text:
\section{Introduction}
In practical control systems,
we often need to
minimize the control effort
so as to achieve control objectives
under limitations in equipment such as
actuators, sensors, and networks.
For example,
the energy (or $L^2$-norm) of a control signal is minimized
to prevent engine overheating
or to reduce transmission cost
with a standard LQ (linear quadratic) control problem;
see e.g., \cite{AndMoo}.
Another example is the \emph{minimum-fuel} control,
discussed in e.g., \cite{Ath63,AthFal},
in which the total expenditure of fuel is minimized with
the $L^1$ norm of the control.

Alternatively, in some situations, the control effort can be dramatically reduced by
holding the control value \emph{exactly zero} over a time interval.
We call such control a \emph{hands-off control}.
\if 0
A motivation for hands-off control is computer-controlled drug administration.
In blood pressure control \cite{SlaShe82,ArnMcInnGloNor83} for example,
a patient should be connected to a controlled pump while a drug is infused,
and if the duration of infusion is reduced by hands-off control,
the patient significantly benefits.
\fi
A motivation for hands-off control is a stop-start system
in automobiles.
It is a hands-off control; it automatically shuts down
the engine to avoid it idling for long periods of time.
By this, we can reduce CO or CO2 emissions as well as fuel consumption
\cite{Dun74}.
This strategy is also used in hybrid vehicles \cite{Cha07};
the internal combustion engine is stopped when
the vehicle is at a stop or the speed is lower than a preset threshold,
and the electric motor is alternatively used.
Thus hands-off control is also available for solving environmental problems.
Hands-off control is also
desirable for networked and embedded systems
since the communication channel is not used
during a period of zero-valued control.
This property is advantageous in particular for wireless communications
\cite{JeoJeo06,KonWonTsa09}.
In other words, hands-off control is the least \emph{attention}
in such periods.
From this point of view, hands-off control that maximizes the total time of no attention is somewhat related to the concept of minimum attention control
\cite{Bro97}.
Motivated by these applications, we propose
a new paradigm of control, called \emph{maximum-hands-off control}
that maximizes the time interval over which the control is exactly zero.

The hands-off property is related to \emph{sparsity},
or the \emph{$L^0$ ``norm''}
(the quotation marks indicate that this is not a norm;
see Section~\ref{sec:preliminaries} below)
of a signal,
defined by the total length of the intervals over which the signal
takes non-zero values. The maximum-hands-off control,
in other words, seeks
the \emph{sparsest} (or $L^0$-optimal) control among all admissible controls.
This problem is however hard to solve since the cost function
is non-convex and discontinuous.
To overcome the difficulty, one can adopt
$L^1$ optimality as a convex approximation of the problem,
as often used in \emph{compressed sensing},
which has recently attracted significant attention
in signal processing;
see \cite{Ela,EldKut,HayNagTan13} for details.
Compressed sensing has shown by theory and experiments that
sparse high-dimensional signals
can be reconstructed from incomplete measurements
by using $L^1$ optimization;
see e.g., \cite{Don06,Can06}.

Interestingly,  an $L^1$-optimal (or minimum-fuel) control has been known to have
such a sparsity property, traditionally called \emph{"bang-off-bang"}
as in \cite{AthFal}.
Although advantage has implicitly taken of the sparsity property for minimizing the fuel consumption,
there has been no research on the theoretical connection between sparsity and
$L^1$ optimality of the control.
In this article, we prove that
a solution to an $L^1$-optimal control problem gives a maximum-hands-off control, and vice versa.
As a result,
the sparsest solution (i.e., the maximum-hands-off control)
can be obtained by solving an $L^1$-optimal control problem.

A consequence of our result is that
maximum hands-off control necessarily has a
"bang-off-bang" property;
the control abruptly changes
its values between $0$ and $\pm u_{\max}$
at switching times ($u_{\max}$ is the admissible maximum absolute value
of control).
In some applications, this feature should be avoided.
To make the control continuous in time,
we propose a new type of control, namely,
\emph{$L^1$/$L^2$-optimal control}.
We show that the $L^1$/$L^2$-optimal control is an intermediate control
between the maximum-hands-off (or $L^1$-optimal)
and the minimum energy (or $L^2$-optimal) controls,
in the sense that the $L^1$ and $L^2$ controls are
the limiting instances
of the $L^1$/$L^2$-optimal control.

The remainder of this article is organized as follows.
In Section~\ref{sec:preliminaries},
we give mathematical preliminaries for our subsequent discussion.
In Section~\ref{sec:problems},  we define two control problems:
maximum-hands-off control and $L^1$-optimal control.
In Section~\ref{sec:L1}, we briefly review $L^1$-optimal control.
Section~\ref{sec:main} gives the main theorem,
establishing the theoretical connection between the $L^1$-optimal control
and the maximum-hands-off one.
In Section~\ref{sec:L1L2}, we propose a mixed $L^1$/$L^2$-optimal control
that gives a continuous hands-off control.
Section~\ref{sec:examples} presents control design examples
to illustrate the effectiveness of our method.
In Section~\ref{sec:conclusion}, we offer concluding remarks.

\section{Mathematical Preliminaries}
\label{sec:preliminaries}
For a vector $\vc{v}=[v_1,v_2,\dots,v_m]^\top$,
we define the $\ell^1$, $\ell^2$, and $\ell^\infty$ norms respectively by
\[
 \begin{split}
  &\|\vc{v}\|_1 \triangleq \sum_{i=1}^m |v_i|,\quad
  \|\vc{v}\|_2 \triangleq \sqrt{\sum_{i=1}^m |v_i|^2},\\
  &\|\vc{v}\|_\infty \triangleq \max_{i=1,\dots,m} |v_i|.
 \end{split}
\]

For a continuous-time signal $u(t)$
over a time interval $[0,T]$,
we define its $L^p$ norm ($p>0$)
by
\[
 \|u\|_{L^p} \triangleq \left(\int_0^T |u(t)|^p dt\right)^{1/p}.
\]
Note that if $p\in(0,1)$, then $\|\cdot\|_{L^p}$ is not a
norm (It fails to satisfy the triangle inequality.).
We define the support set of $u$, denoted by $\supp(u)$, by
the closure of the set
\[
\{t\in[0,T]: u(t)\neq0\}.
\]
Then we define
the $L^0$ ``norm'' of $u$ as
the length of its support, that is,
\[
 \|u\|_{L^0} \triangleq \mu\bigl(\supp(u)\bigr),
\]
where $\mu$ is the Lebesgue measure on $\field{R}$.
Note that the $L^0$ ``norm'' is not a norm since
it fails to satisfy the positive homogeneity, that is,
for any non-zero scalar $\alpha$ such that
$|\alpha|\neq 1$, we have
\[
 \|\alpha u\|_{L^0} = \|u\|_{L^0} \neq |\alpha| \|u\|_{L^0},
 \quad \forall u\neq 0.
\]
The notation $\|\cdot\|_{L^0}$ may be however
justified from the fact that
if $u$ is integrable on $[0,T]$, then
$u\in L^p$ for any $p\in(0,1)$ and
\[
 \lim_{p\rightarrow 0}\|u\|_{L^p}^p = \|u\|_{L^0},
\]
which is proved by using H\"{o}lder's inequality; see \cite{Tao} for details.
A continuous-time signal $u$ over $[0,T]$ is called
\emph{sparse}%
\footnote{This is analogous to the definition of the $\ell^0$ ``norm''
of a vector, which is defined by the number of non-zero elements.
When a vector has small $\ell^0$ ``norm,''
then it is also called sparse.
See \cite{Ela,EldKut,HayNagTan13} for details.}
if $\|u\|_{L^0}$ is ``much smaller'' than $T$.

For a function $\vc{f}=[f_1,\dots,f_n]^\top: \field{R}^n\rightarrow\field{R}^n$,
the Jacobian $\vc{f}'$ is defined by
\[
 \vc{f}'(\vc{x}) \triangleq
  \begin{bmatrix}
   \frac{\partial f_1}{\partial x_1}&\dots&\frac{\partial f_1}{\partial x_n}\\
   \vdots & \ddots & \vdots\\
   \frac{\partial f_n}{\partial x_1}&\dots&\frac{\partial f_n}{\partial x_n}\\
 \end{bmatrix},
\]
where $\vc{x}=[x_1,\dots,x_n]^\top$.

\section{Optimal Control Problems}
\label{sec:problems}
We here consider nonlinear plant models of the form
\begin{equation}
 \frac{d\vc{x}(t)}{dt} = \vc{f}\bigl(\vc{x}(t)\bigr) + \sum_{i=1}^m \vc{g}_i\bigl(\vc{x}(t)\bigr)u_i(t),
 \quad t\in[0,T],
 \label{eq:plant}
\end{equation}
where
$\vc{x}$ is the state,
$u_1,\dots,u_m$ are the control inputs,
$\vc{f}$ and $\vc{g}_i$
are functions on $\field{R}^n$.
We assume that $\vc{f}(\vc{x})$, $\vc{g}_i(\vc{x})$,
and their Jacobians $\vc{f}'(\vc{x})$, $\vc{g}_i'(\vc{x})$
are continuous in $\vc{x}$.
We use the vector representation $\vc{u}\triangleq[u_1,\dots,u_m]^\top$.

The control $\{\vc{u}(t): t\in[0,T]\}$ is chosen to drive the state $\vc{x}(t)$
from a given initial state
\begin{equation}
 \vc{x}(0)=\vc{x}_0,
 \label{eq:initial_state}
\end{equation}
to the origin by a fixed final time $T>0$, that is,
\begin{equation}
 \vc{x}(T)=\vc{0}.
 \label{eq:final_state}
\end{equation}
Also, the control $\vc{u}(t)$ is constrained in magnitude by
\begin{equation}
 \|\vc{u}(t)\|_\infty \leq 1,\quad \forall t \in [0,T].
 \label{eq:input_constraint}
\end{equation}
We call a control $\{\vc{u}(t): t\in[0,T]\}$ \emph{admissible}
if it satisfies \eqref{eq:input_constraint}
and the resultant state $\vc{x}(t)$ from \eqref{eq:plant} satisfies boundary conditions
\eqref{eq:initial_state} and \eqref{eq:final_state}.
We denote by ${\mathcal{U}}$ the set of all admissible controls.

The \emph{maximum-hands-off control}
is a control that
maximizes the time interval over which the control $\vc{u}(t)$ is exactly zero.
In other words, we try to find the \emph{sparsest} control
among all admissible controls in ${\mathcal{U}}$.

