \leq \eps  \srs \exists \tilde{t} \in(0,\tau(\eps, \eps)):\ \|\phi(\tilde{t},x,d)\|_{\A} \leq \eps.
\label{eq:ULIM_UGATT_Section2}
\end{eqnarray}
This means, that trajectories, emanating from $B_{\eps}(\A)$ return to this ball in time not larger than $\tau(\eps, \eps)$.
Hence $\A_\eps =\Rc^{\tau(\eps,\eps)}\big(B_\eps(\A)\big)$.
As $\Sigma$ is RFC, $\A_\eps$ is bounded for any $\eps>0$.

Now the cocycle property implies for any $t>\bar{t}$ that
\begin{eqnarray*}
\phi(t,x,d) = \phi\big(t-\bar{t},\phi(\bar{t},x,d),d(\cdot + \bar{t})\big).
\end{eqnarray*}
This ensures that
\begin{eqnarray*}
\|x\|_{\A} \leq R ,\ d\in\Dc,\ t\geq \bar{t} \srs \phi(t,x,d)  \in \A_\eps,
\end{eqnarray*}
which means that for each $d\in\Dc$
\begin{eqnarray}
\hspace{-5mm} \|x\|_{\A} \leq R,\ t\geq \tau(R,\eps) \srs \|\phi(t,x,d)\|_{\A_\eps} = 0.
\label{eq:UGATT_wrt_Aeps}
\end{eqnarray}
Since $\A \subset \A_\eps$ for any $\eps>0$ it follows that
$\|x\|_{\A} \leq \|x\|_{\A_\eps} + C$, where
$C:=\sup_{y\in\partial\A_\eps,\ z\in\partial\A,\ [y,z]\bigcap\A=\emptyset}\|y-z\|$ and
$[y,z]:=\{ty+(1-t)z:t\in(0,1)\}$.

Define $\tilde\tau(R,\eps):=\tau(R+C,\eps)$. We have:
\begin{eqnarray*}
\hspace{-9mm} \|x\|_{\A_\eps} \leq R ,\ d\in \Dc ,\ t\geq \tilde\tau(R,\eps) \srs \|\phi(t,x,d)\|_{\A_\eps} =0,
\end{eqnarray*}
which shows that $\A_\eps$ is UGATT.
\end{proof}

We call a function $h: \R_+^2 \to \R_+$ increasing, if $(r_1,R_1) \leq (r_2,R_2)$
implies that $h(r_1,R_1) \leq h(r_2,R_2)$, where we use the component-wise
partial order on $\R_+^2$. We call $h$ strictly increasing if $(r_1,R_1)
\leq (r_2,R_2)$ and $(r_1,R_1) \neq (r_2,R_2)$ imply $h(r_1,R_1) <
h(r_2,R_2)$.

We need the following characterization of RFC in terms of comparison-like functions, shown in \cite{MiW17a}:
\begin{lemma}
\label{lem:RFC_criterion}
Consider a  forward complete system $\Sigma=(X,\Dc,\phi)$. The following statements are equivalent:
\begin{enumerate}
	\item[(i)] $\Sigma$ is RFC.
	\item[(ii)] there exists a continuous, increasing function $\mu: \R_+^2 \to \R_+$, such that
 \begin{equation}
    \label{eq:8}
\hspace{-11mm}    x\in X,\ d\in \Dc,\ t \geq 0 \srs \| \phi(t,x,d) \| \leq \mu( \|x\|,t).
\end{equation}
\end{enumerate}
\end{lemma}

The next lemma ensures that existence of a bounded uniform globally weakly attractive set for an RFC system implies its Lagrange stability. It is close in spirit to \cite[Proposition 7]{MiW17b}.
\begin{lemma}
\label{lem:Weak_attr_plus_RFC_imply_pUGS}
Let $\Sigma$ be an RFC system. If there is a bounded globally uniformly weakly attractive set $\A \subset X$, then $\Sigma$ is Lagrange stable.
\end{lemma}

\begin{proof}
Pick any $r>0$ and set $\eps:=\frac{r}{2}$.
Since there is a set $\A\subset X$ which is uniformly globally weakly attractive, there exists a $\tau=\tau(r)$ so that
\begin{eqnarray}
\hspace{-9mm}\|x\|_{\A} \leq r,\ d\in\Dc \srs \exists t\leq \tau(r):\ \|\phi(t,x,d)\|_\A \leq \eps.
\label{eq:ULIM_pGS_theorem1_v2}
\end{eqnarray}
Without loss of generality we can assume that $\tau$ is increasing in $r$, in particular, it is locally integrable.
Defining $\bar\tau(r):=\frac{1}{r}\int_r^{2r}\tau(s)ds$ we see that $\bar\tau(r) \geq \tau(r)$ and $\bar\tau$ is continuous.

Since $\Sigma$ is RFC, by Lemma~\ref{lem:RFC_criterion} there exists a continuous, increasing function
$\mu: \R_+^2 \to \R_+$, such that for all $x\in X, d\in \Dc$ and all $t \geq 0$ we have \eqref{eq:8}.
In particular,
\begin{eqnarray}
\hspace{-9mm}\|x\|_\A {\leq} r,\, d{\in}\Dc,\, t{\in}[0,\tau(r)] \ \Rightarrow\  \|\phi(t{,}x{,}d)\|_\A \leq \mu(r,\bar\tau(r)).
\label{eq:ULIM_pGS_theorem2_v2}
\end{eqnarray}
Since $\mu$ and $\bar\tau$ are both continuous and increasing, $\tilde\sigma:r\mapsto \mu(r,\bar\tau(r))$ is again continuous and increasing. Also from \eqref{eq:ULIM_pGS_theorem1_v2} and \eqref{eq:ULIM_pGS_theorem2_v2} it is clear that $\tilde\sigma(r) \geq \frac{r}{2}$ for any $r>0$.

Seeking a contradiction, assume that there exist $x\in B_r(\A)$, $d\in\Dc$ and $t\geq0$ so that $\|\phi(t,x,d)\|_\A  > \tilde
\sigma(r)$. Define $t_m:=\sup\{s \in[0,t]: \|\phi(s,x,d)\|_\A \leq r\}$ which is well-defined, since ($\Sigma$\ref{axiom:Identity}) implies that $\|\phi(0,x,d)\|_\A = \|x\|_\A \leq r$.

In view of the cocycle property, it holds that
\[
\phi(t,x,d) = \phi\big(t-t_m,\phi(t_m,x,d),d(\cdot + t_m)\big),
\]
and $d(\cdot + t_m) \in\Dc$ due to the axiom of shift invariance.

Assume that $t-t_m \leq\tau(r)$. Since $\|\phi(t_m,x,d)\|_\A\leq r$,
\eqref{eq:ULIM_pGS_theorem2_v2} implies that $\|\phi(s,x,d)\|_\A \leq \tilde \sigma(r)$ for all $s \in [t_m,t]$.
Otherwise, if $t-t_m >\tau(r)$, we may apply
\eqref{eq:ULIM_pGS_theorem1_v2} and there exists $t^*<\tau(r)$, so that
\[
\|\phi\big(t^*,\phi(t_m,x,d),d(\cdot + t_m)\big)\|_\A =
\|\phi(t^*+t_m,x,d)\|_\A \leq \eps = \frac{r}{2}.
\]
This contradicts the definition of $t_m$, since $t_m+t^*<t$.
Hence
\begin{eqnarray}
\hspace{-9mm}\|x\|_\A \leq r,\ d\in\Dc,\ t\geq 0 \srs  \|\phi(t,x,d)\|_\A \leq \tilde\sigma(r).
\label{eq:ULIM_pGS_theorem3_v2}
\end{eqnarray}
Denote $\sigma(r):=\tilde\sigma(r) - \tilde\sigma(0)$, for any $r\geq 0$. Clearly, $\sigma\in\Kinf$.
Then from \eqref{eq:ULIM_pGS_theorem3_v2} we have for all $x\in X,\ d\in\Dc,\
t\geq 0$ that
\begin{eqnarray}
 \|\phi(t,x,d)\|_\A \leq \sigma(\|x\|_\A) + \tilde\sigma(0),
\label{eq:ULIM_pGS_theorem4_v2}
\end{eqnarray}
which shows Lagrange stability of $\Sigma$.
\end{proof}

