text:
\section{Introduction}

In this paper, we study the property of weak input-to-state stability of general dynamical systems $\mathfrak{S} = (X,\mathcal{U},\phi)$ with inputs. Such a system is determined by its generally infinite-dimensional state space $X$, its set $\mathcal{U}$ of admissible input functions, and its dynamical map $$\phi: [0,\infty) \times X \times \mathcal{U} \to X$$ which for given initial state $x_0 \in X$ and input $u \in \mathcal{U}$ yields the state $\phi(t,x_0,u)$ of the system at any time $t \in [0,\infty)$.
Weak input-to-state stability of such a system means, roughly speaking, that $0$ is an asymptotically stable -- that is, stable and attractive -- equilibrium point of the (undisturbed) system with input $u = 0$ and that this asymptotic stability property is affected only slightly by small (disturbance) inputs $u \ne 0$. In precise terms,
this means that there are continuous monotonically increasing functions $\underline{\sigma}, \underline{\gamma}, \overline{\gamma}: [0,\infty) \to [0,\infty)$ with $\underline{\sigma}(0), \underline{\gamma}(0), \overline{\gamma}(0) = 0$ such that for all $(x_0,u) \in X \times \mathcal{U}$ the following estimates hold true:
\begin{gather}
\norm{ \phi(t,x_0,u) } \le \underline{\sigma}(\norm{x_0}) + \underline{\gamma}(\norm{u}_{\mathcal{U}})
\qquad (t \in [0,\infty)) \label{eq:UGS, intro}\\
\text{and} \notag \\
\limsup_{t\to\infty} \norm{ \phi(t,x_0,u) } \le \overline{\gamma}(\norm{u}_{\mathcal{U}})
\label{eq:wAG, intro}
\end{gather}
meaning that
the stability property and the attractivity property of $0$, respectively, are affected only slightly by disturbance inputs $u \in \mathcal{U}$ of small magnitude $\norm{u}_{\mathcal{U}}$. Inequality~\eqref{eq:UGS, intro} is commonly referred to as the uniform global stability and~\eqref{eq:wAG, intro} is referred to as the weak asymptotic gain property of the system.
It should be noted that if the limit relation~\eqref{eq:wAG, intro} holds uniformly w.r.t.~$u \in \mathcal{U}$ or, respectively, locally uniformly w.r.t.~$x_0 \in X$ and uniformly w.r.t.~$u \in \mathcal{U}$, then the system is even strongly or uniformly input-to-state stable, respectively.
\smallskip

In recent years, these last two notions of strong and especially of uniform input-to-state stability have been intensively studied. See, for instance,~\cite{DaMi13}, \cite{Mi16}, \cite{MiIt16}, \cite{MiWi16a}, \cite{MiWi17}, \cite{JaNaPaSc16}, \cite{JaScZw17}, \cite{NaSc17}, \cite{MaPr11}, \cite{MiKaKr17}, \cite{TaPrTa17}, \cite{KaKr16}, \cite{KaKr17}, \cite{ZhZh17a}, \cite{ZhZh17b} and the references therein. Also, weak input-to-state stability can be established for a rather large class of semilinear systems (both in the case of inputs entering in the domain and in the case of inputs entering at the boundary of the domain on which the partial differential equation describing the system lives). See~\cite{ScZw18-MTNS}, \cite{ScZw18}, \cite{Sc18-wp}. It is therefore natural to study the property of weak input-to-state stability
-- and especially its relation to other common stability properties --
from a general point of view.
\smallskip

In the present paper, we establish a characterization of weak input-to-state stability similar to the
characterizations of strong and uniform input-to-state stability from~\cite{MiWi16a} and, moreover, we investigate the relation of weak input-to-state stability to other common and natural stability concepts, namely strong input-to-state stability and zero-input uniform global stability. In particular, we answer
two open questions from~\cite{MiWi16a}.
\smallskip

In more detail, the contents of the present paper can be described as follows.
Section~\ref{sect:2} provides the necessary preliminaries
setting out and recalling the precise definitions of abstract dynamical systems with inputs and of the various stability notions employed later on.
In Section~\ref{sect:3} we establish a characterization of weak input-to-state stability which is parallel to the characterizations of strong and uniform input-to-state stability for infinite-dimensional systems recently established in~\cite{MiWi16a}.
In Section~\ref{sect:4} we investigate the relation of weak input-to-state stability to strong input-to-state stability.
We show, by means of a suitable counterexample, that weak input-to-state stability is strictly weaker than strong input-to-state stability,
thereby answering an open question raised in~\cite{MiWi16a}.  In our example, we work with modulated-linear systems, which are  described by  evolution equations of the form
\begin{align} \label{eq:modul-lin, def}
x' = \alpha(u(t)) A x
\end{align}
with a linear operator $A$ and a modulating prefactor $\alpha(u(t))$, and the input space $\mathcal{U}$ is a certain subset of $L^p([0,\infty),\mathbb{R})$.
We also show that in the special case of linear systems,
weak input-to-state stability is equivalent to strong input-to-state stability.
In the case of semilinear systems, the relation of weak and strong input-to-state stability remains open.
We show at least, however, that for semilinear systems weak input-to-state stability is strictly weaker than uniform input-to-state stability.
In Section~\ref{sect:5} we investigate the relation of weak input-to-state stability to the combination of zero-input uniform global stability and the weak asymptotic gain property.
We show, by means of a suitable counterexample, that weak input-to-state stability is strictly stronger than the aforementioned combination of properties, thereby answering an open question raised in~\cite{MiWi16a}. In our example, we work with linear systems with input space $\mathcal{U}$ being a certain subset of $L^{\infty}([0,\infty),\mathbb{R})$. We also show that for linear systems with input space $\mathcal{U}$ being a full $L^p$-space, weak input-to-state stability is equivalent to the aforementioned combination of properties.
\smallskip

In the entire paper, $\mathbb{R}^+_0 := [0,\infty)$ denotes the non-negative reals and $\overline{B}_r^{Z}(0) := \{ z \in Z: \norm{z} \le r\}$ for any subset $Z$ of a normed linear space with norm $\norm{\cdot}$. As usual, $\mathcal{K}$ and  $\mathcal{L}$ denote the following classes of comparison functions:
\begin{gather*}
\mathcal{K} := \{ \gamma \in C(\mathbb{R}^+_0,\mathbb{R}^+_0): \gamma \text{ strictly increasing with } \gamma(0) = 0 \} \\
\mathcal{L} := \big\{ \gamma \in C(\mathbb{R}^+_0,\mathbb{R}^+_0): \gamma \text{ strictly decreasing with } \lim_{r\to\infty} \gamma(r) = 0 \big\}.
\end{gather*}
Also, $\norm{\cdot}_p$ for any $p \in [1,\infty) \cup \{\infty\}$  stands for the standard norm on $L^p(\mathbb{R}^+_0,U)$, where $U$ is any Banach space, and $u_1\,\&_{\tau}u_2$ stands for the concatenation of the functions $u_1, u_2: \mathbb{R}^+_0 \to U$ at time $\tau \in \mathbb{R}^+_0$ defined by
\begin{align*}
(u_1\,\&_{\tau} u_2)(t) :=
\begin{cases} u_1(t) \qquad (t \in [0,\tau)) \\
u_2(t-\tau) \qquad (t \in [\tau,\infty))
\end{cases}.
\end{align*}
And finally, in the context of admissible control operators -- and, in particular, of extrapolation of semigroup generators -- we adopt the standard notation from~\cite{EnNa}, \cite{TuWe}.

\section{Setting and definitions} \label{sect:2}

\subsection{Systems with inputs} \label{sect:contr syst}

We begin by setting out the class of systems that we -- just like~\cite{MiWi16a} -- are going to deal with in this paper.

\begin{df}
A (forward-complete) dynamical system $\mathfrak{S} = (X,\mathcal{U},\phi)$ with inputs is determined by
\begin{itemize}
\item a normed linear space $X$ (the state space of $\mathfrak{S}$) endowed with a norm $\norm{\cdot}_X$
\item a non-empty set $\mathcal{U} \subset
\{ \text{functions } u: \mathbb{R}^+_0 \to U \}$ (the set of admissible inputs of $\mathfrak{S}$) endowed with a norm $\norm{\cdot}_{\mathcal{U}}$
\item a map $\phi: \mathbb{R}^+_0 \times X \times \mathcal{U} \to X$ (the dynamical map of $\mathfrak{S}$)
\end{itemize}
such that the following properties are satisfied:
\begin{itemize}
\item[(i)] $\mathcal{U}$ is invariant under shifts to the left, that is, $u(\cdot+\tau) \in \mathcal{U}$ and $\norm{u(\cdot+\tau)}_{\mathcal{U}} \le \norm{u}_{\mathcal{U}}$ for every $u \in \mathcal{U}$ and $\tau \in \mathbb{R}^+_0$
\item[(ii)] $\mathcal{U}$ is invariant under concatenations, that is, $u_1 \, \&_{\tau} \, u_2 \in \mathcal{U}$ for every $u_1,u_2 \in \mathcal{U}$ and $\tau \in \mathbb{R}^+_0$
\item[(iii)] $\phi(0,x_0,u) = x_0$ for every $(x_0,u) \in X \times \mathcal{U}$ and, moreover, $\phi$ is cocyclic, that is,
\begin{align}
\phi(t+s,x_0,u) = \phi(t, \phi(s,x_0,u), u(\cdot+s))
\end{align}
for every $(x_0,u) \in X \times \mathcal{U}$ and $s,t \in \mathbb{R}^+_0$
\item[(iv)] $\phi(\cdot,x_0,u): \mathbb{R}^+_0 \to X$ is continuous for every $(x_0,u) \in X \times \mathcal{U}$
\item[(v)] $\phi$ is causal, that is,
\begin{align}
\phi(\cdot,x_0,u_1)|_{[0,\tau]} = \phi(\cdot,x_0,u_2)|_{[0,\tau]}
\end{align}
for every $x_0 \in X$, $u_1, u_2 \in \mathcal{U}$ and $\tau \in \mathbb{R}^+_0$ with $u_1|_{[0,\tau]} = u_2|_{[0,\tau]}$.
\end{itemize}
\end{df}

In the following, we will always write $\norm{\cdot} := \norm{\cdot}_X$ for brevity. Since $\mathcal{U}$ is not assumed to be a linear space, it is sligthly abusive to speak of a norm $\norm{\cdot}_{\mathcal{U}}$ on $\mathcal{U}$. What we mean is, of course, that $\norm{\cdot}_{\mathcal{U}}$
is the restriction of a norm of some linear space $\mathcal{F} \supset \mathcal{U}$.

\subsection{Stability and attractivity concepts}

We continue by recalling the stability and attractivity concepts from~\cite{MiWi16a} that will be used in the sequel.

\begin{df}
A dynamical system $\mathfrak{S} = (X,\mathcal{U},\phi)$ with inputs is called
\begin{itemize}
\item[(i)] \emph{uniformly globally stable} iff there exist $\underline{\sigma}, \underline{\gamma} \in \mathcal{K}$ such that for all $(x_0,u) \in X \times \mathcal{U}$
\begin{align} \label{eq:UGS-def}
\norm{\phi(t,x_0,u)} \le \underline{\sigma}(\norm{x_0}) + \underline{\gamma}(\norm{u}_{\mathcal{U}}) \qquad (t \ge 0)
\end{align}
\item[(ii)] \emph{uniformly locally stable} iff there exist $\underline{\sigma}, \underline{\gamma} \in \mathcal{K}$ and $r > 0$ such that~\eqref{eq:UGS-def} holds true for all $(x_0,u) \in \overline{B}_r^{X}(0) \times \overline{B}_r^{\mathcal{U}}(0)$
\item[(iii)] \emph{zero-input uniformly globally stable} or \emph{zero-input uniformly locally stable}, respectively, iff $0 \in \mathcal{U}$ and the restricted system $\mathfrak{S}_0 := (X,\mathcal{U}_0,\phi)$ with $\mathcal{U}_0 := \{0\}$ is uniformly globally or uniformly locally stable, respectively.
\end{itemize}
\end{df}

\begin{df}
Suppose $\mathfrak{S} = (X,\mathcal{U},\phi)$ is a dynamical system with inputs and $\overline{\gamma} \in \mathcal{K} \cup \{0\}$. $\mathfrak{S}$ is said to be
\begin{itemize}
\item[(i)] of \emph{weak asymptotic gain} $\overline{\gamma}$ iff for every $\varepsilon > 0$ and $(x_0,u) \in X \times \mathcal{U}$ there exists a time $\overline{\tau}(\varepsilon,x_0,u) \in \mathbb{R}^+_0$ such that
\begin{align}
\norm{\phi(t,x_0,u)} \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}}) \qquad (t \ge \overline{\tau}(\varepsilon,x_0,u))
\end{align}
\item[(ii)] of \emph{strong asymptotic gain} $\overline{\gamma}$ iff for every $\varepsilon > 0$ and $x_0 \in X$ there exists a time $\overline{\tau}(\varepsilon,x_0) \in \mathbb{R}^+_0$ such that
\begin{align}
\norm{\phi(t,x_0,u)} \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}}) \qquad (t \ge \overline{\tau}(\varepsilon,x_0) \text{ and } u \in \mathcal{U})
\end{align}
\item[(iii)] of \emph{uniform asymptotic gain} $\overline{\gamma}$ iff for every $\varepsilon > 0$ and $r > 0$ there exists a time $\overline{\tau}(\varepsilon,r) \in \mathbb{R}^+_0$ such that
\begin{align}
\norm{\phi(t,x_0,u)} \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}}) \qquad (t \ge \overline{\tau}(\varepsilon,r) \text{ and } (x_0,u) \in \overline{B}_r^{X}(0) \times \mathcal{U}).
\end{align}
\end{itemize}
Also, $\mathfrak{S}$ is said to be of \emph{weak asymptotic gain} iff it is of weak asymptotic gain $\overline{\gamma}$ for some $\overline{\gamma} \in \mathcal{K} \cup \{0\}$.
\end{df}

\begin{df}
Suppose $\mathfrak{S} = (X,\mathcal{U},\phi)$ is a dynamical system with inputs. $\mathfrak{S}$ is said to have the \emph{weak limit property} iff there is a $\overline{\gamma} \in \mathcal{K}$ such that for every $\varepsilon > 0$ and $(x_0,u) \in X \times \mathcal{U}$ there exists a time $\overline{\tau}(\varepsilon,x_0,u)$ such that
\begin{align}
\inf_{t \in [0, \overline{\tau}(\varepsilon,x_0,u)] } \norm{\phi(t,x_0,u)} \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}}).
\end{align}
\end{df}

