text:
\section{Introduction}
\label{intro} State-of-the-art quantum technology can trap and experiment with individual atoms, image brains as well as generate structural and dynamical information of biological macromolecules. Numerous applications arising from such emerging techniques involve controlling a large quantum ensemble, e.g., on the order of Avogadro number ($6\times 10^{23}$), by use of the same control field \cite{Li_PRA, Li_preprint, Li_thesis, Li_CDC2006, Li_NOLCOS2007, Li_IEEE, Brent1, Brent2, Skinner1, Kobzar, Pattern}. In many cases, the elements of the ensemble show variations in the values of parameters characterizing the system dynamics, and hence the system Hamiltonians of different elements of the ensemble are distinct. This phenomenon results in a dispersion in the system dynamics. For example, in magnetic resonance experiments, the spins of an ensemble may have large dispersions in their natural frequencies (Larmor dispersion), strength of the applied radio frequency (RF) field (RF inhomogeneity), and the dissipation rates of the spins \cite{Levitt, Tycko, Clare}. In solid state NMR spectroscopy of powder, the random distribution of the orientations of internuclear vectors of coupled spins within an ensemble leads to a distribution of coupling strengths \cite{Rohr}.

A canonical problem among these applications is to develop excitations (control signals) that will steer such an ensemble of systems with different dynamics from an initial state to a desired final state in finite time using a time-varying electromagnetic pulse. From the perspective of mathematical control theory, this is a very challenging state transfer problem because it requires steering a continuum of dynamical systems between points of interest in an infinite dimensional state space with the same control function. This motivates the study of \textbf{Ensemble Control} and the notion of ensemble controllability \cite{Li_PRA, Li_preprint, Li_thesis, Li_CDC2006, Li_NOLCOS2007, Li_IEEE} described as follows.

Consider a parameterized family of control systems
\begin{align}
\label{eq:dxs}& \frac{d}{dt}X(t,s)=F(X(t,s), u(t), t, s),\\
& X\in M\subset\mathbb{R}^{\rm n},\quad s\in D\subset\mathbb{R}^{\rm d},\quad u\in U\subset\mathbb{R}^{\rm m},\nonumber
\end{align}
where $F$ is a smooth function of its arguments and $D$ is a compact subset of $\mathbb{R}^{\rm d}$. Different values of the parameter $s$ in (\ref{eq:dxs}) correspond distinct members of the ensemble $X(t,s)$ showing variations, but we are constrained to use the same open-loop control $u(t)$ to steer the whole ensemble. The existence of such a control raises fundamental questions of \emph{ensemble controllability}. The formal definition will be given in Section \ref{sec:basics}. In practice, such control designs are called \emph{compensating pulse sequences} as they can compensate for or are insensitive to the dispersion in system dynamics. Typical applications include the design of excitation and inversion pulses in NMR spectroscopy in the presence of Larmor dispersion and RF inhomogeneity \cite{Skinner1, Kobzar, Pattern, Levitt, Tycko, Levitt1, Garwood}, the transfer of coherence between a coupled spin ensemble with variations in the coupling strengths \cite{Chingas}, and the construction of slice selective pulses in MRI, where some spins of the ensemble are excited or inverted while the others remain unaffected \cite{Silver, Rourke, Shinnar, Roux, Conolly, Mao, Rosenfeld}. Practical considerations, such as power constraints and signal or information losses due to relaxation effects, make it desirable to construct pulses that achieve a desired level of compensation with minimum energy or in the shortest possible time. These considerations give rise to problems in \emph{optimal control of ensembles}. These pulse design problems are widely studied in NMR spectroscopy on the subject of composite pulses that correct the dispersion in system dynamics \cite{Levitt, Levitt1, Garwood, Tycko1, Shaka2, Levitt2}. However, a systematic study of the design of compensating pulse sequences has been missing. The research in Ensemble Control will give explicit answers to the questions of when a compensation for the system dynamics is possible and how, if possible, to achieve the desired level of compensation.

