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

Safety-critical control has garnered a lot of attention in the
controls and robotics communities motivated by applications to many
different classes of engineered and natural systems. Safety refers to
the ability to ensure by design that the evolution of the dynamics
stays within a desired set.  Control barrier functions (CBFs) are a
useful tool to deal with safety specifications that do not require
addressing the difficult task of computing the system's reachable set.
In many scenarios, safety must be achieved together with some
stabilization goal, and this raises interesting challenges for control
design in order to ensure that both are achieved via feedback
controllers that are easily implemented and have appropriate
smoothness guarantees. These challenges motivate us to develop here an
optimization with penalty-based feedback design framework for safe
stabilization of control affine systems.

\subsubsection*{Literature Review}
We rely on ideas from two different bodies of work.  The first one is
CLFs~\cite{ZA:83,EDS:98}, which have been successfully used in the
control design for stabilization of nonlinear systems. Of particular
interest to this work is the pointwise-minimum norm (PMN)
formula~\cite{RAF-PVK:96a}, that uses a CLF to compute a stabilizing
controller. The second relevant body of work pertains to
CBFs~\cite{PW-FA:07,ADA-XX-JWG-PT:17,ADA-SC-ME-GN-KS-PT:19}, whose aim
is to render a certain predefined safe set forward invariant.
However, in applications where both safety and stability must be
certified, CBFs fall short of providing provable stability
guarantees. To tackle this issue,~\cite{MZR-BJ:16} combines a CLF and
a CBF into a so-called CLBF, and then uses Sontag's universal formula
to derive a smooth controller. However, in general it might be
difficult to satisfy the conditions required for the existence of such
a CLBF~\cite{PB-CMK:17}. Another approach is the universal formula for
smooth safe stabilization from~\cite{PO-JC:19-cdc}.  However, this
formula is only applicable in a set where both the CLF and the CBF are
compatible (i.e., there exists a control satisfying their associated
inequalities at every point of the set).  An alternative
approach~\cite{ADA-XX-JWG-PT:17,ADA-SC-ME-GN-KS-PT:19} to tackle joint
safety and stability specifications is to combine the CLF and the CBF
in a quadratic program (QP). To guarantee the feasibility of the
program when the functions are not compatible and to avoid the
resulting controller to be non-Lipschitz when they
are~\cite{BJM-MJP-ADA:15}, the stability constraint is often
relaxed. This results in a lack of guarantee of stability, even for
arbitrarily large penalties in the relaxation parameter~\cite{MJ:18}.
Moreover, as shown in~\cite{MFR-APA-PT:21,XT-DVD:21}, this QP
formulation can introduce undesired equilibria beyond the original
equilibrium, which can even be asymptotically stable. This line of
work~\cite{MJ:18,XT-DVD:21} then identifies conditions under which
local stability guarantees of the equilibrium can be given.  An
alternative design, e.g.,~\cite{LW-ADA-ME:17,UB-LW-ADA-ME:15}, assumes
a priori knowledge of a CBF and a nominal (possibly unsafe)
stabilizing controller. Then, a safety filter is applied to this
nominal controller. As a result, the filtered controller generally
lacks stability guarantees. The recent paper~\cite{WSC-DVD:21} gives
an estimate of the region of attraction of the closed-loop system
obtained by using such a filtered controller.

\subsubsection*{Statement of Contributions}
We consider\footnote{We denote by $\mathbb{Z}_{> 0}, \mathbb{R}$ and
  $\mathbb{R}_{\geq0}$ the set of positive integers, real, and nonnegative
  real numbers, resp. We write
  $\text{int}(\mathcal{S}), \partial\mathcal{S}$ for the interior and
  the boundary of the set $\mathcal{S}$, resp.

  Given $x\in\mathbb{R}^{n}$, $\norm{x}$ denotes its Euclidean norm.  Given
  $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$, $g:\mathbb{R}^{n}\to\mathbb{R}^{n\times m}$ and a
  smooth function $W:\mathbb{R}^{n}\to\mathbb{R}$, the notation
  $\map{L_{f}W}{\mathbb{R}^n}{\mathbb{R}}$
  (resp. $\map{L_gW}{\mathbb{R}^n}{\mathbb{R}^{m}}$) denotes the Lie derivative
  of $W$ with respect to $f$ (resp. $g$), that is
  $L_{f}W=\nabla W^{T}f$ (resp.  $\nabla W^{T} g$).  Given
  $a \in \mathbb{R}^n$ and $b\in \mathbb{R}$, let $H$ denote the hyperplane
  defined by $H = \setdef{x\in\mathbb{R}^n}{\langle a, x\rangle =b}$. We
  denote the projection of $v\in\mathbb{R}^{n}$ onto $H$ by
  $P_{H}(v) =v-\frac{\langle a, v\rangle-b}{\norm{a}^2}a$.  A function
  $\beta:\mathbb{R}\to\mathbb{R}$ is of class $\mathcal{K}$ if $\beta(0)=0$ and
  $\beta$ is strictly increasing. If moreover
  $\lim\limits_{t\to\infty}\beta(t)=\infty$, $\beta$ is of class
  $\mathcal{K}_{\infty}$.  A function $V:\mathbb{R}^{n}\to\mathbb{R}$ is
  positive definite if $V(0)=0$ and $V(x)>0$ for $x \neq 0$.}  the
problem of safe stabilization of control affine systems.  Given a
control Lyapunov function and a control barrier function, we aim to
synthesize a safe, stabilizing feedback and identify the region of
attraction of the origin for the resulting closed-loop system. In
particular, we study under what conditions such region of attraction
contains the safe set. The contributions of this paper are the
following.  Given the safety and stability objectives, our first
contribution designs an optimization with penalty-based controller
that has one of the objectives as a hard constraint and the other as a
soft constraint.  The controller depends on a penalty parameter that
can be tuned to enhance the soft objective at the cost of reduced
optimality, while guaranteeing the satisfaction of the hard
constraint. An advantage of the proposed design is that the controller
is automatically Lipschitz and has a closed-form expression.  Our
second contribution shows that the controller can introduce undesired
equilibrium points different from the origin.  By choosing the penalty
parameter appropriately, and under some technical conditions, these
undesired equilibria can be eliminated. Finally, our third
contribution shows that the proposed controller can be tuned to
provide an inner approximation of the region of attraction of the
origin for the closed-loop system. As a consequence of this analysis,
we provide conditions under which all of the safe set belongs to the
region of attraction of the origin for the closed-loop
system. Simulations on a planar system compare our design with other
approaches from the literature.

\section{Preliminaries on CLFs and CBFs}

This section presents preliminaries on control Lyapunov and barrier
functions.  Consider a control-affine system
\begin{align}\label{eq:control-affine-sys}
  \dot{x}=f(x)+g(x)u,
\end{align}
where $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and $g:\mathbb{R}^{n}\to\mathbb{R}^{n\times m}$
are locally Lipschitz functions, with $x\in\mathbb{R}^{n}$ the state and
$u\in\mathbb{R}^{m}$ the input. Throughout the paper, and without loss of
generality, we assume $f(0)=0$, so that the origin $x=0$ is the
desired equilibrium point of the (unforced) system.

We start by recalling the notion of Control Lyapunov function
(CLF)~\cite{EDS:98,RAF-PVK:96a}.
\begin{definition}\longthmtitle{Control Lyapunov Function}\label{def:clf}
  Given an open set $\mathcal{D}\subseteq\mathbb{R}^{n}$, with
  $0\in\mathcal{D}$, a continuously differentiable function
  $V:\mathbb{R}^{n}\to\mathbb{R}$ is a \textbf{CLF} on $\mathcal{D}$ for the
  system~\eqref{eq:control-affine-sys} if
  \begin{itemize}
  \item $V$ is proper in $\mathcal{D}$, i.e.,
    $\setdef{x\in\mathcal{D}}{V(x)\leq c}$ is a compact set for all
    $c>0$,
  \item $V$ is positive definite,
  \item there exists a continuous positive function
    $W:\mathbb{R}^{n}\to\mathbb{R}$ such that, for each
    $x\in\mathcal{D}\backslash \{0\}$, there exists a control
    $u\in\mathbb{R}^{m}$ satisfying
    \begin{align}\label{eq:clf-ineq}
      L_fV(x)+L_gV(x)u \leq -W(x).
  \end{align}
\end{itemize}
\end{definition}
\medskip

CLFs provide a way to guarantee asymptotic stability of the
origin. Namely, if a Lipschitz controller $u$
satisfies~\eqref{eq:clf-ineq} for all
$x\in\mathcal{D} \setminus \{0\}$, then the origin of the closed-loop
system is asymptotically stable~\cite{EDS:98}.  If $W(x)$
in~\eqref{eq:clf-ineq} is replaced by $\gamma(V(x))$, where $\gamma$
is a class $\mathcal{K}$ function, then such Lipschitz controller
makes the origin exponentially stable. Such controllers can be
synthesized by means of the pointwise minimum-norm (PMN) control
optimization~\cite[Chapter 4.2]{RAF-PVK:96a},
\begin{align*}
  u(x)
  & = \argmin{u \in\mathbb{R}^{m}}{\frac{1}{2}\norm{u}^2}
  \\
  \notag
    & \qquad \text{s.t.~\eqref{eq:clf-ineq} holds}.
\end{align*}
Note that, at each $x\in \mathbb{R}^n$, this is a quadratic program in~$u$,
and the resulting controller is Lipschitz~\cite[Theorem
2]{XX-PT-JWG-ADA:15}.

Next we recall the notion of Control Barrier Function
(CBF)~\cite{ADA-XX-JWG-PT:17}. Let $\mathcal{C}\subseteq\mathbb{R}^{n}$ be
a closed set describing the safe states for the
system~\eqref{eq:control-affine-sys}.

\begin{definition}\longthmtitle{Control Barrier
    Function}
  Let $h:\mathbb{R}^{n}\to\mathbb{R}$ be a continuously differentiable function
  such that $ \mathcal{C}=\setdef{x\in\mathbb{R}^n}{h(x)\geq0}$.  The
  function $h$ is a \textbf{CBF} of $\mathcal{C}$ for the
  system~\eqref{eq:control-affine-sys} if there exists a class
  $\mathcal{K}_{\infty}$ function $\alpha$ such that, for all
  $x\in \mathcal{C}$, there exists a control $u\in\mathbb{R}^{m}$ satisfying
  \begin{align}\label{eq:cbf-ineq}
    L_fh(x)+L_gh(x)u+\alpha(h(x))\geq0.
  \end{align}
\end{definition}
\smallskip

CBFs can be used to guarantee safety, i.e., forward invariance
of~$\mathcal{C}$ under the
dynamics~\eqref{eq:control-affine-sys}. Namely, if a Lipschitz
continuous controller satisfies~\eqref{eq:cbf-ineq} for all
$x\in\mathcal{C}$, then $\mathcal{C}$ is forward
invariant~\cite[Theorem 2]{ADA-SC-ME-GN-KS-PT:19}. Similar to the PMN
controller above, a common design
methodology~\cite{ADA-SC-ME-GN-KS-PT:19} is via the optimization
\begin{align*}
  u(x)
  & = \argmin{u \in\mathbb{R}^{m}}{\frac{1}{2}\norm{u}^2}
  \\
  \notag
    & \qquad \text{s.t.~\eqref{eq:cbf-ineq} holds},
\end{align*}
which results in a Lipschitz controller~\cite[Theorem
2]{XX-PT-JWG-ADA:15}.