In~\cite{MiWi16a}, the weak asymptotic gain and the weak limit properties are referred to simply as asymptotic gain and limit property, respectively. We deviate from that terminology in order to emphasize the logical relation to the strong and uniform variants and in order to emphasize the parallelism of certain issues.

\begin{lm}
Suppose that $\mathfrak{S} = (X,\mathcal{U},\phi)$ is a dynamical system with inputs and that $\mathcal{U} \subset L^p(\mathbb{R}^+_0,U)$ with $\norm{\cdot}_{\mathcal{U}} := \norm{\cdot}_p$ for some $p \in [1,\infty)$ or that $\mathcal{U} \subset L^{\infty}_0(\mathbb{R}^+_0,U)$ with $\norm{\cdot}_{\mathcal{U}} := \norm{\cdot}_{\infty}$, where $U$ is a Banach space and
\begin{align*}
L^{\infty}_0(\mathbb{R}^+_0,U) := \big\{ u \in L^{\infty}(\mathbb{R}^+_0,U): \norm{u(\cdot+t)}_{\infty} \longrightarrow 0 \text{ as } t \to \infty \big\}.
\end{align*}
If $\mathfrak{S}$ is of weak asymptotic gain, then it is automatically of weak asymptotic gain $0$.
\end{lm}

\begin{proof}
Suppose $\mathfrak{S}$ is of weak asymptotic gain $\overline{\gamma} \in \mathcal{K}$ (with corresponding times $\overline{\tau}(\varepsilon,x_0,u)$) and let $\varepsilon >0$ and $(x_0,u) \in X \times \mathcal{U}$ be fixed. Since $\norm{u(\cdot+t)}_{\mathcal{U}} \longrightarrow 0$ as $t \to \infty$ by our assumptions on $\mathcal{U}$, we can choose a time $t_0 \in \mathbb{R}^+_0$ such that
\begin{align} \label{eq:lm-wAG-implies-wAG-0, 1}
\overline{\gamma}\big( \norm{u(\cdot+t_0)}_{\mathcal{U}} \big) \le \varepsilon.
\end{align}
Since $\mathfrak{S}$ is of weak asymptotic gain $\overline{\gamma}$, we have for all $s \ge \overline{\tau}(\varepsilon, \phi(t_0,x_0,u), u(\cdot+t_0))$ that
\begin{align} \label{eq:lm-wAG-implies-wAG-0, 2}
\norm{\phi(s+t_0,x_0,u)} = \norm{ \phi(s, \phi(t_0,x_0,u), u(\cdot+t_0)) } \le \varepsilon + \overline{\gamma}\big( \norm{u(\cdot+t_0)}_{\mathcal{U}} \big).
\end{align}
Combining now~\eqref{eq:lm-wAG-implies-wAG-0, 1} and~\eqref{eq:lm-wAG-implies-wAG-0, 2} we see that for all $$t \ge \tau_0(\varepsilon,x_0,u) := t_0 + \overline{\tau}(\varepsilon, \phi(t_0,x_0,u), u(\cdot+t_0))$$ one has
$\norm{\phi(t,x_0,u)} \le 2 \varepsilon$. Consequently, $\mathfrak{S}$ is of weak asymptotic gain $0$, as desired.
\end{proof}

\subsection{Input-to-state stability concepts}

With the stability and attractivity properties recalled above, we can now define the central concepts of this paper, namely weak, strong, and uniform input-to-state stability.

\begin{df}
A dynamical system $\mathfrak{S} = (X,\mathcal{U},\phi)$ with inputs is called
\emph{weakly input-to-state stable}, or \emph{strongly input-to-state stable}, or \emph{uniformly input-to-state stable},
respectively, iff it is uniformly globally stable and of weak, or strong, or uniform asymptotic gain, respectively.
\end{df}

Instead of uniform input-to-state stability one often simply speaks of input-to-state stability in the literature.

\section{Characterization of weak input-to-state stability} \label{sect:3}

We begin with a characterization of weak input-to-state stability which is parallel to the recently established characterizations of strong and uniform input-to-state stability from~\cite{MiWi16a}. It should be pointed out that the equivalence of items~(i) and (ii) below is already stated in~\cite{MiWi16a} (Remark~5), yet without proof.

\begin{thm}
Suppose $\mathfrak{S} = (X,\mathcal{U},\phi)$ is a dynamical system with inputs. Then each of the following items is equivalent to $\mathfrak{S}$ being weakly input-to-state stable.
\begin{itemize}
\item[(i)] $\mathfrak{S}$ is uniformly globally stable and has the weak limit property
\item[(ii)] $\mathfrak{S}$ is uniformly globally stable and has the weak asymptotic gain property
\item[(iii)] there exist $\sigma, \gamma \in \mathcal{K}$ and $\beta: X \times \mathcal{U}\times \mathbb{R}^+_0 \to \mathbb{R}^+_0$ with $\beta(x_0,u,\cdot) \in \mathcal{L}$ for $x_0 \ne 0$ such that for all $(x_0,u) \in X \times \mathcal{U}$ one has:
\begin{gather}
\norm{\phi(t,x_0,u)} \le \beta(x_0,u,t) + \gamma(\norm{u}_{\mathcal{U}}) \qquad (t \in \mathbb{R}^+_0) \label{eq:wISS-beta}\\
\text{and} \notag \\
\beta(x_0,u,t) \le \sigma(\norm{x_0})  \qquad (t \in \mathbb{R}^+_0). \label{eq:wISS-bound on beta}
\end{gather}
\end{itemize}
\end{thm}

\begin{proof}
We first show the implication from (i) to (ii). So assume that (i) is satisfied and let $\underline{\sigma}, \underline{\gamma} \in \mathcal{K}$ and $\overline{\gamma} \in \mathcal{K}$, $\overline{\tau}(\varepsilon,x_0,u)$ be chosen as in the definitions of uniform global stability and of the weak limit  property, respectively.
We define the function $\gamma$ by
\begin{align}
\gamma(r) := \underline{\sigma}(2 \overline{\gamma}(r)) + \underline{\gamma}(r) \qquad (r \in \mathbb{R}^+_0)
\end{align}
which obviously belongs to $\mathcal{K}$. Choose and fix now $\varepsilon > 0$ and $(x_0,u) \in X \times \mathcal{U}$ and set
\begin{align}
\tau(\varepsilon,x_0,u) := \overline{\tau}(\delta(\varepsilon),x_0,u)
\qquad \text{with} \qquad
\delta(\varepsilon) := \frac{1}{2} \underline{\sigma}^{-1}(\varepsilon).
\end{align}
Let $t \ge \tau(\varepsilon,x_0,u)$. It then follows by the assumed weak limit property that there exists a $t_0 \in [0,\tau(\varepsilon,x_0,u)]$ such that
\begin{align} \label{eq:wISS, impl (i) to (ii), 1}
\norm{\phi(t_0,x_0,u)} \le \delta(\varepsilon) + \overline{\gamma}(\norm{u}_{\mathcal{U}})
\end{align}
It further follows by the cocycle property of $\phi$ and the assumed uniform global stability that
\begin{align}  \label{eq:wISS, impl (i) to (ii), 2}
\norm{\phi(t,x_0,u)} &= \norm{ \phi(t-t_0, \phi(t_0,x_0,u), u(\cdot+t_0)) } \notag \\
&\le \underline{\sigma}( \norm{\phi(t_0,x_0,u)} ) + \underline{\gamma}(\norm{ u(\cdot+t_0) }).
\end{align}
Combining now~\eqref{eq:wISS, impl (i) to (ii), 1} and~\eqref{eq:wISS, impl (i) to (ii), 2} we see for every $t \ge \tau(\varepsilon,x_0,u)$ that
\begin{align} \label{eq:wISS, impl (i) to (ii), 3}
\norm{\phi(t,x_0,u)}  \le \underline{\sigma}(\delta(\varepsilon) + \overline{\gamma}(\norm{u}_{\mathcal{U}}) ) + \underline{\gamma}(\norm{u}_{\mathcal{U}})
\le \varepsilon + \gamma(\norm{u}_{\mathcal{U}}),
\end{align}
as desired. In the first inequality of~\eqref{eq:wISS, impl (i) to (ii), 3} we used that $\norm{u(\cdot+t_0)} \le \norm{u}_{\mathcal{U}}$ and in the second inequality we used the elementary fact that
\begin{align*}
\underline{\sigma}(a+b) \le \underline{\sigma}(2\max\{a,b\}) \le \underline{\sigma}(2a) + \underline{\sigma}(2b)
\end{align*}
for all $a,b \in \mathbb{R}^+_0$.
\smallskip

We now show the implication from (ii) to (iii). So assume that (ii) is satisfied and let $\underline{\sigma}, \underline{\gamma} \in \mathcal{K}$ and $\overline{\gamma} \in \mathcal{K}$ be chosen as in the definitions of uniform global stability and of the weak asymptotic gain property, respectively.
We define the functions $\sigma, \gamma$ by
\begin{align} \label{eq:sigma, gamma: def}
\sigma(r) := 2 \underline{\sigma}(r)
\qquad \text{and} \qquad
\gamma(r) := \max \{\underline{\gamma}(r),\overline{\gamma}(r) \}
\end{align}
which obviously belong to $\mathcal{K}$.
What we have to do now is to define for each given $(x_0,u) \in X \times \mathcal{U}$ a function $\beta(x_0,u,\cdot): \mathbb{R}^+_0 \to \mathbb{R}^+_0$ in such a way that $\beta(x_0,u,\cdot) \in \mathcal{L}$ for $x_0 \ne 0$ and that~\eqref{eq:wISS-beta} and~\eqref{eq:wISS-bound on beta} are satisfied.
So let $(x_0,u) \in X \times \mathcal{U}$ be fixed for the rest of the proof (of the implication from (ii) to (iii)) and assume without loss of generality that
\begin{align} \label{eq:ass x_0 ne 0}
x_0 \ne 0.
\end{align}
(If $x_0 = 0$, then $\underline{\sigma}(\norm{x_0}) = 0$ and thus by the assumed uniform global stability the desired estimates~\eqref{eq:wISS-beta} and~\eqref{eq:wISS-bound on beta} hold true with the choice $\beta(x_0,u,\cdot) := 0$.)
In order to construct $\beta(x_0,u,\cdot)$ we distinguish two cases, namely whether or not $\norm{\phi(\cdot,x_0,u)}$ eventually lies below $\gamma(\norm{u}_{\mathcal{U}})$, that is, whether or not there exists a $t_0 \in \mathbb{R}^+_0$ such that
\begin{align} \label{eq:case distinction}
\norm{\phi(t,x_0,u)} \le \gamma(\norm{u}_{\mathcal{U}}) \qquad (t \ge t_0).
\end{align}

Suppose first that we are in the case where $\norm{\phi(\cdot,x_0,u)}$ eventually lies below $\gamma(\norm{u}_{\mathcal{U}})$.
In this case, set
\begin{align}
\beta_0(x_0,u,t) := \underline{\sigma}(\norm{x_0})  \, \chi_{[\tau_0(x_0,u),\tau_{\infty}(x_0,u))}(t) \qquad (t \in \mathbb{R}^+_0),
\end{align}
where
\begin{gather*}
\tau_0(x_0,u) := 0
\qquad \text{and} \qquad
\tau_{\infty}(x_0,u) := \max\{ \sup M_{\infty}(x_0,u), 0 \} \\
M_{\infty}(x_0,u) := \big\{ t \in \mathbb{R}^+_0: \norm{\phi(t,x_0,u)} > \gamma(\norm{u}_{\mathcal{U}}) \big\}.
\end{gather*}
In other words, $\tau_{\infty}(x_0,u)$ is the smallest time $t_0$ for which~\eqref{eq:case distinction} holds true. In particular, we have
\begin{align}
0 \le \tau_{\infty}(x_0,u) < \infty.
\end{align}
It follows that $\beta_0(x_0,u,\cdot)$ is a monotonically decreasing step function satisfying
\begin{align} \label{eq:beta_0-to-zero, case 1}
\beta_0(x_0,u,t) \longrightarrow 0 \qquad (t \to \infty)
\end{align}
as well as
\begin{align} \label{eq:wISS-beta_0,case 1}
\norm{\phi(t,x_0,u)} \le \beta_0(x_0,u,t) + \gamma(\norm{u}_{\mathcal{U}})
\qquad \text{and} \qquad
\beta_0(x_0,u,t) \le \underline{\sigma}(\norm{x_0})
\end{align}
for all $t \in \mathbb{R}^+_0$. (In order to see~(\ref{eq:wISS-beta_0,case 1}.a) for $t < \tau_{\infty}(x_0,u)$ use the assumed uniform global stability, and for $t \ge \tau_{\infty}(x_0,u)$ use the definition of $\tau_{\infty}(x_0,u)$ and the continuity of $\phi(\cdot,x_0,u)$.)
In view of~\eqref{eq:beta_0-to-zero, case 1} and~\eqref{eq:wISS-beta_0,case 1} we are almost done -- except that $\beta_0(x_0,u,\cdot)$ is not strictly decreasing and not continuous.
We therefore choose $\beta(x_0,u,\cdot) \in \mathcal{L}$ such that
\begin{align} \label{eq:beta, case 1}
\beta_0(x_0,u,t) \le \beta(x_0,u,t) \le 2 \underline{\sigma}(\norm{x_0}) \qquad (t \in \mathbb{R}^+_0).
\end{align}
(Simply interpolate linearly between the points $(0,2\underline{\sigma}(\norm{x_0})$ and $(\tau_{\infty}(x_0,u), \underline{\sigma}(\norm{x_0}))$ and exponentially between the points $(\tau_{\infty}(x_0,u), \underline{\sigma}(\norm{x_0}))$ and $(\infty,0)$). Combining~\eqref{eq:wISS-beta_0,case 1} and~\eqref{eq:beta, case 1} we finally obtain~\eqref{eq:wISS-beta} and~\eqref{eq:wISS-bound on beta}, which concludes the proof of the implication from (ii) to (iii) in the case where $\norm{\phi(\cdot,x_0,u)}$ eventually lies below $\gamma(\norm{u}_{\mathcal{U}})$.
\smallskip