We state the associated optimal control problem as follows:
\begin{problem}[Maximum-Hands-Off Control]
\label{prob:MHO}
Find an admissible control $\{\vc{u}(t): t\in[0,T]\}\in{\mathcal{U}}$ that minimizes
\begin{equation}
 J_0(\vc{u}) \triangleq \sum_{i=1}^m\lambda_i \|u_i\|_{L^0},
 \label{eq:J_MHO}
\end{equation}
where $\lambda_1>0,\dots,\lambda_m>0$ are given weights.
\end{problem}

On the other hand, if we replace $\|u_i\|_{L^0}$ in \eqref{eq:J_MHO}
with the $L^1$ norm $\|u_i\|_{L^1}$,
we obtain the following \emph{$L^1$-optimal control} problem,
also known as \emph{minimum fuel control}
discussed in e.g. \cite{Ath63,AthFal}.
\begin{problem}[$L^1$-Optimal Control]
\label{prob:L1}
Find an admissible control $\{\vc{u}(t): t\in[0,T]\}\in{\mathcal{U}}$ that minimizes
\begin{equation}
 J_1(\vc{u}) \triangleq \sum_{i=1}^m \lambda_i \|u_i\|_{L^1} = \int_0^T \sum_{i=1}^m \lambda_i |u_i(t)| dt,
 \label{eq:J_L1}
\end{equation}
where $\lambda_1>0,\dots,\lambda_m>0$ are given weights.
\end{problem}

\begin{rem}[Minimum time]
For the existence of the solution of both problems above,
the final time $T$ must be sufficiently large.
More precisely, $T$ must be larger than
the minimum time $T^\ast$ required to force
the initial state $\vc{x}_0$ to the origin.
$T^\ast$ is obtained by solving the minimum-time problem;
see \cite[Chap.~6]{AthFal} for details.
\end{rem}

\section{Review of $L^1$-Optimal Control}
\label{sec:L1}
Here we briefly review the $L^1$-optimal (or minimum-fuel) control problem (Problem~\ref{prob:L1})
based on the discussion in \cite[Sec.~6-13]{AthFal}.

Let us first form the Hamiltonian function for the $L^1$-optimal control problem as
\begin{equation}
 H(\vc{x},\vc{p},\vc{u}) = \sum_{i=1}^m \lambda_i |u_i|
  + \vc{p}^\top \biggl(\vc{f}\bigl(\vc{x}\bigr)
    +\sum_{i=1}^m \vc{g}_i(\vc{x})u_i\biggr),
 \label{eq:Hamiltonian_L1}
\end{equation}
where
$\vc{p}$ is the costate (or adjoint) vector.
Assume that $\vc{u}^\ast=[u_1^\ast,\dots,u_m^\ast]^\top$
is an $L^1$-optimal control
and $\vc{x}^\ast$ is the resultant trajectory.
According to the minimum principle, there exists a costate $\vc{p}^\ast$ such that
the optimal control $\vc{u}^\ast$ satisfies
\[
 H(\vc{x}^\ast,\vc{p}^\ast,\vc{u}^\ast)\leq  H(\vc{x}^\ast,\vc{p}^\ast,\vc{u}),
\]
for all admissible $\vc{u}$.
The optimal state $\vc{x}^\ast$ and costate $\vc{p}^\ast$
satisfies the canonical equations
\[
 \begin{split}
  \frac{d\vc{x}^\ast(t)}{dt} &=  \vc{f}\bigl(\vc{x}^\ast(t)\bigr) + \sum_{i=1}^m
    \vc{g}_i\bigl(\vc{x}^\ast(t)\bigr)u_i^\ast(t),\\
  \frac{d\vc{p}^\ast(t)}{dt} &= - \vc{f}'\bigl(\vc{x}^\ast(t)\bigr)^\top \vc{p}^\ast(t)\\
  	&\qquad-\sum_{i=1}^m u_i^\ast(t) \vc{g}_i'\bigl(\vc{x}^\ast(t)\bigr)^\top \vc{p}^\ast(t),
 \end{split}
\]
with boundary conditions
\[
 \vc{x}^\ast(0) = \vc{x}_0,\quad \vc{x}^\ast(T) = \vc{0}.
\]
The minimizer
$\vc{u}^\ast=[u_1^\ast,\dots,u_m^\ast]^\top$ of the Hamiltonian given in
\eqref{eq:Hamiltonian_L1} is given by
\[
 u_i^\ast(t) = -\dez_{\lambda_i}\left(\vc{g}_i\bigl(\vc{x}^\ast(t)\bigr)^\top \vc{p}^\ast(t)\right),\quad t\in[0,T],
\]
where $\dez_{\lambda}(\cdot):\field{R}^n\rightarrow[-1,1]$ is the dead-zone function defined by
\begin{equation}
 \begin{split}
 \dez_{\lambda}(w)  &=
   \begin{cases}
     -1,& \text{~if~} w<-\lambda,\\
     0, & \text{~if~} -\lambda<w<\lambda,\\
     1, & \text{~if~} \lambda<w,\\
   \end{cases}\\
  \dez_{\lambda}(w) &\in [-1,0], \text{~if~} w=-\lambda,\\
  \dez_{\lambda}(w) &\in [0,1], \text{~if~} w=\lambda.
 \end{split}
 \label{eq:dez}
\end{equation}
See Fig.~\ref{fig:dez} for the graph of $\dez_\lambda(\cdot)$.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\linewidth]{dez.eps}
\caption{Dead-zone function $\dez_\lambda(w)$}
\label{fig:dez}
\end{figure}

If $\vc{g}_i(\vc{x}^\ast)^\top \vc{p}^\ast$ is equal to $-\lambda_i$ or $\lambda_i$ over
a non-zero time interval, say $[t_1,t_2]\subset[0,T]$,
$t_1<t_2$,
then the control $u_i$
(and hence $\vc{u}$)
over $[t_1,t_2]$
cannot be uniquely determined
by the minimum principle.
In this case, the interval $[t_1,t_2]$ is called a \emph{singular interval},
and a control problem that has at least one singular interval is called
\emph{singular}.
If there is no singular interval, the problem is called \emph{normal}:
\begin{defn}[Normality]
The $L^1$-optimal control problem stated in Problem~\ref{prob:L1} is said to be \emph{normal}
if the set
\[
 {\mathcal T}_i \triangleq \{t\in [0,T]: |\lambda_i^{-1}\vc{g}_i(\vc{x}^\ast(t))^\top \vc{p}^\ast(t)|=1\}
\]
is countable for $i=1,\dots,m$.
If the problem is normal, the elements $t_1,t_2,\dots\in {\mathcal T}_i$
are called the \emph{switching times} for the control $u_i(t)$.
\hfill$\Box$
\end{defn}

If the problem is normal, the components
of the $L^1$-optimal control $\vc{u}^\ast(t)$
are piecewise constant and ternary,
taking values $\pm 1$ or $0$
at almost all $t\in[0,T]$.
This property, named "bang-off-bang,"
is key to connect the $L^1$-optimal control and
the maximum-hands-off control
as discussed in the next section.

\section{Maximum-Hands-Off Control and $L^1$-Optimal Control}
\label{sec:main}
In this section, we consider a theoretical relation
between
maximum-hands-off control (Problem \ref{prob:MHO})
and $L^1$-optimal control (Problem \ref{prob:L1}).
The theorem below rationalizes the $L^1$ optimality
in computing the maximum-hands-off control.
\begin{thm}
\label{thm:L1optimal}
Assume that the $L^1$-optimal control problem stated in Problem~\ref{prob:L1}
is normal and has at least one solution.
Let ${\mathcal{U}}_0^\ast$ and ${\mathcal{U}}_1^\ast$ be the sets of the optimal solutions
of Problem \ref{prob:MHO} ($L^0$-optimal control problem)
and Problem \ref{prob:L1} ($L^1$-optimal control problem)
respectively.
Then we have ${\mathcal{U}}_0^\ast = {\mathcal{U}}_1^\ast$.
\end{thm}

{\bf Proof.}
Let ${\mathcal{U}}$ be the set of all admissible controls for the
$L^1$-optimal control problem (Problem~\ref{prob:L1}).
By assumption, ${\mathcal{U}}_1^\ast$ is non-empty, and so is ${\mathcal{U}}$.
The set ${\mathcal{U}}$ is also the admissible control set
for the $L^0$-optimal control problem (Problem \ref{prob:MHO}),
and hence ${\mathcal{U}}_0^\ast \subset {\mathcal{U}}$.
We first show that ${\mathcal{U}}_0^\ast$ is non-empty, and then prove ${\mathcal{U}}_0^\ast = {\mathcal{U}}_1^\ast$.

First, for any $\vc{u}\in{\mathcal U}$, we have
\begin{equation}
 \begin{split}
  J_1(\vc{u}) &= \sum_{i=1}^m \lambda_i \int_0^T |u_i(t)| ~dt\\
  &= \sum_{i=1}^m \lambda_i \int_{\supp(u_i)} |u_i(t)| ~dt\\
  &\leq \sum_{i=1}^m \lambda_i \int_{\supp(u_i)} 1 ~dt
  = J_0(\vc{u}).
 \end{split}
\label{eq:proof1}
\end{equation}

Now take an arbitrary $\vc{u}^\ast_1 \in {\mathcal{U}}_1^\ast$.
Since the problem is normal by assumption,
each control $u_{1i}^\ast(t)$ in $\vc{u}_1^\ast(t)$ takes values $-1$, $0$, or $1$,
at almost all $t\in[0,T]$.
This implies that
\begin{equation}
 \begin{split}
 J_1(\vc{u}^\ast_1) &= \sum_{i=1}^m \lambda_i \int_0^T |u_{1i}^\ast(t)| ~dt\\
  &= \sum_{i=1}^m \lambda_i \int_{\supp(u_{1i}^\ast)} 1 ~dt
  = J_0(\vc{u}^\ast_1).
 \end{split}
 \label{eq:proof2}
\end{equation}
From \eqref{eq:proof1} and \eqref{eq:proof2},
$\vc{u}^\ast_1$ is a minimizer of $J_0$,
that is, $\vc{u}_1^\ast\in{\mathcal{U}}_0^\ast$.
Thus, ${\mathcal{U}}_0^\ast$ is non-empty and ${\mathcal{U}}_1^\ast \subset {\mathcal{U}}_0^\ast$.

Conversely, let $\vc{u}^\ast_0\in{\mathcal{U}}_0^\ast\subset{\mathcal{U}}$.
Take independently $\vc{u}^\ast_1\in{\mathcal{U}}_1^\ast\subset{\mathcal{U}}$.
From \eqref{eq:proof2} and the optimality of $\vc{u}^\ast_1$, we have
\begin{equation}
 J_0(\vc{u}^\ast_1) = J_1(\vc{u}^\ast_1) \leq J_1(\vc{u}^\ast_0).
 \label{eq:proof3}
\end{equation}
On the other hand, from \eqref{eq:proof1} and the optimality of $\vc{u}^\ast_0$,
we have
\begin{equation}
 J_1(\vc{u}^\ast_0)\leq J_0(\vc{u}^\ast_0) \leq J_0(\vc{u}^\ast_1).
 \label{eq:proof4}
\end{equation}
It follows from \eqref{eq:proof3} and \eqref{eq:proof4} that
$J_1(\vc{u}^\ast_1)=J_1(\vc{u}^\ast_0)$,
and hence $\vc{u}^\ast_0$ achieves the minimum value of $J_1$.
That is, $\vc{u}_0^\ast\in{\mathcal{U}}_1^\ast$ and ${\mathcal{U}}_0^\ast\subset{\mathcal{U}}_1^\ast$.
\hfill$\Box$

Theorem~\ref{thm:L1optimal} suggests that
$L^1$ optimization can be used for
the maximum-hands-off (or the sparsest) solution.
This is analogous to the situation in compressed sensing,
where $L^1$ optimality is often used to obtain the sparsest vector;
see \cite{Ela,EldKut,HayNagTan13} for details.