When dealing with both the stability and safety of system trajectories
under the dynamics~\eqref{eq:control-affine-sys}, it is important to
note that an input $u$ might satisfy~\eqref{eq:clf-ineq} but
not~\eqref{eq:cbf-ineq}, or vice versa. The following notion, adapted
from~\cite[Definition 2.3]{PO-JC:19-cdc}, captures when the CLF and
the CBF are compatible.

\begin{definition}\longthmtitle{Compatibility of CLF-CBF pair}
  Let $\mathcal{D} \subseteq \mathbb{R}^{n}$ be open, $\mathcal{C} \subset \mathcal{D}$
  closed, $V$ a CLF on $\mathcal{D}$ and $h$ a CBF of~$\mathcal{C}$. Then, $V$
  and $h$ are \textbf{compatible at $x \in \mathcal{C}$} if there exists
  $u\in\mathbb{R}^{m}$ satisfying~\eqref{eq:clf-ineq}
  and~\eqref{eq:cbf-ineq} simultaneously. We refer to both functions
  as compatible if $V$ and $h$ are compatible at every point of $\mathcal{C}$.
\end{definition}

\section{Problem Statement}\label{sec:problem-statement}

We are interested in designing controllers that are both stabilizing
and safe. We also require them to be Lipschitz in order to guarantee
existence and uniqueness of solutions of the closed-loop system.
Formally, consider a control-affine system of the
form~\eqref{eq:control-affine-sys}.  Let $\map{V}{\mathbb{R}^n}{\mathbb{R}}$ be
a CLF on the open set $\mathcal{D} \subseteq \mathbb{R}^{n}$ and
$\map{h}{\mathbb{R}^n}{\mathbb{R}}$ be a CBF of the closed set
$\mathcal{C} \subset \mathcal{D}$.  We assume the origin belongs to
$\mathcal{C}$.  Given the availability of these functions, it seems
reasonable to employ $V$ to ensure the stabilizing aspect of the
controller and $h$ to ensure safety. We also seek to provide formal
characterizations of the region of attraction of the equilibrium for
the resulting closed-loop system.  If $V$ and $h$ are compatible at
every point in the safe set, one option is to find the control through
pointwise optimization with~\eqref{eq:clf-ineq}
and~\eqref{eq:cbf-ineq} as constraints. However,~\cite{BJM-MJP-ADA:15}
gives a counterexample that shows that this pointwise minimization can
result in a non-Lipschitz controller. To remedy this, and also to
extend the design to scenarios where $V$ and $h$ might not be
compatible at some points in the safe set, a popular
approach~\cite{ADA-XX-JWG-PT:17} is to relax one of the
inequalities~\eqref{eq:clf-ineq},~\eqref{eq:cbf-ineq} (in
safety-critical applications, the CLF constraint~\eqref{eq:clf-ineq}),
and formulate a QP that penalizes the relaxation parameter:
\begin{align}\label{eq:clf-cbf-qp}
  u(x)
  & = \argmin{(u,\delta)\in\mathbb{R}^{m+1}}{\frac{1}{2}\norm{u}^2+p\delta^2},
  \\
  \notag
    & \qquad \text{s.t.} \quad \eqref{eq:cbf-ineq}, \ L_fV(x)+L_gV(x) u \leq -W(x)+\delta.
\end{align}
Nevertheless, even in the case where the CLF and the CBF are compatible at
all points in the safe set, the resulting controller might not be
stabilizing even for arbitrarily large values of $p$~\cite{MJ:18}.
Moreover, as pointed out in~\cite{MFR-APA-PT:21,XT-DVD:21},
this design might introduce undesired equilibria in the closed-loop
system, which can even be asymptotically stable. To the best of the
authors' knowledge, only local stability guarantees exist in this line
of work~\cite[Theorem 3]{XT-DVD:21},\cite[Theorem 1]{MJ:18}, and no
estimates of the region of attraction have been given in the
literature.

An alternative design, e.g.,~\cite{LW-ADA-ME:17,UB-LW-ADA-ME:15},
assumes a nominal (possibly unsafe) stabilizing controller $u_{\operatorname{nom}}$
is available, and seeks to modify it as little as possible while
guaranteeing safety. This can be done by solving the following QP:
\begin{align}\label{eq:unom-qp}
  u(x)
  & =  \argmin{u\in\mathbb{R}^{m}}{\frac{1}{2}\norm{u-u_{\operatorname{nom}}(x)}^2},
  \\
  \notag
  & \qquad \text{s.t.} \quad \eqref{eq:cbf-ineq}.
\end{align}
In general, the resulting modified controller might not retain the
stability properties of the original nominal controller but, under
certain conditions~\cite{WSC-DVD:21}, one can provide an estimate of
the region of attraction of the equilibrium. Interestingly, nominal
controllers other than the given one might result in larger regions of
attraction, so in this sense the design directly with the CLF offers
greater flexibility.

We are interested in building an alternative to the
designs~\eqref{eq:clf-cbf-qp},~\eqref{eq:unom-qp} to solve the
aforementioned issues.  In particular, we tackle the following two
problems:

\begin{problem}\label{pb:problem-1}
  Determine a Lipschitz control law $u$ and a region of
  attraction $\Gamma\subseteq\mathbb{R}^{n}$, $\Gamma\cap\mathcal{C}\neq0$
  such that for all $x(0)\in\Gamma\cap\mathcal{C}$,
  $x(t)\in\mathcal{C}$ for all $t\geq0$ and the
  system~\eqref{eq:control-affine-sys} in closed-loop with $u$ is
  asymptotically stable with respect to the origin.  \problemfinal
\end{problem}

\begin{problem}\label{pb:problem-2}
  Determine sufficient conditions on the
  dynamics~\eqref{eq:control-affine-sys}, $V$, $h$ and $\mathcal{C}$ such that
  Problem~\ref{pb:problem-1} is solvable with $\mathcal{C}\subseteq \Gamma$.
  \problemfinal
\end{problem}

For simplicity of presentation, we assume that there exists a
neighborhood $\bar{\mathcal{C}}$ of $\mathcal{C}$ such that the CBF satisfies
$L_gh(x)\neq0$ for all $x\in\bar{\mathcal{C}}$ and the CLF satisfies
$L_{g}V(x)\neq0$ for all $x\in\bar{\mathcal{C}}\setminus\{0\}$.  These two
conditions imply that the input $u$ appears in the Lie derivative of
$h$ and $V$ with respect to~\eqref{eq:control-affine-sys}.

\section{Safety and Stability via QP with
  Penalty}\label{sec:design}

In this section we design a candidate control law to solve
Problems~\ref{pb:problem-1} and~\ref{pb:problem-2} by leveraging the
CLF $V$ and the CBF~$h$. We first present our exposition in a general
context, then particularize to our setting.  Consider general
Lipschitz functions $a,c : \mathbb{R}^{n}\to\mathbb{R}$ and
$b,d : \mathbb{R}^{n}\to\mathbb{R}^{m}$, with $b(x),d(x)\neq0$ for all
$x\in\mathcal{C}\backslash\{0\}$. Consider the following two affine
inequalities in $u\in\mathbb{R}^{m}$,
\begin{align*}
  a(x)+b(x)u\leq0 , \quad
  c(x)+d(x)u\leq0 .
\end{align*}
To select $u$, we regard at the first inequality as a \textit{soft
  constraint} and the second as a \textit{hard
  constraint}.
Inspired by the theory of penalty methods for constrained
optimization~\cite[Chapter 13]{DGL:84}, we formulate a QP where we
include the soft constraint in the objective function with a penalty
parameter ($\epsilon>0$) and enforce the hard constraint.  The
resulting solution of the QP is parametrized by $x\in\mathbb{R}^{n}$ and
$\epsilon$:
\begin{align}\label{eq:usafe-qp}
  \notag
  u_{\epsilon}(x)
  & :=  \argmin{u\in\mathbb{R}^m}
  {\frac{1}{2}\norm{u}^2}+\frac{1}{\epsilon}(a(x)+b(x)u),
  \\
  & \qquad \text{s.t.} \quad c(x)+d(x)u\leq0.
\end{align}
Since this optimization problem is a QP, it is convex.  The following
result gives a closed-form expression for $u_{\epsilon}$ and
establishes that it is Lipschitz.

\begin{proposition}\longthmtitle{Closed-form expression for Lipschitz
    controller}\label{prop:usafe-formula}
  Let $a,c : \mathbb{R}^{n}\to\mathbb{R}$ and $b,d : \mathbb{R}^{n}\to\mathbb{R}^{m}$ be
  Lipschitz, with $b(x),d(x)\neq0$ for all
  $x\in\bar{\mathcal{C}}\backslash\{0\}$, where $\bar{\mathcal{C}}$ is a neighborhood
  of $\mathcal{C}$. For each $x \in \mathcal{C}$, let
  $H(x):=\setdef{u\in\mathbb{R}^m}{c(x)+d(x)u=0}$ and
  $e(x) := c(x)-\frac{1}{\epsilon}d(x)b(x)$.  Then,
  \begin{align}\label{eq:usafe}
    u_{\epsilon}(x)=
    \begin{cases}
      -\frac{1}{\epsilon}b(x) & \text{if } e(x)\leq0,
      \\
      P_{H(x)}(-\frac{1}{\epsilon}b(x)) & \text{if } e(x) >0,
    \end{cases}
  \end{align}
  and $u_{\epsilon}$ is Lipschitz on
  $\bar{\mathcal{C}}\backslash\{0\}$. Moreover, if $d(0)\neq0$, $u_{\epsilon}$
  is Lipschitz at $0$.
\end{proposition}
\begin{proof}
  The expression~\eqref{eq:usafe} follows by calculating the KKT
  points of~\eqref{eq:usafe-qp}.

  Lipschitzness of $u_{\epsilon}(x)$ follows from~\cite[Section 3.10,
  Theorem 2]{DGL:69}, which as a special case includes the
  minimization of a quadratic cost function subject to affine
  inequality constraints.
\end{proof}

We next particularize the general design~\eqref{eq:usafe-qp} to our
setup. We consider two cases:
\\
\textbf{Safety QP with stability penalty:} The selection
$a(x)=L_{f}V(x)+W(x)$, $b(x)=L_{g}V(x)$,
$c(x)=-L_{f}h(x)-\alpha(h(x))$, and $d(x)=-L_{g}h(x)$ makes the CLF
inequality~\eqref{eq:clf-ineq} a soft constraint and the CBF
inequality~\eqref{eq:cbf-ineq} a hard one. We denote by
$u_{\epsilon}^{\operatorname{safe}}$ the controller resulting
from~\eqref{eq:usafe-qp}. If $L_gh(0)\neq0$,
Proposition~\ref{prop:usafe-formula} guarantees that
$u_{\epsilon}^{\operatorname{safe}}$ is Lipschitz on $\mathcal{C}$. Moreover, since it
satisfies the CBF inequality~\eqref{eq:cbf-ineq} for all $x \in \mathcal{C}$,
the resulting closed-loop system is safe for all $\epsilon>0$;
\\
\textbf{Stability QP with safety penalty:} Alternatively, the selection
  $a(x)=-L_{f}h(x)-\alpha(h(x))$, $b(x)=-L_{g}h(x)$,
  $c(x)=L_{f}V(x)+W(x)$, and $d(x)=L_{g}V(x)$, makes the CBF
  inequality~\eqref{eq:cbf-ineq} a soft constraint and the CLF
  inequality~\eqref{eq:clf-ineq} a hard one.  We denote by
  $u_{\epsilon}^{\operatorname{stable}}$ the resulting controller
  from~\eqref{eq:usafe-qp}. In this case, $d(0)=0$ and hence
  Proposition~\ref{prop:usafe-formula} only guarantees that
  $u_{\epsilon}^{\operatorname{stable}}$ is Lipschitz in
  $\bar{\mathcal{C}}\backslash\{0\}$. Moreover, since~\eqref{eq:clf-ineq} is
  satisfied for all $x\in\bar{\mathcal{C}}\backslash\{0\}$, the origin is
  asymptotically stable for the resulting closed-loop system.

From this point onwards, we formulate the results for the controller
$u_{\epsilon}^{\operatorname{safe}}$. With minor modifications, similar results can
be stated for $u_{\epsilon}^{\operatorname{stable}}$.  Note also that
Proposition~\ref{prop:usafe-formula} provides a closed-form expression
for the controllers. This allows the closed-loop system to be
implemented without having to continuously solve the
optimization~\eqref{eq:usafe-qp}, which is something one faces when
dealing with~\eqref{eq:clf-cbf-qp},
e.g.~\cite{SCH-XX-ADA:15,ADA-XX-JWG-PT:17}.

\begin{remark}\longthmtitle{Nominal Controller}
  Our framework can be adapted to the scenario described
  in~\eqref{eq:unom-qp}, where instead of a CLF, one has access to a
  nominal stabilizing controller $u_{\operatorname{nom}}$ and a certificate of
  stability in the form of a Lyapunov function $V$ satisfying
  $L_{f}V(x)+L_{g}V(x)u_{\operatorname{nom}}(x)+W(x)\leq0$ for $x\in\mathcal{D}$, with $\mathcal{D}$
  some open set.  To design a control $u$ as close as possible to
  $u_{\operatorname{nom}}$ that is safe and stabilizing, one can set
  $v=u-u_{\operatorname{nom}}$. Then, it is easy to check that $V$ is a CLF for
  $\dot{x}=\bar{f}(x)+g(x)v$, where
  $\bar{f}(x)=f(x)+g(x)u_{\operatorname{nom}}(x)$. In this case, one could use the
  safety QP with stability penalty setting
  $a(x)=L_{\bar{f}}V(x)+W(x)$, $b(x)=L_{g}V(x)$,
  $c(x)=-L_{\bar{f}}h(x)-\alpha(h(x))$, and $d(x)=-L_{g}h(x)$.