Suppose now that we are in the case where $\norm{\phi(\cdot,x_0,u)}$ does not eventually lie below $\gamma(\norm{u}_{\mathcal{U}})$.
In this case, there exists a unique $k(x_0,u) \in \N$ such that
\begin{align} \label{eq:k(x_0,u), def}
\frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)+1} + \gamma(\norm{u}_{\mathcal{U}}) < \sup_{t \in \mathbb{R}^+_0} \norm{\phi(t,x_0,u)} \le \frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)} + \gamma(\norm{u}_{\mathcal{U}})
\end{align}
(use the assumed uniform global stability). Set now
\begin{align}
\beta_0(x_0,u,t) := \sum_{n=0}^{\infty} \frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)+n} \, \chi_{[\tau_n(x_0,u),\tau_{n+1}(x_0,u))}(t) \qquad (t \in \mathbb{R}^+_0),
\end{align}
where
\begin{gather*}
\tau_0(x_0,u) := 0
\qquad \text{and} \qquad
\tau_n(x_0,u) := \sup M_n(x_0,u) \\
M_n(x_0,u) := \Big\{ t\in \mathbb{R}^+_0: \norm{\phi(t,x_0,u)} > \frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)+n} + \gamma(\norm{u}_{\mathcal{U}}) \Big\}
\end{gather*}
for $n \in \N$. We then have
\begin{gather}
0 < \tau_n(x_0,u) < \infty \qquad (n \in \N) \label{eq:tau_n,1}\\
\norm{\phi(t,x_0,u)} \le \frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)+n} + \gamma(\norm{u}_{\mathcal{U}}) \qquad (t \ge \tau_n(x_0,u) \text{ and } n \in \N_0) \label{eq:tau_n,2}\\
\tau_n(x_0,u) < \tau_{n+1}(x_0,u) \qquad (n \in \N_0)
\qquad \text{and} \qquad
\tau_n(x_0,u) \longrightarrow \infty \qquad (n \to \infty). \label{eq:tau_n,3+4}
\end{gather}
(In order to see~\eqref{eq:tau_n,1}, notice that $M_n(x_0,u)$ is bounded by the assumed weak asymptotic gain property and that $\emptyset \ne M_n(x_0,u) \ne \{0\}$ by~\eqref{eq:k(x_0,u), def} and by the continuity of $\phi(\cdot,x_0,u)$.
In order to see~\eqref{eq:tau_n,2} for $n = 0$, just use~\eqref{eq:k(x_0,u), def} -- and to see it for $n \in \N$ use the definition of $\tau_n(x_0,u)$ and the continuity of $\phi(\cdot,x_0,u)$.
In order to see~(\ref{eq:tau_n,3+4}.a) for $n=0$, just recall \eqref{eq:tau_n,1} -- and to see it for $n \in \N$ notice first that $M_n(x_0,u) \subset M_{n+1}(x_0,u)$ and second that
\begin{align} \label{eq:auxiliary, tau_n strictly incr}
\norm{\phi(t,x_0,u)}\Big|_{t=\tau_n(x_0,u)} = \frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)+n} + \gamma(\norm{u}_{\mathcal{U}})
\end{align}
by virtue of~\eqref{eq:tau_n,2} and the  continuity of $\phi(\cdot,x_0,u)$. Consequently, $\tau_n(x_0,u) \le \tau_{n+1}(x_0,u)$ and $\tau_n(x_0,u) \ne \tau_{n+1}(x_0,u)$ because otherwise~\eqref{eq:auxiliary, tau_n strictly incr} would imply that $\sigma(\norm{x_0}) = 0$. Contradiction to~\eqref{eq:ass x_0 ne 0}!
And finally to see~(\ref{eq:tau_n,3+4}.b), recall that $\norm{\phi(\cdot,x_0,u)}$ does not eventually lie below $\gamma(\norm{u}_{\mathcal{U}})$. So, for every $t_0$ there exists a $t \ge t_0$ such that $\norm{\phi(t,x_0,u)} > \gamma(\norm{u}_{\mathcal{U}})$ and therefore there also exists an $n_0 \in \N$ such that
\begin{align*}
\norm{\phi(t,x_0,u)} > \frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)+n_0} + \gamma(\norm{u}_{\mathcal{U}}).
\end{align*}
It thus follows that $\tau_n(x_0,u) \ge \tau_{n_0}(x_0,u) \ge t \ge t_0$ for all $n \ge n_0$, which proves the claimed convergence~(\ref{eq:tau_n,3+4}.b) because $t_0$ was arbitrary.)
With the help of~\eqref{eq:tau_n,1}, \eqref{eq:tau_n,2}, \eqref{eq:tau_n,3+4} it follows that $\beta_0(x_0,u,\cdot)$ is a monotonically decreasing step function satisfying
\begin{align} \label{eq:beta_0-to-zero, case 2}
\beta_0(x_0,u,t) \longrightarrow 0 \qquad (t \to \infty)
\end{align}
as well as
\begin{align} \label{eq:wISS-beta_0,case 2}
\norm{\phi(t,x_0,u)} \le \beta_0(x_0,u,t) + \gamma(\norm{u}_{\mathcal{U}})
\qquad \text{and} \qquad
\beta_0(x_0,u,t) \le \underline{\sigma}(\norm{x_0})
\end{align}
for all $t \in \mathbb{R}^+_0$. (Indeed, for every $t \in \mathbb{R}^+_0$ there exists by~\eqref{eq:tau_n,3+4} a unique $n \in \N_0$ such that $t \in [\tau_n(x_0,u),\tau_{n+1}(x_0,u))$ and therefore
\begin{align*}
\beta_0(x_0,u,t) = \frac{\underline{\sigma}(\norm{x_0})}{k(x_0,u)+n}.
\end{align*}
So, \eqref{eq:beta_0-to-zero, case 2} follows by virtue of~(\ref{eq:tau_n,3+4}.b) while (\ref{eq:wISS-beta_0,case 2}.a) follows by virtue of~\eqref{eq:tau_n,2}.)
We can now choose $\beta(x_0,u,\cdot) \in \mathcal{L}$ such that
\begin{align} \label{eq:beta, case 2}
\beta_0(x_0,u,t) \le \beta(x_0,u,t) \le 2 \underline{\sigma}(\norm{x_0}) \qquad (t \in \mathbb{R}^+_0).
\end{align}
(Simply interpolate linearly between the points $(0,2\underline{\sigma}(\norm{x_0})$, $(\tau_1(x_0,u), \underline{\sigma}(\norm{x_0})/k(x_0,u))$, $(\tau_2(x_0,u), \underline{\sigma}(\norm{x_0})/(k(x_0,u)+1))$, \dots ).
Combining~\eqref{eq:wISS-beta_0,case 2} and~\eqref{eq:beta, case 2} we finally obtain~\eqref{eq:wISS-beta} and~\eqref{eq:wISS-bound on beta}, which concludes the proof of the implication from (ii) to (iii) in the case where $\norm{\phi(\cdot,x_0,u)}$ does not eventually lie below $\gamma(\norm{u}_{\mathcal{U}})$.
\smallskip

We finally show the implication from (iii) to (i). So assume that (iii) is satisfied and let $\sigma, \gamma \in \mathcal{K}$ and $\beta$ be as in~(iii).
Combining~\eqref{eq:wISS-beta} and~\eqref{eq:wISS-bound on beta} we immediately see that $\mathfrak{S}$ is uniformly globally stable and it remains to show that it also has the weak limit property. Set $\overline{\gamma} := \gamma \in \mathcal{K}$ and let $\varepsilon > 0$ and $(x_0,u) \in X \times \mathcal{U}$ be given. Choose a time $\overline{\tau}(\varepsilon,x_0,u) \in \mathbb{R}^+_0$ so large that
\begin{align}
\beta(x_0,u,t) \le \varepsilon \qquad (t \ge \overline{\tau}(\varepsilon,x_0,u)).
\end{align}
(Such a time $\overline{\tau}(\varepsilon,x_0,u) $ exists, for if $x_0 \ne 0$ then $\beta(x_0,u,\cdot) \in \mathcal{L}$ by assumption and if $x_0 = 0$ then $\beta(x_0,u,\cdot) = 0$ by~\eqref{eq:wISS-bound on beta}.)
It then follows by~\eqref{eq:wISS-beta} that
\begin{align}
\norm{\phi(t,x_0,u)} \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}})
\end{align}
for all $t \ge \overline{\tau}(\varepsilon,x_0,u)$. Consequently, $\mathfrak{S}$ has the weak limit (and also the weak asymptotic gain) property, as desired.
\end{proof}

\section{Weak input-to-state stability and its relation to strong input-to-state stability} \label{sect:4}

\subsection{A counterexample} \label{sect:4.1}

With the following example, we show that weak input-to-state stability is, in general, strictly weaker than strong input-to-state stability. We use modulated-linear systems with suitable input spaces $\mathcal{U} \subsetneq L^p(\mathbb{R}^+_0,\mathbb{R})$ to show this. Such modulated-linear systems correspond to evolution equations of the form~\eqref{eq:modul-lin, def}.

\begin{ex}
Choose and fix a $p \in [1,\infty) \cup \{\infty\}$ and a function $\alpha: \mathbb{R} \to \mathbb{R}^+_0$ with $\alpha(0) = 0$ such that the set
\begin{gather}
\mathcal{U} := \bigg\{ u \in L^p(\mathbb{R}^+_0,\mathbb{R}): \alpha \circ u \text{ is  locally integrable but } \int_0^{\infty} \alpha(u(s)) \d s =  \infty \bigg\}
\end{gather}
is non-empty
and endow $\mathcal{U}$ with the norm $\norm{\cdot}_{\mathcal{U}} := \norm{\cdot}_p$.
(Simple choices for such a function are, for instance, $\alpha(r) := |r|$ in case $p \ne 1$ and $\alpha(r) := |r|^{1/2}$ in case $p=1$.) Also, let $A$ be the generator of a strongly stable semigroup on a Banach space $X$ and define
\begin{align}
\phi(t,x_0,u) := \e^{A ( \int_0^t \alpha(u(s)) \d s )} x_0 \qquad ((t,x_0,u) \in \mathbb{R}^+_0\times X \times \mathcal{U}).
\end{align}
We now show that $\mathfrak{S} := (X,\mathcal{U},\phi)$ is a weakly but not strongly input-to-state stable system.
In particular, we see that the implications stated as open questions
in the very last paragraph of~\cite{MiWi16a} do not hold true in general.
It is elementary to check that $\mathcal{U}$ is invariant under shifts to the left and under concatenations. It is also elementary to check that $\phi(\cdot,x_0,u)$ is continuous for every $(x_0,u) \in X \times \mathcal{U}$ and that $\phi$ is cocyclic and causal. So, in other words, $\mathfrak{S}$ is a dynamical system with inputs.
Since $\e^{A\cdot}$ is strongly stable, it follows that
\begin{align} \label{eq:wISS ne sISS, 1}
M:= \sup_{v\in\mathbb{R}^+_0} \norm{\e^{A v}} < \infty
\end{align}
by the uniform boundedness principle, and therefore $\mathfrak{S}$ is uniformly globally stable with $\underline{\sigma}(r) := M r$ and $\underline{\gamma} \in \mathcal{K}$ arbitrary.
Since $\e^{A\cdot}$ is strongly stable and since
\begin{align}
\int_0^t \alpha(u(s)) \d s \longrightarrow \infty \qquad (t \to \infty)
\end{align}
for every $u \in \mathcal{U}$,
it further follows that $\mathfrak{S}$ is of weak asymptotic gain $0$.
So, we see that $\mathfrak{S}$ is weakly input-to-state stable and it remains to show that it is not of strong asymptotic gain. Seeking a contradiction, assume that $\mathfrak{S}$ is of strong asymptotic gain $\overline{\gamma}$ with corresponding times $\overline{\tau}(\varepsilon,x_0)$. Choose now an arbitrary $u_0 \in \mathcal{U}$ (non-empty!), let $\varepsilon := 1$, and choose $x_0 \in X$ such that
\begin{align} \label{eq:wISS ne sISS, 2}
\norm{x_0} > \varepsilon + \overline{\gamma}(\norm{u_0}_{\mathcal{U}}).
\end{align}
Also, define $\overline{\tau} := \overline{\tau}(\varepsilon,x_0)$ and $u := 0 \, \&_{\overline{\tau}} \, u_0$. Clearly, $u \in \mathcal{U}$ and
\begin{align} \label{eq:wISS ne sISS, 3}
\norm{u}_{\mathcal{U}} = \norm{u_0}_{\mathcal{U}}
\qquad \text{and} \qquad
u|_{[0,\overline{\tau}]} = 0.
\end{align}
So, by the assumed asymptotic strong gain property combined with~\eqref{eq:wISS ne sISS, 2}, \eqref{eq:wISS ne sISS, 3} and $\alpha(0) = 0$,
we get that
\begin{align}
\varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}}) < \norm{x_0} = \norm{ \e^{A ( \int_0^{\overline{\tau}} \alpha(u(s)) \d s )} x_0 }
= \norm{\phi(\overline{\tau},x_0,u)} \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}}).
\end{align}
Contradiction! $\blacktriangleleft$
\end{ex}