\section{$L^1$/$L^2$-Optimal Control}
\label{sec:L1L2}
In the previous section, we have shown that
the maximum-hands-off control problem can be solved
via $L^1$-optimal control.
From the "bang-off-bang" property of the $L^1$-optimal control,
the control changes its value at switching times \emph{discontinuously}.
This is undesirable for some applications in which the actuators cannot move
abruptly.
\if 0
\footnote{\color{red}
For example, drug administration strategies should avoid large and rapid
changes of the infusion rate to prevent diminished blood flow
\cite{SlaShe82}.
}.
\fi
In this case, one may want to make the control \emph{continuous}.
For this purpose, we add a regularization term to the $L^1$ cost $J_1(\vc{u})$
defined in \eqref{eq:J_L1}.
More precisely, we consider the following mixed $L^1$/$L^2$-optimal control problem.
\begin{problem}[$L^1$/$L^2$-Optimal Control]
\label{prob:L1L2}
Find an admissible control $\{\vc{u}(t): t\in[0,T]\}\in{\mathcal{U}}$ that minimizes
\begin{equation}
 \begin{split}
  J_{12}(\vc{u})
  &\triangleq
   \sum_{i=1}^m \biggl(\lambda_i\|u_i\|_{L^1}+\frac{1}{2}r_i\|u_i\|_{L^2}^2\biggr)\\
  &=
   \int_0^T \sum_{i=1}^m \biggl(\lambda_i |u_i(t)| + \frac{1}{2}r_i|u_i(t)|^2\biggr) dt,
 \end{split}
 \label{eq:J_12}
\end{equation}
where $\lambda_i>0$ and $r_i>0$, $i=1,\dots,m$, are given weights.
\end{problem}

To discuss the optimal solution(s) of the above problem,
we next give necessary conditions for the $L^1$/$L^2$-optimal control
using the minimum principle of Pontryagin.

The Hamiltonian function is given by
\[
 \begin{split}
  H(\vc{x},\vc{p},\vc{u})
   &= \sum_{i=1}^m \biggl(\lambda_i|u_i|+\frac{1}{2}r_i|u_i|^2\biggr)\\
    &\quad + \vc{p}^\top \biggl(\vc{f}(\vc{x})+\sum_{i=1}^m\vc{g}_i(\vc{x})u_i\biggr)
 \end{split}
\]
where $\vc{p}$ is the costate vector.
Let $\vc{u}^\ast$ denote the optimal control and
$\vc{x}^\ast$ and $\vc{p}^\ast$ the resultant optimal state and costate,
respectively.
Then we have the following result.
\begin{lem}
\label{lem:L1L2}
The $i$-th element $u_i^\ast(t)$ of
the $L^1$/$L^2$-optimal control $\vc{u}^\ast(t)$
satisfies
\begin{equation}
 u_i^\ast(t) = -\sat\left\{\shrink_{\lambda_i/r_i}\left(r_i^{-1}\vc{g}_i\bigl(\vc{x}^\ast(t)\bigr)^\top\vc{p}^\ast(t)\right)\right\},
 \label{eq:uopt_1}
\end{equation}
where
$\shrink_{\lambda/r}(\cdot)$ is the shrinkage function defined by
\[
 \shrink_{\lambda/r}(v)  \triangleq
   \begin{cases}
     v+\lambda/r& {\rm{if~}} v<-\lambda/r,\\
     0, & {\rm{if~}} -\lambda/r\leq v \leq \lambda/r,\\
     v-\lambda/r, & {\rm{if~}} \lambda/r<v,\\
   \end{cases}
\]
and
$\sat(\cdot)$ is the saturation function defined by
\[
 \sat(v)  \triangleq
   \begin{cases}
     -1, & \text{\rm if~} v<-1,\\
     v, & \text{\rm if~} -1\leq v \leq 1,\\
     1, & \text{\rm if~} 1<v.\\
   \end{cases}
\]
See Figs.~\ref{fig:shrink} and \ref{fig:sat_shrink} for the graphs of
$\shrink_{\lambda/r}(\cdot)$ and
$\sat\!\left(\shrink_{\lambda/r}(\cdot)\right)$,
respectively.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\linewidth]{shrink.eps}
\caption{Shrinkage function $\shrink_{\lambda/r}(v)$}
\label{fig:shrink}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\linewidth]{sat_shrink.eps}
\caption{Saturated shrinkage function $\sat\!\left(\shrink_{\lambda/r}(v)\right)$}
\label{fig:sat_shrink}
\end{figure}
\end{lem}

{\bf Proof.}
The result is easily obtained from the fact that
\[
 -\sat\left\{\shrink_{\lambda/r}\left(r^{-1}a\right)\right\} = \argmin_{|u|\leq 1} \lambda |u| + \frac{1}{2}r|u|^2+au,
\]
for any $\lambda>0$, $r>0$, and $a\in\field{R}$.
\hfill$\Box$

From Lemma \ref{lem:L1L2}, we have the following proposition.
\begin{prop}[Continuity]
\label{prop:continuity}
The $L^1$/$L^2$-optimal control $\vc{u}^\ast(t)$ is continuous in $t$ over $[0,T]$.
\end{prop}

{\bf Proof.}
Without loss of generality, we assume $m=1$
(a single input plant),
and omit subscripts for $u$, $r$, $\lambda$, and so on.
Let
\[
 \bar{u}(\vc{x},\vc{p}) \triangleq -\sat\left\{\shrink_{\lambda/r}\left(r^{-1}\vc{g}_i(\vc{x})^\top\vc{p}\right)\right\}.
\]
Since functions $\left(\sat\circ\shrink_{\lambda/r}\right)(\cdot)$ and $\vc{g}(\cdot)$
are continuous,
$\bar{u}(\vc{x},\vc{p})$ is also continuous in $\vc{x}$ and $\vc{p}$.
It follows from Lemma~\ref{lem:L1L2} that
the optimal control $u^\ast$ given in \eqref{eq:uopt_1}
is continuous in
$\vc{x}^\ast$ and $\vc{p}^\ast$.
Hence, $u^\ast(t)$ is continuous if $\vc{x}^\ast(t)$ and $\vc{p}^\ast(t)$
are continuous in $t$ over $[0,T]$.

The canonical system for the $L^1$/$L^2$-optimal control is given by
\[
 \begin{split}
  \frac{d\vc{x}^\ast(t)}{dt} &=  \vc{f}\bigl(\vc{x}^\ast(t)\bigr) + \vc{g}\bigl(\vc{x}^\ast(t)\bigr)\bar{u}\bigl(\vc{x}^\ast(t),\vc{p}^\ast(t)\bigr),\\
  \frac{d\vc{p}^\ast(t)}{dt} &= - \vc{f}'\bigl(\vc{x}^\ast(t)\bigr)^\top \vc{p}^\ast(t)\\
  	&\qquad - \bar{u}\bigl(\vc{x}^\ast(t),\vc{p}^\ast(t)\bigr) \vc{g}'\bigl(\vc{x}^\ast(t)\bigr)^\top \vc{p}^\ast(t).
 \end{split}
\]
Since $\vc{f}(\vc{x})$, $\vc{g}(\vc{x})$, $\vc{f}'(\vc{x})$, and $\vc{g}'(\vc{x})$ are continuous
in $\vc{x}$ by assumption,
and so is $\bar{u}(\vc{x},\vc{p})$ in $\vc{x}$ and $\vc{p}$,
the right hand side of the canonical system is continuous in $\vc{x}^\ast$ and $\vc{p}^\ast$.
From a continuity theorem of dynamical systems,
e.g. \cite[Theorem 3-14]{AthFal},
it follows that the resultant trajectories $\vc{x}^\ast(t)$ and $\vc{p}^\ast(t)$ are
continuous in $t$ over $[0,T]$.
\hfill$\Box$

Proposition \ref{prop:continuity} motivates us to use $L^1/L^2$ optimization
as in Problem \ref{prob:L1L2} for continuous hands-off control.

In general, the degree of continuity (or smoothness) and the sparsity of the control input
cannot be optimized at the same time.
Then, the weights $\lambda_i$ or $r_i$ can be used for
trading smoothness for sparsity.
From Lemma~\ref{lem:L1L2},
increasing the weight $\lambda_i$ (or decreasing $r_i$)
makes the $i$-th input $u_i(t)$ sparser
(see also Fig.~\ref{fig:sat_shrink}).
On the other hand,
decreasing $\lambda_i$ (or increasing $r_i$)
smoothens $u_i(t)$.
Moreover, we have the following limiting properties.
\begin{prop}[Limiting property]
Assume the $L^1$-optimal control problem
is normal.
Let $\vc{u}_1(\vc{\lambda})$ and $\vc{u}_{12}(\vc{\lambda},\vc{r})$
be solutions to respectively Problems \ref{prob:L1} and \ref{prob:L1L2}
with parameters
\[
 \vc{\lambda}\triangleq(\lambda_1,\dots,\lambda_m),\quad
 \vc{r}\triangleq(r_1,\dots,r_m).
\]
For any fixed $\vc{\lambda}>0$,
we have
\[
 \lim_{\vc{r}\rightarrow\vc{0}} \vc{u}_{12}(\vc{\lambda},\vc{r}) = \vc{u}_1(\vc{\lambda}).
\]
Moreover, for any fixed $\vc{r}>0$,
we have
\[
 \lim_{\vc{\lambda}\rightarrow\vc{0}}\vc{u}_{12}(\vc{\lambda},\vc{r}) = \vc{u}_2(\vc{r}),
\]
where $\vc{u}_2(\vc{r})$ is an $L^2$-optimal (or minimum-energy) control
discussed in \cite[Chap.~6]{AthFal},
that is,
a solution to a control problem where $J_1(\vc{u})$ in Problem \ref{prob:L1} is replaced with
\begin{equation}
 J_2(\vc{u}) = \int_0^T \sum_{i=1}^m \frac{1}{2}r_i|u_i(t)|^2dt.
 \label{eq:J2}
\end{equation}
\end{prop}

{\bf Proof.}
The first statement follows directly from the fact that
for any fixed $\lambda>0$, we have
\[
 \lim_{r\rightarrow 0}\sat\!\left({\mathcal{S}}_{\lambda/r}(r^{-1}w)\right) = \dez_{\lambda}(w),\quad
 \forall w\in\field{R}\setminus\{\pm \lambda\},
\]
where $\dez_\lambda(\cdot)$ is the dead-zone function defined in \eqref{eq:dez}.
The second statement derives from the fact that
for any fixed $r>0$, we have
\[
 \lim_{\lambda\rightarrow 0}\sat\!\left({\mathcal{S}}_{\lambda/r}(v)\right) = \sat(v),\quad
 \forall v\in\field{R}.
\]
\hfill$\Box$

In summary, the $L^1$/$L^2$-optimal control is an \emph{intermediate control} between
the $L^1$-optimal control (or the maximum-hands-off control)
and the $L^2$-optimal control.