  \demo
\end{remark}

\section{Analysis of Safety QP with Stability
  Penalty}\label{sec:analysis}

Here, we analyze the closed-loop properties
of~\eqref{eq:control-affine-sys} under $u_{\epsilon}^{\operatorname{safe}}$. We
first show how to choose $\epsilon$ to avoid undesired equilibria of
the closed-loop system and then go on to solve
Problems~\ref{pb:problem-1} and \ref{pb:problem-2}. Throughout the
section,
\begin{align*}
  e(x) = -L_{f}h(x)+\frac{1}{\epsilon}L_gh(x)^T L_gV(x)-\alpha(h(x)) .
\end{align*}

\subsection{Ruling out Undesired Equilibrium Points}

Here we show that the closed-loop implementation of the safety QP with
stability penalty controller might introduce new equilibria other than
the origin.  The next result characterizes such equilibria and shows
that, under some conditions, they can be confined to an arbitrarily
small neighborhood of the origin for small enough~$\epsilon$.

\begin{proposition}\longthmtitle{Characterization of
    Equilibria}\label{prop:undes-eq}
  For $\epsilon>0$, the set of equilibrium points of the closed-loop
  system $\dot{x}=f(x)+g(x)u_{\epsilon}^{\operatorname{safe}}(x)$ in $\mathcal{C}$
  is
  $\mathcal{Q}=\mathcal{Q}_{1}^\epsilon\cup\mathcal{Q}_{2}^\epsilon$,
  with
  \begin{align*}
    \mathcal{Q}_1^{\epsilon}
    &:=\setdef{x\in\mathcal{C}}{e(x)\leq0, \
      f(x)=\frac{1}{\epsilon}g(x)L_gV(x)},
    \\
    \mathcal{Q}_2^{\epsilon}
    &:=\setdef{x\in\partial\mathcal{C}}{{ e(x)}>0, \ f(x) =
      \frac{L_fh(x)}{\norm{L_gh(x)}^2}g(x) L_gh (x)
    \\
    &
      +
      \frac{g(x)}{\epsilon}(L_gV(x)-\frac{L_gh(x)^T L_gV(x)}{\norm{L_gh(x)}^2}
      L_gh(x))},
  \end{align*}
  and $0\in\mathcal{Q}_{1}^{\epsilon}$.  Moreover, let $N_{1}, N_{2}$
  and $N_4$ be defined by
  %
  \begin{align*}
    N_{1}&:=\sup_{x\in\mathcal{C}} \norm{f(x)},
    \\
    N_{2}&:=\sup_{x\in\mathcal{\partial C}}
             \norm{f(x)-\frac{L_fh(x)}{\norm{L_gh(x)}^2}g(x)L_gh(x)},
    \\
    N_{4}&:=\inf_{x\in\mathcal{\partial C}}
           \norm{g(x)(L_gV(x) -
           \frac{L_gh(x)^T L_gV(x)}{\norm{L_gh(x)}^2}L_gh(x))}.
  \end{align*}
  then,
  \begin{itemize}
    \item if $N_{1}$ and $N_{2}$ are finite, and
          there exists a neighborhood $\mathcal{V} \subseteq\mathcal{C}$ of
          the origin such that
          \begin{align*}
            N_{3}^{\mathcal{V}}:=\inf_{x\in\mathcal{C\backslash\mathcal{V}}}
            \norm{g(x)L_gV(x)}>0 ,
          \end{align*}
          for $0<\epsilon<\frac{N_3^{\mathcal{V}}}{N_1}$, then
          $\mathcal{Q}_1^\epsilon\subseteq\mathcal{V}$,
        \item if $N_{4}$ is strictly positive, for
          $0<\epsilon<\frac{N_4}{N_2}$, then
          $\mathcal{Q}_{2}^{\epsilon}=\emptyset$.
  \end{itemize}
\end{proposition}
\begin{proof}
  Since $u_{\epsilon}^{\operatorname{safe}}(x)$ takes a different form depending on
  the sign of $e(x)$, we distinguish two cases:
  \\
  \textbf{Case 1: $e(x)\leq0$}: In this case the equilibrium points of
  the closed-loop system satisfy
  $f(x)=\frac{1}{\epsilon}g(x)L_{g}V(x)$.  The choice
  $\epsilon<\frac{N_3^{\mathcal{V}}}{N_1}$ rules out any equilibrium
  of this kind in $\mathcal{C}\backslash\mathcal{V}$. Note that, since
  $f(0)=0$ and $\nabla V(0)=0$, we have $e(0)=-\alpha(h(0))\leq0$, and
  hence $0\in\mathcal{Q}_{1}^{\epsilon}$.
  \\
  \textbf{Case 2: $e(x)>0$}: In this case the equilibrium points of
  the closed-loop system satisfy
  \begin{multline}\label{eq:case2-eq}
    f(x) -\frac{L_fh(x)+\alpha(h(x))}{\norm{L_gh(x)}^2}g(x)L_gh(x) =
    \\
    = \frac{g(x)}{\epsilon}(L_gV(x)-\frac{L_gh(x)^T
      L_gV(x)}{\norm{L_gh(x)}^2}L_gh(x)) .
  \end{multline}
  Let us show that these equilibria can only occur in
  $\partial\mathcal{C}$.  Multiplying both sides
  of~\eqref{eq:case2-eq} by $\nabla h(x)^{T}$, we obtain
  $-\alpha(h(x))=0$.  Since $\alpha$ is a class $\mathcal{K}_{\infty}$
  function, this can only occur when $h(x)=0$, i.e.,
  $x\in\partial\mathcal{C}$. Now, note that by taking
  $\epsilon<\frac{N_4}{N_2}$, all equilibrium points of these kind are
  ruled out.
\end{proof}

Note also the assumption that $N_{1}$ and $N_{2}$ are finite in
Proposition~\ref{prop:undes-eq} is satisfied if $\mathcal{C}$ is
bounded. Also, if $g(x)$ is full column rank for all
$x\in\mathcal{C}$, then there exist arbitrarily small neighborhoods
$\mathcal{V}$ of the origin for which $N_{3}^{\mathcal{V}}$ is
strictly positive. Thus, $N_4$ is positive, the controller
$u_{\epsilon}^{\operatorname{safe}}$ with $\epsilon$ small enough confines the
equilibria of the closed-loop system arbitrarily close to the
origin. However, as $\mathcal{V}$ gets arbitrarily small,
$N_3^{\mathcal{V}}$ (and hence $\epsilon$) could also get arbitrarily
small. In Corollary~\ref{cor:conv-origin} later, we give sufficient
conditions to ensure that this does not happen.

\begin{remark}\longthmtitle{Existence of boundary
    equilibria}\label{rem:bdy-eq}
  The assumption in Proposition~\ref{prop:undes-eq} that $N_{4}$ is
  strictly positive is not satisfied if $g(x)L_{g}V(x)$ and
  $g(x)L_{g}h(x)$ are linearly dependent at some
  $x\in\partial \mathcal{C}$. In this scenario,
  condition~\eqref{eq:case2-eq} reduces to
  $f(x)=\frac{L_fh(x)}{\norm{L_gh(x)}^2}g(x)L_{g}h(x)$. Thus, the
  equilibrium points in $\partial\mathcal{C}$ that cannot be removed
  by tuning the parameter $\epsilon$ are those where $f(x)$,
  $g(x)L_{g}V(x)$ and $g(x)L_{g}h(x)$ are collinear and $e(x)\geq 0$
  for all $\epsilon$.
  \demo
\end{remark}

\subsection{Incompatibility and Region of Attraction}
Here we show that $u_{\epsilon}^{\operatorname{safe}}$ solves
Problems~\ref{pb:problem-1} and~\ref{pb:problem-2}. The flexibility
provided by the design parameter $\epsilon$ is instrumental in doing
so.  We first introduce a characterization of points where the CLF and
the CBF are incompatible.

\begin{lemma}\longthmtitle{Characterization of incompatible
    points}\label{lem:incompat-points}
  Let $\mathcal{D} \subseteq \mathbb{R}^{n}$ be open, $\mathcal{C} \subset \mathcal{D}$
  closed, $V$ a CLF on $\mathcal{D}$ and $h$ a CBF of~$\mathcal{C}$. $V$ and
  $h$ are incompatible at $x\in\mathcal{C}$ if and only if $L_{g}V(x)$ and
  $L_{g}h(x)$ are linearly dependent, $L_{g}V(x)^T L_{g}h(x)>0$ and
  $L_{f}V(x)+W(x)>\frac{L_gV(x)^T L_gh(x)}{\norm{L_gh(x)}^2}(L_fh(x)
  +\alpha(h(x)))$.
\end{lemma}
\begin{proof}
  It is clear that if $L_{g}V(x)$ and $L_{g}h(x)$ are linearly
  independent,~\eqref{eq:clf-ineq} and~\eqref{eq:cbf-ineq} are
  simultaneously feasible for some $u$,
  since hyperplanes with linearly independent normal vectors have
  non-empty intersection and $u$ is unconstrained. Suppose instead
  that $L_{g}V(x)=\mu(x) L_{g}h(x)$, with
  $\mu(x)=\frac{L_gV(x)^T L_gh(x)}{\norm{L_gh(x)}^2}$. If
  $L_{g}V(x)^T L_{g}h(x)<0$, by multiplying~\eqref{eq:cbf-ineq} by
  $\mu(x)$ we obtain $L_{g}V(x) u\leq-\mu(x)(L_fh(x)+\alpha(h(x)))$,
  and~\eqref{eq:clf-ineq} yields $L_gV(x)u\leq-L_{f}V(x)-W(x)$. Since
  $u$ is unconstrained, we can find a $u$ that satisfies both
  inequalities simultaneously. Instead, if $L_{g}V(x)^T L_{g}h(x)>0$,
  the two inequalities read
  $L_{g}V(x) u\geq-\mu(x)(L_fh(x)+\alpha(h(x)))$,
  $L_gV(x)u\leq-L_{f}V(x)-W(x)$, which can not be simultaneously
  solved if and only if
  $-L_{f}V(x)-W(x)<-\mu(x)(L_fh(x)+\alpha(h(x)))$.
\end{proof}

The next result shows that, under certain technical assumptions, by
taking $\epsilon$ sufficiently small, any level set of $V$ that does
not contain points incompatible points is a region of attraction of a
small neighborhood of the origin for the closed-loop system.

\begin{theorem}\longthmtitle{Parameter tuning for guaranteed region of
    attraction}\label{thm:epsilon-roa}
  Let $\mathcal{D} \subseteq \mathbb{R}^{n}$ be open, $\mathcal{C} \subset \mathcal{D}$
  closed, $V$ a CLF on $\mathcal{D}$ and $h$ a CBF of~$\mathcal{C}$. Let
  $\nu>0$ be such that the sublevel set
  $\Gamma_{\nu}=\setdef{x\in \mathbb{R}^n}{V(x)\leq\nu}$ does not contain
  any incompatible points. Define functions $B$ and $C$ by

  {\relsize{-0.95}
    \begin{align*}
      B(x)&:=L_fV(x)\!+\!W(x)\!-\!\frac{L_fh(x)+\alpha(h(x))}{\norm{L_gh(x)}^2}L_gV(x)^T
            L_gh(x),
      \\
      C(x)&:=\frac{(L_gV(x)^T
            L_gh(x))^2}{\norm{L_gh(x)}^2}-\norm{L_gV(x)}^2.
    \end{align*}
  } Assume $\setdef{x\in\Gamma_\nu\backslash\{0\}}{B(x)=0, \ C(x)=0}=\emptyset$.
  Let $\mathcal{V}\subseteq\Gamma_{\nu}$ be a neighborhood of the
  origin and $\mathcal{W}$ a neighborhood of
  $P_{\nu}=\setdef{x\in\Gamma_\nu}{C(x)=0}$ such that $B(x)<0$ for all
  $x\in\mathcal{W}\backslash\{0\}$. Define constants $M_{1}^{\nu}$, $M_{2}^{\nu}$,
  $M_{3}^{\nu,\mathcal{V}}$ and $M_{4}^{\nu,\mathcal{V},\mathcal{W}}$
  by
  \begin{align*}
    M_{1}^{\nu}&:=\sup_{x\in\Gamma_\nu}|L_fV(x)+W(x)|,
    \\
    M_{2}^{\nu}
               &:=\sup_{x\in\Gamma_\nu}|\frac{L_fh(x)+\alpha(h(x))}{\norm{L_gh(x)}^2}L_gh(x)^T
                 L_gV(x)|,
    \\
    M_{3}^{\nu,\mathcal{V}}
    &:=\inf_{x\in\Gamma_\nu\backslash\mathcal{V}}\norm{L_gV(x)}^2,
      \\
    M_{4}^{\nu,\mathcal{V},\mathcal{W}}
    &:=\inf_{x\in\Gamma_\nu\backslash(\mathcal{W}\cup\mathcal{V})}
      |C(x)|.
  \end{align*}
  Then, for
  $\epsilon<\bar{\epsilon}:=
  \min\{\frac{M_{4}^{\nu,\mathcal{V},\mathcal{W}}}{M_1^\nu+M_{2}^\nu}
  , \frac{M_{3}^{\nu,\mathcal{V}}}{M_{1}^\nu}\}$, $\mathcal{V}$ is
  asymptotically stable under the closed-loop dynamics and
  $\Gamma_{\nu}\cap \mathcal{C}$ is forward invariant and a subset of the
  region of attraction of~$\mathcal{V}$.
\end{theorem}
\begin{proof}
  Let $z_{\epsilon}(x):=L_fV(x)+L_gV(x)u_{\epsilon}^{\operatorname{safe}}(x)+W(x)$.
  It follows from~\eqref{eq:usafe} that
  \begin{align*}
    z_{\epsilon}(x)=\begin{cases}
      L_fV(x)+W(x)-\frac{1}{\epsilon}\norm{L_gV(x)}^2 \quad \text{if} \ e(x)\leq0,& \\
      B(x)+\frac{1}{\epsilon}C(x) \qquad \qquad \qquad \qquad \quad \ \ \text{else},
    \end{cases}
  \end{align*}
  We show that $z_{\epsilon}(x)\leq0$ for all
  $x\in\mathcal{C}\backslash\mathcal{V}$ if $\epsilon<\bar{\epsilon}$,
  from which the result follows.  First of all, note that
  $\mathcal{W}$ as required in the statement exists because
  $\Gamma_{\nu}$ does not contain any point where $V$ and $h$ are
  incompatible and therefore by Lemma~\ref{lem:incompat-points}, all
  points in $\Gamma_{\nu}$ satisfying $C(x)=0$ necessarily also
  satisfy $B(x)<0$ ($B(x)=0$ can not hold since
  $\setdef{x\in\Gamma_\nu\backslash\{0\}}{B(x)=0, \ C(x)=0}=\emptyset$ by assumption)
  and hence by continuity of $z_{\epsilon}(x)$ we can take a
  neighborhood $\mathcal{W}$ around them for which $B(x)<0$ for all
  $x\in\mathcal{W}$. As a consequence, $B(x)+\frac{1}{\epsilon}C(x)<0$
  everywhere in $\mathcal{W}$ for any $\epsilon$ (since by
  Cauchy-Schwartz's inequality, $C(x)\leq0$), and by taking
  $\epsilon<\bar{\epsilon}$ we ensure that $z_{\epsilon}(x)\leq0$ for
  all $x\in\mathcal{W}$.