The following lemma is routine:
\begin{lemma}
\label{lem:UGATT_plus_LagrangeSt_imply_pUGAS}
Let $\Sigma$ be a system and let $\A \subset X$ be a bounded invariant set of $\Sigma$. Then:
\begin{enumerate}[label=(\roman*)]
   \item If $\A$ is UGATT and Lagrange stable, then $\Sigma$ is pUGAS.
   \item If $\A$ is UGATT and globally stable, then $\Sigma$ is UGAS.
\end{enumerate}
\end{lemma}

\begin{proof}
Let us start with (i).
Since $\A$ is Lagrange stable, there is a $\sigma\in\Kinf$ and $c>0$:
\begin{eqnarray}
\hspace{-9mm}t\geq 0,\, \|x\|_\A \leq \delta,\, d \in \Dc \srs \|\phi(t,x,d)\|_\A \leq \sigma(\delta) + c.
\label{eq:TmpEstimate}
\end{eqnarray}
Define $\eps_n:=\frac{1}{2^n} \sigma(\delta)$, for $n \in \N$.
UGATT of $\A$ ensures that there is a sequence of times $\tau_n:=\tau(\eps_n,\delta)$ which we assume without loss of generality to be strictly increasing, such that
\[
t \geq \tau_n,\ \|x\|_\A \leq \delta,\ d \in \Dc \srs \|\phi(t,x,d)\|_\A \leq \eps_n.
\]
Define $\omega(\delta,\tau_n):=\eps_{n-1}$, for $n \in \N$, $n \neq 0$ and extend the function $\omega(\delta,\cdot)$ for $t \in \R_+ \backslash \{\tau_n: n \in \N\}$ so that $\omega(\delta,\cdot) \in \LL$.
Note that for any $n\in\N$ and for all $t \in (\tau_n,\tau_{n+1})$ it holds that $\|\phi(t,x,d)\|_\A \leq \eps_n < \omega(\delta,t)$.
Doing this for all $\delta \in \R_+$ we obtain function $\omega$.

Now choose $\tilde\beta(r,t)=\sup_{0 \leq s \leq r}\omega(s,t) \geq \omega(r,t)$. Obviously, $\tilde\beta$ is non-decreasing w.r.t. the first argument and decreasing w.r.t. the second one. It can be estimated from above by $\beta \in\KL$ and
\eqref{eq:practical_UGAS_wrt_A_estimate} is satisfied with such a $\beta$.

Let us show (ii). Since $\A$ is globally stable, \eqref{eq:TmpEstimate} holds with $c=0$. The same argument as above shows now UGAS of $\A$.
\end{proof}

Finally, we can prove Theorem~\ref{thm:RFC_weak_attr_imply_pUGAS}.
\begin{proof} (of Theorem~\ref{thm:RFC_weak_attr_imply_pUGAS}.)
\ref{enum:pUGAS_item1} $\Rightarrow$ \ref{enum:pUGAS_item11}.
Assume, that $\Sigma$ is a pUGAS system.
Then there is a $\beta\in\KL$ and $c>0$ so that for all $x\in X$, $t\geq 0$ and $d\in\Dc$ we have
\begin{equation}
\label{eq:practical_UGAS_Xnorm}
\hspace{-1mm}\|\phi(t,x,d)\| \leq \beta(\|x\|,t) +c.
\end{equation}

Then $\overline{B_c(0)}$ is uniformly globally attractive (but may be not invariant).
According to Proposition~\ref{prop:from_ULIM_to_UGATT} there is a bounded invariant UGATT set $\A$ (which can be chosen e.g. as $(B_c(0))_\eps$ for any $\eps>0$).

At the same time \eqref{eq:practical_UGAS_Xnorm} implies Lagrange stability since $\beta(\|x\|,t)\leq \beta(\|x\|,0)$ for any $x \in X,\ t\geq0$.

\ref{enum:pUGAS_item11} $\Rightarrow$ \ref{enum:pUGAS_item2}. Clear.

\ref{enum:pUGAS_item2} $\Rightarrow$ \ref{enum:pUGAS_item21}. Follows by Remark~\ref{rem:UGATT_implies_Unif_Boundedness}.

\ref{enum:pUGAS_item21} $\Rightarrow$ \ref{enum:pUGAS_item3}. Evident, since the set $B_K$ in Definition~\ref{def:Unif_Ultimate_Boundedness}
 is uniformly globally weakly attractive.

\ref{enum:pUGAS_item3} $\Rightarrow$ \ref{enum:pUGAS_item1}.
Let $\Sigma$ be an RFC system and there is a bounded uniformly globally weakly attractive set for $\Sigma$.
By Proposition~\ref{prop:from_ULIM_to_UGATT} there is a bounded invariant UGATT set for $\Sigma$.
Lemma~\ref{lem:Weak_attr_plus_RFC_imply_pUGS} implies that $\Sigma$ is Lagrange stable and finally
Lemma~\ref{lem:UGATT_plus_LagrangeSt_imply_pUGAS} proves that $\Sigma$ is pUGAS.
\end{proof}

\begin{remark}
\label{rem:Forward_completeness_is_not_enough}
Theorem~\ref{thm:RFC_weak_attr_imply_pUGAS} does not hold, if we require instead of RFC merely forward completeness, see an example in \cite{MiW17a}.
\end{remark}

\subsection{Existence of a non-coercive Lyapunov function implies pUGAS}

As an application of Theorem~\ref{thm:RFC_weak_attr_imply_pUGAS} we obtain the following non-coercive Lyapunov sufficient condition for pUGAS of RFC systems.

\begin{corollary}
Let $\Sigma=(X,\Dc,\phi)$ be an RFC system.
If there is a bounded set $\A\subset X$ so that $\Sigma$ possesses a non-coercive Lyapunov function w.r.t. $\A$, then $\Sigma$ is practically UGAS.
\end{corollary}

\begin{proof}
Follows by Theorem~\ref{thm:RFC_weak_attr_imply_pUGAS} and Proposition~\ref{prop:noncoeLT_plus_FC_implies_ULIM_v2} which is proved next.
\end{proof}

The following relation between non-coercive Lyapunov functions and uniform weak attractivity is valid:
\begin{proposition}
    \label{prop:noncoeLT_plus_FC_implies_ULIM_v2}
    Consider a forward complete system $\Sigma=(X,\Dc,\phi)$ and let $\A \subset X$ be bounded. If there
    exists a non-coercive Lyapunov function $V$ for $\A$, then $\A$ is uniformly globally weakly attractive.
\end{proposition}

This proposition has been shown in \cite{MiW17a} for $\A=\{0\}$. The proof is completely analogous for general bounded $\A$.
We include it here merely for the sake of completeness.

\begin{proof}
Let $V$ be a non-coercive Lyapunov function and let $\alpha \in \mathcal{K}$ be such that the decay estimate \eqref{DissipationIneq_UGAS_With_Disturbances} is true.
Along any trajectory $\phi$ of a forward complete system $\Sigma$ we have the inequality
\begin{equation}
\dot V_{d(t+\cdot)}(\phi(t,x,d)) \leq -\alpha(\|\phi(t,x,d)\|_{\A}), \quad
\forall t\geq 0.
\label{eq:JustEq1}
\end{equation}
Since $\alpha \in\K$, it follows that
\begin{equation}
V(\phi(t,x,d)) - V(x) \leq -\int_0^t \alpha(\|\phi(s,x,d)\|_{\A})ds,
\label{eq:JustEq3}
\end{equation}
which implies that for all $t\geq 0$ we have
\begin{eqnarray}
\int_0^t \alpha(\|\phi(s,x,d)\|_{\A})ds
\leq
V(x)  \leq \psi_2(\|x\|_{\A}).
\label{eq:Vintbound_2_v2}
\end{eqnarray}