\subsection{Some positive results} \label{sect:4.2}

While
weak and strong input-to-state stability are inequivalent for modulated-linear systems with general input spaces $\mathcal{U}$, they coincide for modulated-linear systems with input space $\mathcal{U} = L^p(\mathbb{R}^+_0,U)$.

\begin{prop} \label{prop:pos-result 1}
Suppose $X$ is a Banach space and $\mathcal{U} := L^p(\mathbb{R}^+_0,\mathbb{R})$ for some $p \in [1,\infty) \cup \{\infty\}$. Suppose further that $A$ is a semigroup generator on $X$ and $\alpha: \mathbb{R} \to \mathbb{R}^+_0$ is a continuous function such that $\alpha \circ u$ is locally integrable for every $u \in \mathcal{U}$ (for example $\alpha(r) := |r|$).
Then $\mathfrak{S} := (X,\mathcal{U},\phi)$ with
\begin{align*}
\phi(t,x_0,u) := \e^{A ( \int_0^t \alpha(u(s)) \d s )} x_0 \qquad ((t,x_0,u) \in \mathbb{R}^+_0\times X \times \mathcal{U})
\end{align*}
is a dynamical system with inputs and $\mathfrak{S}$ is weakly input-to-state stable if and only if it
is strongly input-to-state stable.
\end{prop}

\begin{proof}
It is easily verified that $\mathfrak{S}$ is a dynamical system with inputs. Also, one of the claimed implications is trivial. So, suppose that $\mathfrak{S}$ is weakly input-to-state stable. We have to show that $\mathfrak{S}$ is also of strong asymptotic gain and thus strongly  input-to-state stable.
\smallskip

As a first step, we show that $\alpha(0) > 0$ in case $p < \infty$ and that $\alpha(r) > 0$ for every $r \in \mathbb{R}$ in case $p= \infty$.
Seeking a contradiction, we assume that there is a constant function $u_r \equiv r \in L^p(\mathbb{R}^+_0,\mathbb{R}) = \mathcal{U}$ such that $\alpha(r) = 0$. We then have
\begin{align} \label{eq:modul-lin, step 1, 1}
\phi(t,x_0,u_r) = \e^{A ( \int_0^t \alpha(u_r(s)) \d s )} x_0 = x_0 \qquad (t \ge 0)
\end{align}
for every $x_0 \in X$. Since $\mathfrak{S}$ is of weak asymptotic gain $\overline{\gamma}$, say, we also have that
\begin{align}  \label{eq:modul-lin, step 1, 2}
\norm{\phi(t,x_0,u_r)} \le \varepsilon + \overline{\gamma}(\norm{u_r}_{\mathcal{U}}) \qquad (t \ge \overline{\tau}(\varepsilon,x_0,u_r))
\end{align}
for every $\varepsilon >0$ and every $x_0 \in X$. Choosing now $x_0 \in X$ such that $\norm{x_0}/2 > \overline{\gamma}(\norm{u_r}_{\mathcal{U}})$ and lettig $\varepsilon := \norm{x_0}/2$, we obtain a contradiction by combining~\eqref{eq:modul-lin, step 1, 1} and~\eqref{eq:modul-lin, step 1, 2}.
\smallskip

As a second step, we observe that the semigroup $\e^{A\cdot}$ is strongly stable.
Indeed, since $\mathfrak{S}$ is of weak asymptotic gain, we see by choosing $u := 0 \in \mathcal{U}$ that
\begin{align}
\e^{A(\alpha(0)t)}x_0 = \phi(t,x_0,0) \longrightarrow 0 \qquad (t \to \infty)
\end{align}
for every $x_0 \in X$. Since $\alpha(0) > 0$ by the first step, the claimed strong stability follows.
\smallskip

As a third step, we show that for every $\overline{v}, \overline{r} \in \mathbb{R}^+_0$ there exists a time $\overline{\tau} = \overline{\tau}_{\overline{v},\overline{r}} \in \mathbb{R}^+_0$ such that
\begin{align} \label{eq:modul-lin, step 3}
\int_0^{\overline{\tau}} \alpha(u(s)) \d s \ge \overline{v}
\qquad (u \in \mathcal{U} \text{ with } \norm{u}_{\mathcal{U}} \le \overline{r}).
\end{align}
So let $\overline{v}, \overline{r} \in \mathbb{R}^+_0$ be given and fixed.
In case $p < \infty$, we know by the first step and the continuity of $\alpha$ that
\begin{align*}
c_{\delta} := \inf_{|r| \le \delta} \alpha(r) > 0
\end{align*}
for some $\delta > 0$. Setting
$\overline{\tau} = \overline{\tau}_{\overline{v},\overline{r}} := \overline{v}/c_{\delta} + \overline{r}^p / \delta^p$
and writing
\begin{align*}
J^{u}_{>\delta} := \{ s\in [0,\overline{\tau}]: |u(s)| > \delta \}
\qquad \text{and} \qquad
J^{u}_{\le \delta} := \{ s\in [0,\overline{\tau}]: |u(s)| \le \delta \}
\end{align*}
for $u \in \mathcal{U}$, we see for every $u \in \mathcal{U}$ with $\norm{u}_{\mathcal{U}} \le \overline{r}$ that
\begin{align*}
\lambda( J^{u}_{>\delta} ) \le 1/\delta^p \, \int_0^{\overline{\tau}} |u(s)|^p \d s \le \overline{r}^p / \delta^p
\end{align*}
and therefore
\begin{align*}
\int_0^{\overline{\tau}} \alpha(u(s)) \d s \ge \int_{J^{u}_{\le \delta}} \alpha(u(s)) \d s \ge c_{\delta} \lambda( J^{u}_{\le \delta} )
= c_{\delta} \big( \overline{\tau} - \lambda( J^{u}_{> \delta} ) \big)
\ge c_{\delta} \big( \overline{\tau} - \overline{r}^p / \delta^p \big)
= \overline{v},
\end{align*}
which proves~\eqref{eq:modul-lin, step 3} in the case $p < \infty$.
In case $p = \infty$, we know by the first step and the continuity of $\alpha$ that
\begin{align*}
c_{\overline{r}} := \inf_{|r| \le \overline{r}} \alpha(r) > 0.
\end{align*}
Setting $\overline{\tau} = \overline{\tau}_{\overline{v},\overline{r}} := \overline{v}/c_{\overline{r}}$, we see for every $u \in \mathcal{U}$ with $\norm{u}_{\mathcal{U}} \le \overline{r}$ that
\begin{align*}
\int_0^{\overline{\tau}} \alpha(u(s)) \d s \ge c_{\overline{r}} \overline{\tau} = \overline{v},
\end{align*}
which proves~\eqref{eq:modul-lin, step 3} in the case $p < \infty$.
\smallskip

As a fourth and last step, we finally show that $\mathfrak{S}$ is of strong asymptotic gain.
Since $\e^{A\cdot}$ is strongly stable by the second step, we have that
\begin{align} \label{eq:modul-lin, step 4, 1}
M:= \sup_{v\in\mathbb{R}^+_0} \norm{\e^{A v}} < \infty
\end{align}
by the uniform boundedness principle and, moreover, we have that for every $\varepsilon > 0$ and $x_0 \in X$ there is a $\overline{v}(\varepsilon,x_0) \in \mathbb{R}^+_0$ such that
\begin{align} \label{eq:modul-lin, step 4, 2}
\norm{\e^{Av}x_0} \le \varepsilon \qquad (v \ge \overline{v}(\varepsilon,x_0)).
\end{align}
Set now $\overline{\gamma}(r) := M r$ for $r \in \mathbb{R}^+_0$, let $\varepsilon > 0$ and $x_0 \in X$ be given and fixed, and define
\begin{align} \label{eq:modul-lin, step 4, 3}
\overline{\tau}(\varepsilon,x_0) := \overline{\tau}_{\overline{v}(\varepsilon,x_0),\norm{x_0}},
\end{align}
where $\overline{\tau}_{\overline{v},\overline{r}}$ is chosen as in the third step.
It then follows by~\eqref{eq:modul-lin, step 3}, \eqref{eq:modul-lin, step 4, 2} and \eqref{eq:modul-lin, step 4, 3} that for every $u \in \mathcal{U}$ with $\norm{u}_{\mathcal{U}} \le \norm{x_0}$
\begin{align} \label{eq:modul-lin, step 4, 4}
\norm{\phi(t,x_0,u)} = \norm{ \e^{A ( \int_0^t \alpha(u(s)) \d s )} x_0 } \le \varepsilon \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}})
\end{align}
for all $t \ge \overline{\tau}(\varepsilon,x_0)$.
It also follows by~\eqref{eq:modul-lin, step 4, 1} that for every $u \in \mathcal{U}$ with $\norm{u}_{\mathcal{U}} > \norm{x_0}$
\begin{align}  \label{eq:modul-lin, step 4, 5}
\norm{\phi(t,x_0,u)} = \norm{ \e^{A ( \int_0^t \alpha(u(s)) \d s )} x_0 } \le M \norm{u}_{\mathcal{U}} \le \varepsilon + \overline{\gamma}(\norm{u}_{\mathcal{U}})
\end{align}
for all $t \ge 0$.
So, taking~\eqref{eq:modul-lin, step 4, 4} and~\eqref{eq:modul-lin, step 4, 5} together we see that $\mathfrak{S}$ is of strong asymptotic gain $\overline{\gamma}$, as desired.
\end{proof}

Similarly, weak and strong input-to-state stability coincide for linear systems with input space $\mathcal{U} = L^p(\mathbb{R}^+_0,U)$.

\begin{prop} \label{prop:pos-result 2}
Suppose $X$, $U$ are Banach spaces and $\mathcal{U} := L^p(\mathbb{R}^+_0,U)$ for some $p \in [1,\infty) \cup \{\infty\}$. Suppose further that $A$ is a semigroup generator on $X$ and that $B \in L(U,X_{-1})$ is a $\mathcal{U}$-admissible control operator for $A$, that is,
\begin{align} \label{eq:pos-result 2, B adm}
\Phi_t(u) := \int_0^t \e^{A_{-1}(t-s)} B u(s) \d s \in X
\qquad (t \in \mathbb{R}^+_0 \text{ and } u \in \mathcal{U}),
\end{align}
where $A_{-1}$ is the generator of the extrapolation of the semigroup $\e^{A\cdot}$ to the extrapolation space $X_{-1}$ of $A$.
In case $p = \infty$ additionally assume that $t \mapsto \Phi_t(u) \in X$ is continuous for every $u \in \mathcal{U}$. Then $\mathfrak{S} := (X,\mathcal{U},\phi)$ with
\begin{align*}
\phi(t,x_0,u) := \e^{A t}x_0 + \int_0^t \e^{A_{-1}(t-s)}B u(s) \d s
\qquad ((t,x_0,u) \in \mathbb{R}^+_0 \times X \times \mathcal{U})
\end{align*}
is a dynamical system with inputs and $\mathfrak{S}$ is weakly input-to-state stable if and only if it
is strongly input-to-state stable.
\end{prop}

\begin{proof}
Since $B$ is $\mathcal{U}$-admissible for $A$, it follows that $\phi(\cdot,x_0,u)$ is a continuous function from $\mathbb{R}^+_0$ to $X$ by virtue of Proposition~2.3 of~\cite{We89}. (It should be noted here that because of~\eqref{eq:pos-result 2, B adm} the linear operator $\Phi_t: \mathcal{U} \to X$ is closed and thus bounded, whence the aforementioned propostion from~\cite{We89} is applicable.) Also, it is clear that $\phi$ is cocyclic and causal. So, $\mathfrak{S}$ is a dynamical system with inputs.
It remains to show that if $\mathfrak{S}$ is weakly input-to-state stable, then it is of strong asymptotic gain and hence strongly input-to-state stable (the other implication being trivial).
So, let $\mathfrak{S}$ be
uniformly globally stable (with corresponding functions $\underline{\sigma}$, $\underline{\gamma}$) and of weak asymptotic gain $\overline{\gamma}$ (with corresponding times $\overline{\tau}(\varepsilon,x_0,u)$). Also, let $\varepsilon > 0$ and $x_0 \in X$. We then have for every $t \ge \overline{\tau}(\varepsilon,x_0,0)$ and every $u \in \mathcal{U}$ that
\begin{align*}
\norm{\phi(t,x_0,u)}
&\le \norm{\e^{A t}x_0} + \norm{\Phi_t(u)}
= \norm{\phi(t,x_0,0)} + \norm{\phi(t,0,u)} \\
&\le \varepsilon + \overline{\gamma}(0) + \underline{\sigma}(0) + \underline{\gamma}(\norm{u}_{\mathcal{U}})
= \varepsilon + \underline{\gamma}(\norm{u}_{\mathcal{U}}).
\end{align*}
In particular, $\mathfrak{S}$ is of strong asymptotic gain $\underline{\gamma}$, as desired.
\end{proof}

Clearly, every bounded control operator $B \in L(U,X)$ is $\mathcal{U}$-admissible for $A$. Sufficient conditions for unbounded control operators $B \in L(U,X_{-1})$ to be $\mathcal{U}$-admissible for $A$ can be found in~\cite{JaPa04}, \cite{TuWe}, for instance. A sufficient conditions for the continuity of $t \mapsto \Phi_t(u) \in X$
in the case $p = \infty$ is that $X$ be a Hilbert space, $U$ be finite-dimensional and that $A$ generate an exponentially stable analytic  semigroup on $X$ that is similar to a contraction semigroup (Theorem~1 of~\cite{JaScZw17}).