More generally, ensemble control provides a framework to devise open-loop controls that are robust to parameter uncertainties, namely, they are insensitive to parameters. In many control applications, an accurate model is not available and systems have unknown parameters.  They either have some level of uncertainty or are not deterministic, however, there are instead bounds or distributions that describe these parameters. For example, systems biology models have numerous parameters, such as kinetic constants, which are unknown or only weakly constrained by existing experimental knowledge \cite{Gutenkunst}. In chaotic dynamics, the problem of parameter uncertainties is unavoidable in synchronizing chaotic systems \cite{Etemadi}. In such scenarios, one aims to design controllers that are robust to these parameter uncertainties. Subjects on robust control theory and sliding mode control are well studied to design controllers that can control or stabilize such systems using ``feedback'' \cite{Byrnes, Edwards}. While effective, these controllers are closed-loop and dependent on measurement of the system state - at times a difficult, or impossible requirement. Ensemble control, rather, provides a systematic framework for the design of open-loop controls that are immune to parameters and robust to uncertainties.

This paper is organized as follows. In the following section, we review ensemble control problems and the notion of ensemble controllability. We summarize our previous work of steering an ensemble of systems evolving on SO(3) to highlight the basics of ensemble control. Simple examples of ensemble control of linear systems are addressed to motivate the need of developing ensemble controllability conditions. In Section \ref{sec:theory}, we present our main results of necessary and sufficient controllability conditions for ensemble control of finite-dimensional time-varying linear systems. Finally, we study optimal control of an ensemble of harmonic oscillators as a demonstration of our main results as well as provide an insight of how ensemble control framework can be adopted to deal with systems with parameter uncertainties. We show that this system is ensemble controllable and the optimal control of such a system is pertinent to the study of time-frequency limited signals and prolate spheroidal wave functions (pswf). We present both analytical and numerical solutions.

\section{Basics of Ensemble Control}
\label{sec:basics} In this section, we review the basics of the ensemble control and the notion of ensemble controllability \cite{Li_thesis, Li_IEEE}. Ensemble Control involves problems of simultaneously manipulating a continuum of dynamical systems with different internal and external dynamics by use of the same open-loop control input. The general form of an ensemble control system is shown as in (\ref{eq:dxs}), and standard linear and bilinear ensemble control systems are of the respective forms
\begin{align*}
\frac{d}{dt}X(t,s) &= A(t,s)X(t,s)+B(t,s)u(t),\\
\frac{d}{dt}X(t,s) &= \Big[A(t,s)+\sum_{i}u_i(t) B_i(t,s)\Big]X(t,s).
\end{align*}
The investigation of what kind of dispersions $s$ in the system dynamics can and cannot be corrected is a subject of fundamental and practical importance. This raises interesting questions of ensemble controllability.

\begin{figure}[t]
\centering
\begin{tabular}{cc}
{\small (a)}\includegraphics[scale=0.7]{ec.eps}&\,
{\small (b)}\includegraphics[scale=0.70]{poly1.eps}
\end{tabular}
\caption{\small (a) Illustration of the idea of ensemble controllability. $X_0(s)$ and $X(T,s)$ are two points on the function space $L_{\infty}(D,M)$, which correspond to two functions on the $s-X$ domain. If there exists a $u(t)$ that steers the system (\ref{eq:dxs}) from an initial point $X_0(s)$ to $X(T,s)\in\mathbf{B}_{\varepsilon}(X_F(s))$, for some finite time $T$, then the system is ensemble controllable. (b) The idea of the polynomial approximation, where the constant function $\theta(\epsilon)=1$ is approximated by an odd polynomial of degree $2n+1$, i.e., $\sum_{k=0}^{n}c_k\varepsilon^{2k+1}\approx 1$ for all $\varepsilon\in[1-\delta,1+\delta]$.
\label{fig:ec}
\end{figure}