  Now we argue that if $\epsilon<\bar{\epsilon}$, $z_{\epsilon}(x)<0$
  for all
  $x\in\Gamma_{\nu}\backslash(\mathcal{W}\cup\mathcal{V})$. Note that
  $\Gamma_{\nu}\backslash(\mathcal{W}\cup\mathcal{V})$ does not
  contain any points where $L_{g}V(x)$ and $L_{g}W(x)$ are linearly
  dependent, since that would imply $C(x)=0$ and hence
  $x\in\mathcal{W}$. Thus, by Cauchy-Schwartz's inequality, $C(x)<0$
  for all
  $x\in\Gamma_{\nu}\backslash(\mathcal{W}\cup\mathcal{V})$. Hence,
  $M_{4}^{\nu,\mathcal{V},\mathcal{W}}>0$.  Note also that since
  $L_{g}V(x)=0$ if and only if $x=0$,
  $M_3^{\nu,\mathcal{V}}>0$. Therefore, regardless of whether
  $e(x)\leq0$ or $e(x)>0$, by taking $\epsilon<\bar{\epsilon}$ we
  ensure that $z_{\epsilon}(x)\leq0$ for all
  $x\in\Gamma_{\nu}\backslash(\mathcal{V}\cup\mathcal{W})$, as
  claimed.  Moreover, since by construction $u_{\epsilon}^{\operatorname{safe}}$
  satisfies~\eqref{eq:cbf-ineq} and is Lipschitz, by~\cite[Theorem
  2]{ADA-SC-ME-GN-KS-PT:19}, trajectories stay inside $\mathcal{C}$
  for all $t\geq0$.
\end{proof}

Similar to Proposition~\ref{prop:undes-eq}, since by assumption
$\norm{L_gV(x)}^{2}=0\Leftrightarrow x=0$, we can pick $\mathcal{V}$
arbitrarily small, although this might require $\epsilon$ to be
arbitrarily small. The next result states that, under some
additional assumptions, $\epsilon$ does not converge to zero as
$\mathcal{V}$ is taken arbitrarily small, and hence there exists a
finite $\epsilon$ for which trajectories converge to the origin.

\begin{corollary}\longthmtitle{Convergence to the
    origin}\label{cor:conv-origin}
  Under the same assumptions and notation of
  Theorem~\ref{thm:epsilon-roa}, assume additionally that
  \begin{gather}\label{eq:small-control}
    \lim_{x\to0}\frac{\norm{L_gV(x)}^2}{|L_fV(x)\!+\!W(x)|}\! =\! l_1>0,
    \; \lim_{x\to0}\frac{|C(x)|}{|B(x)|} \!=\! l_2>0 .
  \end{gather}
  Then, there exists $\tilde{\epsilon}>0$ and a choice of
  $\mathcal{V}$ such that $V$ is decreasing also on $\mathcal{V}$
  under the closed-loop dynamics.  Moreover, for
  $\epsilon<\hat{\epsilon}:=
  \min\{\frac{M_{4}^{\nu,\mathcal{V},\mathcal{W}}}{M_1^\nu+M_{2}^\nu},
  \frac{M_{3}^{\nu,\mathcal{V}}}{M_{1}^\nu}, \tilde{\epsilon}\}$, the
  origin is asymptotically stable and the set $\Gamma_{\nu}\cap C$ is
  forward invariant and is a region of attraction of the origin.
\end{corollary}
\begin{proof}
  Note that because of the limit conditions~\eqref{eq:small-control},
  there exists a neighborhood $\bar{\mathcal{V}}$ of the origin and a
  constant $L>0$ such that $\norm{L_gV(x)}^2 \geq L|L_fV(x)+W(x)|$,
  $|C(x)|\geq L|B(x)|$ for all $x\in\bar{\mathcal{V}}$. Let us pick
  $\mathcal{V}$ from Theorem~\ref{thm:epsilon-roa} to be
  $\bar{\mathcal{V}}$. Now by taking $\tilde{\epsilon}<L$ we can
  achieve $z_{\tilde{\epsilon}}(x)\leq0$ for all $x\in\mathcal{V}$ and
  hence $V$ is decreasing on $\mathcal{V}$. Now, following the proof
  of Theorem~\ref{thm:epsilon-roa}, one can show that $V$ is
  decreasing in $\Gamma_{\nu}\backslash\mathcal{V}$, and the result
  follows.
\end{proof}

Note that by ensuring that the origin is asymptotically stable in
$\Gamma_{\nu}$ we rule out the existence of equilibrium points in
$\Gamma_{\nu}$ other than the origin.  Theorem~\ref{thm:epsilon-roa}
and Corollary~\ref{cor:conv-origin} solve
Problem~\ref{pb:problem-1}. Under the stated assumptions, by taking
$u_{\epsilon}^{\operatorname{safe}}$ with $\epsilon<\hat{\epsilon}$ as a safe
stabilizing controller, an inner approximation of the region of
attraction of the origin is the largest level set of $V$ that does not
contain any incompatible points inside it.
Theorem~\ref{thm:epsilon-roa} and Corollary~\ref{cor:conv-origin} also
provide a solution to Problem~\ref{pb:problem-2}. Namely, if there
exists a sublevel set of $V$ that contains $\mathcal{C}$,
$u_{\epsilon}^{\operatorname{safe}}$ with $\epsilon<\hat{\epsilon}$ safely
stabilizes the origin and $\mathcal{C}$ is in its region of
attraction.

\section{Simulations}\label{sec:sims}
In this section, we compare the stability QP with safety penalty
controller with the CLF-CBF QP~\eqref{eq:clf-cbf-qp} and its
modification, M-CLF-CBF QP, introduced in~\cite[Theorem~3]{XT-DVD:21}
to avoid undesired equilibria. We focus on the following simple
two-dimensional system
\begin{align}\label{eq:example}
  \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix}
  =\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} +
  \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}u.
\end{align}
For this system,
$V(x_1,x_2) = \frac{1}{2}x_{1}^{2}+\frac{1}{2}x_{2}^{2}$ is a CLF with
$W(x_1,x_2) = x_{1}^{2}+x_{2}^{2}$ (this can be checked by taking
$u(x_1,x_2)=(-2x_1,-2x_2)$).  The safe set $\mathcal{C}$ is the complement of
the ball $\setdef{x\in\mathbb{R}^2}{\norm{x-(0,4)}\leq 2}$, and we use the
CBF $h(x_1,x_2)=x_{1}^{2}+(x_2-4)^{2}-4$, with $\alpha(s)=s$.
According to~\cite{XT-DVD:21}, the CLF-CBF QP~\eqref{eq:clf-cbf-qp}
creates undesired equilibria in $\text{int}(\mathcal{C})$ for all
values of $p$, although their norm is bounded by the inverse of
$p$. Instead, both M-CLF-CBF QP and the stability QP with safety
penalty controller $u_{\epsilon}^{\operatorname{safe}}$, with $\epsilon\neq1$, do
not introduce undesired equilibria in $\text{int}(\mathcal{C})$.  The
latter can be checked from the definition of
$\mathcal{Q}_{1}^{\epsilon}$ given in Proposition~\ref{prop:undes-eq}.
In this example, the incompatible points are given by
$\setdef{(x_1,x_2)\in\mathbb{R}^2}{x_1=0,x_2>4}$. With the notation of
Theorem~\ref{thm:epsilon-roa}, we also have
$\setdef{x\in\mathbb{R}^2\backslash\{0\}}{B(x)=0,C(x)=0}=\partial\mathcal{C}$. Therefore, the best
inner approximation of the region of attraction given by
Theorem~\ref{thm:epsilon-roa} is
$\Gamma_{2} = \setdef{(x_1,x_2)\in\mathbb{R}^2}{x_1^2+x_2^2<4}$.  Moreover,
condition~\eqref{eq:small-control} holds and hence by taking
$\epsilon<0.032<\hat{\epsilon}$ as given in Corollary~\ref{cor:conv-origin},
the origin is asymptotically stable with region of attraction given by
$\Gamma_{2}$.  Figure~\ref{fig:example} shows the closed-loop
trajectories of the three approaches for $11$ different initial
conditions.  The stability QP with safety penalty controller and
M-CLF-CBF QP behave similarly, whereas CLF-CBF
QP~\eqref{eq:clf-cbf-qp} fails to stabilize the origin. The plot also
illustrates that trajectories starting at $(0,9)$ converge to the
boundary equilibrium point at $(0,6)$ for all three approaches (this
corresponds to a point where $f, gL_{g}V$, and $gL_{g}h$ are
collinear, cf. Remark~\ref{rem:bdy-eq}).  The main advantage of the
approach proposed here is the explicit inner approximation of the
region of attraction which, as Figure~\ref{fig:example} shows, is
conservative.

\begin{figure}[htb]
  \centering
  %
  \includegraphics[scale=0.21]{figures/comp2-fi.jpg}
  \caption{Safe stabilization of a planar system.  The green ball is
    the set of unsafe states and the small dots display ten initial
    conditions for the system trajectories under the CLF-CBF QP, the
    M-CLF-CBF QP, and the safety QP with stability penalty
    controllers. The orange dotted curve marks the boundary of the
    estimate $\Gamma_{2}$ of the region of attraction. The CLF-CBF QP
    controller (with $p=1$) preserves safety but does not reach the
    origin because of undesired equilibrium points.  The safety QP
    with stability penalty (with $\epsilon=0.01$) and the M-CLF-CBF QP
    (with $p=1$) preserve safety and have trajectories converge to the
    origin, except for the one starting at $(0,9)$, which gets stuck
    at a boundary equilibria for all three
    designs.}\label{fig:example}
  \vspace*{-1ex}
\end{figure}

\section{Conclusions}\label{sec:conclusions}
We have addressed the problem of safe stabilization of nonlinear
affine control systems by proposing an optimization-based feedback
design framework inspired by penalty methods for constrained
optimization. Our design enforces strictly either stability or safety
via a hard constraint while promoting the satisfaction of the other
property via a soft constraint.  We have characterized the equilibria
of the closed-loop system under the proposed controllers.  Under
suitable assumptions on the dynamics and the safe set, we have shown
how to tune the penalty parameter to eliminate spurious equilibria and
to increase the region of attraction to all Lyapunov level sets that
do not include points where the CLF and the CBF are not
compatible. Future work will develop tighter estimates of the region
of attraction by designing a controller that makes the closed-loop
system avoid incompatible points and explore the extension of the
proposed framework to input constraints, partial knowledge of the
dynamics, multiple CBFs, and higher-order CBFs.

redpajama_set_name: RedPajamaArXiv


Another great review came in today for my novel The Soul Bearers by Maria Snell, blogger and book reviewer. Her blog is called "Flair".
of rejection and abuse, hope resulting from true unconditional love. Hope that heals, hope that forgives, hope that is the way to rebirth. Love as the only important thing worth living for. Alex will find love - both spiritual and physical - through the love shared by Steve and Matthew, a love that will grow to embrace her too and help her exorcise the demons of her very abusive childhood. And healing will come on the wings of a butterfly.
This book is a definite MUST READ. No holds barred, no caveats. This book will show you pure, unadulterated love and hope through tears and loss.
Sylvia Says: Thank you, Maria, for your candid review of my novel. By the way, this review was not solicited. Maria asked for a copy of the book to review, and knowing she's an excellent reviewer, I sent her a copy hoping she would like it. Well, she did, and she's made my day! Thank you, thank you, thank you!