Fix $\eps>0$ and $r>0$ and define $\tau(\eps,r):=\frac{\psi_2(r) +
  1}{\alpha(\eps)}$. We claim that this choice of $\tau = \tau(\eps,r)$
yields an appropriate time to conclude uniform weak attractivity of
$\A$. Assume to the contrary that there exist $x\in B_r(\A)$ and $d\in \Dc$ with the property
$\|\phi(t,x,d)\|_{\A} \geq \eps$ for all $t\in[0,\tau(\eps,r)]$. In view of \eqref{eq:Vintbound_2_v2} and since $\alpha\in\K$, we obtain
\[
\psi_2(r) + 1 = \tau(\eps,r) \alpha(\eps) \leq  \int_0^{\tau(\eps,r)}\hspace{-4mm} \alpha(\|\phi(s,x,d)\|_{\A})ds \leq \psi_2(r),
\]
a contradiction.
\end{proof}

\subsection{Strengthening of Proposition~\ref{prop:from_ULIM_to_UGATT}}

Proposition~\ref{prop:from_ULIM_to_UGATT} shows the existence of bounded UGATT sets for RFC systems possessing bounded uniform global attractors. A mere existence result was satisfactory for characterizing pUGAS systems, but for other purposes it is of interest to find tight estimates of the attraction region.

Next proposition, motivated by \cite[Theorem 1.25 in Chapter V]{BhS02}, shows under additional assumptions a result of such kind.
\begin{proposition}
\label{prop:from_ULIM_to_UGATT_strengthening}
Let $\Sigma$ be an RFC system and assume that $\A$ is a bounded  uniform global weak attractor
for $\Sigma$. If
\begin{eqnarray}
\forall\eps>0\  \exists \delta >0:\ \sup_{y\in \A_\delta}\|y\|_{P_+(\A)}<\eps,
\label{eq:Continuity_reachability_sets}
\end{eqnarray}
then $P_+(\A)$ is a bounded invariant UGATT set.

If additionally $\A$ is a robust invariant set, then there is no UGATT set $S$ so that $\A\subset S \subset P_+(\A)$ and $S$ is not dense in $P_+(\A)$ w.r.t. topology defined by $\|\cdot\|$.
\end{proposition}

\begin{proof}
Let \eqref{eq:Continuity_reachability_sets} holds. For any $\eps>0$ pick $\delta=\delta(\eps)$ as in \eqref{eq:Continuity_reachability_sets}. According to \eqref{eq:UGATT_wrt_Aeps} for any $R>0$ we have
\[
\|x\|_{\A} \leq R ,\ d\in \Dc ,\ t\geq \tau(R,\delta(\eps)) \srs \phi(t,x,d) \in \A_\delta
\]
and in view of \eqref{eq:Continuity_reachability_sets}:
\[
\|x\|_{\A} \leq R ,\ d\in \Dc ,\ t\geq \tau(R,\delta(\eps)) \srs \|\phi(t,x,d)\|_{P_+(\A)}\leq \eps.
\]
Since $\A \subset P_+(\A)$ it follows that $\|x\|_{\A} \leq \|x\|_{P_+(\A)}+C_1$, for a certain $C_1>0$. Denoting
$\bar\tau(R,\eps):=\tau(R+C_1,\eps)$ we finally obtain
\[
\|x\|_{P_+(\A)} {\leq} R ,\, d\in \Dc ,\, t{\geq} \bar\tau(R,\delta(\eps)) \srs \|\phi(t,x,d)\|_{P_+(\A)}{\leq} \eps.
\]
This shows, that $P_+(\A)$ is UGATT. $P_+(\A)$ is bounded since $\A_\eps$ are bounded for any $\eps>0$. Invariance of $P_+(\A)$ follows from
Lemma~\ref{lem:properties_A_epsilon_0}.
The first part of the proposition is proved.

Now assume that $\A$ is a robust invariant set. Assume that the claim of the proposition does not hold, and there exists a UGATT set $S$ which is not dense in $P_+(\A)$ (w.r.t. topology generated by $\|\cdot\|$) and so that $\A\subset S \subset P_+(\A)$.

Then for any $r,\eps >0$ there is a $\tau=\tau(r,\eps)$ satisfying
\begin{eqnarray}
\hspace{-8mm} \|x\|_{S} \leq r,\ d\in\Dc,\ t \geq \tau(r,\eps) \ \Rightarrow \ \|\phi(t,x,d)\|_{S} \leq \eps.
\label{eq:UGATT_proff}
\end{eqnarray}

Since $S$ is not dense in $P_+(\A)$, there is some $y\in P_+(\A)$: $\|y\|_S>0$. Set $\eps:=\frac{\|y\|_S}{2}$ and $r:=1$.
Pick $h>\tau(1,\eps)$.

As $\A$ is a robust invariant set, for these $\eps,h$ there is a $\delta >0$ (without loss of generality let $\delta<1$) so that \eqref{eq:RobEqPoint} holds.
Since $\A \subset S$, $\|\phi(t,x,d)\|_S\leq \|\phi(t,x,d)\|_\A$ and thus
\begin{eqnarray}
\hspace{-8mm} t\in[0,h],\ \|x\|_{\A} \leq \delta,\ d\in\Dc \; \Rightarrow \;  \|\phi(t,x,d)\|_{S} \leq \eps.
\label{eq:RobEqPoint_proof}
\end{eqnarray}

As $y\in P_+(\A)$, then also $y\in \A_\delta = \Rc (B_\delta(\A))$.
Pick any $t_\delta>0$, $x_\delta \in B_\delta(\A)$, $d_\delta\in\Dc$ so that $\phi(t_\delta,x_\delta,d_\delta)=y$.

Applying \eqref{eq:RobEqPoint_proof} for $x:=x_\delta$ we see that
\[
\|\phi(t,x_\delta,d_\delta)\|_{S} \leq \eps = \frac{\|y\|_S}{2}
\]
for $t\in[0,h]$, and in particular for $t\in[0,\tau(1,\eps)]$.

But $\|\phi(t_\delta,x_\delta,d_\delta)\|_S=\|y\|_S = 2\eps$ which means that $t_\delta>h>\tau(1,\eps)$.

On the other hand, $\A \subset S$ and thus $\|x_\delta\|_S \leq \|x_\delta\|_\A \leq \delta <1=r$ and according to \eqref{eq:UGATT_proff} it holds \mbox{$\|\phi(t_\delta,x_\delta,d_\delta)\|_{S} \leq \eps$}, a contradiction.
This finishes the proof.
\end{proof}

In the next example, we are going to show that the assumption of robust invariance of $\A$ is necessary
for the second part of Proposition~\ref{prop:from_ULIM_to_UGATT_strengthening} to hold.

\begin{example}
\label{exam:UGATT_Robustness_Invariant_Sets}
Let $\Dc :=L^\infty(\R_+,\R)$, $x(t), y(t) \in\R$ and consider a planar system
\begin{subequations}
\begin{align}
\dot{x}(t) =&\ |d(t)| \big(1-x(t)\big) |y(t)| - x^3(t) - x^{1/3}(t), \\
\dot{y}(t) =&\ - y^3(t) - y^{1/3}(t).
\end{align}
\label{eq:Counterexample_Robust_Inv_Sets}
\end{subequations}
It is easy to see that \eqref{eq:Counterexample_Robust_Inv_Sets} is a forward complete system, $\A:=(0,0)$ is its equilibrium which is however not a robust equilibrium. All solutions of \eqref{eq:Counterexample_Robust_Inv_Sets} converge in finite time to $\A$ which makes $\A$ a uniform global attractor (more details you can find in \cite{MiW17a}).
Moreover, \eqref{eq:Counterexample_Robust_Inv_Sets} is RFC since $\dot{x}(t) > 0$ whenever $x(t) < 0$ and $\dot{x}(t) < 0$ for $x(t) > 1$.