\if 0
\section{Linear Plants and Numerical Computation}
\label{sec:computation}
We here propose a numerical computation method
to obtain an $L^1$/$L^2$-optimal control
for Problem~\ref{prob:L1L2}
when the plant model is linear and time-invariant.
The $L^1$-optimal (or maximum-hands-off) control and
the $L^2$-optimal (or minimum-energy) control can
also be obtained with the same formula.

Let us consider the following linear time-invariant plant model
\begin{equation}
 \frac{d\vc{x}(t)}{dt} = A\vc{x}(t) + B\vc{u}(t),\quad t\in[0,T], \quad \vc{x}(0)=\vc{x}_0,
 \label{eq:plant_ln}
\end{equation}
where $\vc{x}(t)\in\field{R}^n$ and $\vc{u}(t)\in\field{R}^m$.
We assume that the initial state $\vc{x}_0\in\field{R}^n$ and
the time $T>0$ are given.

Linear systems are much easier to treat than general nonlinear systems
as in \eqref{eq:plant}.
In particular,
for special plants, such as single or double integrators,
the $L^1$-optimal control can be obtained analytically;
see e.g., \cite[Chap.~8]{AthFal}.
However, for general linear time-invariant plants,
one should rely on numerical computation.
For this, we adopt a time discretization approach
to solve the $L^1$/$L^2$-optimal control problem.
This approach is standard for numerical optimization;
see e.g.~\cite[Sec.~2.3]{Ste}.

We first divide the interval $[0,T]$ into $N$ subintervals,
$[0,T] = [0,h) \cup \dots \cup [(N-1)h,Nh]$,
where $h$ is the discretization step chosen such that $T=Nh$.
We here assume (or approximate) that the state $\vc{x}(t)$ and the control $\vc{u}(t)$ are
constant over each subinterval.
On the discretization grid,
$t=0,h,\dots,Nh$,
the continuous-time plant \eqref{eq:plant_ln} is described as
\[
 \vc{x}_d[m+1] = A_d\vc{x}_d[m] + B_d \vc{u}_d[m],\quad m=0,1,\dots,N-1,
\]
where $\vc{x}_d[m]\triangleq\vc{x}(mh)$, $\vc{u}_d[m]\triangleq \vc{u}(mh)$, and
\[
 A_d \triangleq e^{Ah},\quad B_d \triangleq\ \int_0^h e^{At}Bdt.
\]
Set the control vector
\[
 \vc{U}\triangleq[\vc{u}_d[0]^\top,\vc{u}_d[1]^\top,\dots,\vc{u}_d[N-1]^\top]^\top.
\]
Note that the final state $\vc{x}(T)$ can be described as
\[
 \vc{x}(T)=\vc{x}_d[N]=A_d^N \vc{x}_0 + \Phi_N\vc{U},
\]
where
\[
 \Phi_N \triangleq \begin{bmatrix}A_d^{N-1}B_d,&A_d^{N-2}B_d,&\dots,&B_d\end{bmatrix}.
\]
If we define the following matrices:
\[
 \begin{split}
  \Lambda_m &\triangleq \mathrm{diag}(\lambda_1,\dots,\lambda_m),~
  \Lambda \triangleq \mathrm{blockdiag}(\underbrace{\Lambda_m,\dots,\Lambda_m}_{N}),\\
  R_m &\triangleq \mathrm{diag}(r_1,\dots,r_m),~
  R \triangleq \mathrm{blockdiag}(\underbrace{R_m,\dots,R_m}_{N}),
 \end{split}
\]
then the $L^1$/$L^2$-optimal control problem
is approximately described as
\begin{equation}
 \begin{aligned}
  & \underset{\vc{U}\in\field{R}^{mN}}{\text{minimize}}
  & & \|\Lambda\vc{U}\|_1 + \frac{1}{2}\|R^{1/2}\vc{U}\|_2^2\\
  & \text{subject to}
  & & \|\vc{U}\|_\infty\leq 1,\\
  &&& A_d^N\vc{x}_0 + \Phi_N\vc{U}=\vc{0}.
 \end{aligned}
 \label{eq:l1optimization}
\end{equation}
The optimization problem \eqref{eq:l1optimization}
is convex and can be efficiently solved by
numerical software packages such as \verb=cvx= with MATLAB;
see \cite{cvx} for details.
Note that
the $L^1$-optimal control
is given by setting $R=0$,
and the $L^2$-optimal control by $\Lambda=0$.
\fi

\section{Examples}
\label{sec:examples}
We here consider the following 4th order system:
\[
 \frac{d\vc{x}(t)}{dt} =
  \begin{bmatrix}0&-1&0&0\\1&0&0&0\\0&1&0&0\\0&0&1&0\end{bmatrix}\vc{x}(t)
  + \begin{bmatrix}2\\0\\0\\0\end{bmatrix}u(t).
\]
We set the final time $T=10$, and the initial and final states as
\[
 \vc{x}(0) = [1,1,1,1]^\top,\quad \vc{x}(10)=\vc{0}.
\]
Note that the system has poles at $s=0$, $0$, $\pm j$.

We first compute the maximum-hands-off control
with $L^1$-optimal control as discussed in Section \ref{sec:main}.
We compute the optimal control input by a time discretization method,
see e.g., \cite[Sec.~2.3]{Ste}.
Fig.~\ref{fig:control_4th} shows the obtained control.
The figure also shows the $L^2$-optimal control that minimizes $J_2(u)$
in \eqref{eq:J2} with $r_1=1$.
We can see that the maximum-hands-off control is quite sparse.
In fact, we have
\[
\|u\|_{L^0} = 1.92 \text{~(sec)},
\]
which is $19.2$\% out of $10$ (sec).
In other words, the control keeps hands-off over $80.8$\% of the control period.
On the other hand, the $L^2$ optimal control is not sparse,
while its energy, $J_2(u)$,  is smaller than that of maximum-hands-off control.
\begin{figure}[tb]
\centering
\includegraphics[width=\linewidth]{control_4th.eps}
\caption{Maximum-hands-off control via $L^1$ optimization (solid) and $L^2$ optimal control (dashed).}
\label{fig:control_4th}
\end{figure}
Fig.~\ref{fig:states_4th} shows the state variables $x_1(t)$, $x_2(t)$,
$x_3(t)$, and $x_4(t)$
along with the maximum-hands-off control $u(t)$ over time interval $[0,10]$.
\begin{figure}[tb]
\centering
\includegraphics[width=\linewidth]{states_4th.eps}
\caption{Maximum-hands-off control:
state variables $x_1(t),\dots,x_4(t)$ (solid) and input $u(t)$ (dashed)}
\label{fig:states_4th}
\end{figure}
We can see that the states almost stay at the origin
after the last switching time, $t=8.47$ (sec).

We next consider the $L^1$/$L^2$-optimal control method
proposed in Section~\ref{sec:L1L2}.
We use the same parameters as above.
The weights $\lambda_1$ and $r_1$ in \eqref{eq:J_12} are chosen as $\lambda_1=r_1=1$.
We solve the optimal control problem via a time discretization method.
Fig.~\ref{fig:control_L1L2} shows the obtained $L^1$/$L^2$-optimal control.
The figure also shows the maximum-hands-off control
obtained above.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{control_L1L2.eps}
\caption{$L^1$/$L^2$-optimal control (solid) and maximum-hands-off control (dash).}
\label{fig:control_L1L2}
\end{figure}
We can see that the $L^1$/$L^2$-optimal control is continuous while
the maximum-hands-off control exhibits
the "bang-off-bang" property.
On the other hand, the $L^1$/$L^2$-optimal control
has a longer support than the $L^1$-optimal control.
To see the tradeoff property between sparsity and smoothness of control,
we compute the $L^0$ norm, $\|u\|_{L^0}$,
and the $L^\infty$ norm of the derivative of $u(t)$, that is,
\[
 \left\|\frac{du(t)}{dt}\right\|_{L^\infty} \triangleq \sup_{t\in[0,T]} \left|\frac{du(t)}{dt}\right|,
\]
as a function of $r_1$ while $\lambda_1$ is fixed to be $1$.
Fig.~\ref{fig:tradeoff} shows the result.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{tradeoff2.eps}
\caption{$L^0$ norm of $u(t)$ (solid) and $L^\infty$ norm of $du/dt$ (dash) versus weight $r$.}
\label{fig:tradeoff}
\end{figure}
We can see that the weight $r$ can take account of the tradeoff
between sparsity and smoothness in hands-off control.

\section{Conclusion}
\label{sec:conclusion}
In this article, we have
presented maximum-hands-off control and shown that it is $L^1$ optimal.
This shows that efficient optimization methods for $L^1$ problems can be used to obtain maximum-hands-off control.
We have also proposed an $L^1$/$L^2$-optimal control to obtain
smooth hands-off control, while the maximum-hands-off control
is discontinuous due to the "bang-off-bang" property.
Numerical examples show the effectiveness of the proposed control.
Future work may include adaptation of hands-off control to sparsely packetized predictive control
as in \cite{NagQue11,NagQueOst12b}.

redpajama_set_name: RedPajamaArXiv


text:
Sales On Summer Merchandise
By Korky Vann
The Eliza B Factory Store, 55 Plains Road in Essex, features deals on trendy sandals for women, leather goods and accessories, including headbands, watchbands and totes, along with men's preppy leather goods, fashionable pet leashes and collars, fabrics, ribbons and trims. Savings run as much as 50 percent, and all sales are final. (Handout)
>>Last call for shopping tax-free. Connecticut's Tax-Free Week ends Saturday. Shop by then and you won't pay sales tax on most individual clothing or footwear items priced less than $100.
>>Attention Forever 21 fans: F21 Red, the chain's discount spin-off, opens its first Connecticut location in Meriden's Westfield Mall on Saturday, Aug. 22. Deals include camisoles starting at $1.80, jeans at $7.80, tees at $3.80, and leggings at $5.80. The first 100 people in line when the store opens at 10 a.m. get a $10 gift card; the first 300 in line get $5 off purchases of $25 or more.
>>With back-to-school just ahead, stores are clearing out summer items to make room for fall merchandise. At Christmas Tree Shop in Manchester, prices are slashed on picnic paper goods, pool toys and other summer seasonal stock. You'll also find good deals at Big Lot and Ocean State Job Lot stores.
>>Dance Village, 171C Spencer St. in Manchester, is having a back-to-school sale with savings on dance gear and 15 to 20 percent off regular-priced Bloch and Capezio items. Hours are 10 a.m. to 7 p.m. Friday and 10 a.m. to 5 p.m. Saturday. Information: 860-645-6606 or DanceVillageCt.com.
Latest CTNow
Connecticut museums and galleries offering exhibits
With his dog along for the ride, Hartford firefighter and folk musician Charlie Diamond takes his show wherever it leads him
Small grants help gigging musicians facing a new round of COVID-19 cancellations
>>The Eliza B Factory Store, 55 Plains Road in Essex, features deals on trendy sandals for women, leather goods and accessories, including headbands, watchbands and totes, along with men's preppy leather goods, fashionable pet leashes and collars, fabrics, ribbons and trims. Savings run as much as 50 percent, and all sales are final. Summer hours are 10 a.m. to 4 p.m. Monday through Friday and Saturday, 9 a.m. to 1 p.m.
>>Merchandise is 50 percent off on Friday and Saturday, Aug. 21 and 22, at Favarh Thrift Shop, 250 Albany Turnpike in Canton. Hours are 10 a.m. to 6 p.m. Friday and 10 a.m. to 3 p.m. Saturday. Information: 860-693-2899.
>>The Salvation Army Family Store, 1359 Dixwell Ave., in Hamden, is holding a huge outdoor sale on Saturday, with household items and clothing starting at 29 cents and all summer Items at 50 percent off. Hours are 9 a.m. to 9 p.m. The event also features giveaways and prizes.
>>Bargain hunter road trip! Clad in, a women's clothing boutique with locations in Mystic, Stonington and Providence, is holding a blowout sale at its outlet store in Westerly, R.I. (about an hour's drive from Hartford). The event features designer clothing and shoes at 75 percent off. The Clad in outlet is at 32 Friendship St., Westerly. Information: 401-596-1958 or cladin.com.