text:
Pakistan And 9/11
Jon Gold Mon, 08/21/2006 - 6:28pm
I thought I would put a collection together of all of the Pakistan - 9/11 articles I have. As well as articles related to our relationship with Pakistan.
This is in no particular order...
The 9/11 Commission
Did Pakistan Influence The 9/11 Commission Report?
Members Of The 9/11 Commission Bribed?
Pakistan Bribed 9/11 Commission
Thousands Of Dollars Spent By Pakistan To Get 9/11 Findings Dropped
Pakistan Military Brass Knew In Advance That 9/11 Would Take Place
Pakistan Embassy Official's Claims Triggers U.S. Media Interest
DIA Says Pakistan Backed Al Qaeda
Pakistan Blames West For Terrorism
Former ISI Chief Says 9/11 An Inside Job (Interview on 9/26/01)
Alex Jones' Team Confronts 9/11 Commission Chairman Thomas Kean - Video Inside
Pakistani Defense Minister: We Have No Control Over ISI
What Comes First, The Chicken Or The Egg?
New Offer On Bin Laden
India Should Ask For International Monitoring Of ISI: Rajnath
"Pakistan Did Not Influence 9/11 Probe"
Ex-State Department Security Officer Charges Pre-9/11 Cover-Up
HBO Documentary Traces Daniel Pearl Case
Some Correspondence I Had With Michael Meacher, Former Member Of Parliament
Letter To Mariane Pearl (Daniel Pearl's Wife)
This War On Terrorism Is Bogus
The Real Culprit Of 9/11?
India Wants Terror Spotlight On Kashmir
FBI Seeks Pearl Video Ban On Net
Bin Laden Denies Being Behind Attacks
FBI Penetrated; Again!
White House Sells F-16s To 9/11 Co-Conspirator Pakistan
US 'to offer F-16s to Pakistan'
U.S. Approves Sale of Missiles to Pakistan
FBI Weaving Web In Pakistan To Launch Attack On Iran
Pakistan Tests Nuclear Capable Missile
U.S. Hopes To Win Hearts In Pakistan
U.S. Gives Pakistan 8 Military Aircraft Free Of Charge
Suspect In Pearl Killing Arrested In Pakistan
Militant Sentenced To Death In Pearl Case
Iranian Nuclear Chief Admits Ties To Pakistan
Pakistan Admits Khan Gave Iran Nuke Material
9/11 CitizensWatch Co-Founder Confronts 9/11 Commissioner - Video Inside
Did Pearl Die Because Pakistan Deceived CIA?
President Musharraf: West Responsible For "Breeding Terrorism In His Country"
Cover-Up Or Complicity Of The Bush Administration?
Terrorism Defendant: Pakistani ISI Threatening Family
Musharraf: U.S. Threatened To Bomb Pakistan After 9/11
Musharraf Suggests 9/11 Paymaster Was British Agent
Ex-Pakistan Spy Chief Knew Of 9/11: Report
Musharraf Defends Pakistani Intel Agency
Blair, Musharraf Pledge to Fight Terror
Key quotes from the document
Musharraf defends his spy service
President dubs alleged Pearl killer MI6 spy
Pakistan 'role in Mumbai attacks'
ISI, Lashkar behind 7/11 blasts: Mumbai police
US asks India to stop blaming Pakistan
India to give names of 7/11 suspects to Pak
Pakistan And The Terror Nexus
Pakistan In Review
Bush Urges Allies To Unite In Terror War
Explosion near Musharraf residence, no casualties or damage
Britain says Pakistan is hiding Taliban chief
Who is planning to terrorize Pakistan's ISI now?
Nato's top brass accuse Pakistan over Taliban aid
NATO to confront General over ISI support to Taliban
Police did not want to 'jump the gun'on train blast probe
Unlikely suspect
"Clearly, the ISI link is no mere conspiracy theory." — Lorie Van Auken
Jon Gold's blog
That's a really long list.
{{jaw drops}}
Reprehensor on Mon, 08/21/2006 - 8:30pm.
It's one of the things that caught my attention from the get-go...
Jon Gold on Mon, 08/21/2006 - 8:33pm.
Oooh Oooh Mista Kotter!!!
Why is it ok for Pakistan, a proven co-conspirator in the attacks of 9/11, to have nuclear weapons, but not Iran, who is only alleged to have ties to 9/11?
A State Within A State
You may have heard a lot of talk from Bush recently regarding Hezbollah being a "State Within A State" in Lebanon. Why is it ok for the Pakistani ISI to be a "State Within A State" in Pakistan, and not ok for Hezbollah and Lebanon?
So many double standards in the Middle East.
Can Pakistan be the country with a "get out of jail free" card for GIVING nuclear material to Iran, but Iran, who is 5-10 years away from having a nuclear weapon, is PROHIBITED from using it for energy?
Where are the trolls when it
Where are the trolls when it comes to this stuff?
Anonymous (not verified) on Mon, 08/21/2006 - 9:14pm.
You know...
You make an EXCELLENT point.
As a matter of fact...
Do you see the word Pakistan anywhere on these sites?
Picture One
Picture Two
Picture Three
How can they say...
They are for the truth regarding an event that Osama Bin Laden, according to them, was responsible for, and not even mention Pakistan?
Even MSNBC mentions Pakistan's relationship to Osama bin Laden, and the CIA.
MSNBC's version of the story is down it seems. Here is the archive of it on my site.
911Myths.com
Has taken on the Pakistani ISI connection.
Tell me how they well they did. :)
http://911myths.com/html/pakistan_s_isi_link_to_9_11_fu.html
"It was all about finding a way to do it. That was the tone of it. The president saying 'Go find me a way to do this."
Jon Gold on Thu, 09/21/2006 - 6:05pm.
Fantastic work, Jon!
Jon, that list (+ links) is amazing!!! You really should write a book or at least a booklet or adobe document of this research! I'm very serious, we need to get your material out into the mainstream!
Anonymous (not verified) on Tue, 08/22/2006 - 10:04pm.
A lot of it
Is in 9/11 Press For Truth. Thanks.
Jon Gold on Wed, 08/23/2006 - 8:19am.
To pockybot for the latest entry.
"An error does not become truth by reason of multiplied propagation, nor does truth become error because nobody sees it." - Gandhi
A new...
Addition... interesting for Musharraf to say this now, eh?
Jon Gold on Wed, 09/13/2006 - 8:40pm.
Kasuri spoke of the close cooperation between intelligence officials in the United States and Pakistan, saying it was "inconceivable" that Pakistan would know about the presence of al-Qaida and Taliban leaders in Pakistan without U.S. intelligence also knowing.
And it is "inconceivable" that intelligence officials of Pakistan knew that Lt. General Mahmoud Ahmad sent $100,000 to Mohammad Atta "without U.S. intelligence also knowing."
Jon Gold on Sun, 09/17/2006 - 6:33pm.
Thanks Jon, this is a mighty big list.
I wish we had a way to mark our favorite blog entries.
MikeJr
MikeJr on Thu, 09/21/2006 - 8:46am.
No problem...
After Ian's contribution yesterday, this thread should start receiving a lot of attention.
Jon Gold on Thu, 09/21/2006 - 9:44am.
The Pakistani ISI's connection to 9/11 is THE "Smoking Gun" for the 9/11 Truth Movement.
Jon Gold on Thu, 09/21/2006 - 12:31pm.
I just added a link...
And I'm going to need some old-timers for this question...
Supposedly Richard Armitage threatened to bomb Pakistan after 9/11...
Since we suspect that the United States, in partnership with the ISI, pulled off 9/11, it doesn't make sense that they would threaten to bomb them afterwards.
However... since it's been proven that the Pakistani ISI was involved with 9/11, and we DIDN'T bomb them, and instead bombed Afghanistan and Iraq, it makes sense that either Musharraf is lying about the threat, or it was given to them for "show."
My question is... I remember hearing a long time ago that Lt. General Mahmoud Ahmad was told by someone, and I don't remember who, that at the time of 9/11, "You're either with us or against us..." or something like that...
Does anyone remember anything like that?
Never mind...
Good ole cooperativeresearch.org...
September 11-16, 2001: Pakistan Threatened; Promises to Support US
ISI Director Lt. Gen. Mahmood Ahmed, extending his Washington visit because of the 9/11 attacks [Japan Economic Newswire, 9/17/2001] , meets with US officials and negotiates Pakistan's cooperation with the US against al-Qaeda. It is rumored that later in the day of 9/11 and again the next day, Deputy Secretary of State Richard Armitage visits Mahmood and offers him the choice: "Help us and breathe in the 21st century along with the international community or be prepared to live in the Stone Age." [Deutsche Presse-Agentur (Hamburg), 9/12/2001; LA Weekly, 11/9/2001] Secretary of State Powell presents Mahmood seven demands as an ultimatum and Pakistan supposedly agrees to all seven. [Washington Post, 1/29/2002] Mahmood also has meetings with Senator Joseph Biden (D), Chairman of the Senate Foreign Relations Committee, and Secretary of State Powell, regarding Pakistan's position. [Miami Herald, 9/16/2001; New York Times, 9/13/2001 ; Reuters, 9/13/2001; Associated Press, 9/13/2001] On September 13, the airport in Islamabad, the capital of Pakistan, is shut down for the day. A government official later says the airport had been closed because of threats made against Pakistan's "strategic assets," but does not elaborate. The next day, Pakistan declares "unstinting" support for the US, and the airport is reopened. It is later suggested that Israel and India threatened to attack Pakistan and take control of its nuclear weapons if Pakistan did not side with the US. [LA Weekly, 11/9/2001] It is later reported that Mahmood's presence in Washington was a lucky blessing; one Western diplomat saying it "must have helped in a crisis situation when the US was clearly very, very angry." [Financial Times, 9/18/2001]
Why no link (clickable)?
Little Pipe on Tue, 11/14/2006 - 9:55pm.
Click...
Jon Gold on Tue, 11/14/2006 - 9:59pm.
The Pakistani ISI Connection And 9/11 Truth...
Is Easy Money...
Sander Hicks sums it up
Sander Hicks sums it up nicely:
"So, it's ironic that this same State Department denounces 9/11 Truth as "conspiracy theories." This same State Department is packed to the gills with the top criminal minds of Iran/Contra. This same State Department's Francis X. Taylor, in July 2001, told an informant from the Joint Terrorism Task Force, Randy Glass, "we know about the threat, the terrorist threat, from Al Qaeda and bin Laden flying air planes into the World Trade Center. Musharraf [the Pakistani president] has guaranteed us—because his ISI behind it—that he can stop it if we support him publicly."
pockybot (not verified) on Thu, 09/21/2006 - 6:10pm.
I added...
Two new articles today. Thank Ian C. for the last one.
Here's a summary...
In the last few weeks, some REALLY interesting information has surfaced regarding the Pakistani ISI's relationship to "terrorism", and Western Intelligence's relationship to the ISI.
I wrote a really big article about this, but I lost it all when dz was paid by the CIA to move the site to a new server (that's a joke).
Anyway, I'm just going to put the important quotes out there for people to see...
President Musharraf of Pakistan said the West was responsible for "breeding terrorism in his country by bringing in thousands of mujahideen to fight the Soviet Union in Afghanistan and then leaving Pakistan alone a decade later to face the armed warriors."
Musharraf also restated the supposed threat he received from Richard Armitage. "Help us and breathe in the 21st century along with the international community or be prepared to live in the Stone Age."
I say restated because it was first reported by [Deutsche Presse-Agentur (Hamburg), 9/12/2001; LA Weekly, 11/9/2001].
Richard Armitage denied ever saying that, "but did not deny the message was a strong one."
Bush "denied knowledge of a phone conversation in which the US threatened to attack Pakistan."
A leaked document from Britain's Ministry of Defense, "accuses Pakistan's intelligence agency of indirectly supporting terrorist groups including al-Qaida and calls on Pakistani President Gen. Pervez Musharraf to disband the agency."
It also specifically accused the ISI of being involved in the 7/7 bombings.
"Indirectly, Pakistan (through the ISI) has been supporting terrorism and extremism _ whether in London on (July 7, 2005) or in Afghanistan or Iraq."
Mumbai (Bombay) police commissioner AN Roy said today, "We have solved the 11 July bombings case. The whole attack was planned by Pakistan's ISI and carried out by Lashkar-e-Toiba and their operatives in India."
So the ISI was allegedly involved in both the 7/7 bombings, and the 7/11 bombings as well.
Musharraf said "I totally, 200% reject it. I reject it from anybody - MoD or anyone who tells me to dismantle ISI."
There was a lot of "hoo-ha" last week, "when a defendant accused Pakistan's intelligence service of threatening his relatives in the South Asian nation after he testified that the spy agency played a role in training Islamic militants."
"We are in uncharted territory here," said Sajjan Gohel of the Asia-Pacific Foundation, a counter-terrorism think tank here. "It hasn't happened before. The ISI has always been a murky organization. There have always been suspicions of a nexus between them, terror groups in Pakistan and Al Qaeda. But here you have a guy from the UK giving testimony that is very relevant because the ISI is supposed to be the key ally in the hunt for [Osama] bin Laden. And it's concerning that a country that is supposed to be an ally in the war on terror is intimidating witnesses, almost 'Godfather'-style."
Paul Thompson reported, "It is believed in some quarters that while Omar Sheikh was at the LSE he was recruited by the British intelligence agency MI6. It is said that MI6 persuaded him to take an active part in demonstrations against Serbian aggression in Bosnia and even sent him to Kosovo to join the jihad. At some point he probably became a rogue or double agent."
"There are many in (Pakistani President) Musharraf's government who believe that Saeed Sheikh's power comes not from the ISI, but from his connections with our own CIA. The theory is that ... Saeed Sheikh was bought and paid for."
Nafeez Ahmed wrote a huge article on the "continuity" of Western/Al-Qaeda relations in the Post-Cold War period.
Lt. General Mahmoud Ahmad, is "accused of authorising British-born Pakistani terrorist Ahmed Omar Saeed Sheikh to make a $100,000 transfer to Mohamed Atta, the operational chief of the September 11 conspiracy, a charge that met vehement denials."
Wanna bet the ISI is 3 for 3?
Think it's time we asked why the financier of the 9/11 attacks, the head of the ISI, was meeting with Principals in Washington D.C. the week prior to and on the day of 9/11?
And why George W. Bush still refers to Pakistan, in light of all of this information, as "Allies" in the "War On Terror?"
Jon Gold on Sat, 09/30/2006 - 5:57pm.
There are a few new articles, including one that says the U.S. has asked India to stop blaming Pakistan for the Mumbai attacks... however Pakistan is going to hand over the evidence that they have.
A few new stories...
Apparently there was an explosion near Musharraf's residence... there was a similar explosive found near the ISI's headquarters... Britain is accusing the ISI of training the Taliban fighters... And there are a few more articles about the Mumbai bombings...
Staff Statements...
http://www.9-11commission.gov/staff_statements/911_TerrFin_Monograph.pdf
"Senior al Qaeda detainee Abu Zubaydeh has commented on the source of the funding; he said that KSM received funds for the 9/11 operation directly from UBL, bypassing al Qaeda Finance Chief, Shayk Said, and suggested that some of the funds came from money that Zubaydeh had provided UBL for use in an operation against Israel. Zubaydeh, however, apparently did not participate in the 9/11 planning, and his statements lack any foundation."
It seems to me that Shayk Said is a different spelling for Omar Sayeed Sheikh. I could be wrong. However, I can't find any mention of Shayk Said being the "Finance Chief" for "Al-Qaeda." Other than from the 9/11 Report.
http://www.cooperativeresearch.org/essay.jsp?article=essaysaeed
"The story is too complicated to greatly detail here, but the FBI and media have variously filled Saeed Sheikh's shoes with an Egyptian named Shaykh Saiid [Sydney Morning Herald, 9/28/01, New York Times, 10/15/01, Los Angeles Times, 10/20/01]"
I'm not suggesting from this that Pakistan is the quote unquote "solution" to 9/11 because it's not. Pakistan is one part in a very complicated story. The question to me is who else was involved with Al-Qaeda? Was Al-Qaeda used as a tool just as in the 1980's the Mujahadeen were basically used by the U.S. Government? - 9/11: Press For Truth
Sounds rather "MIHOPPISH" to me.
According to Dick...
Pakistan is still "great allies" of ours.
"We've been offered a unique opportunity and we must not let this moment pass."
— George W. Bush - State Of The Union Address - January 29th, 2002