Pick $\eps<1$. The following inclusions hold:
\[
\Rc\big((-\sqrt{\eps},\sqrt{\eps})^2\big) \subset \Rc\big(B_\eps((0,0)^T)\big) \subset
\Rc\big((-\eps,\eps)^2\big).
\]
It is not hard to compute that $\Rc\big((-\eps,\eps)^2\big)= [-\eps,1) \times (-\eps,\eps)$,
which implies that $P_+(\A)=[0,1)\times\{0\}$.

Thus, $P_+(\A)$ is UGATT, but it contains a set $S=\A$ which is again UGATT and is not dense in $P_+(\A)$.
~ \hfill{} $\square$
\end{example}

\subsection{Relations of weak attractivity to other stability notions}

For systems without disturbances the concept of weak attractivity has been introduced in \cite{Bha66} and it was one of the basic notions upon which stability theory for systems without disturbances in \cite{BhS02} has been developed.
On the other hand, in the language of \cite{SoW96}, weak attractivity is the limit property with zero
gain.

As long as the author is concerned, the notion of uniform weak attractivity has been introduced in \cite{MiW17a, MiW17b},  but it was implicitly used in \cite[Proposition 2.2]{Tee13} and \cite[Proposition 3]{SuT16} to characterize UGAS of finite-dimensional hybrid and stochastic systems.
The concept of weak attractivity is intimately related to that of recurrent sets:
\begin{definition}{}
    \label{d:stability_new_2}
Consider a system $\Sigma=(X,\Dc,\phi)$. A set $\A \subset X$ is called
\begin{enumerate}[label=(\roman*)]

\item globally recurrent, if
\begin{eqnarray}
\hspace{-5mm} x\in X,\ d \in\Dc \srs \exists t\geq 0: \phi(t,x,d) \in \A.
\label{eq:Glob_recurr}
\end{eqnarray}

\item uniformly globally recurrent, if $\forall R>0$ $\exists \tau=\tau(R)$:
\begin{eqnarray}
\hspace{-15mm} \|x\|_\A \leq R,\ d \in\Dc \srs \exists t\in[0,\tau]: \phi(t,x,d) \in \A.
\label{eq:Unif_Glob_recurr}
\end{eqnarray}

\end{enumerate}
\end{definition}

The notion of a recurrent set is fairly classical, see e.g. \cite[Definition 1.1 in Chapter 3]{BhS02}.
Together with a more recent notion of uniformly recurrent sets (see e.g. \cite[Lemma 6.5]{Kha11}, \cite[Section 2.4]{Tee13}) it is widely used in the theory of hybrid and especially stochastic systems \cite{Kha11,Tee13,SuT16}.
The following relations between weak attractivity and recurrence are valid:
\begin{remark}
\label{rem:Weak_attractivity_and_recurrence}
Let $\Sigma$ be a system and let $\A \subset X$ be arbitrary.
Then the following relations hold:
\vspace{-2mm}
\begin{itemize}
   \item If $\A$ is globally recurrent, then $\A$ is globally weakly attractive.	\vspace{-2mm}
   \item If $\A$ is uniformly globally recurrent, then $\A$ is uniformly globally weakly attractive.	\vspace{-2mm}
   \item $\A$ is globally weakly attractive $\Iff$ any neighborhood of $\A$ is globally recurrent.	\vspace{-2mm}
   \item $\A$ is uniformly globally weakly attractive $\Iff$ any neighborhood of $\A$ is uniformly globally recurrent.
\end{itemize}
\end{remark}

Finally, in some works the term of weak attractivity (or weak stability) is used with a meaning, which is quite different from that used in this paper.
In particular, in infinite-dimensional linear systems theory "weak" stability means convergence of solutions in the weak topology on $X$, see e.g. \cite{EFN07}.
In \cite[Definition 4.1]{LiK12} weak attractivity has been used to study convergence properties of stochastic systems with small parameters.
In \cite[Definition 9.2]{Rox65} weak stability of systems with multiple solutions is understood in the sense that for each neighborhood $W_1$ of $0$ there is a neighborhood $W_2$ of 0, so that for each $x\in W_2$ there is a solution staying within $W_1$ for all times, and analogously weak attractivity could be defined.
Yet another notion hiding behind the name "weak attractivity" has been used in the context of dynamical systems under perturbations in \cite{Gru02b}.

\section{Characterizations of UGAS}

Arguments, exploited in the previous section to characterize pUGAS, can be adapted to obtain criteria of UGAS.

The relation between uniform global attractivity and uniform global weak attractivity is given by:
\begin{proposition}
\label{prop:UGATT_and_Un_approachability_v2}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$.
Then $\A$ is a UGATT robustly invariant set if and only if  $\A$ is uniformly stable and uniformly globally weakly attractive.
\end{proposition}

\begin{proof}
$\Rightarrow$.
Let $\Sigma=(X,\Dc,\phi)$ be a forward complete system.
Due to items \ref{def:UniformGlobalWeakAttractivity} and \ref{def:UniformGlobalAttractivity}
of Definition~\ref{d:stability_new}, UGATT of $\A$ implies uniform global weak attractivity of $\A$. To show uniform
 stability of $\A$ fix $\eps>0$ and
$r>0$. By item\,\ref{def:UniformGlobalAttractivity} of Definition~\ref{d:stability_new} there is a $\tau=\tau(\eps,r)>0$ s.t.
\begin{eqnarray*}
\hspace{-5mm}\|x\|_\A \leq r,\ d\in\Dc,\ t \geq \tau(r,\eps) \srs \|\phi(t,x,d)\|_\A \leq \eps.
\end{eqnarray*}
Since $\A$ is a robust invariant set of $\Sigma$, there is a $\hat\delta>0$ corresponding to $\tau$ so that
\[
t\in[0,\tau],\ \|x\|_\A \leq \hat\delta, \ d \in \Dc \srs  \|\phi(t,x,d)\|_\A \leq \eps.
\]
The combination of the two implications shows that $\A$ is uniformly stable with $\delta:=\min\{r,\hat\delta\}$.

$\Leftarrow$. Fix $\eps>0$. Since $\A$ is uniformly stable, there is a $\delta>0$ so that
\eqref{eq:Uniform_Stability} holds. This shows that $\A$ is a robust invariant set.
Now pick any $r>0$. Since $\A$ is a uniform global weak attractor, for the
above $\delta$ there exists a $\tau=\tau(\delta,r)$ so that
\begin{eqnarray*}
\hspace{-9mm}\|x\|_\A \leq r,\ d\in\Dc \srs \exists \bar{t}(x,d,\delta) \leq \tau:\ \|\phi(t,x,d)\|_\A \leq \delta.
\end{eqnarray*}
Finally, ULS of $\Sigma$ and the cocycle property imply that
\[
\|x\|_\A \leq r,\ d\in\Dc,\ t \geq \bar{t}(x,d,\delta) \srs\|\phi(t,x,d)\|_\A \leq \eps.
\]
Specifying this to $t\geq \tau(\eps,r)$, we see that $\A$ is UGATT.
\end{proof}

Now we are able to characterize UGAS property.
\begin{theorem}
\label{cor:ULIM_RFC_LS_v2}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$. Let $\A \subset X$ be a bounded set.
The following statements are equivalent:
\begin{enumerate}[label=(\roman*)]
 \item \label{enum:UGAS} $\A$ is UGAS.
 \item \label{enum:RFC_UGATT_RIS} $\Sigma$ is RFC and $\A$ is a UGATT robust invariant set.
 \item \label{enum:RFC_ULS_UGWATT} $\Sigma$ is RFC and $\A$ is uniformly stable and uniformly globally weakly attractive.
 \item \label{enum:UGS_UGWATT} $\A$ is uniformly globally stable and uniformly globally weakly attractive.
 \item \label{enum:UGS_UGATT} $\A$ is uniformly globally stable and UGATT.
\end{enumerate}
\end{theorem}

\begin{proof}
\ref{enum:UGAS} $\Rightarrow$ \ref{enum:RFC_UGATT_RIS}. Estimate $\beta(\|x\|_\A,t)\leq \beta(\|x\|_\A,0)$ shows global stability of $\A$ and hence RFC of a system $\Sigma$ and robust invariance of $\A$. UGATT of $\A$ holds with $\tau(r,\eps)$ as a solution of the equation $\beta(r,\tau)=\eps$ provided such a solution exists or $\tau(r,\eps):=0$ if this equation is not solvable.