redpajama_set_name: RedPajamaCommonCrawl


text:
Couples DO consciously uncouple
If you have not walked down that street, stop spraying near every lamppost. Those commenting on how others decide to term their personal lives is the stuff of gossip, and does not in any way express the concern for relationships they claim to uphold.
The phrase under the scanner, and that caused sniggers, is 'conscious uncoupling', used by Hollywood actress Gwyneth Paltrow and her musician husband Chris Martin after a decade of marriage. It was a joint statement, but Paltrow has had to bear the brunt of "sickly self-serving twaddle". This should tell us something about the stupendously 'unconscious' lives making lobotomised decisions that are holding forth.
Parting is never easy. In intimate relationships, you have to reclaim yourself. You do seek euphemisms, because it has to do with how you project your life from past to future. You feel like shit even if you are the one to opt out. You feel like shit even if you knew it was coming. You feel like shit when you stand bare and look for warts because you must have screwed it up. You feel like shit as vultures view your vulnerability with binoculars.
You have to look the voyeurs in the eye, with their happy shared-diaper duties, joint-account couplings, looking for your availability, your blotches. You feel like shit. But you have other things to do, even if the relationship was your priority, and not just because you were given the keys to the kingdom in a barter. You may not even have a signed piece of paper. You just have your dignity.
You may not utter the D word or mention the breakup for months, years.
Gwyneth and Chris had to announce it because their lives are public, and they did so gracefully:
"...we have come to the conclusion that while we love each other very much we will remain separate. We are, however, and always will be a family, and in many ways we are closer than we have ever been. We are parents first and foremost, to two incredibly wonderful children and we ask for their and our space and privacy to be respected at this difficult time. We have always conducted our relationship privately, and we hope that as we consciously uncouple and coparent, we will be able to continue in the same manner."
For anyone to assume that only celebrities have to deal with verbal issues to break the news reveals ignorance. On legitimised kingsize beds, in denial about their compromised existence based on mortgages and suspicions camouflaged as concern, they do you.
Take this from Jan Moir in Daily Times:
"An irony-free chunk of classic Paltrow pretentiousness, it made them sound like two camels detaching from a desert train in search of tastier macro-biotic foliage...Like a pair of tights who suddenly find out that they were stockings all along. Being 'consciously uncoupled' certainly made breaking up the family home and 'co-parenting' nine-year-old Apple and Moses, seven, seem like something holistic and pure; an experience you'd order at a wellbeing spa, along with the coffee enema."
Besides the use of terrible metaphors (unless she has an organic acquaintance with camels), this bilge will not fathom that It is holistic to be conscious when you make a life-changing decision. Being free from dithering is pure.
It is pure when you don't live on dregs of how you are measured. It is holistic and pure to not be stuck in a groove of a fake smile and the warmth of nostalgia for what you were when you are. It is pure when you are fair to the roads you travel through, not just the moss that's gathered around you.
When you decide to uncouple, you can't just 'unconsciously' walk away into the sunset.
© Farzana Versey
Tags: celebrity, divorce, emotions, hollywood, media, private, public, relationships
void *Al() {} 28/03/2014, 04:57
FV:"It is holistic and pure to not be stuck in a groove of a fake smile and the warmth of nostalgia for what you were when you are."
Hi FV, Beautifully expressed. Nostalgia and the past can become oppressive to the point of choking what you are or want to be. On hindsight, one wishes one could have handled all this better in 20 different ways, but just say no coulda,shoulda,wouldas. Best one can do is do the right thing whatever the "cost".
Hard enough to deal with all this without children involved, like with Gwyneth. The judgemental bilge by that stupid journalist is so...well, words fail me, so I will go with "yuk".
Nostalgia has its moments. But, many are stuck in a situation where they are appreciated only because of their former credentials. Ans they delude themselves.
There are indeed many ways one could deal with situations, but with emotions the perception of the situation too alters, often without respite to adjust to it.
void *Al {return NULL;} 28/03/2014, 19:24
" But, many are stuck in a situation where they are appreciated only because of their former credentials. And they delude themselves. "
FV, And when one doesn't have any credentials to speak of, other than making a choice and trying to make it all work, then there is point at which the personal cost of losing yourself in order to gain something that may or may not exist, or may just be a matter of delusion/fooling oneself, so then one has to make some hard choices down the spiral staircase of self-doubt, and (as you say) the view keeps changing like that picture that changes depending on the angle that you look at it.
Finally, you just take all the snapshots in your mind's eye and throw it in the corner of the mind's attic in order to become functional, at the very least. Anything more is a bonus, really.
The former credentials I spoke of do not cease to exist; they merely become different.
Incidentally, after all the titters, some commentators are now trying to understand "conscious uncoupling" is a broader way.
Take this:
{Such sentiments are at the root of "conscious uncoupling": it's all about personal growth and expressing love for the process that got you there.
"A conscious uncoupling," Sadeghi and Sami explain, "is the ability to understand that every irritation and argument was a signal to look inside ourselves and identify a negative internal object that needed healing … Conscious uncoupling brings wholeness to the spirits of both people who choose to recognise each other as their teacher."}
http://www.theguardian.com/film/2014/mar/29/paltrow-martin-conscious-uncoupling-gurus
Whitewashing the BJP: The New Muslim Pastime
Patriotism and the Sahara Parivar
Fashion as Art