redpajama_set_name: RedPajamaCommonCrawl


Your telecommunications equipment needs require dependable and knowledgeable support. Sometimes this involves the headache of many resources no matter how small or large your project. Walker and Associates provides customers a one stop resource for the best in telecommunications equipment and integration services.
Your telecommunications equipment needs are unique. You need someone who understands your needs and who can provide you workable solutions. Walker and Associates empowers you to do this and more with knowledgeable engineering support.
Expect to get the equipment your project requires and at a competitive price? Now you can be sure of it. Walker and Associates' multi-vendor resources and bulk buying power get you exactly what you need within your budget parameters.
Walker and Associates' highly skilled technicians thrive on perfectionism to ensure your equipment gets the best in assembly, testing, and packaging available in the business.
Walker works to ensure that each configuration is integrated, manufactured, and tested exactly like the one before.
Walker provides high-quality packaging to ensure the product arrives damage free.
Walker Integration, (hereinafter referred to as "Walker") warrants to Buyer that goods bearing the Walker Integration name (hereinafter referred to as the "Goods") will conform to Walker's published specifications in effect as of the date of shipment and will be free from substantial defects in materials and workmanship under normal use, given proper installation and maintenance, for the period listed in Attachment A.
Buyer must promptly notify Walker of any claimed defect in the Goods. If the defect was caused by the freight shipper, the notification must be within 48 hours. Walker or its agent may inspect the Goods on the Buyer's premises. Goods returned to Walker under this warranty must be shipped prepaid by Buyer.
Walker assumes no liability with respect to defects caused by modification, repair, installation, operation, negligence, or other improper use of the Goods.
All equipment not bearing the Walker Integration name is supplied with the Original Equipment Manufacturer's (OEM) warranty. Walker will assign or otherwise make available to Buyer any warranty of any item which has been assigned by an OEM and which Walker has the right to so assign or otherwise make available to Buyer. In addition, Walker has no liability for equipment furnished by Buyer nor does this warranty cover any copy of or update to user manuals for the Goods.
Walker's entire liability and Buyer's exclusive remedy, whether in contract, tort or otherwise, for any claim related to or arising out of the Buyer's purchase or use of Goods, shall be correction of defects by repair, replacement, or credit, at Walker's discretion. Refurbished material may be used to repair or replace the Goods. Walker has no liability with respect to claims relating to or arising from the use of equipment not bearing the Walker Integration name.
No agent, distributor, or representative is authorized to make any warranties on behalf of Walker or to assume for Walker any other liability in connection with any Goods.
THERE ARE NO WARRANTIES FOR ANY PURCHASE OR USE OF WALKER'S GOODS BY BUYER WHICH EXTEND BEYOND THE DESCRIPTION OF THE FACE HEREOF. ALL OTHER WARRANTIES, INCLUDING BUT NOT LIMITED TO ANY EXPRESS AND IMPLIED WARRANTIES AND ANY WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE, ARE DISCLAIMED AND EXCLUDED BY WALKER.
Walker will not be liable for injuries or damage to persons or property resulting from any cause whatsoever, with the exception of injuries or damages caused by the gross negligence of Walker. This limitation applies to all Goods during and after the warranty period.
In no event shall Walker be liable for any damages from loss of use or loss of profits and Walker further disclaims any and all liability for indirect, incidental, special, consequential, or other similar damages.
In the event any remedy fails of its essential purpose, or in any other event, Walker's aggregate liability shall not exceed the amount actually paid to Walker by Buyer for the affected Goods.
Contact us today to learn more about our integration services capabilities.


You can also print your UPS shipping label at textbooksolutions.com by signing into your account and clicking the "My Account" link in the top right corner. Click on "Rental Book Status" to find the rentals that have not been returned. Check the box next to all of the books you need to ship and click "Return Selected Books". This will generate a UPS shipping label for you to print and attach to your package.
If you have any other questions regarding how to return your rental, click here. Rental return instructions can be found here.
If you are unable to get your UPS return label, please contact us and we can send one to your email.


text:
Q: Testcafe aborts with ELIFECYCLE npm error and exit status 4 I know this is the most common issue with Testcafe, but I previously posted questions and solutions doesn't match/work to resolve my issue. Testcafe version 1.8.4, with allure reporting. I have configured few test suites running sequential in my project but, after first suite run Testcafe terminates with the following error and aborts the test run.
npm ERR! code ELIFECYCLE
npm ERR! errno 4
npm ERR! user-journey-tests@1.0.0 test:appjourneys:azure: `TARGET_ENV=test DATA_CENTER=azwus2 testcafe 'chromium:headless:emulation' './journeys/app-journeys/*-spec.js' -r allure`
npm ERR! Exit status 4
npm ERR!
npm ERR! Failed at the user-journey-tests@1.0.0 test:appjourneys:azure script.
npm ERR! This is probably not a problem with npm. There is likely additional logging output above.

npm ERR! A complete log of this run can be found in:
npm ERR!     /Users/Venkat/.npm/_logs/2020-05-27T03_13_43_108Z-debug.log
Below is the debug log
0 info it worked if it ends with ok
1 verbose cli [
1 verbose cli   '/usr/local/Cellar/node/12.10.0/bin/node',
1 verbose cli   '/usr/local/bin/npm',
1 verbose cli   'run',
1 verbose cli   'test:appjourneys:azure'
1 verbose cli ]
2 info using npm@6.11.3
3 info using node@v12.10.0
4 verbose run-script [
4 verbose run-script   'pretest:appjourneys:azure',
4 verbose run-script   'test:appjourneys:azure',
4 verbose run-script   'posttest:appjourneys:azure'
4 verbose run-script ]
5 info lifecycle user-journey-tests@1.0.0~pretest:appjourneys:azure: user-journey-tests@1.0.0
6 info lifecycle user-journey-tests@1.0.0~test:appjourneys:azure: user-journey-tests@1.0.0
7 verbose lifecycle user-journey-tests@1.0.0~test:appjourneys:azure: unsafe-perm in lifecycle true
8 verbose lifecycle user-journey-tests@1.0.0~test:appjourneys:azure: PATH: /usr/local/lib/node_modules/npm/node_modules/npm-lifecycle/node-gyp-bin:/Users/Venkat/GAP/store-service/user-journeys-testcafe/node_modules/.bin:/usr/local/Cellar/maven/3.6.0/bin:/Library/Java/JavaVirtualMachines/jdk1.8.0_131.jdk/Contents/Home/bin:/usr/local/bin:/usr/bin:/bin:/usr/sbin:/sbin:/usr/local/Cellar/maven/3.6.0/bin:/Library/Java/JavaVirtualMachines/jdk1.8.0_131.jdk/Contents/Home/bin
9 verbose lifecycle user-journey-tests@1.0.0~test:appjourneys:azure: CWD: /Users/Venkat/GAP/store-service/user-journeys-testcafe
10 silly lifecycle user-journey-tests@1.0.0~test:appjourneys:azure: Args: [
10 silly lifecycle   '-c',
10 silly lifecycle   "TARGET_ENV=test DATA_CENTER=azwus2 testcafe 'chromium:headless:emulation' './journeys/app-journeys/*-spec.js' -r allure"
10 silly lifecycle ]
11 silly lifecycle user-journey-tests@1.0.0~test:appjourneys:azure: Returned: code: 4  signal: null
12 info lifecycle user-journey-tests@1.0.0~test:appjourneys:azure: Failed to exec test:appjourneys:azure script
13 verbose stack Error: user-journey-tests@1.0.0 test:appjourneys:azure: `TARGET_ENV=test DATA_CENTER=azwus2 testcafe 'chromium:headless:emulation' './journeys/app-journeys/*-spec.js' -r allure`
13 verbose stack Exit status 4
13 verbose stack     at EventEmitter.<anonymous> (/usr/local/lib/node_modules/npm/node_modules/npm-lifecycle/index.js:332:16)
13 verbose stack     at EventEmitter.emit (events.js:209:13)
13 verbose stack     at ChildProcess.<anonymous> (/usr/local/lib/node_modules/npm/node_modules/npm-lifecycle/lib/spawn.js:55:14)
13 verbose stack     at ChildProcess.emit (events.js:209:13)
13 verbose stack     at maybeClose (internal/child_process.js:1021:16)
13 verbose stack     at Process.ChildProcess._handle.onexit (internal/child_process.js:283:5)
14 verbose pkgid user-journey-tests@1.0.0
15 verbose cwd /Users/Venkat/GAP/store-service/user-journeys-testcafe
16 verbose Darwin 17.7.0
17 verbose argv "/usr/local/Cellar/node/12.10.0/bin/node" "/usr/local/bin/npm" "run" "test:appjourneys:azure"
18 verbose node v12.10.0
19 verbose npm  v6.11.3
20 error code ELIFECYCLE
21 error errno 4
22 error user-journey-tests@1.0.0 test:appjourneys:azure: `TARGET_ENV=test DATA_CENTER=azwus2 testcafe 'chromium:headless:emulation' './journeys/app-journeys/*-spec.js' -r allure`
22 error Exit status 4
23 error Failed at the user-journey-tests@1.0.0 test:appjourneys:azure script.
23 error This is probably not a problem with npm. There is likely additional logging output above.
24 verbose exit [ 4, true ]

I tried the following:

*

*npm -clean install

*update testcafe to latest version 1.8.4 3.
I merged my suite in to one single npm run <cmd> - This helps to run the tests but still the error continues to show up at the end and build status is unstable. Any thoughts how to over come this?