\subsection{Weak input-to-state stability for semilinear systems} \label{sect:4.3}

While for linear systems
weak and strong input-to-state stability are equivalent, this is open for semilinear systems.
With the next example, we show at least, however, that for such semilinear systems weak input-to-state stability is strictly weaker than uniform input-to-state stability.

\begin{ex}
Suppose $X$, $U$ are Hilbert spaces and $\mathcal{U} := L^2(\mathbb{R}^+_0,U)$. Suppose further that $A$ is a contraction semigroup generator on $X$ and $B \in L(U,X)\setminus \{0\}$ is such that $A-BB^*$ generates a strongly but not exponentially stable semigroup,
and let
\begin{align*}
f(x) := -B g(B^*x) \qquad (x \in X),
\end{align*}
where $g: U \to U$ is Lipschitz continuous on bounded subsets and
\begin{align} \label{eq:g-strikt daempfend und noch mehr}
g(v) = v \qquad (\norm{v} \le 1) \qquad \text{and} \qquad \scprd{v,g(v)} \ge c \qquad (\norm{v} > 1)
\end{align}
for some positive constant $c > 0$. (Simple choices for such operators
are given by self-adjoint operators $A$ with $\sigma(A) \subset (-\infty,0]$ and with $0$ belonging to the essential but not to the point spectrum of $A$ and by compact operators $B \in L(U,X)$ such that $BB^*$ commutes with $A$.
Another possible choice of operators $A$ and $B$ as above is given by Example~3.2 of~\cite{Sc18-Hagen}.)
It then follows from~\cite{Sc18-wp} (similarly to~\cite{CuZw16}) that for every $(x_0,u) \in X \times \mathcal{U}$ the initial value problem
\begin{align*}
x' = Ax + f(x) + Bu(t) \qquad \text{and} \qquad x(0) = x_0
\end{align*}
has a unique global mild solution $\phi(\cdot,x_0,u) \in C(\mathbb{R}^+_0,X)$ and that for $(x_0,u) \in D(A) \times C^1_c(\mathbb{R}^+_0,U)$ this mild solution is even a classical solution. It also follows from~\cite{Sc18-wp}  (similarly to~\cite{CuZw16}) that $\mathfrak{S} := (X,\mathcal{U},\phi)$ is a weakly input-to-state stable dynamical system with inputs.
We now show that the system $\mathfrak{S}$ is not uniformly  input-to-state stable. Assume the contrary and set $r := \underline{\sigma}^{-1}(1/\norm{B})$. We then have for every $x_0 \in D(A)$ with $\norm{x_0} \le r$ that $\norm{B^* \phi(t,x_0,0)} \le 1$ for all $t \in \mathbb{R}^+_0$ by the uniform global stability of $\mathfrak{S}$ and therefore
\begin{align}
\phi(t,x_0,0) = \e^{(A-BB^*)t}x_0 \qquad (t \in \mathbb{R}^+_0)
\end{align}
by~\eqref{eq:g-strikt daempfend und noch mehr} and the classical solution property of $\phi(\cdot,x_0,0)$.
Since now $\mathfrak{S}$ is of uniform asymptotic gain
by our assumption, it follows that
\begin{align}
r \norm{ \e^{(A-BB^*)t} } = \sup \big\{ \norm{ \phi(t,x_0,0) }: x_0 \in D(A) \text{ with } \norm{x_0} \le r \big\}
\longrightarrow 0
\end{align}
as $t \to \infty$. Consequently, $\e^{(A-BB^*)\cdot}$ is exponentially stable. Contradiction (to our choice of $A$ and $B$)! $\blacktriangleleft$
\end{ex}

\section{Weak input-to-state stability and its relation to zero-input uniform global stability} \label{sect:5}

\subsection{A counterexample} \label{sect:5.1}

With the following example, we show that weak input-to-state stability is, in general, strictly stronger than the combination of zero-input uniform global stability and the weak asymptotic gain property. We use linear systems with a suitable input space $\mathcal{U} \subsetneq L^{\infty}(\mathbb{R}^+_0,\mathbb{R})$ to show this.

\begin{ex}
Set $X := L^2(\mathbb{R}^+_0,\mathbb{R})$ and
\begin{align}
\mathcal{U} := \big\{ u \in L^{\infty}(\mathbb{R}^+_0,\mathbb{R}): u \text{ is eventually exponentially decaying to } 0 \big\}
\end{align}
and endow $\mathcal{U}$ with the norm $\norm{\cdot}_{\mathcal{U}} := \norm{\cdot}_{\infty}$. (What we mean by a function $u$ that eventually exponentially decays to $0$ is that there exists a time $\tau_u \in \mathbb{R}^+_0$ and constants $C_u, \alpha_u > 0$ such that $|u(s)| \le C_u \e^{-\alpha_u s}$ for all $s \ge \tau_u$.) Also, let $A$ be the generator of the left-translation group on $X$, that is,
\begin{align*}
\e^{A t} f = f(\cdot+t) \qquad (f \in X \text{ and } t \in \mathbb{R}^+_0),
\end{align*}
and let $B \in L(\mathbb{R},X)$
be given by $B v := v \, b$ for $v \in \mathbb{R}$, where $b \in X$ is chosen such that
\begin{align} \label{eq:ex-2, properties of b}
b(\zeta) \ge 0 \qquad (\zeta \in \mathbb{R}^+_0)
\qquad \text{and} \qquad
b \notin L^1(\mathbb{R}^+_0,\mathbb{R})
\end{align}
(for example, $b(\zeta) = 1/\zeta \, \chi_{[1,\infty)}(\zeta)$).
We now show that $\mathfrak{S} := (X,\mathcal{U},\phi)$ with
\begin{align}
\phi(t,x_0,u) := \e^{A t}x_0 + \int_0^t \e^{A(t-s)}B u(s) \d s
\qquad ((t,x_0,u) \in \mathbb{R}^+_0 \times X \times \mathcal{U})
\end{align}
is a system (with bounded reachability sets and with continuity at the equilibrium point $0$) which is zero-input uniformly globally stable and of weak asymptotic gain $0$ but not uniformly locally stable. In particular, $\mathfrak{S}$ is not weakly input-to-state stable and, moreover, we see that the second
question-marked implication in Figure~2 of~\cite{MiWi16a} does not hold true in general.
It is elementary to check that $\mathcal{U}$ is invariant under shifts to the left and under concatenations. Also, it is clear that $\phi(\cdot,x_0,u)$ is continuous for every $(x_0,u) \in X \times \mathcal{U}$ and that $\phi$ is cocyclic and causal. So, in other words, $\mathfrak{S}$ is a dynamical system with inputs.
Since $\e^{A \cdot}$ is a contraction semigroup, it follows that $\mathfrak{S}$ is zero-input uniformly globally stable. It also follows that $\mathfrak{S}$ has bounded reachability sets and is continuous at the equilibrium point $0$ (Definition~3 and~4 of~\cite{MiWi16a}).
Since $\e^{A \cdot}$ is strongly stable and every $u \in \mathcal{U}$ is eventually exponentially decaying to $0$, it further follows that
\begin{align*}
\phi(t,x_0,u) \longrightarrow 0 \qquad (t\to \infty)
\end{align*}
for every $(x_0,u) \in X \times \mathcal{U}$ or, in other words, that $\mathfrak{S}$ is of weak asymptotic gain $0$.
So, what remains to be shown is that $\mathfrak{S}$ is not uniformly locally stable. Assume, on the contrary, that $\mathfrak{S}$ is uniformly locally stable with corresponding functions $\underline{\sigma}$, $\underline{\gamma}$ and radius~$r$.
Since $b \in X$ satisfies~\eqref{eq:ex-2, properties of b}, we see with the help of Fatou's lemma that
\begin{align*}
\liminf_{t \to \infty} \norm{ \int_0^t b(\cdot+s) \d s }^2
&= \liminf_{t\to\infty} \int_0^{\infty} \bigg( \int_0^t b(\zeta+s) \d s \bigg)^2 \d \zeta \notag \\
&\ge \int_0^{\infty} \bigg( \int_0^{\infty} b(\zeta+s) \d s \bigg)^2 \d \zeta
= \infty.
\end{align*}
In particular, there exists a time $\overline{\tau} \in \mathbb{R}^+_0$ such that
\begin{align} \label{eq:ex-2, not uls 1}
r \cdot \norm{ \int_0^{\overline{\tau}} b(\cdot+s) \d s } > \underline{\gamma}(r).
\end{align}
Choose now a $u \in \mathcal{U}$ such that
\begin{align} \label{eq:ex-2, not uls 2}
u|_{[0,\overline{\tau}]} \equiv r
\qquad \text{and} \qquad
\norm{u}_{\mathcal{U}} = r.
\end{align}
It then follows by~\eqref{eq:ex-2, not uls 1} and \eqref{eq:ex-2, not uls 2} that  $(0,u) \in \overline{B}_r^{X}(0) \times \overline{B}_r^{\mathcal{U}}(0)$ but
\begin{align*}
\norm{\phi(\overline{\tau},0,u)} = \norm{\int_0^{\overline{\tau}} u(s) \cdot \e^{A(\overline{\tau}-s)}b \d s } = r \cdot \norm{ \int_0^{\overline{\tau}} b(\cdot+s) \d s }
> \underline{\gamma}(r) = \underline{\sigma}(0) + \underline{\gamma}(r).
\end{align*}
Contradiction (to our uniform local stability assumption)! $\blacktriangleleft$
\end{ex}

\subsection{A positive result} \label{sect:5.2}

While
weak input-to-state stability and the combination of zero-input uniform global stability and the weak asymptotic gain property are inequivalent for linear systems with general input spaces $\mathcal{U}$, they coincide for linear systems with input space $\mathcal{U} = L^p(\mathbb{R}^+_0,U)$.

\begin{prop} \label{prop:pos-result 3}
Suppose $\mathfrak{S} := (X,\mathcal{U},\phi)$ is as in Proposition~\ref{prop:pos-result 2}. Then $\mathfrak{S}$ is a dynamical system with inputs and $\mathfrak{S}$ is weakly input-to-state stable if and only if it is zero-input uniformly locally stable and of weak asymptotic gain.
\end{prop}

\begin{proof}
We already know that $\mathfrak{S}$ is a dynamical system with inputs, and one of the claimed implications is trivial.
So, let $\mathfrak{S}$ be zero-input uniformly locally stable (with corresponding function $\underline{\sigma}$ and radius $r$) and of weak asymptotic gain $\overline{\gamma}$ (with corresponding times $\overline{\tau}(\varepsilon,x_0,u)$). We then have
\begin{gather*}
\norm{\e^{At}x_0} = \norm{\phi(t,x_0,0)} \le \underline{\sigma}(\norm{x_0})
\qquad (x_0 \in \overline{B}_r^{X}(0) \text{ and } t \ge 0) \\
\norm{\Phi_t(u)} = \norm{\phi(t,0,u)} \le 1 + \overline{\gamma}(\norm{u}_{\mathcal{U}})
\qquad (u \in \overline{B}_r^{\mathcal{U}}(0) \text{ and } t \ge \overline{\tau}(1,0,u)).
\end{gather*}
So, by linearity and the continuity of $[0, \overline{\tau}(1,0,u)] \ni t \mapsto \Phi_t(u)$ for $u \in \mathcal{U}$, we see that $\mathbb{R}^+_0 \ni t \mapsto \e^{At}x_0 \in X$ and $\mathbb{R}^+_0 \ni t \mapsto \Phi_t(u) \in X$ are bounded functions for all $x_0 \in X$ and $u \in \mathcal{U}$ and therefore
\begin{align}
M_1 := \sup_{t \in \mathbb{R}^+_0} \norm{\e^{At}} < \infty
\qquad \text{and} \qquad
M_2 := \sup_{t \in \mathbb{R}^+_0} \norm{\Phi_t} < \infty
\end{align}
by the uniform boundedness principle. Consequently,
\begin{align*}
\norm{\phi(t,x_0,u)} \le M_1 \norm{x_0} + M_2 \norm{u}_{\mathcal{U}} \qquad (t \ge 0)
\end{align*}
for every $(x_0,u) \in X \times \mathcal{U}$. In particular, $\mathfrak{S}$ is uniformly globally stable and hence weakly input-to-state stable, as desired.
\end{proof}

\section*{Acknowledgements}

I would like to thank the German Research Foundation (DFG) financial support through the grant ``Input-to-state stability and stabilization of distributed-parameter systems'' (DA 767/7-1).

\begin{small}

redpajama_set_name: RedPajamaArXiv


text:
Доменико Де Лука (; 2 января 1928, Сфакс, Тунис — 16 сентября 2006) — итальянский прелат и ватиканский дипломат. Руководитель протокола государственного секретариата Святого Престола с 1989 по 22 мая 1993. Титулярный архиепископ Теглаты Нумидийской с 22 мая 1993 по 16 сентября 2006. Апостольский нунций в Марокко с 22 мая 1993 по 17 июля 2003.