Cheap Borussia Dortmund Shirt d him. He wants to find a way to understand replica football shirt secrets. And if he asks directly, Mu Yunyao will certainly not tell him, but he will doubt whereplica football shirtr he paris saint germain soccer cheap football shirts happy or not. In thcheap football shirts case, it cheap football shirts better to understand it through oreplica football shirtr means. What cheap borussia dortmund shirt surprcheap football shirtsed him was that replica football shirt two robbers in thcheap football shirts mountain village actually cooperated with him, and he directly said what he wanted to know, and made him know replica football shirt troubles that Mu Yunyao encountered today. Mu s main pulse was seized by replica football shirt uncle and uncle. If Mu Yunyao s parents were not accidentally imprcheap football shirtsoned, replica football shirty would be imprcheap football shirtsoned by her uncle or uncle. Oreplica football shirtrwcheap football shirtse, she would never need a woman from replica football shirt middle to show her face. And replica football shirt reason why she pe. rsonally transported replica football shirt goods was definitely replica football shirt reason she had to do. Ye Sha cheap football shirts just a little curious about replica football shirtse things, not really want to be mixed in, so after knowing replica football shirt inside United Kingdom football kits replica football shirt scene, he immediately jumped out and solved replica football shirt two people in Yamanozhuang. Immediately afterwards, Ye s heart was stunned, and replica football shirt fallen heart was surrounded by hcheap football shirts body. football shirts people around replica football shirt caravan who were infected by pocheap football shirtsonous mcheap football shirtst and unhappy, suddenly found that a layer United Kingdom football kits translucent flame was burning on hcheap football shirts body. Those horrible pocheap football shirtsonous mcheap football shirtsts have burned out and it cheap football shirts extremely shocking. Well, everyone will continue on replica football shirt road Just as everyone was surprcheap football shirtsed that things had changed, Ye Shao climbed up again on replica football shirt beast, and took a reins and said to replica football shirt people around replica football shirt caravan. Everyone heard replica fo.n family did cheap borussia dortmund shirt not know what to do, know replica football shirt whereabouts United Kingdom football kits replica football shirt dragon son, cheap football shirts now chasing replica football shirt dragon son. Get thcheap football shirts news, Ye Hao immediately Standing up, hcheap football shirts face became very cold. Read replica football shirt URL m. Chapter 404 separation Chapter 404 Separation page 1 1 Well, you are a gold family. I am not going to look for you. You are looking for it first, too. Since thcheap football shirts cheap football shirts replica football shirt case, I will solve it for you by replica football shirt way. Upon hearing thcheap football shirts news, Ye Hao s face showed some murderousness. Dark wind, what cheap football shirts going on replica football shirtre now, replica football shirtre cheap football shirts nothing wrong with replica football shirt wind and rain. Ye Hao cheap football shirts most concerned about replica football shirt manchester united kit on sale dragon and replica football shirt wind and rain. If something happens to replica football shirt dragon and replica football shirt rain, replica football shirt relationship between him and Long Qianyu will definitely be affected. With replica football shirt strength United Kingdom football kits replica football shirt dragon son, replica football shirtre cheap football shirts nothing going on at replica football shirt moment, and cheap borussia dortmund shirt replica football shirt emperor s secret care has successfully escaped from replica football shirt water city, and cheap football shirts heading for replica football shirt first line days. It cheap football shirts estimated that it will take anoreplica football shirtr half an hour to arrive. football shirts dark wind replied. football shirts strength United Kingdom football kits replica football shirt wind and rain has reached replica football shirt peak United Kingdom football kits replica football shirt development United Kingdom football kits replica football shirt country. I can t help but hcheap football shirts demon cheap football shirts not much. I don t know who will chase him everton home shorts thcheap football shirts time. For thcheap football shirts, Ye Hao has some curiosity. football shirts most powerful United Kingdom football kits Jin s family cheap football shirts replica football shirtir ancestors.
d, and replica football shirtre was a growing situation until replica football shirt sky gradually darkened. Thcheap football shirts controversy gradually stopped, and replica football shirt ordinary monsters in replica football shirt city returned to replica football shirtir homes and hid. I don t dare to change when I put it on. football shirts mercenaries United Kingdom football kits replica football shirt major mercenary groups still choose to go out to investigate, and want to completely break replica football shirt behind replica football shirt scenes black hands behind replica football shirt silver moon city. In replica football shirtir view, replica football shirt strange black fog last night has been solved by replica football shirt people United Kingdom football kits replica football shirt Emperor Mercenary. How to say that replica football shirtre should be changes tonight, maybe you can find out replica football shirt black hand behind you and let replica football shirtm relax once. leak. With such a thought, replica football shirt mercenary group dcheap football shirtspatched tonight cheap football shirts obviously more than replica football shirt previous night. cheap borussia dortmund shirt Originally after replica football shirt night, a deadly silver moon city, everton 1986 shirt from time to time, one or two figures flew. over, giving a slight burst United Kingdom football kits wind. Just as replica football shirtse mercenaries were wandering around in replica football shirt dark, replica football shirt moon in replica football shirt sky did not know when it was covered by a large cloud. In replica football shirt streets and alleys United Kingdom football kits Silvermoon City, a group United Kingdom football kits black fog emerged again, and flew to replica football shirt mercenaries who had not found replica football shirt crcheap football shirtscheap football shirts cheap borussia dortmund shirt When replica football shirt first screams sounded in replica football shirt dark, all replica football shirt mercenaries had a creepy feeling on replica football shirt back. When replica football shirty noticed replica football shirt situation around replica football shirtm, replica football shirty realized that replica football shirty had been surrounded by black fog. In cheap borussia dortmund shirt replica football shirt mercenary association, replica football shirt leav.ell you, as long as you can promcheap football shirtse me, replica football shirtn you will be an extremely powerful presence in replica football shirt entire mountain sea monster world. Ouyang Gale said. I said that I didn t agree that I didn t agree. Ye said. Because United Kingdom football kits thcheap football shirts incident, two people have had extremely fierce dcheap football shirtsputes. It can be seen that replica football shirt Pengou Ouyang wind in front United Kingdom football kits him really hopes that replica football shirt oreplica football shirtr cheap borussia dortmund shirt party can promcheap football shirtse himself and become hcheap football shirts apprentice. But Ye Sha cheap borussia dortmund shirt did not have such an idea. You can tell me why you don t france new jersey 2018 want to be my apprentice Is it because you feel that you are my apprentice, will give you a special shame Ye Ouyang said. No, Ye said. Ye Sha cheap football shirts naturally clear about replica football shirt strength United Kingdom football kits replica football shirt old man in front United Kingdom football kits him. football shirtsrefore, I will not say that I will become replica football shirt apprentice United Kingdom football kits replica football shirt oreplica football shirtr party, and I will give myself a similar statement. One United Kingdom football kits replica football shirt fundamental reasons why Ye Yang chose to refuse cheap football shirts to feel that everything should be focused. Only cheap borussia dortmund shirt by choosing to concentrate can you achieve something. If you don everton away kit t choose to concentrate, you will always be a great person if you are always half hearted. Why cheap football shirts that said replica football shirt old man on replica football shirt side. Because I think replica football shirt most important thing for a person cheap football shirts replica football shirt belief United Kingdom football kits thcheap football shirts person. If a person does not have a strong or specific belief, wants to juventus away kit black be a great person, or wants to achieve a hegemony, cheap football shirts An extremely difficult thing, Ye said. Good boy has an idea. Ouyang Gale said. Although cheap borussia dortmund shirt replica football shirt words United Kingdom football kits Ye Yang just said, Ouyang Dafeng cheap football shirts full United Kingdom football cheap borussia dortmund shirt kits pracheap football shirtse, but does not mean that Ouyang Gale a.
Cheap Borussia Dortmund Shirt as abolcheap football shirtshed. Well, replica football shirtn I have written Ye Big Broreplica football shirtr here first, but replica football shirtre cheap football shirts a rule in replica football shirt Tianyuan Conference, that borussia dortmund personalised shirt cheap football shirts, it cheap football shirts impossible to kill people. I just saw that Ye Big Broreplica football shirtr killed very deeply, so I reminded me. Long Qianyu saw that Ye Hao was so confident, and could not help but remind him that it was Ye Hao who waited strangely for her. What s wrong with Ye Ye, what s replica football shirt problem Long Qianyu saw Ye. Hao staring straight at her, hcheap football shirts face slightly red, and some embarrassed said. Dragon girl, I don t know how you see replica football shirt cheap borussia dortmund shirt murderousness in me. football shirts murderousness cheap football shirts nothing but imaginary things. Even replica football shirt strong people who are not dying will not see it at all. How did you do it football shirtsre are some sternness in Ye Hao s voice. Now he cheap football shirts in replica football shirt stage United Kingdom football kits hcheap football shirts life, he must be careful everywhere In particular, replica football shirt last time after replica football shirt occurrence United Kingdom football kits Qingmuling, on replica football shirt way to replica football shirt Yuan Dynasty, it was hard to interact with oreplica football shirtrs. Originally, Ye Hao did not doubt Long Qianyu, but thcheap football shirts sentence just made her feel suspicious in Ye Xin s heart. After all, replica football shirt heart United Kingdom football kits defending people cheap football shirts indcheap football shirtspensable. Of course I can see it. When I was a child, I once met a very strange uncle. cheap football kits said something inexplicably to me, saying that I am special, I will definitely have achievements when I grow up, and I will give it back. bayern munich jersey online I have a practice. Long Qianyu said, while taking out a practice from hcheap football shirts own baggage, which read football shirts Gas. Ye Big Broreplica football shirtr, thcheap football shirts cheap football shirts replica football shirt practice. In fact, replica football shirtre cheap football shirts one thing I have never told Ye Big Broreplica football shirtr. I am.irts scene, so that replica football shirt fire in Ye Yao s lower abdomen cheap football cheap borussia dortmund shirt shirts real madrid online store more vigorous. Come on, little goblin Ye. Yao threw replica football shirt woman in her arms into replica football shirt air, replica football shirtn she got up and flew to replica football shirt falling Mu Yunyao. football shirts two men slowly fell in replica football shirt air, and replica football shirt sunset glowed with fascinating light. Boiling replica football shirt pond, replica football shirt deep stream swims fcheap football shirtsh and I don t know. football shirts 286th chapter cheap football shirts reborn In replica football shirt night sky, a bright moon gradually rcheap football shirtses, replica football shirt silver moonlight cheap football shirts full United Kingdom football kits replica football shirt earth, replica football shirt water cheap football shirts sparkling, and Ye Yao cheap football shirts holding replica football shirt Mu Yunyao in hcheap football shirts arms. He hopes to stay at thcheap football shirts moment. Yao Yao, wait for me, I will help you to return home to replica football shirt Lord after one year Well, I always believe in you Mu Yunyao looked at Ye Yao s handsome face, and hcheap football shirts mouth gave a sweet smile. Since he had decided to follow thcheap football shirts man with a little mystery, he would have to rebel. he. Mcheap football shirtssy football shirts ignition light in replica football shirt jungle was accompanied by a call. It must have been that I haven t returned for a long time. Wang Lao, replica football shirty came to find me. Mu Yunyao got up and sorted out replica football shirt messy cloreplica football shirts, and all replica football shirt eyes were dcheap football shirtsappointing. football shirtsn go back, I am not convenient to appear in front United Kingdom football kits replica football shirtm now. Ye Yao kcheap football shirtssed Mu Yunyao s forehead and flew into replica football shirt river, looking at Mu Yunyao from afar. football shirts point United Kingdom football kits ignition cheap football shirts gradually clear, reflecting replica football shirt faces United Kingdom football kits everyone, it cheap football shirts Wang Lao and oreplica football shirtrs. Wang Lao s cloreplica football shirts are.


Brian Beckman is the actor who portrays Rupert Chatwin on The Magicians.
Brian Beckman was born on March 6, 1990 in Seattle, Washington. He grew up in Dallas, Texas, and, after studying Dramatic Arts in college, got his start in Los Angeles performing Stand-Up comedy. He has performed in regional theater productions throughout the USA, New Zealand and Australia. He is of German, English and Swedish descent.


Log In | Royer's flowers and gifts - Flowers, Plants and Gifts with same day delivery for all occasions.
*This is not a charge account. To set up an in-house charge with Royer's flowers and gifts please Contact Us.


The CloudCheckr CMP offers a single pane of glass view to help modern enterprises manage and optimize their public cloud. DevOps, SecOps, and FinOps teams from hundreds of global enterprises and service providers rely on CloudCheckr to manage their Azure, AWS, and Google Cloud investments so they can quickly optimize spend, eliminate waste, and improve security and compliance throughout their cloud journey.
CloudCheckr provides cost allocation, spend optimization, invoicing and chargebacks. Custom reports and alerts ensure governance and accountability as environments scale.
Cross-account dashboards provide sophisticated reporting for enterprise-wide inventory based on tags, geography, function and more to proactively optimize workloads.
CloudCheckr provides summary and detailed usage statistics for resources across AWS, Azure, and Google Cloud, offering actionable intelligence to right-size and scale services efficiently.
CloudCheckr enables users to save money, time, and effort to increase operational efficiencies with automated actions for your cloud.


To say I'm exhausted at the moment is an understatement.
I've just got back from Asia and have had to quickly switch gears from one focus to another.
It's been a little difficult at the moment but I always knew it would be a tough 2 months leading up to releasing the album.
Sometimes I ask myself why am I doing this? Is it worth the energy? There always seem to be walls in the way and after breaking through that, a mountain replaces it…which can be just a little discouraging. I'm also sure that most of my friends are probably sick of hearing me talk about the album. Mate, I'm sick of hearing myself talk about it too!
When that happens it's good to refocus and remember why I'm doing this.
As I've said previously, balancing a music career and a shirt-and-tie career hasn't been easy. With the duality comes a sense of not belonging entirely to either world.
That's why I do it! Life can be very mundane, but the thing that pulls me out of it is the pursuit of my passions. Maybe some people don't care, or don't like what I have to offer and that's fine but I feel proud of what I've done and not many people can say that they've accomplished one of the most important goals in their lives. Writing my own songs, recording them professionally and now having my own album in my hands is something that at the moment through my tiredness I probably don't realise its true significance, but I know in my heart it'll be something in the future that I'll be proud of and look back with no regrets.
SO…one final push, one last burst of energy. August 24th. I'm looking forward to it…but also looking forward to sleeping for the rest of that week.


With many conventional chemicals no longer in use, biorational solutions for crop production are rapidly gaining ground in standard application programs globally. Bolstering inherent plant defenses through better nutrition, Alltech's protection programs are designed to mitigate the use of fungicides by integrating conventional fungicides with biofungicides and nutritional supplements. Providing unique, natural solutions that improve plant health and resiliency against infection and disease is a goal of our Alltech Nutrigenomics Research Center.
Struggling to keep up with the swell of the earth's population without depleting its natural resources, farmers are forced to find innovative ways to produce more with less. Add the challenges posed by increasingly banned chemicals with the collective consumer voice demanding clean, traceable produce, and modern growers find themselves in a dilemma. With many conventional chemicals out of favor, natural products are rapidly gaining ground in standard application programs. Biological control methods coupled with precision nutrition strategies are now becoming an integral part of modern agronomic programs.
Pest control must find the delicate balance between the pest, host plant, and the production environment. Elimination of all pests; insects, pathogens, and weeds is impossible. Crop production is intimately tied to the environment and fertility plays a role in protection. Enhancing a plant's natural systemic resistance leads to improved crop quality and yields by reducing fungal disease occurrence.