\ref{enum:RFC_UGATT_RIS} $\Iff$ \ref{enum:RFC_ULS_UGWATT}. Follows due to Proposition~\ref{prop:UGATT_and_Un_approachability_v2}.

\ref{enum:RFC_ULS_UGWATT} $\Rightarrow$ \ref{enum:UGS_UGWATT}. Lemma~\ref{lem:Weak_attr_plus_RFC_imply_pUGS} shows that $\A$ is Lagrange stable. Since $\A$ is ULS, Remark~\ref{rem:UGS_equals_LagrStab_and_ULS} ensures UGS of $\A$.

\ref{enum:UGS_UGWATT} $\Rightarrow$ \ref{enum:UGS_UGATT}. Since $\A$ is UGS, $\Sigma$ is RFC. Now already proved equivalence \ref{enum:RFC_UGATT_RIS} $\Iff$ \ref{enum:RFC_ULS_UGWATT} shows the claim.

\ref{enum:UGS_UGATT} $\Rightarrow$ \ref{enum:UGAS}. Follows by item (ii) of Lemma~\ref{lem:UGATT_plus_LagrangeSt_imply_pUGAS}.
\end{proof}

\section{Weak attractivity vs uniform weak attractivity}

We devote this final section to the comparison between weak attractivity and uniform weak attractivity.
This can be shown by looking at the special classes of systems: linear systems over Banach spaces and ordinary differential equations. As we will see, in the first case uniform weak attractivity is much stronger than weak attractivity and is equivalent to UGAS of $\{0\}$. In the ODE case uniform and non-uniform weak attractivity coincide provided the disturbances are uniformly bounded. In both cases, criteria for UGAS and pUGAS become much simpler.

\subsection{Linear systems}
\label{sec:linear_systems}

Let us take a quick look at linear systems over Banach spaces without disturbances of the form
\begin{eqnarray}
\dot{x} = Ax,
\label{eq:LinSys}
\end{eqnarray}
where $A:D(A)\to X$ is the infinitesimal generator of a $C_0$-semigroup $T(\cdot)$ of bounded operators over a Banach space $X$ with a domain of definition $D(A)$.
Mild solutions of \eqref{eq:LinSys} are given by $\phi(t,x)=T(t)x$, for any $x\in X$ and $t\geq 0$.
In this way, \eqref{eq:LinSys} gives rise to a system without disturbances.

For such systems criteria from Theorem~\ref{cor:ULIM_RFC_LS_v2} take a particularly simple form.

\begin{proposition}
\label{prop:Uniform_appr_equals_Exp_Stability_for_linear_systems_v2}
The following statements are equivalent for the system \eqref{eq:LinSys}:
\begin{enumerate}[label=(\roman*)]
  \item \label{item:ExpSt} $T$ is exponentially stable.
	\item \label{item:LinUGAS} $\{0\}$ is UGAS.
	\item \label{item:LinUGATT} $\{0\}$ is uniformly globally attractive.
	\item \label{item:LinUGWATT} $\{0\}$ is uniformly globally weakly attractive.
	\item \label{item:Lin_noncoerc} There is a non-coercive Lyapunov function for \eqref{eq:LinSys}.
\end{enumerate}

\end{proposition}

\begin{proof}
Equivalence  \ref{item:ExpSt} $\Iff$ \ref{item:LinUGAS} is easy to show \cite[Lemma 1]{DaM13}.
Implications \ref{item:LinUGAS} $\Rightarrow$ \ref{item:LinUGATT} $\Rightarrow$ \ref{item:LinUGWATT} holds already for general nonlinear systems.

Let us show that  \ref{item:LinUGWATT} $\Rightarrow$ \ref{item:LinUGAS}. It is well-known that every strongly continuous semigroup satisfies a bound of the type $\|T(t)x\|\leq M e^{\omega t}\|x\|$, for all $t\geq 0, x\in X$ and suitable $M,\omega \in \R$,
    \cite[Theorem 2.1.6]{CuZ95}. Hence it is RFC.  By
    Lemma~\ref{lem:Weak_attr_plus_RFC_imply_pUGS}, $T$ is
    Lagrange stable. In
    particular, for any $x\in X$ it holds that $\sup_{t\geq0}\|T(t)x\| <
    \infty$ and the Banach-Steinhaus theorem implies
		$\sup_{t\geq 0,\, \|x\|=1}\|T(t)x\| < \infty$
    which means that
    $\{0\}$ is globally stable.
		Thus item \ref{enum:UGS_UGWATT} of Theorem~\ref{cor:ULIM_RFC_LS_v2} shows the claim.

Equivalence \ref{item:LinUGAS} $\Iff$ \ref{item:Lin_noncoerc} follows by Proposition~\ref{t:noncoeLT}.
\end{proof}
\begin{remark}
\label{rem:Approachability_weaker_than_Uniform_Approachability_v2}
Proposition~\ref{prop:Uniform_appr_equals_Exp_Stability_for_linear_systems_v2}
shows that uniform weak attractivity is a stronger
requirement than weak attractivity. In fact, every strongly stable
$C_0$-semigroup $T$ (a semigroup, satisfying $T(t)x\to 0$ as $t\to\infty$
for any $x\in X$) is globally weakly attractive, while it is not necessarily exponentially stable (and hence does not have to be uniformly weakly attractive).
\end{remark}

\subsection{Ordinary differential equations}

Here we specify the developments of Section~\ref{sec:pUGAS_equals_ULIM} to the case of ODEs
\begin{eqnarray}
\dot{x}=f(x,d),
\label{eq:ODE_Sys}
\end{eqnarray}
where $f:\R^n\times D \to \R^n$ is locally Lipschitz continuous w.r.t. the first argument uniformly w.r.t. the second one, $D$ is a compact subset of $\R^m$
and disturbances $d$ belong to the set $\Dc:=L_\infty(\R_+,D)$ of Lebesgue measurable globally essentially bounded functions with values in $D$.

In contrast to
Remark~\ref{rem:Approachability_weaker_than_Uniform_Approachability_v2},
for systems of nonlinear ordinary differential equations with uniformly bounded
disturbances the notions of global weak attractivity and uniform global
weak attractivity coincide which is a simple application of \cite[Corollary III.3]{SoW96}:
\begin{proposition}
\label{prop:Approachability_equals_Uniform_Approachability}
Consider a system \eqref{eq:ODE_Sys} with $\Dc$ as above. Let $\A \subset \R^n$ be any bounded set. Then $\A$ is uniformly globally weakly attractive if and only if it is globally weakly attractive.
\end{proposition}

\begin{proof}
Pick any $r,\eps>0$.
An application of \cite[Corollary III.3]{SoW96} with $C:=\overline{B_r(\A)}$, $\Omega:=B_{\eps}(\A)$,
$K:=\overline{B_{\frac{\eps}{2}}(\A)}$ ensures existence of a $\tau(r,\eps)$ such that for all $x\in C$ and
any $d\in \Dc$ there is a $t\leq\tau$ with $|\phi(t,x,d)| <\eps$.
\end{proof}

\begin{remark}
\label{rem:ISS_version_of_LIM_equals_ULIM}
A counterpart of Proposition~\ref{prop:Approachability_equals_Uniform_Approachability}
can be also proved in the context of input-to-state stability theory, see \cite[Proposition 9]{MiW17b}.
\end{remark}

\begin{remark}
\label{rem:Uniform_Boundedness_of_disturbances_is_essential}
It is easy to see that Proposition~\ref{prop:Approachability_equals_Uniform_Approachability} is not valid for
systems with disturbances which are not uniformly bounded.
Indeed, the system $\dot{x} = - \frac{1}{|d| + 1}x$ with $\Dc= L_\infty(\R_+,\R)$ has $0$ as
a global weak attractor but not as a uniform one.
\end{remark}