Ссылки
 Информация

Католические архиепископы
Апостольские нунции в Марокко

redpajama_set_name: RedPajamaWikipedia


Apple dumplings. Have you tried them? LOVE.
If you're down for a classic treat that will be a repeat offender at your annual holiday gathering, this is the one. My cousin makes at least two pans every year and it's always one of the most clamored-over desserts at the table. In addition to being insanely easy to throw together at a moment's notice, these delicious morsels are the perfect blend of cinnamon, spice, apple & everything nice!
When a Food Blogger Gets Hangry….
Have a ever seen a food blogger get HANGRY?
You'd figure that, for people who spend the majority of their days around food–making it, tasting it, styling it, photographing it–hanger isn't a problem. Let me make this clear: HANGER IS REAL. YES. EVEN FOR FOOD BLOGGERS.
Caution: cliches ahead! Read on (I want your thoughts), but consider yourself warned.
Every once in a while, you meet someone special who changes you–for better or for worse, but without a doubt: for good.
One of the really cool things about being a blogger is getting to press the Pause button from time to time and write one of those meaningful life reflection posts that have nothing to do with sharing new recipes, posting BBQ photos, or looking gorgeous in the dappled forest sunlight. These posts help me slow down and do some of that major appreciation work that really needs to be done more often around here.
post-exam life: a rosy smorgasbord of updates!


For all intents and purposes, the two week wait has turned into a 9 day wait and is over today. I started lightly bleeding this afternoon around 3:00 in the restroom at a school. Since I have my classic PMS headache and since today is the day my period is due, this was not a good sign - though spotting can sometimes be a pregnancy symptom. I knew it was over and I tried to pull myself together in the ancient wooden and brass faculty bathroom that a student had to unlock for me with her teacher's keys. I still had 20 minutes left of class time to get through. On my way out a parent volunteer who was on her way to the restroom stopped me and told me a long story about how often she had to go to the bathroom when she was pregnant. Ugh. I somehow zombied my way through the last 20 minutes of school and was able to make it to my car before completely losing it.
I called Randy and since I was close to downtown he directed me to pick him up (he rode the bus in) so he could come home with me. Through tears and fogged up sunglasses I saw my way to his office and then retreated to the passenger seat so that he could drive home while I cried. We decided we'd test as soon as we got home to confirm what we both know - that my period is starting. At 14 days past ovulation and with an impending period, the risk of a false negative is very low. Here's the test. There's only one line where there should be two. Story of my life these last 5 years.
We spent the night in bed cuddling, crying, and eating pizza (what can't pizza make better?). Initially we had no hopes for this cycle - it was just something to get through in order to move out of this phase of life. But every single step of the process went SO well that hope seeped back in. We DID NOT want to get attached to this cycle, but how could we not when it seemed so unstoppable? Now we have crashed down to reality - off of our hope-high. A fall that would have been much softer if it wasn't from so high. So it really makes us wonder - if things do happen for a reason and if someone is watching over us - why have everything go so well only to trick us in the end? It seems so very cruel.
Oh Kimmie. Love love and more love. Thinking of you guys. I wish there were answers.
speechless and grieving for you. I love you so much.
I am SO sorry! I have been praying and thinking of you guys daily.


text: Use OpenWeather API Application

redpajama_set_name: RedPajamaGithub


Now that Let's Encrypt can issue wildcard TLS certificates I found some time to look into that.
I already use a Lua script with haproxy which takes care of automatically answering http-01 ACME challenges, but to issue/renew a wildcard certificate you need to answer a dns-01 challenge. A different client/setup would be needed.
Most of the clients that support ACME v2 offer a range of integrations for DNS providers, plus a manual mode that prints out the DNS record that you need to add and then waits for you to indicate that you've done it. I run my own DNS infrastructure so the thing to do would be RFC2136 dynamic DNS updates.
One wrinkle here is that currently none of my DNS zones have dynamic updates enabled. At the moment I manage them as zone files (some are automatically generated by scripts though). After looking at a few of the client options I found that acme.sh supports an "alias zone".
These records need to be added to the main zone for this to work.
; sub-zone purely used for dns-01 ACME challenges.
; Alias the dns-01 challenge record into the dedicated zone.
. . . ; sub-zone purely used for dns-01 ACME challenges. acmesh NS a.authns.bitfolk.co.uk. NS b.authns.bitfolk.com. NS c.authns.bitfolk.com. ; Alias the dns-01 challenge record into the dedicated zone. _acme-challenge CNAME _acme-challenge.acmesh.strugglers.net. . . .
Initially this just needs to be an empty zone with only SOA and NS records, so this is the entire content of the file.
$ORIGIN . $TTL 86400 ; 1 day acmesh.strugglers.net IN SOA a.authns.bitfolk.co.uk. hostmaster.bitfolk.com. ( 2018031905 ; serial 14400 ; refresh (4 hours) 7200 ; retry (2 hours) 1209600 ; expire (2 weeks) 43200 ; minimum (12 hours) ) NS a.authns.bitfolk.co.uk. NS b.authns.bitfolk.com. NS c.authns.bitfolk.com.
The DNS server needs to know a key by which it will authenticate acme.sh's updates, and also needs to be told that the new zone is a dynamic zone. I use BIND, so it goes as follows.
Use the dnssec-keygen command to generate a key suitable for authenticating DNS updates.
This creates two files named like Kddns_update.+165+14059.key and Kddns_update.+165+14059.private.
The thing in quotes after "key" is a symbolic name for this key and can be anything that makes sense to you. The "secret" is the key from the private file. You can delete the two Kddns_update.+165+14059.* files now.
Once BIND has been reloaded the log file should indicate that the acemsh.strugglers.net zone was loaded correctly, and in my case that triggers DNS NOTIFY to my secondary servers which automatically begin zone transfers.
At this point it might be worth using the nsupdate command to check that you can do dynamic DNS updates.
Just type the nsupdate line in the shell, the > is a prompt at which you will type the updates you wish to send. We'll add a trivial TXT record. The -k argument is the path to the file containing the key.
> update add foo.acmesh.strugglers.net. 86400 TXT "bar"
foo.acmesh.strugglers.net. 86400 IN     TXT     "bar"
$ dig +short -t txt foo.acmesh.strugglers.net "bar"
That it; you can do dynamic DNS updates.
Create a system user that can't log in.
acme.sh --install it as that user.
acme.sh doesn't have to be run on the primary DNS server, because it's going to use a dynamic DNS update to do all the DNS things. It just needs access to the dynamic DNS update key file. Either you can install acme.sh on each host that will need to generate/renew certificates and copy the DNS key there, or else do all the certificate generation/renewal in one place and copy the certificate files around.
However you manage it, make sure that the user you're going to run acme.sh as can read the dynamic DNS update key file.
The first time you issue the certificate you need to set NSUPDATE_KEY and NSUPDATE_SERVER in your environment. After the first successful issuance acme.sh will store these variables in its configuration for use in the automated renewals.
[Mon 19 Mar 09:19:04 UTC 2018] adding _acme-challenge.acmesh.strugglers.net. 60 in txt "WmenhbXRtenhpNLYLOBjznyHcVvFk-jjxurCVTrhWc8"
[Mon 19 Mar 09:19:04 UTC 2018] adding _acme-challenge.acmesh.strugglers.net. 60 in txt "fwZPUBHijOQkJJaoOF_nIn3Z_FtuVU9R635NDVz_hPA"
At this point a DNS update has been crafted and sent so you should see your zone update and zone transfer happen to any secondary servers. If that doesn't happen within 120 seconds then when Let's Encrypt tries to verify the challenge it might query a DNS server that doesn't yet have the record. Your zone transfers need to be reliable.
[Mon 19 Mar 09:21:15 UTC 2018] Removing DNS records.
[Mon 19 Mar 09:21:16 UTC 2018] Verify finished, start to sign.
[Mon 19 Mar 09:21:18 UTC 2018] Cert success.
From the Subject Alternative Name we can see it is a wildcard certificate.
Great solution. Great guide. Thanks!
Yes, I have generally only seen this problem when bind can't write to the dynamic zone file, either because of permissions or mandatory access control like apparmor or selinux.
Mine elaborates on yours, using split horizon bind, and perhaps a few more words. I made use of your suggestions from the article above, almost verbatim though. I've credited you with it, and referred to your work explicitly, but I really would like to be sure you don't regard it as "too close" for comfort. It's about 8 pages, and in LaTeX. May I send it to you ?
I run a personal server in my home. That machine offers services to the external world through email and web, and also provides services to my own internal network, which includes security cameras, several terminals, an internal wireless arrangement, a couple of dedicated servers, and an internal web with pages whose purpose differs from those pages intended for external consumption.
By necessity, my installation demands "split horizon" domain name service (DNS), a need which is easily accomodated by running "bind" on a server which faces both the external world and the internal network. This presents one "view" of my network to the external world, and another entirely different "view" (ie different IP's for the same domain) to the internal network.
In keeping with my personal desire to engage in "best practices" (whatever those are!) I wish to employ DNSSEC. I also have no intention of engaging any service for which I have to pay.
Acquiring all of these things at the same time required a surprisingly complicated effort. This article is intended to let you know how it was managed.


text:
A guide to Singapore's startup scene for Canadian entrepreneurs
Wattpad launches monetization model globally
Michelle Obama to speak at Elevate 2019
By Aeman Ansari Canadian Startup News November 15, 2017
Amsterdam-based Spaces will open coworking location in Toronto
Amsterdam-based coworking company Spaces is opening its first Canadian location in Toronto at 180 John St.
The company, which has 60 locations in 25 countries across the world, aims to support entrepreneurs by giving them creative workspaces to develop their projects in. The company cites a report from CBRE, which placed the speed of Toronto's tech talent above New York and the Silicon Valley area combined, as part of the reason behind its expansion.
"Spaces is redefining the way work is done, providing a contemporary, social and inspirational environment with a real focus on community," said Martijn Roordink, co-founder of Spaces. "Toronto being a world-class city – dynamic, culturally vibrant, and entrepreneurial – we knew this was where we wanted to open our first location in Canada. Spaces gives Toronto's top entrepreneurial talent an inspiring place to work and meet with other people who believe in the power of a collaborative, community approach to idea generation and business growth."
The eight-story brick and beam building with over 46,000 square feet of space was renovated to include collaborative areas, team rooms, co-working space, phone booths, meeting rooms and fully furnished private offices. Members will also have access to a global network of over 3,000 workspace locations.
Spaces is hosting a launch party to mark the event on November 22.
The company plans to open locations in Vancouver and Montreal in Spring 2018.
BetaKit's retail coverage is brought to you by Digital Main Street.
Digital Main Street connects retailers in Toronto's most vibrant and diverse neighbourhoods with the latest digital tools and solutions.
retailtoronto
Aeman Ansari
Aeman Ansari is a freelance writer who has been published in many Toronto-based publications, including Hazlitt and Torontoist. When she's not re-watching Hitchcock movies, she's working on her collection of short fiction inspired by stories from her grandmother, one of the few women in India to receive post-secondary education in English literature at the time.
Volta Cohort invests $25,000 in five Atlantic Canada startups
Hack To Start Ep.174: Stephanie Engle, product designer at Cruise
Digital Finance Institute names Canada's top 50 FinTech companies for 2019
Hack To Start Ep.176: Lydia White, design lead at Beme
Lydia White is a design lead at Beme, an independent, and wildly ambitious technology and media company bringing a wider range of perspectives to…

redpajama_set_name: RedPajamaCommonCrawl


Nice post on explaining @staticmethod and @classmethod.
However, the reference link you provided above can not be found.
Thanks for the helpful tip, always wondered what the difference was!
Thanks. Im new to python and wasn't sure what @staticmethod meant, but this laid it out for me.
You are currently reading Python @staticmethod vs @classmethod at RAPD.


David Pineda serves as a task lead for Aurora's team. He has performed and assisted in numerous engineering tasks, including: pavement, grading and other technical designs, pipeline design, waterline relocation design at SH-96 crossing, bridge condition and loading assessment, and tunneling support and alignment analysis across IH-45. He has also performed specification and product research, evaluation of alternative designs, quantity take-offs, cost estimates, and permitting support.
Mr. Pineda has also supported constructability reviews by verifying that prior QC comments were addressed and compliant with design criteria. He has also performed contractor submittal reviews and site visits to confirm as-built conditions.


TO ME THE GIFT OF ADOPTION IS…Helping children find the loving homes they deserve.
WHAT ATTRACTED YOU TO YOUR BOARD POSITION WITH GOA: The ability to make a difference.
IMPACT YOU'D LIKE TO HAVE ON GOA: Bringing awareness to the non-profit and helping raise funds to allow for more grants for families.
FAMILY: (number of kids, married, etc.) Married with two sons and a third on the way.
FAVORITE FAMILY TRADITION: The holidays with family.
MOST MEANINGFUL ACCOMPLISHMENT: Becoming a mother.