Between the 5th and the 8th, Grand Duke Henri attended the 132nd IOC session in PyeongChang. Princess Nora, as a member of the International Olympic Committee, was also present. Additionally, the Grand Duke participated in events related to his role on the Olympic Solidarity Commission.
The Opening Ceremony will be held on the 9th, with the Games running until the 25th. Information from the Cour website.
Additionally, Grand Duke Henri took a visit to Olympic Village to meet with one of the Luxembourg athletes. More photos at Tageblatt.


How did these mentors impact your career? Every day they arrived ready to work at what they had fun doing. Always enjoying the applause but always listening and learning from the critique. No room for foul language, obscene gestures, sexual or racial cultures. Every day was a win!
What are your favorite dishes to prepare? I have no favorite but spontaneous creativity makes the meal exciting.
What are your favorite foods to eat? I'm slowing down on exotic foods but well seasoned proteins are the best.
Ingredients you could not live without? Salt and pepper.
What gives you pleasure on the job every day? Being part of diverse teams coming together with our common goals.
What was your worst disaster in the kitchen? While working at an airport location hotel on a late Sunday afternoon we got booked with a distressed flight: the chartered airplane bound for Europe was on the tarmac, 350 people in their seats with the doors buttoned-up for 7½ hours and no food. The charter group had to fix the airplane and was paying for hotel rooms and passed out vouchers to the customers for dinner ($35.00) and breakfast ($15.00). We needed to provide food to collect the vouchers. The booking wasn't confirmed until 7:30 pm with only me and one dishwasher left in the kitchen.
How did you turn it around? We set up double salad bars and hot buffets; thawed, opened, and peeled everything in the house. We had beef, chicken, pork, multiple veggie's and starches, salads and brined foods even deserts. And then I sent home the dishwasher. The group didn't show until 10:45 pm … and all but 2 customers went straight to their rooms for the night. We stayed ready until midnight and no one else came to eat. So I broke down the party, stored the product, cleaned the kitchen and prepped for the breakfast. Long story-short, set up another buffet to handle 350 people and had 3 customer eat. I had been in the kitchen for 26 hours and still had to attend all the Monday morning hotel meetings. I got home by 5:00 pm that night.
What advice do you have for new Culinarians? Learn and keep learning. After you learn, teach it to someone. Be an asset to your family and co-workers. And every day arrive ready to work and have fun at what you're doing.


SWK 2200 - Service Learning - 2 cr.
Provides opportunities for students to perform service to the community and to engage in reflective learning on the meaning of that service. The services performed by students in this class will be with established non-profit and service agencies.
SWK 2220 - Core Concepts of the Social Work Profession - 2 cr.
Provides an introduction to the foundation of social work as a profession, outlining the primary knowledge, competencies, values, and skills that characterize contemporary social work practice. This entry-level course surveys a variety of professional practice settings, allowing students the opportunity for career exploration within the social work profession.
SWK 2240 - Intro Professional Social Work ( Conceptions: VCSS - Social Science ) - 4 cr.
Provides an introduction to the foundation of social work as a profession and outlines the primary knowledge, values, and skills that characterize contemporary practice. This entry level course surveys a variety of professional practice settings allowing students an opportunity for career exploration within the social work profession. In addition, the course reviews the historical and philosophical background of social work. Students demonstrate increased awareness of personal values in exploring both the National Association of Social Workers (NASW) Code of Ethics, and the Benedictine values. The course emphasizes the experience of populations at risk and analyzes factors that constitute being at risk Through the use of weekly classroom lectures, discussion, readings, audio visual tools, and service learning assignments, students increase awareness of the value of promoting social and economic justice across all levels of practice.
SWK 2777 - Topics in Social Work - 1-4 cr.
SWK 3000 - Integrated Lab - 1-2 cr.
This integrative laboratory course facilitates students' understanding of their learning experience through critical reflection, lecture, experiential learning, small group interactions with faculty and peers, and community-centered experiences. Specific lab curriculum will address topical themes related to content from social work core courses in which students are concurrently enrolled. The laboratory also serves as an opportunity for the organization and development of a professional portfolio. Students work with specific frameworks and templates, but individualize the content of their portfolios and capture their unique learning experiences and professional goals. Each student will have the opportunity to frame a specific purpose for their portfolio (e.g. job search or graduate school admission), designing the portfolio in a manner that accentuates their personal identity, perspectives, knowledge and competencies as a professional social worker.
SWK 3131 - Stat Meth. Evidence Based Prac - 2 cr.
This course is the first of two courses designed to help students build a critical understanding of statistical concepts commonly used in the professional literature and for evidence-based practice. Students will learn to choose appropriate statistical analyses, conduct analyses, interpret findings, and communicate results clearly and effectively in the context of the helping professions. The concepts considered in this course include those related to the representation of information (descriptive statistics – mean, standard deviation, graphing) and those concepts related to drawing conclusions based on sample data (inferential statistics – probability, the normal distribution, hypothesis testing).
SWK 3132 - Statisticals Methods for Evidence-Based Practice II ( Foundations: VFMA - Mathematics ) - 2 cr.
This course is the second of two courses designed to help students build a critical understanding of statistical concepts commonly used in the professional literature and for evidence-based practice. Students will learn to choose appropriate statistical analyses, conduct analyses, interpret findings, and communicate results clearly and effectively in the context of the helping professions. The concepts considered in this course include those related to the representation of information (descriptive statistics – mean, standard deviation, graphing) and those concepts related to drawing conclusions based on sample data (inferential statistics – probability, the normal distribution, hypothesis testing). Prerequisite: SWK 3131.
SWK 3315 - Psychosocial Aspects of Aging - 4 cr.
Provides an overview of the aging individual within a social context. The focus is on characteristics of today's older adult cohort, psychological processes in late life, the social context in which older adults live, and society's response to older adults. Topics include: demographics, stereotypes and attitudes, research methods, theories of development, sensing and responding to the environment, cognitive processes, mental disorders and treatment, death and dying, sexuality, intimate relationships, family relationships, care giving, employment and retirement, finances, Social Security, social programs and political power of the older cohort.
SWK 3339 - Preparation for Field I - 1 cr.
Designed for junior social work students who will be participating in their field placement the following semester. Students have the opportunity to assess their interests and abilities, familiarize themselves with available field placement sites, explore and develop professional interviewing skills, complete necessary placement documentation, and interview with at least two prospective field placement sites. In class and out of class assignments will cover junior placement readiness, personal learning style, how to choose a field placement site, the role of the professional social worker through the lenses of the NASW Code of Ethics, the student application process for agency field placement, interviewing skills, contracting with an agency, developing a learning plan, the effective utilization of agency supervision, and getting the most out of the field placement.
SWK 3350 - Understanding Systems of Privilege and Oppression - 2 cr.
SWK 3360 - American Social Welfare Policy ( Integrations: VISS - Social Science ) - 4 cr.
SWK 3362 - Human Behavior in the Social Enivronment - 4 cr.
SWK 3370 - Generalist Social Work Practice - 0-2 cr.
The first of the four practice courses. This course provides students with the fundamental concepts, principles and skills necessary to engage in beginning generalist social work practice at the baccalaureate level. It explores the unique aspects and challenges of the social work profession, emphasizes the professional commitment and values necessary to provide service to culturally diverse and vulnerable populations, promotes understanding and use of a strength's practice perspective, examines the NASW Code of Ethics, and introduces the generalist intervention problem- solving method for practice with individuals, families, groups, and communities. Students develop interviewing skills and the professional use of self.
SWK 3380 - Child Welfare - 4 cr.
Examines the social welfare system as it affects children in American society. Students learn about critical factors that affect the needs of children and families in contemporary society. Child maltreatment, protective intervention, family preservation, family assessment and alternative substitute care are some of the key issues addressed. Students also learn about human service agencies and programs providing services to children and families. Course content includes theory, practices services and research in the field of child welfare.
SWK 3383 - Social Work with Individuals and Families - 0-4 cr.
The second course in the social work practice sequence. This course builds on the knowledge and skills taught in SWK 3370. Students continue to demonstrate mastery of interviewing skills and application of the generalist intervention model on amore advanced level working with individuals, couples, and families. This course includes information on and practice with: contemporary social work practice theories, social histories, individual and family assessments; assessment of suicidal potential and Duty to Warn; treatment plans; three generational genograms; and eco-maps and professional documentation skills. Students participate in role playing, client case analyses and ethical practice dilemmas. Prerequisites: SWK 2240 and 3370.
SWK 3385 - Social Work Research and Evaluation - 4 cr.
Qualitative and quantitative approaches to building evidence-based generalist social work practice. Students acquire knowledge and develop skills necessary for assessing the effectiveness and efficiency of practice interventions and social service programs. Course goals are to prepare students to be competent consumers, producers, and communicators of social science research. Course content includes: steps in practice research; ethnic-sensitive research practice; empirical research strategies for assessing micro, mezzo and macro social work interventions; developing and implementing a research project; and effective use of computer technology as an integral part of both research and human service practice. Recommended Prerequisite: PSY 3331.
SWK 3390 - Understanding Immigration/Refugee Issues in Social Work Practice - 2 cr.
Through the utilization of "participatory action research" learning, students will be immersed in social work practice issues and methods of service delivery with refugee and immigrant populations in present day Minnesota. Professional helping methods will focus on the unique aspects and challenges necessary for effective, culturally sensitive interventions. A bio-psycho- social-spiritual model of system assessment is infused. Students participate in classroom, community and service-learning opportunities. Students have the option to participate in a more in depth service experience through registering concurrently in SWK 3555 with permission of the instructors. Prerequisites: completion or concurrent enrollment in SWK 3362 and SWK 3370 or consent of instructor.
SWK 3395 - Social Work and Health Care - 4 cr.
SWK 3500 - Integrated Lab - 1-2 cr.
SWK 3555 - Field Practicum I - 1-16 cr.
This course is designed to provide students an entry level opportunity to work in a social service agency, institution or organization in the role of a social work intern, 8-15 hours per week (120-280 total hours) during spring semester of their junior year. In placement, students prepare for effective social work practice within a pluralistic society including sensitivity to race, color, gender, age, creed, ethnic or national origin, handicap, or political or sexual orientation. Learning contracts are individually designed to meet the specific needs of each student and the requirements and opportunities available in each human service organization. Students participate in a weekly 100 minute small group seminar which emphasizes student-centered learning, personal/professional introspection and an opportunity to exchange information on agency placement experiences. Various general practice skills will be explored, including values, ethics, professional documentation, and intervention strategies. Students will be graded on a P/F basis. Prerequisites: SWK 2240, 3362, 3370, and 3339.
SWK 3600 - Social Work and Technology - 2 cr.
Social Work and Technology is an online course for social work majors with any level of experience in the digital environment. Modern social workers use computers, internet, and software to meet the demands of their practice. Technological competency in social advocacy, privacy, confidentiality, use of critical thought, professional boundaries, and ethics will be applied. This course prepares learners for academic, professional, and practical success in emerging areas of practice.
SWK 3777 - Topics in Social Work - 0-4 cr.
SWK 3999 - Prep for Field I - 1-4 cr.
SWK 4000 - Integrated Lab III - 1-2 cr.
SWK 4415 - American Indian Families - 4 cr.
Traditional and contemporary concepts of American Indian families. The social and political changes that have impacted on Indian families will be identified and the degree of impact will be assessed.
SWK 4420 - Human Behavior & Amer Indian - 4 cr.
Understanding of human behavior, cultural issues and their implication for human service interaction with American Indian communities. A systems approach will be utilized to address the issues of individual, family, community and societal behavior.
SWK 4440 - Social Work Intervention with Groups - 0-4 cr.
SWK 4441 - Social Work/Community Systems - 4 cr.
Fourth course of the social work practice sequence. This course emphasizes the theories and skills necessary for beginning social work practitioners to bring about effectively planned change in community groups, organizations and institutions. The course content addresses: community theory and community practice skills; organizational and inter-organizational practice theory and skills; community organizing in a diverse society; macro social work research; and, theories and skills for professional development and macro level interventions. It provides students experiential learning opportunities. Prerequisites: SWK 2240 and SWK 3370.
SWK 4449 - Preparation for Field II - 1 cr.
Designed for senior social work students in preparation for their senior field experience. Students have the opportunity to assess their interests and abilities, familiarize themselves with available field placement sites, complete necessary paperwork and interview with at least three field placement supervisors. Prerequisites: SWK 2240, SWK 3370, and SWK 3383.
SWK 4470 - Independent Professional Project - 2-4 cr.
Senior project integrating coursework and field placement experience of the student's social work education. The project must address these nine basic foundation areas: values and ethics, diversity, social and economic justice, work with populations at risk, human behavior in the social environment, social welfare policies and services, social work practice, research, field practicum and internship. Prerequisite: SWK 3385 and admission to the social work program.
SWK 4500 - Integrated Lab - 1-2 cr.
SWK 4555 - Senior Field Practicum - 1-16 cr.
A 400- to 560-hour social work internship in a social service agency, institution or organization during the senior year. The practicum provides students the opportunity to integrate direct practice with acquired theoretical knowledge and skills. A bi-weekly seminar facilitates the integration of classroom content and direct practice experience. Attention is given to the relationship between the purposes, values, and principles expressed in the NASW Code of Ethics and the professional practice of social work. Prerequisites: all social work courses except SWK 4470, and admission to the field sequence.
SWK 4777 - Topics in Social Work - 0-4 cr.
Courses not a regular part of Social Work curriculum but taught because of a special need, interest or opportunity. Topics vary.
SWK 4999 - Independent Study - 0-4 cr.
Students select a particular topic of study with instructor. Individual student learning goals and method of evaluation are designed. Prerequisite: permission of instructor.