The criterion for practical UGAS of a system \eqref{eq:ODE_Sys} takes a particularly simple form.
We have the following corollary of Proposition~\ref{prop:Approachability_equals_Uniform_Approachability}:
\begin{corollary}
\label{cor:ODE_pUGAS}
\eqref{eq:ODE_Sys} is pUGAS if and only if there is a bounded globally weakly attractive set $\A\subset \R^n$ for \eqref{eq:ODE_Sys}.
\end{corollary}

\begin{proof}
Follows from Theorem~\ref{thm:RFC_weak_attr_imply_pUGAS}, Proposition~\ref{prop:Approachability_equals_Uniform_Approachability} and
\cite[Proposition 5.1]{LSW96} (showing that forward completeness and robust forward completeness are equivalent notions for ODEs with uniformly bounded disturbances).
\end{proof}

\vspace{-7mm}

\section{Conclusions}

For robustly forward complete systems we have shown that practical uniform global asymptotic stability is equivalent to the existence of a global uniform weak attractor.
For the same class of systems, we proved that existence of a non-coercive Lyapunov function implies practical UGAS.
Relations between weak attractivity and uniform weak attractivity have been discussed for the ODEs and linear infinite-dimensional systems without disturbances.

\vspace{-4mm}

\section*{Acknowledgements}

This research has been supported by the German Research Foundation (DFG) within the project "Input-to-state stability and stabilization of distributed parameter systems" (grant Wi 1458/13-1).
The author thanks Fabian R. Wirth for fruitful conversations and his feedback concerning this paper.
The author is also thankful to the anonymous reviewers for the careful
evaluation of the paper and their valuable suggestions.

\bibliographystyle{abbrv}

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text:
Caecum clarum is een slakkensoort uit de familie van de Caecidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1910 door Lamy.

Caecidae

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The local coal and wood-powered plant, along with PSNH's Merrimack Station in Bow, is included in a report released by Toxic Action Network naming the power company as one of top 12 toxic polluters in New England.
The public health and environmental nonprofit selected PSNH as one of the recipients of the "Dirty Dozen Awards," making it one of the companies it says has failed to take appropriate action to address their pollution problems.
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It has been five years since the organization released its last Dirty Dozen Awards. Claire Miller, senior community organizer for Toxics Action Network, said this year celebrates the organization's 25th anniversary. This year, Miller said the organization decided to start fresh and take a new look at the companies in New England.
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Jonathan Peress of the Conservation Law Foundation said PSNH is worthy of the award.
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Peress said PSNH's Schiller Station is the least-efficient coal unit in all of New England, a fact he said makes its power more costly for consumers.
"In a nutshell, PSNH sells the dirtiest and most expensive power in New England," he said.
Jerry Curran of the New Hampshire Sierra Club said PSNH has polluted the air and water for decades.
"It's time PSNH moved beyond coal, beyond mercury, beyond sulfur and nitrous oxides — not to mention climate change," he said. "Our communities and neighborhoods can no longer bear the costs on our health and environment."
In response to the report released Tuesday, PSNH spokesman Martin Murray said the company has been committed to giving its customers energy that is clean, reliable and as economic as possible.
"Both of the plants named have operated reliably for years, and have provided significant benefit to our customers," he said. "We constantly seek innovative measures to reduce our environmental impact, and both plants have earned awards for their environmental initiatives."
As proof, Murray said both plants have previously earned environmental initiative awards from the U.S. Environmental Protection Agency and the state of New Hampshire.
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El comtat de Flers fou una jurisdicció de l'antic règim a França centrada a Flers (Orne) i a l'antic castell de Flers. Fou una senyoria al . Al  segons la tradició, dos germans de noms Folc d'Aunou i Guillem de Gasprée, es van casar amb dues germanes, senyores de Flers i altres dominis. La capital, Flers, va quedar en la part de Folc d'Aunou. El castell es va construir al . El 1356 durant la Guerra dels Cent Anys, el castell no és esmentat el que voldria dir que havia perdut importància almenys estratègica. Junt amb altres senyories va formar la baronia de Flers. Nicolas de Grosparmy governava la bornia la primera meitat del  (1527-1541); més tard el domini fou transmès a la família de Pellevé. Al  fou elevat a comtat. Lluís de Pellevé era comte quan va construir la nova façana del castell el 1698. Després Jacint Lluís de Pellevé va fer arranjar el parc. Al  va passar a la família de La Motte-Ango pel matrimoni d'Antonieta de Pellevé, senyora i comtessa de Flers, amb Felip Ramon de La Motte-Ango. El seu fill Àngel Jacint de La Motte-Ango, comte de Flers, va encomanar treball de reforma del castell. El comtat fou suprimit amb la revolució restant només el títol.

Flers


In all of the circumstances described, it is important to remain calm and follow the steps as indicated. If there are any questions concerning these procedures, please refer to the Emergency Phone Numbers listed.
MU subscribes to a mass notification system to help expedite communications with the campus community in the event of an emergency. All MU employees should access the registration site to make sure that their contact information is up to date. (You will need your Pawprint and Pawprint password to access the site.) Students can access the registration site by logging onto myZou, going to "Self Service", then "Campus Personal Information" and then "Phone Numbers."


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Cannabaudes or Cannabas (died 271) was a third-century leader of the Gothic tribe of the Tervings, who died in a battle against the Roman emperor Aurelian.

Life

In the third century many Germanic peoples invaded the Roman Empire and plundered the border regions. Among these, the Goths were the largest group crossing the Roman borders and devastating the Balkan peninsula.  They also raided the coasts of the Black Sea and the Aegean Sea.  Emperor Aurelian, the former magister equitum of emperor Claudius Gothicus, who had died of the plague, was occupied fighting the dead emperor's brother Quintillus for the Roman throne and later fighting against the Vandals and Juthungi in Italy.  The Tervings under their leader Cannabaudes took advantage of this situation and devastated the provinces of the lower Danube and sacked some cities.

On his march to the East to reintegrate the Palmyrene Empire to the Roman Empire, Aurelian drove the Tervings out of the Roman empire and, unlike others before him, followed them over the Danube. In the ensuing battle, Cannabaudes died along with 5,000 of his men.

For this victory Aurelian received the surname Gothicus Maximus. At his triumph after his victory over the Palmyrene Empire, he paraded Gothic women, dressed as Amazons, and a chariot, pulled by four stags, that was said to have belonged to Cannabaudes.

Despite his victory over Cannabaudes, Aurelian abandoned the province of Dacia, which was populated by barbaric tribes such as the Goths.

Through his military achievements, Aurelian averted a further Gothic intrusions into Roman lands for a century.

While some historians call Cannabaudes a king of the Goths, others consider that he was only one of the more powerful leaders of the Goths.

The only ancient source mentioning his name is the Historia Augusta, which is not a generally reliable source. Nevertheless, Cannabaudes is seen as a historic person by most historians. Some historians want to identify Cannabaudes with Cniva, the Gothic leader who defeated and killed emperor Decius at the Battle of Abrittus twenty years earlier, others historians consider that he was Cniva's son.

Notes

Ancient sources
Historia Augusta, vita Aureliani.