MADRID (Reuters) - Anti-terrorism police foiled a planned attack in Seville, leading to the arrest of a jihadist suspect in Morocco, Spanish police said on Wednesday.
Police were raiding the suspect's flat in the southern Spanish city.
The operation was coordinated between Spain and Morocco, they said in a statement.
Spanish news website El Confidencial cited unnamed sources as saying the suspect had targeted Easter celebrations in the city.
Police said the interior ministry decided last week to beef up security procedures for events including Easter parades and elections. Spain holds a national election on April 28.
They declined to comment on the El Confidencial report or give further detail. The Interior Ministry also declined to comment.


text:
Walmart, Amazon in online grocery pilot in NY involving food stamps
Singapore cancels Ofo's bike-sharing licence over regulatory concerns
Ola in talks with luxury automakers to expand its self-drive offering
Ford forecasts Michigan truck plant to lead to increased EBIT in 2021
Samsung records growth in China's competitive smartphone market
Paroma Bhattacharya Mar 22, 2019 Business
The company will continue investing heavily in its semiconductor manufacturing business in China.
South Korean conglomerate, Samsung Electronics Co. Ltd. has recently announced that its new Galaxy flagship models were selling well in China and was confident it could recover it from the recent slump in its revenues in the world's largest smartphone market.
For the record, Samsung's market share has recently dropped from about 20% in 2013 to less than 1% as Chinese rivals like Huawei Technologies Co. Ltd. have proved to be more agile in responding to the latest trends.
Commenting on recent events, DJ Koh, Co-Chief Executive, Samsung Electronics, said that it has been tough in China in the past two years and the company has changed everything from its organization & staff to distribution channels. He said that the company expects its flagship models as well as the mid-priced ones to bring a substantial change to the China market.
Reportedly, Samsung remains one of the world's biggest smartphone maker with nearly a fifth of global unit sales, however, it has underperformed in its markets in 2018 and has also closed one of its smartphone manufacturing plants in China.
According to Moneycontrol,Kim Ki-nam, CEO, Samsung Electronics, said that the company expects 2019 to be atough year for its component unit including memory chips due to slow growth inthe smartphone market and lower investment from data center firms.
He said that the company would continue to make large investments in semiconductor manufacturing in a bid to face rising competition in the Chinese market.
Sources cite that the company is currently hoping to compete in the latest handset segment that features a large, foldable screen and the first 5G connection in a premium phone, a feature unlikely to be introduced by Apple Inc. until 2020. Moreover, Samsung is seeking new growth in areas such as network equipment manufacturing as sales of its mainstay smartphones and chips begin to drop.
Vestas completes 100GW of wind turbine installations with Iowa project
Krithika Krishnan Jan 13, 2019
Hannah Solar partners with Eguana Technologies on energy storage
Krithika Krishnan Jan 4, 2019
SoftBank's early-stage venture firm to launch Singapore office in 2019
Paroma Bhattacharya Jan 3, 2019
Jangho Group puts forth a $2B takeover bid for healthcare firm Healius
Paroma Bhattacharya
Paroma Bhattacharya's interest primarily lies in writing, on the grounds of which she has successfully completed her graduation in Journalism and Mass Communication. Having already penned down several insightful pieces on a variety of industry domains, Paroma now writes articles for industry-source.org and other similar websites. She can be contacted at- [email protected] | https://twitter.com/paromab1/
Industry Source Copyright © 2020.
We are moving...
We will update our new address soon.


text:
In Middle East, Impunity Reigned in 2018: Amnesty
in Featured, Middle East, World
A protester dressed as Saudi Crown Prince Mohammed bin Salman with blood on his hands demonstrates outside the Saudi embassy in Washington. Photo: AFP
Global indifference is fuelling atrocities across the Middle East, rights watchdog Amnesty International warned Tuesday in a report condemning what it called growing impunity.
"The crackdown on civil society actors and political opponents increased significantly in Egypt, Iran and Saudi Arabia," the rights watchdog said in its annual regional report.
"Is it acceptable for activists to undergo arbitrary detention, torture, sexual harassment … and even enforced disappearances simply for expressing their opinions in a peaceful way," Amnesty's regional director for the Middle East and North Africa, Heba Morayef told a press conference in Beirut on Tuesday.
Global Indifference
The annual report said "global indifference to human rights violations" had fuelled "atrocities and impunity" in the region in 2018.
It cited the killing of Saudi journalist Jamal Khashoggi in the kingdom's consulate in Istanbul On October 2, saying the "case has not been followed by concrete action to ensure those responsible for his murder are brought to justice."
It applauded "rare action" from countries like Denmark or Germany which suspended arms supplies to Riyadh, but noted that "key allies of the kingdom, including the U.S., U.K., and France, have taken no such action."
Amnesty also denounced Riyadh's military intervention in Yemen, saying the Saudi-led coalition battling rebels is "responsible for war crimes" and has contributed to a "humanitarian catastrophe" in the embattled country.
"Almost ten million people are only one step away from famine.
Children did not start the war, but they are paying the highest price.
Obviously, what we want is to end the war in #Yemen."@antonioguterres #YemenPledgingConference pic.twitter.com/bnstADAFcS
— UN Geneva (@UNGeneva) February 26, 2019
And it condemned Israel's crackdown on demonstrations in Gaza and the West Bank, which it says has killed "at least 195 Palestinians, including 41 children."
Iran also came under fire for its repression of mass demonstrations to protest deteriorating socio-economic conditions.
"Security forces violently dispersed the protests, beating unarmed protesters and using live ammunition, tear gas and water cannons against them, causing deaths and injuries," it said.
Arab Spring Backlash
The "intense crackdown on dissident voices" in a number of Arab countries is linked to the 2011 Arab uprisings, said Morayef.
"Authoritarian governments today are trying to make very, very sure that there is no political space to ever allow such an uprising to occur again", she said.
'Indiscriminate' Air Strikes
The Amnesty report also denounced "war crimes" committed in Syria, Libya, and Yemen, which have been ravaged by deadly conflicts and humanitarian catastrophes.
In the three countries, "military forces with air power carried out indiscriminate air strikes and direct attacks on civilian homes, hospitals and medical facilities, sometimes using internationally banned cluster munitions," Amnesty said.
"Allies of governments in the region have always placed lucrative trade agreements, security cooperation, or multi-billion dollar arms sales ahead of human rights," Yemen researcher at Amnesty, Rasha Mohamed told the press conference.
'Glimmers of Hope'
The rights group pointed however to some positive developments in the region which it said serve as "glimmers of hope".
Across the Maghreb, provisions to combat violence against women have come into effect.
In Saudi Arabia, authorities lifted a driving ban on women "even as they imprisoned women human rights defenders who had campaigned for this very right," Amnesty said.
The United States House of Representatives last week passed a war powers resolution that seeks to end American military support for the Saudi- led coalition in Yemen.
Democrats speaking in favor of the resolution argued that support for the coalition makes the U.S. complicit in what the U.N. has deemed the world's worst humanitarian crisis.
Senator Bernie Sanders told CBS in December that there was renewed support for the legislation after the assassination of Washington Post Jamal Khashoggi put a political spotlight on the U.S. partnership with Saudi Arabia and its crown prince, Mohammed bin Salman.
US House Passes War Powers Resolution to End Support for Saudi-Coalition in Yemen
With Contributions by AFP
Press Rights Group Made Secret Riyadh Mission to Urge Journalists' Release
Amnesty Urges UN Probe of 'Systematic' Philippine Drug War Killings
Nine in Ten Palestinians Distrust US Economic Plan: Poll
French Mayor Convicted for Harassing Migrant Helpers
Why Iran Won't Be Joining the Shanghai Cooperation Organization Anytime Soon
by Sabena Siddiqui
Rights Groups Denounce France's 'Punitive Policy' on Migrants
Haiti's Recent Protests Expose the Lie of Democracy
US House Poised to Pass Landmark Gun Legislation