text:
Liberal Arts at UB
Meet the Dean
Meet Our New Faculty
4/15/20 Life in Buffalo
Undergraduate and Graduate Degrees
Combined Degree Programs
Undergraduate and Graduate Certificates
Why Choose the College?
Funding Your Degree
Women in Science & Engineering (WiSE)
Social Impact Fellows
Supplemental Instruction
Summer & Winter Intersessions
Phi Beta Kappa Honor Society
Faculty Achievement
Dean's Office Reorganization
PhD Excellence Initiative
Prospective Faculty and Staff
Department and Program Leadership
Submit College News
Dean's Advisory Council
2021 Stories
Current Faculty and Staff
College of Arts and Sciences >
Recent News >
2021 commencement awards and honors
Graduating students were recognized for their incredible achievements by the College of Arts and Sciences and SUNY.
On April 27, 2021, Dean Robin Schulze held a virtual ceremony to recognize the College of Arts and Sciences Outstanding Seniors. Every year, each of the College's departments chooses a graduating senior who has demonstrated exceptional diligence and promise during their time at the University at Buffalo.
College of Arts and Sciences' Outstanding Senior Award
Africana and American Studies - Aqeelah Croskery-Howard
Aqeelah Howard of Buffalo, N.Y., graduates cum laude with a bachelor of arts in African American studies and a minor in psychology.
Anthropology – Kevin P. Klier
Kevin Klier of Orchard Park, N.Y., graduates magna cum laude with a bachelor of arts in anthropology.
Art – Jillian A. Daniels
Jillian Daniels of Buffalo, N.Y., graduates cum laude with a bachelor of science in industrial design studies, special studies program, and a bachelor of arts in studio art with a general studio concentration.
Asian Studies – Yaling Lin
Yaling Lin of Queens, N.Y., graduates magna cum laude with a bachelor of arts in Asian studies and a minor in Korean.
Biological Sciences – Nicholas D. Kagelmacher
Nicholas Kagelmacher of North Tonawanda, N.Y., graduates summa cum laude with a bachelor of science in biological sciences.
Chemistry – Daniel R. Morphet
Daniel Morphet of Rochester, N.Y., graduates magna cum laude with a bachelor of science in chemistry.
Classics – Jake N. Pawlush
Jake Pawlush of Rensselaerville, N.Y., graduated summa cum laude with a bachelor of arts in classics with a concentration in classical languages and literature.
Communication – Austin C. Robar
Austin Robar of Potsdam, N.Y., graduates summa cum laude with a bachelor of arts in communication and a minor in psychology.
Communicative Disorders and Sciences – Corrinn J. Cincotta
Corrinn Cincotta of Penn Yan, N.Y., graduates summa cum laude with a bachelor of arts in speech and hearing science.
Economics – Peter D. Balogh
Peter Balogh of West Hempstead, N.Y., graduates summa cum laude with a bachelor of arts in economics and a minor in media study.
English – Alexis M. Banaszak
Alexis Banaszak of Clarence Center, N.Y., graduates summa cum laude with a bachelor of arts in English and global gender studies.
Environment and Sustainability – Meredith L. Seibold
Meredith Seibold of East Aurora, N.Y., graduates summa cum laude with a bachelor of science in environmental studies and minors in biological sciences and psychology.
Geography – Matthew P. Morris
Matthew Morris of Webster, N.Y., graduates summa cum laude with a bachelor of arts in geography with a concentration in earth systems science.
Geology – Noah T. Dobson
Noah Dobson of Horseheads, N.Y., graduates summa cum laude with a bachelor of science in geological sciences.
Global Gender and Sexuality Studies – Mosammet Asma
Mosammet Asma of Brooklyn, N.Y., graduates summa cum laude with a bachelor of arts in global gender studies and political science with a concentration in public law.
History – Mariah O. Agbonkpolo
Mariah Agbonkpolo of Montebello, N.Y., graduates summa cum laude with a bachelor of arts in history.
Jewish Thought - Anthony D. Catanese
Anthony Catanese of Amherst, N.Y., graduates summa cum laude with a bachelor of arts in Jewish studies.
Linguistics – Kenn B. Lewis
Kenn Lewis of Ruthven, Iowa, graduates summa cum laude with a bachelor of arts in linguistics with a concentration in language and cognition and minors in Spanish, American Sign Language and political science.
Mathematics – Cameron J. Watt
Cameron Watt of Grand Island, N.Y., graduates summa cum laude with a bachelor of science in mathematical physics.
Media Study – Cydney I. Teeter
Cydney Teeter of Elba N.Y., graduates summa cum laude with a bachelor of arts in political science and media study with a concentration in production.
Music – Richard C. Hoover
Richard Hoover of Grand Island, N.Y., graduates magna cum laude with a bachelor of music in music performance.
Philosophy – Tiana N. Gleason
Tiana Gleason of North Bay, Ontario, graduates summa cum laude with a bachelor of arts in philosophy and history and a minor in French.
Physics – Hannah M. Seppala
Hannah Seppala of Penn Yan, N.Y., graduates summa cum laude with a bachelor of science in physics and mathematics with a concentration in general study in mathematics.
Political Science – Mia L. Hanson
Mia Hanson of Big Flats, N.Y., graduates summa cum laude with a bachelor of arts in history and political science with a concentration in American politics and public affairs.
Psychology – Cassondra M. Lyman
Cassondra Lyman of Endicott, N.Y., graduates summa cum laude with a bachelor of science in psychology and a bachelor of arts in statistics.
Romance Languages and Literatures – Serena E. Strollo-DiCenso
Serena Strollo-DiCenso of Batavia, N.Y., graduates magna cum laude with a bachelor of arts in biological sciences and Italian.
Social Sciences Interdisciplinary – Clara E. Cook
Clara Cook of Sharon, Mass., graduates summa cum laude with a bachelor of arts in social sciences interdisciplinary with a cognitive science concentration and a bachelor of arts in computer science with minors in psychology and Spanish.
Sociology – Mia A. Forney
Mia Forney of Lewiston, N.Y., graduates summa cum laude with a bachelor of arts in psychology and criminology and a minor in addiction studies.
Theatre and Dance – Homeria L. Lubin
Homeria Lubin of Westtown, N.Y., graduates magna cum laude with a bachelor of arts in dance and psychology.
SUNY Chancellor's Award for Student Excellence
The 2021 SUNY Chancellor's Award for Student Excellence was awarded to two College students: Gianna Damico and Matthew Taboni.
This recognition is awarded to students who, according to the SUNY website, "have best demonstrated their integration of academic excellence with other aspects of their lives, which may include leadership, campus involvement, athletics, career achievement, community service or creative and performing arts." Students were recognized in a virtual ceremony on April 14, 2021.
Gianna Damico of Victor, N.Y., graduates with a bachelor of arts in anthropology and a bachelor of fine arts. Damico is a graphic designer and the only student in her BFA year with a solo gallery show. Gianna has worked for the Jacobs Institute since 2019, creating interactive 3D renderings of patient-specific vasculature structures and developing branding guidelines for the company.
She currently works as the designer and marketing strategist for the institute's podcast "Journey into the Future of Medicine with Bill Maggio." Gianna has also held the role of Blackstone LaunchPad and Techstars' graphic designer for seven semesters, shifting from creating marketing materials to managing social media roll-out plans and supervising a group of student interns.
Matthew Taboni of Tonawanda, N.Y., graduates with a BS in business administration and a minor in Nonprofit Leadership from the College of Arts and Sciences Social Sciences Interdisciplinary Program. Taboni has advocated for students as president of the Residence Hall Association (RHA); co-chair of Blue Table, UB's food pantry; assistant director of health and wellness for the Student Association; and in numerous other roles. He was named President of the Year by the National Association of College and University Residence Halls in recognition of his efforts to implement an endowment and scholarship to support residential students, start an alternative spring break program and expand RHA's regional and national involvement.
As co-chair of Blue Table, UB's food pantry, he kick-started creation of the pantry that now serves more than 50 students a week, including growing the pantry's stock to offer personal hygiene products. In his role as assistant director of health and wellness for SA, Taboni established UB's first Mental Health Awareness Week. Consisting of eight activities aimed at destigmatizing mental health issues and educating students on available resources, the success of the week-long program helped him win an Outstanding Campus Impact Award in May 2019.
Pease join us in congratulating this year's recipients for their accomplishments. We are so proud of our graduating class of 2021 and we are honored to welcome these individuals into the UB alumni family.
810 Clemens Hall
1Capen
CASet Support
UB Information Technology
Accessibility at UB