A: I see the following error in the NPM log:
Error: user-journey-tests@1.0.0 test:appjourneys:azure: \\`TARGET_ENV=test DATA_CENTER=azwus2 testcafe 'chromium:headless:emulation' './journeys/app-journeys/*-spec.js' -r allure\\`
Exit status 4
     at EventEmitter.<anonymous>

It means that TestCafe finished tests with exit code 4.
TestCafe finishes with a non-zero exit code when some tests fail during your test session. In this particular case, 4 tests failed.
If you want to disable test failure indication via exit codes, you can append || exit 0 to the TestCafe command.

redpajama_set_name: RedPajamaStackExchange


text: Q: Connecting OneDrive data to Azure Databricks I have created an Azure Databricks cluster and would like to connect to a SharePoint folder to read and upload files. I cannot seem to find any solution to this. Please advise.


text:
Campus Headlines
St. Patrick's Day At UD: A Look At What Happened And Why
March 27, 2018 March 27, 2018 fn_admin
A portion of this article was originally placed in the print issue of Flyer News under the title: "University Officials Respond To St. Paddy's Day Celebration." This online edition adds to the in-depth account to the University's response, what occurred and contains video and images (one graphic) to highlight the events of the holiday.
Roberto De La Rosa-Finch
"St. Patrick's Day, around here, seems to have some magical meaning. Which is the celebration of complete intoxication. I'm not going to say St. Patrick's Day. Maybe at some point I will, but today it's too wrong."
In large part due to the incidents of last weekend, University President Eric Spina will continue to refer to St. Patrick's Day as March 17. In this response to a student's question at a public SGA meeting March 18, Spina encapsulated his emotions on the events of the day, as well as, his attitude moving forward.
His avoidance of even naming the centuries old Irish holiday stems from the behavior of UD's student body, specifically on Lowes Street, Saturday afternoon.
>>The Madness Of March: Writer Reflects On March Madness Upsets
>>Opinion: Gun Violence, Separating Truth From "Untruth" Part II
>>ArtStreet Thursday Night Live: Guitarists On Display
Four University students were arrested for a variety of misdemeanor alcohol offenses, public safety officers were hit in the face with full cans of beer, large rocks and other projectiles — causing them to withdraw and call for approximately 50 officers (some in riot gear) to control the crowd, cars were damaged and at least one student was transported to the hospital, according to University officials.
"We had a student. Unconscious," Christine Schramm, University dean of students, said at the SGA meeting.
"We couldn't get to her. It took eight officers, because the officers had to protect each other so that they could go attend to your student. We had to watch that for four hours."
Dayton Officers Retreating Due To Celebration BY UD Students
Submitted video of officers retreating on St. Patrick's Day.
Posted by Flyer News on Tuesday, March 27, 2018
Students forced officers in riot gear to retreat. They stood on top of roofs. They jumped on top of parked and moving cars. And the calamity, or jubilance, did not disperse until 6:30 p.m.
A quiet and desolate Lowes Street gave relief to University officials but also sparked a rush of emotions.
At the meeting, Spina began listing off the names of 12 cities. Each, as of 9 o'clock on March 18 saw footage of the fracas on UD's campus.
"All have in their mind that the University of Dayton is about what happened on March 17… that's really crappy news," he said.
"As we go to recruit students from those places, as we go to recruit faculty and staff, as we spread the good news that is the University of Dayton, that's what we have to overcome: what folks saw yesterday."
What quite a few are wondering, from higher-ups to freshmen is, why did it reach the level that it did? What sparked this? While there is no clear and definitive answer, there are numerous factors that impacted the outcome.
A UD student bleeding after accidentally getting hit in the face with a fist during the uproar on Lowes Street.
The Commitment To Just Talking About Community
As officers headed to an unconscious girl, they told a sea of students that they were on their way to assist someone in need. These students stood their ground, scoffed and said they didn't care, according to Schramm.
Countless students, whether they attended UD or not, were seen throwing beer cans or other objects like, a broken flip phone, a rock or whatever they deemed necessary to fling. What was not apparent were fellow students intervening on their actions.
"What happened to our bystander intervention?," Schramm said. "What happened to our Green Dot? What happened to what we know — what kind of be together and community — I didn't see it…What happened?"
Projectiles were thrown when the streets were open and continued when drivers could only put a toe to the pedal. As students plopped themselves on top of cars — sitting, standing or jumping — they were met with cheers, a beer or the back of iPhones. There was no student mediation, and if there was it wasn't widespread enough.
"Where do students begin to take responsibility for themselves even in their irresponsibility?," Schramm said.
Dusty Chairs And Uneaten Food
"We didn't get any calls yesterday for help. Not one," Bill Fischer, vice president of student development, said at the meeting. "Do something. Call public safety … Some of this is in your control. You have to work with us. We have to work with you. This is your university."
SGA held a Q&A Feb. 14 for students to speak with public safety police chief Rodney Chatman and Spina. If readers see Flyer News's story for the talk, the first thing they'll notice is the picture of the two speakers and empty seats.
Chatman has constantly professed that his door is always open and proactively speaks on campus. There have been numerous events like these where Schramm, Fischer, Chatman, or Spina and others have been available for questions and conversation. Yet, what countless University officials — and the students, too — will tell you is that the attendance is measly.
"No one will come," Schramm said in response to opening up the conversation about St. Patrick's Day to the student body.
Selling Girl Scout cookies on a college campus on St. Patrick's day. GENIUS. @oldrowofficial
A post shared by Barstool Sports (@barstoolsports) on Mar 18, 2018 at 2:01pm PDT
Poking The Passive Aggressive Bear
Public safety officers are strolling through Lowes Street. The students are drinking and listening to music blaring from inside the house. There's a girl scout cashing in on thin mints. Crossing the street isn't a hassle. It's the early afternoon.
Officers are taking pictures with students. Posing. Smiling. Having a quick laugh or a discussion about the aspects of the job.
One officer is standing, observing and gets a full Natural Light can to his heel. The population of the crowd has picked up a bit, but nothing too dangerous. The sidewalks were still quite walkable. He turns around to see the student — not very conspicuously — walking away toward a house with his head down.
The officer stares for a bit, but then continues to walk and mingle.
Incidents like this set the tone for those who partook in the ransacking of the neighborhood and a firm step over the line. That student and the others around him witnessed and understood that the officers were going to let them have fun. And some people's idea of fun can be very different from others.
After that incident, Lowes gradually grew and with the growth came more destructive behavior before the street was packed and rules were no more. Which in turn, caused the removal of officers.
No arrests were made before officers were asked to leave, according to Chatman. But more importantly no arrests were made during the gradual growth of students.
Indeed as Chatman said, when there was plethora of green, "the crowd quickly swelled to a point where those committing criminal acts benefited from the anonymity they had as an unidentifiable member of the crowd."
Nevertheless, when inappropriate actions were acted out before the untamed activity, they were met with no response virtually fueling the fire inside students with no regard for others.
The running of no souls @barstoolflyers
A post shared by 5th Year (@5thyear) on Mar 19, 2018 at 5:59am PDT
"They have orange hair. It's like their holiday today."
Chatman has been at UD for two years. Schramm, for around 20. Both had no idea what the Ginger Run was.
"I found out about the Ginger Run this year," Chatman said.
For those who have not seen, the Ginger Run or A Running Of The Gingers is when UD students with red hair run through the most crowded street that day in extra celebration of the Irish holiday. The participants even made shirts this year.
With a combined total of roughly 22 St. Patrick Day celebrations at UD under their belts, the two were ignorant of an event that has been around for several years.
UD has a history of ruckuses on March 17 and the lack of knowledge around the "traditional" festivities places University officials at a disadvantage. Hence, when Chatman and Schramm spoke to the students who participated before they took off and urged them to walk, the students agreed they would do so, but then went on to cheerfully sprint.
There are other factors, including the potential misuse of neighborhood fellows and the schedule change of the 2019 spring break that extends over St. Patrick's Day that affected this outcome.
But, moving forward, the University is searching for ways to "build bridges where gaps appear," as SGA president Jamie Vieson said in a letter to all students. And while that bridge is under construction, UD will avoid the color green.
"I think you now have an answer about where that spring break is going to fall year after year after year," Spina said. "We're going to avoid that date like a plague. I don't know what it is about Ohio, Dayton, and this university. I don't know of other places that have that kind of 'tradition.' But it's going to be something we're going to move away from."
Photos Taken by Christian Cubacub/Director of Digital Media
← The Madness Of March: Writer Reflects On March Madness Upsets
"Deeply Disappointed": Opinion Editor Responds To President Spina →


Abstract: Since Hyogoken-nanbu earthquake in 1995, many studies about strength and ductility of cantilever type steel bridge piers have been conducted. However, most of them have used constant thickness plates. When horizontal loads act on cantilever type steel bridge piers, bending moments of these piers become maximum at bottom of piers, and they show triangular distribution. Therefore, piers with uniform cross section in direction of height are not efficient. To use tapered plate can improve this inadequacy. In this study, tapered plates are employed at bottom parts of cantilever type steel bridge piers, and seismic resistance performance of these piers are investigated by experimental and analytical method. From the investigation, a good seismic resistance performance in strength and ductility are obtained for steel bridge piers with tapered plates, comparing with constant thickness plates.