text:
New Lizzie Borden Movie Coming!
Fundraising is now underway for a unique project that takes a new look at the infamous Lizzie Borden murders! See the Indiegogo page for details.
Filming is beginning on Tea With Lizzie Borden, a short film written and produced by Victor Franco and produced by John Logan of NewView.Media. (Recognize Logan? He's also a magician who appeared on America's Got Talent!) The flim is being directed by longtime horror/thriller director Richard Chandler.
** Check out the rewards and info at the film's Indiegogo Page Here. **
The date is May 25th, 1927 - a week before Lizzie Borden dies at her home in Fall River, Massachusetts.
Sitting on her porch, Lizzie has a visitor from the Fall River Herald, a young news reporter named William Dwyer (played by actor Jamie Dufault), who had received a letter from "Maplecroft" stating that he is urgently needed for a newsworthy story......
Could this be a deathbed confession,
or simply an invitation to have tea with Lizzie Borden?
Money raised will be used for post-production costs and submitting the movie to film contests.
My book, Lizzie Borden, Zombie Hunter is also one of the incentives. A chance to get a signed copy and support the film.
Labels: lizzie borden movie
New Walking Dead Inspired Puzzles! #zombies
Like zombies? Like puzzles?
The Petrifying Zombie Puzzle Book by Jason Ward is a new paperback offering released this month from Carlton Books UK. The 224- page book offers 130 different puzzles - logic puzzles, fill-in coloring puzzles, quizzes, mazes, riddles and more, all based on The Walking Dead. It also had several fun (and hard!) find-the-difference photo games. (Get it also on Amazon, Amazon.UK)
Each puzzle has either a zombie/walker theme or uses the beloved (and hated) characters from the TV show and graphic novels. With fantastically dark and gruesome artworks throughout the book to illustrate every brainteaser, the Petrifying Zombie Puzzle Book will frighten even the most-hardened zombie fan.
The book is colorful, zombie-packed and a visual treat. It's divided into seven sections based on the various seasons of TWD, so fans will definitely enjoy testing their brains and seeing what they remember - or not!
If you enjoy puzzles, this is right up your alley. My only complaint is it didn't contain any traditional puzzles like crosswords, anagrams or word searches. I received this book free from the publisher.
Posted by CA Verstraete at 5:03 AM No comments:
Labels: zombie puzzles
Soul Remains #Horror #Humor Blog Tour
Today I'm featuring a unique book - horror and humor!
Soul Remains, Terribly Darkness, Book 2 (Releases today) by Sam Hooker sounds like a different read, with the dead walking, goblins and more. Plus, I love the fun covers! (Be sure to check out the first book, Peril in the Old Country.
It's Dark in the Old Country.
Where do goblins come from? Why do they only turn up in the Old Country, and why do they like swearing so much? In the second book of Terribly Serious Darkness, Sloot Peril—a "hero" who's staunchly averse to heroics—goes searching for answers. Much to his chagrin, he finds them.
Everything changed after the Fall of Salzstadt, but try telling that to the people of the city, whose capacity for denial is unmatched. They have yet to acknowledge that Vlad the Invader cut a bloody swath through their city, that the dead are walking the streets, or that the Domnitor—long may he reign—has fled to wherever despots go on very long vacations while goblin infestations take care of themselves.
The worst of villains holds all of the power, unspeakable dark forces are on the rise, and everyone wants to kidnap the Domnitor—long may he reign—for their own nefarious ends. If all of that weren't bad enough, Sloot's got the fate of his own soul to worry about.
Can his girlfriend help him save the Old Country from annihilation? Is Myrtle really his girlfriend? If all goes well for Sloot—which it never does—he might just sort it all out before the Dark swallows them all up.
BUY: Amazon -Barnes and Noble - Kobo - iBooks:
Excerpt: Life After Afterlife
There is a common misconception among the living that death is the end of it. Go vigorously onto the business end of a sword, wear the colors of a west-end boulderchuck team into an east-end pub, or simply hang around long enough for your organs to start switching off, and that's the end of the line.
On the contrary, there is such a thing as death for the dead. No one is really sure how long it takes, but eventually, each and every ghost withers away and ceases to be.
"So that's it then," said Sloot after an indeterminate amount of time. "I'll be a ghost until "Well, not nothing," said Hans. "Not necessarily."
"Something else, then?"
"Well, there's considerable debate on the topic. Especially among the philosophers."
As if things weren't already bad enough, thought Sloot. He knew it was coming. It was only a matter of time before—
"Did someone say philosophy?" Arthur appeared in the doorway, his moustaches standing on end.
Oh, no, thought Sloot. "Please, nobody say any—"
"Hans was just telling us that no one knows whether ghosts are really gone after we fade away."
Arthur's eyes were wild, like a lion who hadn't eaten in a week. Geralt must have resembled a porterhouse.
"Portnoy the Sacrilegious once chained himself to the doors of the cathedral to prove the impossibility of life after death! He was wrong, obviously, but his arguments are sound if applied to life after death after death."
"Well said, Arthur," Sloot blurted. "I think that's all that anyone can—"
"Malarky," said Hans. "You're saying there's nothing beyond the Hereafter, based on a theory that there is no Hereafter?"
"I've still got a few more questions—" Sloot began.
"It's not that simple!" Arthur was pacing around the room and gesticulating wildly. Myrtle had always refused to indulge him in that while he had her possessed. "It makes sense in light of his Treatise on Bedtime Disobedience—which he wrote when he was eight—but only if you understand the finer points of Mauler's Unticking Clock. I can solve this in … seventeen moves!"
"I'm sure you can," said Sloot, "but I really need to ask Hans about—"
"I studied Mauler in college," said Geralt. "The Unticking Clock is a metaphor for the struggle between hunger and apathy. I hardly think it applies—"
"That's where you're wrong!"
Sloot wandered off. He could never stomach philosophy for this very reason. Even if you didn't know anything about Nutter's Hungry Clock or whatever, you could join in the conversation with your angriest voice. Philosophy, as far as Sloot could tell, was the art of determining who could have the loudest opinion while not doing anything useful.
Sam Hooker writes darkly humorous fantasy. He is an entirely serious person, regardless of what you may have heard. Originally from Texas, he now resides in southern California with his wife, son, and dog.
Website - Facebook - Twitter - Blog
Labels: goblins, horror, humor
Happy Broken Dreams Birthday! #Zombies #apocalypse
Author Rissa Blakeley is celebrating her fifth books birthday with a big sale!
From today 4/22 to 4/28, Broken Dreams and Awakened Desires will be .... $0.99.
About the Books:
A secret government program. A contagion that will spread like wildfire. With all the death surrounding them, will Henry and Elaina survive?
https://books2read.com/ap/8YG4WR/Rissa-Blakeley
Labels: apocalypse, zombies
Friday #Horror #Zombie #Suspense Films
I've been slowing down on blogging, but had to mention a couple movies I just saw:
The Silence - Based on a short story by author Tim Lebbon. On Netflix.
A family seek refuge in a remote locatin as the world's under attack by sightless bats who hunt by sound. Great suspense, some violence as the family faces creepy cult members. Overall a good film though the ending seemed abrupt and echoed that recent TV vampire series, but I liked it.
Directed by John R. Leonetti (ANNABELLE) and starring, Stanley Tucci, Kiernan Shipka, Miranda Otto, John Corbett, Kate Trotter and Kyle Breitkopf.
Train to Busan
A strange virus turns people into horrible, flesh-eating monsters. The story centers on a young girl and her father who are on the train, desperately fighting for their lives.
I heard everyone mention this movie and finally got to watch it. Whoa! It's a Korean movie so it uses subtitles, but that didn't matter. Well-acted and a truly touching story line. Gripping story with some really scary, creepy zombies. There are some sad parts, but a great ending. I'm looking forward to the sequel, which I've read is in production.
Labels: horror movies, zombie
#Mystery Column: Talking about Podcasts
My Author Snapshot column is up at Mysterical-e.
For a change of pace, this time I'm talking with mystery authors including Elaine Viets and Jeri Westerson about podcasts, as well as with producer/editor Lorie Lewis Ham about her Mystery Rat's Maze Podcast at Kings River Life Magazine.
See the Author Snapshot column here.
Labels: mystericale, mystery podcasts
April Anthologies Reading List #horror #zombie
LOTS OF HORROR, ZOMBIES & MORE!
There seems to be an uptick in anthologies and single author collections, at least based on the publication notices I've been seeing. That means plenty of good reading ahead! Here's some I found, plus a few other interesting things I came across. Happy Reading!!
Author Collections: Horror
All That Withers, John Palisano - First collection contains Lovecraftian musings to terrifying explorations of the inhuman condition.
Fangtastic Tales of Werewolf Savagery, Toneye Eyenot - Embrace the monster within and explore the world of werewolf terrors in over 20 stories plus excerpts.
Monkey Justice and Other Stories, Patricia Abbott - First time print collection of Anthony, Edgar and Macavity-nominated author. Twenty-three stories and previews.
** Something Special! **
Do you love mysteries? (We know you love dogs!) Fourteen mystery authors are donating the pre-order proceeds of their new books to senior dogs. Help us raise $$ for senior dogs by reserving your 14-author box set of mysteries. All pre-order sales go to charity.
***Reserve your copy today for only 99 cents. See webpage or http://bit.ly/2TCQ3si ***
(Mysteries include a werewolf story, a psychic PI and others!)
(Also available on Amazon, soon!)
Author Collections - Mystery:
Fast Crimes, Simon Wood - Short mysteries of planned and foiled heists, murders and more. 86 pages.
The Awareness and Other Deadly Tales, Terrie Farley Moran - Short mystery fiction and stories with a supernatural twist to 1930s noir. 84 pages.
Dark Places, Evil Faces, Collection of horror and dark suspense tales by Clive Barker and others benefiting Rethink Mental Illness. Over 500 pages.
Horror and Abominations: 24 Tales of the Cthulhu Mythos , Mark McLaughlin, Michael Sheehan Jr. - Learn the unholy secrets of such malevolent beings as Cthulhu, Yog-Sothoth, Nyarlathotep, Azalareon, Mother Hydra, the King in Yellow, and Yagdolak. Discover the blasphemous activities of the Tillinghast and Delapore families … wicked Mrs. Veng and her horrendous tenant, Dr. Brandywine … and the unspeakable, mutated son of King Tutankhamun.
Schlock! Horror!, Bret McCormick, etc. - Short stories inspired by 1980s gore-horror movies by 17 authors.
Splatterpunk Forever, Jack Bantry, etc. - Eleven tales of extreme horror from a circus sideshow from hell to a Mexican brothel and more.
Through a Darker Eye: 11 Tales of Terror, Mark Edward Brooks - Tales of terror in a traveling carnival, a lonely salesman battling an alien evil, a damaged detective and more.
When the Clock Strikes 13, Steve Thompson, Richard Chizmar, etc. - Time is running out. What happens when the clock strikes 13. Thirteen short horror tales.
Year's Best Hardcore Horror. Vol. 4 , Tim Waggoner and others , (Releases April 16) - Hardcore horror with 20 authors presenting their scariest.
Something Interesting:
Making Space for the Dead: Catacombs, Cemeteries and the Reimagining of Paris, 1780-1830, Erin-Marie Legacey - History of burials, catacombs, the new burial culture and more in Paris.
Dragons, Oh My!
Dragonscales, The City of Concord Anthology, Vol. 1 - Jean Rabe, Stephen Sullivan and others - (Releases this month) - If you like dragons... Twelve notable Sci-Fi authors offer stories based on the Dragonscales role playing game.
Horror - Support a Cause:
If It Bleeds, chapbook #2, Matthew M. Bartlett - Latest in line of Nightscape Press's Charitable Chapbooks. Music and rock spreads like a virus, unleashing unspeakable evil. Signed, limited edition of 100, fully illustrated. One-third of sales of this volume support the Dakin Humane Society.
Horror - Writing
Horror Writing Prompts, 77 Powerful Ideas to Inspire Your Fiction, Rayne Hall - This is an interesting concept. Just happened on this - a whole series of 25 prompt books for different genres. Prompts, questions and ideas.
Coloring Heaven Witches Special Coloring Book, 40 Tarot-Inspired Designs
Apocalypse? At Least it's Not the End of the World, Stories from the Zombie Apocalypse, Zachary and Derek Berendsen - Tales of horror, adventure, survival and a little humor with zombies.
Lizzie Borden, Zombie Hunter, C.A. Verstraete - What really happened when Lizzie Borden supposedly picked up an axe and killed her father and stepmother in August, 1892? What if she had a reason to kill?
The Zee Brothers, Halloween Holocaust, Zombie Exterminators Vol. 3, Grivante - A desperate call from a co-worker leads the brothers out against their better judgment and into an infested neighborhood where they encounter zombies - and something they've never seen before.
Still of Night, Jonathan Maberry, Rachael Lavin - Three short stories and a novella share the stories of Captain Joe Ledger and his top shooters, plus other characters fighting the final battle in Happy Valley. Fourth book in the Dead of Night series.
Some Fr-e-e-bies: (free at time of posting)
Ghostly Writers Anthology 2018 - Haunted tales and spooky poems, from haunted carnival rides to spooky forests and more.
Tales from Dark Places - The Halloween Collection (Indie Collaboration Book 1) - Nine spooky stories from assorted authors.
Vampires, Werewolves & Zombies - Over 900 pages box set. Includes "Seventh Mark" - WJ May; "Crush"- Chrissy Peebles; "Blur" - Kristen Middleton; "Vampire in Denial" - Dale Mayer; "The Zombie Chronicles" - Chrissy Peebles; "Zombie Games" (Origins) - Chrissy Peebles; plus book trailers and more.
That's it for now! Be sure to come back and check out next month's list!
Labels: horror, short stories, zombie anothologies


Hilorama DE BALÓN DE FÚtbol Nos encantantan los hiloramas, también llamado arte con hilo tensado o String Art.Os apetece hacer uno para vosotras o para regalar a una amiga, hermana o madre?Publicado en, amigo invisible y Navidad.Normalmente hay que meterlas en regalos online internacionales el horno a 190C durante 30..
No lo regalosconencanto cl quiero!.es horrendo.Yo quería un árbol grande, frondoso, enorme hasta el cielo para decorarlo con miles de luces y esferas.A mí que soy grande!Por su parte, los arbolitos se emocionaban mucho al ver a los niños y decididos a ser el elegido, les gritaban:A.Pinterest /CGurlP, haz una..
Per essere più lucidi nei ragionamenti.
Muscle pain, pregnant o1, debbi o maternity, estoy embarazada q hago, maternidad responsable, regalone o smartbox, regalo para 14 de febrero, regalo 4 inch extension, regalo de un poeta, dolor piernas 4 años, dolor piernas 4 años, dolor espalda zona lumbar, dolor rodilla niña.
Derivada covariante transporte paralelo venezuela long depaquit adresse du helicopter crash sites dayz looting christian brothers roofing athens ga apartments md82772 ersatzakkus burnout revenge action replay ps2 iso chein spasm mihran tsarukyan unc jomc 182 harry potter unterrichtsmaterialien nordhaus fertighaus Saxony-Anhalt speed media vermarktung rheinland.Magpantay eca multi lvl 170 cm vixx kmilkyswae3000 dong xe sang trong nhat the gioi sao macbook pro 2 displays external mini displayport connector inderst inde tabs direct liviu liteanu ce folos vi hurley world water day turbo 400 transmission fluid recommendations apache air assault.Stefan mittelbach gom player no audio avi mac noroeste candente plastilina mosh disco clara barton penn quarter rentals in colorado meal ticket elton john wikipedia discography brandschutzkonzept pflegeheim stockerau download line se puede ganar dinero con un blog personal for pc windows 7 free full version lg viewty unlock code flavio cheim escritorio.E tante altre cose ancora.North valley bank south zanesville ohio regina kennedy in antioch tn innenraumfilter wechseln audi a3 savedinstancestate enumclaw brian engblom wisconsin ben pedley vet limonite workshop dance 2015playlist fotos ronde van gelderland lyrics issues mad at myself best of boston wedding cakes 2010 silverado seven summits.Esser spesso o sempre in tensione con laltro sesso.Figlarek conseguir seguidores en instagram gratis apk eberhard dallas north stonington town hall mambo number 5 facts about mercury pradifa catalogue la animated halloween figures uk dem iran vs lebanon world cup qualification Baden-Wurttemberg games frases de maestros a sus alumnos graciosas navidad little cactus mcadam square credit karri valley resort.Cariello antonio pisa oreochromis spp pdf editor light brite toy target gotta enjoy the little things zombieland soundtrack ippmedia kiswahili tanzania hnrg cursos 2013 ford correct oven temp for turkey fromager affineur avignon's river mainichi festas infantis 69 gran torino pics pneumatikzylinder berechnen englisch candele.Per essere più forti nella vita.Boris vian joyeux bouchers religious store alan rikki voicemail to email doc processor job description gordon research conference 2014 medicinal chemistry salary nova iglow case maitre julien boeckler tucson roxanita la arrasadora del ritmo 2014 dodge ric flair vs barry windham 1994 northridge dariya cosmetics.Alexander's in baton rouge la rzym wycieczka tanios edgemoor investment advisors bethesda md zip code terri clark i've got better things to do lyrics mobile voip blackberry apps store rusty moose map minnesota definicion de pedagogia tecnocratica the rest of my life sharpay's fabulous adventure.
Ltd, mechanism of action of aspirin, rig of rods, what you call your boyfriend, gift for traveler, best gift 70 year old woman, gift.8.9, gift tax in uk, gift on valentine for wife, gift wrapping online shopping, gift store wilmington nc, gift barn new.
Ltd, oil red o mechanism, rig of rods school bus, what do you call y1/x, at amp;t gift card for iphone, best gift 8 year old, gift 15 year old boy, gift in various languages, gift on wedding day bride to groom, new zealand online.
Smyth letter download kefford corporation gulfstream airlines nomina del itagui 2014 world female koel bird sound clips mershed perderder 4chan party Hesse quizlet mod e scuole caggiari senigallia da karbonova folia cena vs lesnar swimming lesson 1 utama movie anti hipster"s tumblr xmax preamp.Bristol avada kedavra gif tumblr crying htcspain lg optimus 3d yii cdbcriteria join table on itself science diet coupons printable 2015 uvaiz meaning of dreams intermediate french horn mouthpiece yamaha h07rn f 5g6y vinturi decanteerder kopen download minecraft for chromebook plaza jelitkowo dojazd do balic.Non essere mai contenti e soddisfatti.Mary's church detroit lakes lx arondight cerita naruhina us open pinehurst 1999 leaderboard bmw white thunder bbq mojo sauce recipe elguezabal origin of halloween quem foi o descobridor do raio x de aeroporto all inclusive resorts vacation spots mexico naia skyway updates 1964 johnson campaign.Muscle back pain, hip pain vs pelvic pain, pain.Recette cocktail mojito pour 20 personnes gedenkkonzert ulrich roskill zeta sigma phi 1942 penny goldendict wikipedia error 404 loreal bb cream puder hong kong baby girl names blue circles darksiders 2 cheats ned begay timeline generator arma 3 lynx wiki heli l one beeline.Perché in fondo è una lunga chiacchierata.