Modern literature
Wolf von Barloewen: Geschichte der Germanen bis 376 n.Chr., in: Wolf von Barloewen [ed.]: Abriss der Geschichte antiker Randkulturen, Oldenbourg, Munich 1961.
Timothy Barnes: The Sources of the Historia Augusta, Ed. Latomus, Brüssel 1978.
Thomas Burns: A History of the Ostrogoths, Indiana Univ.Press, Bloomington 1984.
Thomas Gerhardt, Udo Hartmann: Fasti, Die germanischen Herrscher, in: Klaus-Peter Johne [ed.]: Die Zeit der Soldatenkaiser, Berlin 2008, p. 1192-1198.
Andreas Goltz: Die Völker an der mittleren und nordöstlichen Reichsgrenze, in: Klaus-Peter Johne [ed.]: Die Zeit der Soldatenkaiser, Berlin 2008, p. 449-464.
Edmund Groag: Domitius [36] Aurelian, Realencyclopädie der Classischen Altertumswissenschaft V 1, Stuttgart 1903, p. 1347-1419.
Udo Hartmann: Claudius Gothicus und Aurelian, in: Klaus-Peter Johne [ed.]: Die Zeit der Soldatenkaiser, Berlin 2008, p. 297-324.
Peter Heather: The Goths, Blackwell, Oxford, Cambridge/Massachusetts 1996.
Michael Kulikowski: Rome's Gothic Wars, Cambridge Univ. Press, Cambridge u.a. 2007.
Bruno Rappaport: Die Einfälle der Goten in das Römische Reich bis auf Constantin, Hirschfeld, Leipzig 1899.
Alaric Watson: Aurelian and the third Century, London, New York 1998.
Herwig Wolfram: Die Goten, Beck, Munich ³1990.
Herwig Wolfram: Kniva, Reallexikon der Germanischen Altertumskunde 17, Walter de Gruyter, Berlin, New York 2001, p. 34-37.
Herwig Wolfram: Die Goten und ihre Geschichte, Beck, Munich ²2005.
Herwig Wolfram: Gotische Studien, Beck, Munich 2005.

See also
Thervingi
Crisis of the Third Century
Aurelian
Cniva

3rd-century Gothic people
Gothic kings
Gothic warriors
271 deaths
3rd-century monarchs in Europe
Crisis of the Third Century
Monarchs killed in action
Year of birth missing


Wall Mount Tv Wiring Guide - A array of flat-panel LCD and Plasma television universal wall-mount solutions have been brought into the market by Sanus Methods. These flat-panel TV mounts are of state-of-the-art technologies and are elegantly engineered being compatible with the FDMI, VESA- TV viewing experience more pleasurable, significantly safer and comfortable would be made by MIS standards. Tilting kind wall mounts are usually handy when there is unwanted reflections of light or a problem of glare on the TV set.
Hence the glare may be reduced and adjusted and the Tilting Wall Mounts facilitate the swiveling of the TV set accordingly. It is universally accepted that Tilting sort Wall Mounts bring out sleek and elegance to the drawing-room. One must keep in your mind that the dimensions and weight of the flat panel TV set need to be taken into account while selecting the proper wall mount since the costly TV set down onto the floor might be dropped by picking an incorrect mount. Sanus Methods, as a part of Tilting Wall Mounts, offers wall-mounting solutions extensively under various categories.
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Related Post "Wall Mount Tv Wiring Guide"


Communications Internship | Upward - Find a Job. Move Upward.
The Major League Baseball Players Association's communications intern will gain valuable communications/public relations experience working within a preeminent sports labor and marketing organization. Reporting to the director and assistant director of communications, the communications intern will be involved with the MLBPA's day-to-day media, public relations and marketing communications activities, supporting the communications objectives of the Association's labor, licensing, and not-for-profit activities. These activities will include, but not be limited to, copy writing for MLBPLAYERS.com, PlayersTrust.org and ActionTeam.org, news monitoring, press release and PSA distribution, social media account coordination, story development, coordination and placement in various media, and marketing communications and promotion in support of the Players Choice licensing program and the Major League Baseball Players Trust.
Eligible candidates should be on track to obtain a degree in public relations, communications and/or sports management, and should possess excellent written and verbal communications skills, outstanding interpersonal skills, experience working with the Microsoft suite of programs (Word, PowerPoint, Excel, Publisher), graphic design/image editing using PhotoShop, working knowledge of various social networking sites and applications including Twitter, Facebook, YouTube, etc. and experience using the Internet as a communications and research tool. Experience with content management systems (e.g., Titan, Word Press), video editing software and knowledge of Major League Baseball is helpful. Recent graduates may apply. References and transcript are required.


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INTERVIEW: Writer/Director Theo Francocci Became The Demon In NYCTOPHOBIC
Nyctophobic is a horror short film by writer/director Theo Francocci (Rick & Morty) about a woman battling inner demons while real demons, both human and other, run amuck.
The film begins on a dialogue-less scene that delivers a gruesome reality through creepy camera work, great performances, and an ominous painting hanging in the background. The painting is The Nightmare by Harry Fuseli, and in the short sequence, the film delivers its antagonist, the demon inside the artwork. Alex Knell (Baker's Dozen) plays Stacy, a woman with a fear of the dark. She's spending a quiet night with her ex-boyfriend Mark played by Theo Francocci. The dark story plays out between Mark's tough love and supernatural horrors.
PopAxiom spoke with Theo Francocci about going from being an audio description writer on projects like Rick and Morty and The Royals to creating the horror of Nyctophobic.
Sound & Story
"One of my dreams was to become a singer/songwriter," filmmaker Theo Francocci begins. "So, I moved to London in 2010 to study sound engineering. While there, I discovered sound design."
Theo fell in love with creating sound for pictures. "I met the right person at the right time and had the right connections and ended up at Studio Marinelli in Rome, Italy. They were famous sound designers with a massive history working with Fellini, Antonioni, and Leone. Masters of the time. I learned the job there and so much about the history of film."
"Sound plays such an important role in making a story unique."
About Nyctophobic
Years, projects, and some film schooling later, Theo conceived Nyctophobic. "It connects to COVID very much. The pandemic made our fears emerge. We were lonelier and surrounded by a world that seemed more hostile. I like to get into the psychology of things, so I reassessed my traumas. That's where the characters came from; my fear of darkness as a kid. I attached that fear to the protagonist Stacy. That fear of darkness is so relatable; I think everyone had it at some point. It's profound."
"So, I asked," he continues, "'What if this character is so traumatized that it affects her life; her relationships and work?'"
The process started around October of 2020 while Theo was in school. "We had an exercise to try and describe a scene with dialogue only. No action lines. That was the beginning of their script and the dialogue while they were sitting on the floor."
The exercise was complete, but Theo "felt that dialogue could be developed more, so that's what I did. I finished it on Halloween of 2020. I made a pact with myself to start the production and have a premiere by next Halloween." Mission accomplished.
Making Nyctophobic
"I wanted that scene to be visual and live on those sound effects of the footsteps and the crackling," Theo says about the haunting opening scene. "It wasn't complicated to shoot, but it was tricky to get the sensation. So, we worked with the actors to get the movement. Then later in post-production with the sound and camera effects."
Theo says, "being on set is problem-solving from the moment you arrive to the moment you leave. So, rehearsals are so important for me because I want to get to set and just know that the actors are comfortable with what they're doing."
"Delegating and trusting your people is vital," he continues," You have to trust the people you're working with and make sure they trust you and see your vision."
Theo plays one of the two primary characters in the film. But that wasn't the plan. "I didn't want to be the lead. Being the director is already so much work. It's a lot."
So, what changed? "I had an actor that I liked, but the timing didn't work out. He got a better offer for a TV show, so we were happy for him. But it was two weeks before shooting, and I was trying to recast quickly. So, I had a conversation with Matt [Richardson], my co-producer. He said, 'no one knows the character like the writer.'
"So, I thought," he concludes, "'well, this character's going to have an Italian accent.'"
The star of Nyctophobic went to Alexandra Knell, who Theo says "was perfect for Stacy. She's got so much experience and talent."
The Creeps
Nyctophobic is the best kind of horror that blends real fears with supernatural frights. It all starts with the painting. "I love that painting; The Nightmare by Henry Fuseli that was there since the script. It gives me the creeps. I fought to have that painting."
"We got in touch with the LA School of Makeup," he says about creating the demon's design, "we had a sort of casting. So we asked everyone to submit what a demon of darkness would look like. We received a lot of incredible designs. But Yun Nam blew us away."
The final look "had that Voldemort nose but was very much like a scaly reptile for Theo. She understood all that."
In Nyctophobic, there's a moment that's a clear reference to The Shining, "Why not? I love Kubrick, and I wanted to do that as precisely as possible."
Inspirational Cinema
We discuss cinema worldwide, and Theo shares his thoughts about his homeland "Italian cinema is a lot about style, especially in horror. We're able to make good movies with very low budgets. It's a necessity that I think we mastered along the way."
"Korean films," he continues, "like Parasite blew my mind. They have a huge history and tradition in films. Now they've blown everyone away with Squid Game."
As for the US, Theo says it "is the driving force of the entertainment industry. Whatever gets made here sells all over the world. I love the direction into more diversity and embracing different styles."
"It's always great to let these incredible pieces of art from these incredible places inspire you."
The art is an inspiration, as are the people creating it. "We spoke a lot about Italian masters, but Roberto Benigni is one of those actors/directors that I carry in my heart. Life is Beautiful is one of my favorite movies ever made."
"Katherine Bigelow … Strange Days and Zero Dark Thirty were so intense," he adds, "I love the realism and darkness."
"Martin Scorsese. I need to say, Jordan Peele. He's a beast. He can do comedy, horror, whatever he wants. Tim Burton. Christopher Nolan. Ridley Scott. I'll stop there," Theo laughs.
Asking Theo about a dream project prompts a quick answer. "I wrote a pilot for a TV show last year, and it's about the death of Julius Cesar. That would be one of my dream projects. I love culture and ancient Rome. Building a set that would be authentic and the costumes. At the same time, having VFX, beasts, animals, ghosts, gladiators. Make it feel like fantasy, but it's history."
Nyctophobic is making the rounds at film festivals. So, what's next for Theo? "I'm always working on something else. That's how creativity works. You're focused on this, but your brain is like, 'but what if we do that?' So I'm working on another short film called Living in Color.
Is Nyctophobic on your watch list?
Thanks to Theo Francocci and Lumos PR
for making this interview possible.
Read more PopAxiom interviews from Ruben R. Diaz here!