Now you've seen the great pictures and are itching to add some of these special perennials to your garden beds–why not consider joining the Society for Siberian Irises. $10.00 for US citizens, $15.00 for people overseas will bring you twice yearly info. on the latest cultivars, and a connection with others who grow these lovely plants!
Its great to finish the Siberian Iris Feature month with what was once considered unobtainable, a Yellow Siberian and this 2009 introduction 'LUCY LOCKET,' shows the considerable advance made recently by Hybridisers in this colour tone.Congratulations to and Well done Jan and Marty.
LUCY LOCKET . Schafer/Sacks (Seedling No. S01-91-10), 26", E-M. Small, sweet, golden yellow bitone flowers with lovely flaring form. Two branches with six buds and a long season of bloom. Foliage remains beautiful late in the Fall. Very vigorous. (((sib x Pleasures of May) x sib to Sarah Tiffney) x ((Star Cluster x Ruffled Velvet) x Isabelle) x (Kenbee x Alter Ego))) X Lady's Chain.
Hybridisers are somewhat shy when it comes to show and tell what is in the future breeding patterns so we are really lucky when Jeff Dunlop gave his consent for me to post these brilliantly bright new seedlings of his. Jeff had this to say "As pointed out earlier 'Another Pretty Face' is not only a lovely and pleasing new tet flower but is also proving to be a parent with interesting kids too. 'Another Pretty Face' really has a lot of red in it from both parents: a nice red Dunlop seedling, 96102-3T X Strawberry Fair (Hollingworth '94). Here are just a couple of seedlings out of 'Another Pretty Face' which appear to be especially exciting. Both of these I think of as red/blues as a quick look indicates".
Above is the stunning 02527-8T, a cross of a converted lavender McEwen seedling, S89/108F X Another Pretty Face. The aqua style arms with raspberry rims are especially prominent and increase the wow factor of this seedling. Nice red standards and very wide blended red/blue falls with small white signals are a special plus too. This one has been lined out and may be ready within a few years if it performs well.
The second seedling is 03092-1T, a superb All American colored seedling. Falls and standards are blends of reds and blues with white signals and white wire rims. Styles have tones of light red, aqua and near white. Elements of the flower have nice width and great ruffling making for a pleasing package. This is a cross of Another Pretty Face X Miss Portland (Dunlop '06) a wonderful starchy blue with a white signal and a white wire rim.
Please note that in the 2 photos, all 3 of the flowers have been crossed. The evidence is clearly seen by anthers inserted in the stigmatic lips of all 3 blossoms, to make the new cross, if the pollen takes.
Another species look alike variety which I have gathered by page viewed statistics to date are the most popular Siberian Irises with visitors to this blog.
Look its really great to show the latest and greatest Siberian Iris but it would be wrong not to point out to readers that, listed within the 2009 catalogues of Siberian Irises from all growers contains some fantastic varieties with reasonable price tags .(Check out Siberian Links for a grower in your area).
Sweet Little Susie is one of many great selections available in the six to eight dollar price range, an amazing shorter growing iris for the front of the border. So if you are new to growing Siberians or have never tried these Irises you can buy a great perennial priced for way less than a flat of Annuals.
Joe Pye Weeds Garden have an great range which includes a few special priced Siberian Irises which are available for Spring shipping only, and will not be available again after that. These are marked in the catalogue with *Spring only, and all are very fine Siberians. Catalogue can be downloaded.
SWEET LITTLE SUSIE - '01, EM, 22". Nice short habit makes it perfect for a smaller garden. Periwinkle blue flowers - lighter in the standards and deeper in the falls. Large cheerful white signals. Medium to small flowers have soft ruffles and archingform.
SWEET LITTLE SUSIE Schafer, Marty and Sacks, Jan Reg. 2001 Sdlg. S94-30-1 SIB (dip.) (22" 56 cm) EM. Medium hyacinth to wisteria blue (RHS 91 A, 92A) self, F. with large round white signal veined violet-blue; ruffled; slight fragrance. S90-45-2: (S86-16-1: (Percheron x Isabelle) x Sailor's Fancy) X S92-65-1: (S89-9-2: (Isabelle x Silver Illusion) x Snow Prince). Joe Pye Weed's Garden 2001.
An unusual shade of lavender-blue is presented in a ruffled flower with elegant wide, overlapping form. The falls are decorated with a small white signal, veined violet, and surrounded by a bluer glow. Beautiful upright foliage and stems.
LOVE ALWAYS Calvin Helsley, Reg. 2008 Sdlg. 05-03. SIB, 35" (89 cm), M ; S. medium lavender-blue (RHS 94B) veined violet; style arms lavender blue, turquoise midrib; F. same as S., bluer area around small violet veined white signal; ruffled, flat form. Where Eagles Dare X Rosy Bows. Iris City 2009.
Introduced this year by Iris City Gardens who have an very extensive catalogue of Siberian Irises with a very good price range.
This website has voluntarily been taken down in protest against this law, which will be used to disconnect New Zealanders from the internet based on accusations of copyright infringement, without a trial and without evidence held up to court scrutiny.
May we be very clear: we do not support or condone copyright infringement or illegal downloads.
But this blatant disregard towards the basic human right to a fair trial is completely unjust and unworkable and it has the potential to punish New Zealand businesses and individuals were in fact no laws have been broken.
Similar laws have been rejected in the EU as being against "a fair balance between various fundamental rights", rejected in the UK due to "impracticalities", and rejected in Germany as being 'Unfit for Germany, Unfit For Europe'.
We don't care who voted for the law in the first place. We just want it stopped. We call on the Minister responsible, National's Simon Power, to do the right thing and repeal Section 92A immediately.
THIS BLOG HAS BEEN BLACKED OUT FOR 24 HOURS AS OUR CONTRIBUTION TO THE PROTEST.
style arms white, turquoise midrib, feathered at tip; F. bluish violet, pale blue wire edge. Sib to So Be It; seed from M. Schafer cross S97-43. Tempo Two 2005/06.
More Barry Blyth Introductions just catalogue and Checklist descriptions and I'll let the photos do the talking.
Fond Kiss sib; seed from M. Schafer cross S97-62-A. Tempo Two 2004/05.
My congratulations to Jennifer Hewitt for her Dykes Medal Award for the Siberian Iris 'Peter Hewitt' which continues a proud British Iris Society tradition of awarding DM to beardless Irises. Available in England from Alden Farms who have a great catalogue of Iris.
Alden Farm Catalogue 2009 Iris sibirica 'Peter Hewitt' £ 5.00 A dependable iris producing plentiful flowers and deserving its Dykes Medal. A combination of violet-blue against an eye catching deep gold signal. The falls are usually bounded by a gold edge. An attractive addition to any garden.
This iris, bred by Jennifer Hewitt and named for her late husband, is the first British cultivar to receive this highest and most prestigious award since TB 'Alexia' in 2006 and, before that, TB 'Darley Dale' in 2001. Often the award is not given unless an iris is considered worthy of receiving it and 'Peter Hewitt' certainly is. Registered in 2003 this iris blooms mid-late season at a height of 37ins (94cm). It is described as having violet blue standards, fading to wisteria blue with darker veins; turquoise style arms and falls of deep blue violet, bluer around the deep gold signal and edged gold. Its parentage is 'Coronation Anthem' x 'Gold Edge'.
It is probably an opportune time to note that the first ever Iris awarded a Dykes Memorial Medal was formally made at the General Meeting of the Iris Society (BIS) on November 30th 1927 to Mr Amos Perry for his hybrid between I.Chrysographes and I. Douglasiana named MARGOT HOLMES.
It was a major logistical exercise to piece this post together. I offer my sincere thanks to the following people for their help, which has allowed the sharing of their information with the visitors to this blog. Jennifer Hewitt, Olga Wells, Editor BIS Yearbook, Clive Russell, Chairman BIS, Julian Browse of Seagate Irises.
To Alun and Jill Whitehead of Alden Farm who have recently become the guardians for the old National Collection of Siberian Iris in England, all the very best for getting the status renewed and a Huge thanks for the Photo and information.
Massive hat tip to Anne Milner for her guidance and steering me in the right direction around English Iris Circles.
The Siberian Iris PARASOL, formerly known as 'HARASUGATA' clumps up well. The style will probably not appeal to traditional Siberian Iris purist, but these spectacular forms of Siberians can add novelty to the perennial border.
PARASOL Shidara, Ho Reg. 1999 SIB (dip.) (30" 76 cm) S. absent; style arms white; F. light to medium lavender-pink, small yellow-green signal, multiple (6). Unknown parentage. Ensata Gardens 1997.
Another extraordinary hat tip to Mike Unser for sharing the photo which is an outstanding display of this variety.
Extraordinary hat tip to Mike Unser for sharing the photo which is an outstanding display of this variety.
Emphasis is on form with this new white Siberian iris. Horizontally flaring, flat flowers are ruffled to the extreme! Newly opened flowers have pale yellow falls that become creamy white by the second day with yellow veining streaming from the small lemon-yellow signals. You will believe!
BELIEVE IN ANGELS Calvin Helsley, R. 2005 , Sdlg. 04-17. SIB, 33" (84 cm), ML ; White self, signal bright lemon yellow, some veining half way down F., all petals edged with small ruffles; flat form. Ships Are Sailing X Where Eagles Dare. Iris City 2009.
A clump of this iris will draw you from across the garden. Well branched stems have flowers of violet blue with reddish tones. The style arms are red violet with copper tinges. Very slight white signals are veined violet.
Calvin Helsley has always been a hybridiser that I have admired for his talent of consistenly producing no fuss Siberians that form good clumps with copious bloom stalks. His 1984 introduction 'Mabel Coday' is still a standout Siberian in my Garden and much admired by garden visitors. The current 2009 introductions are a continuation of a 28 year legacy and long may it continue.
THEN SINGS MY SOUL Calvin Helsley, Reg. 2005 Sdlg. 05-02. SIB, 32" (81 cm), EM ; S. red violet; style arms same, copper tints; F. violet blue, reddish tints, signal white, very small, veined violet blue; rounded form. Ragtime Dance X Where Eagles Dare. Iris City 2009.
GATINEAU by Miss Isabella Preston of the Central Experimental Farm, Ottawa, Canada, was a significant advance Siberian Iris hybridising in the 192o's by Crossing Iris sibirica maxima with Snow Queen and is a good example of the Hybrid Vigour that can be achieved when crossing two inbred varieties. The more recent use of 'Snow Prince' is a similar example. Available in USA and Canada from Chuck Chapman Iris.
The orientalis variety Snow Queen grows very well at the experimental Farm, Ottawa, Canada and it seemed a pity that these beautiful flowers could not be grown on longer stems. So crosses were made between Iris sibirica maxima and I. orientalis var. Snow Queen. The cross was made both ways and a number of seedlings obtained. They were all blue but varied a great deal in the shade, markings and size of flower. Several of these were named after Canadian rivers. GATINEAU is one of the best of the pale blues. Others that have been named are Abitibi, Chaudière, Kootenay, Madawaska, Ottawa, and Rideau. Two of the seedlings were crossed with the pollen of Snow Queen and one of the seedlings was named Mattawin. It is a medium shade of blue. There were some white amongst the seedlings but none was considered an improvement on Snow Queen.
I am speculating but feel seed from Isabella Preston was taken to England by Lady Byng of Vimy. ( Lord Byng was the Governor General of Canada 1921-1926) I know for certain that Lady Byng exhibited Canadian introductions 'Caesar' and 'Ottawa' , at the Iris Society Show, Vincent Square, June 8th 1928. Ottawa is illustrated in "Gardeners Chronicle" of June 16th 1928.
Top hat tip to Chuck Chapman for sharing another classy Historic Iris photo.
With seven Morgan Wood Medals to his name and Countless Awards of Merit and Honourable Mentions, Bob Hollingworth's 2009 Siberian Iris ' HOW AUDACIOUS' is another award winning Iris in the making. A tetraploid out of Ruffles and Flourishes and a seedling out of Currier x Bright and Beautiful. Tetraploid Siberians have features that include bigger plants, foliage and flowers.
HOW AUDACIOUS Robert Hollingworth, Reg. 2009 Sdlg. 02B10B10. SIB (tet.), 34" (86 cm), ML ; S. red violet; style arms blue violet, brown pink tips and edges; F. blue violet shading to red violet at edge, cream wire rim, cream dashes over outer petals give plicata effect, yellow signal area covers half of petal. Ruffles and Flourishes X 97B1A2, Bright and Beautiful sib. Ensata 2009.
Extraordinary hat tip to Judy Hollingworth for sharing the photo which is outstanding.
This is another colour advance from Jan Sacks and Marty Schafer. These smaller flowering species like Siberians are very popular when seen by Garden Visitors and after first viewing rapidly become a must have plant.
SO VAN GOGH Marty Schafer/Jan Sacks, Reg. 2005 Sdlg. S97-20-10. SIB, 30" (76 cm), EM ; S. and style arms medium blue-violet (RHS 90B to 91B), darker veining and edges (89B at darkest); F. yellow (13C), lighter at edge, darkest at signal, veined blue violet (89A to 90B), darkest at tip, signal yellow, veined deep blue-violet, blends into F. Sarah Tiffney X Banish Misfortune. Joe Pye Weed 2005. HM 2008, AM 2010.
Considered to be the first 'Yellow' Siberian and possibly the most important colour break in Siberian Iris breeding at the time.
Hybridised by Currier McEwen the father of modern day Siberian Irises.
Falls of bright butter yellow between neat white standards. The large, shapely flowers are carried on 70cm stems, and in our garden 'Butter and Sugar' blends in well with other plants in the perennial border. Awarded the Morgan Award and the Morgan-Wood Medal. As a 33 year old it has become a relatively new Historic Iris.
Available World Wide and very reasonably priced. Check Siberian link list for a grower in your region.
GRANDIS an Historic Siberian Iris that is just as elegant today, as the day it was introduced 97 years ago. A great example of classic form.It is appropriate to feature an heirloom Siberian iris introduced by Bertrand H. Farr . Considered to be the first person in the United States to take up seriously the breeding of Irises, his first introduction being in 1909.Indication of I. sibirica parentage are the bud pattern, height, and form.
GRANDIS This variety has rich violet standards and styles with white falls beautifully veined rich violet and bronze veining at the throat. Very unusual on account of these markings on the falls which are prominent because they hang straight down Height 45 inches.
GRANDIS Listed SIB (dip.) (36" 91 cm) E B3L. Blue-violet standards and style arms, white area on falls; F. reticulated with blue-violet lines that coalesce at the tips. Listed as superceded in 1939 CL. Synonyms: Sibirica Grandesse; sibirica grandis. Unknown parentage. Farr 1912.
Big hat tip to Chuck Chapman for sharing the photo which is pure class.
I think that this variety is Truly Amazing!
PROPHET'S KISS (Blyth, 04 Aust) (24" 61 cm) EML.Standards are a soft greyed lavender with deeper veining and lemon edges. Falls are a washed brownish rose with a large gold signal. Stylearms are greyed lemon with with a deeper turquoise midrib. A long season with lots of flowers over a compact clump. A new colour combination in Siberian Irises Sdlg. J245-1 Sib to Huntress. Seed from M. Schafer cross S97-23.
PROPHET'S KISS Barry Blyth, Reg. 2004 Sdlg. J245-1. SIB, 24" (61 cm), EML ; S. greyed violet lemon; style arms grey lemon, turquoise midrib; F. brownish rose, gold signal extending 1/3 way down and heavily veined brownish rose. Seed from M. Schafer cross S97-23. Sib to Huntress. Tempo Two 2004/05.
TROUBLED WATERS Barry Blyth, Reg. 2005 Sdlg. J252-5. SIB, 36" (91 cm), EM ; S. blue to blue lavender; style arms blue white hazed turquoise; F. deep blue-lavender, ¹⁄₁₆˝ edge of S. color; signal absent. Seed from M. Schafer cross S97-43. Sib to So Be It. Tempo Two 2005/06.
Barry Blyth is a world class hybridizer with a half century of introduction's and has made a fantastic contribution to the Iris world which thankfully included Siberian Irises. However he is not listing, selling or growing Siberian Iris any longer. Barry and his partner Lesley are cutting way back and taking it easy, and are now just concentrating on TB's.
It give's me great delight to display another standout Siberian from the talented hybridizing duo, Dunlop & Cole. Introduced last season 2008 by Fieldstone Gardens Maine,USA.
DREAMING OF YOU (Jeff Dunlop,R. 2007). Sdlg. 994336-1T. SIB (tet), 26" (66cm), Mid to Late. Ruffled light yellow near amoena. S. near white, slightly darker veins near center base; style arms very light yellow with darker midribs. Falls light yellow (RHS 4B) with deep yellow hafts, signal area and throat. Extravagant ruffling overall, and probably the most ruffled Siberian we know. Flower fades to very light yellow (RHS 4D) at the end of the bloom period. The first color fast yellow tetraploid with good form and great ruffling. One branch with 3-4 blooms. McEwen T9/38-1 X McEwen T9 95/94-9. Dunlop & Cole, Fieldstone Gardens, 2008.
DREAMING OF YOU Jeff Dunlop, Reg. 2007 Sdlg. 99436-1T. SIB (tet.), 26" (66 cm), ML ; S. near white, slightly darker veins near center base; style arms very light yellow, darker midribs; F. light yellow (RHS 4B) fading to near white (4D), deep yellow hafts, signal area and throat; ruffled. McEwen T9 95/38-1 X McEwen T9 95/94-9. Fieldstone 2008.