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Q: Find the tension and angle from a vertical of a rope hooked up to an object Below is the homework problem I am working on:

A 18.3-kg object hangs in equilibrium at the end of a rope (taken as massless) while a wind pushes the object with a 63.9-N horizontal force. Find the tension in the rope and the rope's angle from the vertical. The acceleration due to gravity is $9.81 \space m/s^2$.

I am having a hard time correctly defining the system of two equations to solve for tension and angle. I came up with $$T\cos{(\theta)} -F_H=0 \\T\sin{(\theta)}-(-F_g)=0$$ where $F_g$ is the force due to gravity and $F_H$ is the horizontal force. I use a free body diagram where $F_H$ is on the x-axis and $F_g$ is on the y-axis. Apparently, my setup is incorrect. Could you put me on the right track. Let me know if more clarification is needed.

A: Assuming $\theta$ is the angle measured from the vertical, then $Tcos(\theta)$ will give you the vertical component of the tension, whereas $Tsin(\theta)$ will give you the horizontal component. Therefore, the expression $Tsin(\theta)$ should be equal and opposite to $F_H$, and $Tcos(\theta)$ should be equal and opposite to $F_g$. I believe your system has this backwards.
Also depending on exactly what you are plugging in for $F_g$ (meaning, whether you are plugging the absolute value of the gravitational force or not), this could be causing you problems with the signs you are using. Personally, I would write the second equation like this:
$$ T_y-F_g=0$$ Then plug in the magnitude of the gravitational force for $F_g$.

redpajama_set_name: RedPajamaStackExchange


Work to Achieve Your Highest Level: Be the New Prosperity.
Learning to live in the physical world is a lifetime of learning that . . . you have come . . . and you will go. One of the fundamental laws of this world is the impermanence of the physical body; it displays the permanence of spirit and if you are upset by the impermanence of this world, you are constantly struggling against spirit. You are un-tuning your physical instrument and fighting what is. Each of us has a core melody and a core rhythm. It is our song of this life and we came into this world to play it. You didn't come into this world to copy mine, or play anyone else's. But if your instrument is out of tune — even if you know your song very well — you will not be able to play your song harmoniously. This is the importance of health and consciousness.
When you begin to align with the harmonic convergence of your song, you become intuitive. Intuition is not good guessing . . . intuition knows beyond the rational. You know because you know and that is that. When your instrument is fully tuned, you not only live harmoniously in time space, but you know beyond timespace.
With an un-tuned instrument, there tends to be more aggression and frustration, which leads to a further need for dominance. This creates the malfunctions of societies with their main emphasis of life becoming the struggle for law and order. Just like today's world . . . we have now entered the era where nothing will work out until the instruments of humanity are tuned.
Those reading this are perhaps more in tune than most, but that doesn't make us better, it only makes us more in tune. We may feel less in tune at times, but that is because we are more aware. It is time we further tune our own instruments.
We all have a core value. We all have a core song at the center of our being; it governs our tendencies and our expectations. We need to move through the debris that has surrounded this song for generations . . . for lifetimes . . . the debris that has kept us out of tune and out of touch.
When we become more in tune, we will then have additional responsibility. The word is actually response ability . . . moving through timespace with the ability to respond. This is the sign of leadership, the sign of a teacher and of mastery.
This leadership has confidence, is deeply benevolent, and is completely centered and highly intuitive . . . this leadership is able to guide all life forward. The instrument of this leadership plays the songs of life in harmony, because the instruments are well tuned. This is the leadership of inspiration with the coming era . . . the era of the new economics in the next evolution. This is a change we can all bank on, we can invest in. This is the new prosperity . . . advancing the greatest possibilities of life.


Being a Realtor (in my past life) and Real Estate Photographer over the last 30 years, I know how important good images are in a marketplace with almost 90% of Buyers are online. The internet has given power to Buyers and Sellers with lots of information that some people believe that the days of the Real Estate Agent are almost over. But I think that agent's role in assisting people buying and selling homes is merely evolving rather than diminishing altogether. Despite our future, one area that each listing agent should attend to now is the quality of the media they use to promote homes and draw buyers. Therefore, here we discuss the prerequisites for hiring a great photographer.
Regardless of cameras, what is important is the ability to communicate "the look" you are after and then be able to ask the correct questions to discern if your prospective photos can deliver.
Does your equipment capable of capturing a scene in the 16-24mm dynamic focal length range? Though the effective focal length will differ depending on the camera. Ensure the photographer is equipped to shoot anywhere close to this range. It is imperative that the photographer not only can shoot ultra wide but also know how and when to use it.
Do you do post processing or digital editing to optimize the images? All photos have the potential to be far better than what came right out of the camera. In some situation, fantastically better. Whether or not you decide to make it so is your responsibility in whom you choose to hire. Just know that been skillful in this area of digital photography is a real game changer.
Do you capture them "As-Is" or you "Stage" the scenes? Taking the time Staging is a tedious and severe element to getting the best promising image. Either full-blown home staging or just minor decluttering, ability to stage a photo allows photograph reach new potential.
Do you incorporate a website or virtual tour in addition to the images? Some photographers include a turn-key custom website and virtual tour linkage that displays your photographs in a Hi-Def Photo gallery and Slideshow format. These sites can function as a virtual tour link for the MLS also as a single property website containing all the key information about the property. If you are in need of a whole online marketing package, as contrasting to just pictures, this is the best solution.
There is much more to Real Estate photograph than ultra wide-angle shots. Lifestyle images, Twilight and Dusk images, Time-Lapse, Panoramas, Creative Light Painting, Pole Aerial, or Photography. Professional photography services can render some or all of those in addition to just images.