\begin{defn}
\label{def:controllability}
Consider a family of control systems as in (\ref{eq:dxs}). This family is called {\it ensemble controllable} on the function space $L_{\infty}(D,M)$ if and only if for all $\varepsilon>0$ and for all $X_0,X_F\in L_{\infty}(D,M)$, there exists $T>0$ and an open-loop piecewise-continuous control $u:[0,T]\rightarrow U$ such that starting from any initial state $X_0(s)=X(0,s)$, the final state $X_T(s)=X(T,s)\in L_{\infty}(D,M)$ satisfies $\|X_T-X_F\|_{\infty}\leq\varepsilon$.
\end{defn}

Note that $T\in(0,\infty)$ may depend on $\varepsilon$, $D$, and the bound of the control amplitude. The idea of ensemble controllability is illustrated in Figure \ref{fig:ec}(a).

\begin{rmk}
Note that ensemble control systems are different from distributed parameter systems where the systems are governed by partial differential equations \cite{Stavroulakis}. The difference can be seen from the fact that in (\ref{eq:dxs}) there is no partial derivative term with respect to the second variable, $\frac{\partial x}{\partial s}$. Ensemble control is also different from robust control in the sense that in ensemble control one is interested in devising an open-loop control signal, $u(t)$, that is insensitive to the state variable $x$ and the parameter $s$, while in most cases of robust control designs, constructing a closed-loop feedback control is of interest \cite{Feintuch}.
\end{rmk}

\subsection{The Prototype of Ensemble Control Problem}
The prototype of ensemble control problem was presented in our previous work, that is, control of a continuum of systems on SO(3) \cite{Li_thesis, Li_IEEE},
$$\dot{X}(t,\omega,\epsilon)=\Big[\omega\Omega_z+\epsilon u\Omega_y+\epsilon v\Omega_x\Big]X(t,\omega,\epsilon),\quad X(0,\omega,\epsilon)=I,$$ where $X\in SO(3)$, $(\omega,\epsilon)\in D=[a,b]\times[c,d]\in\mathbb{R}\times\mathbb{R}^{+}$, and $\Omega_x,\Omega_y,\Omega_z$ are generators of rotation around $x,y$, and $z$ axis, respectively. This system is ensemble controllable on the function space $\mathcal{S}(D)$, the set of all SO(3) valued measurable functions defined on $D$. We also showed through the study of this system that generating higher order Lie brackets by use of the control vector fields which carry higher order powers of the dispersion parameters is a key to investigating ensemble controllability. As a result, the infinite dimensional control problem can be translated to the problem of polynomial approximation. In this example, we can synthesize new generators of rotations by successive Lie bracketing of $\{\omega\Omega_z,\epsilon\Omega_y,\epsilon\Omega_x\}$ and then produce $(\omega,\epsilon)$-dependent evolutions of the form,
$$\exp\bigg\{\sum_{l}\Big(\sum_{k}c_{kl}\omega^{k}\Big)\epsilon^{2l+1}\Omega_x\bigg\}.$$
This evolution can then be used to approximate the desired evolution, $\exp\{\theta(\omega,\epsilon)\Omega_x\}$, with dependence on $(\omega,\epsilon)$ by appropriate choice of $c_{kl}$'s, where $\theta(\omega,\epsilon)$ is a measurable function over $D$.

The idea of polynomial approximation is shown in Figure \ref{fig:ec}(b). Based on this concept, a result on ensemble controllability of a simple linear system is immediately evident.
\begin{ex}
\rm An ensemble of time-invariant linear systems $$\dot{X}(t,s)=A X(t,s) + sBu(t),$$
where $s\in[s_1,s_2]\subset\mathbb{R}$, $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times m}$, is not ensemble controllable \cite{Li_IEEE}.\\
\end{ex}
{\it Proof.} Observe that $$sBu=\sum_{k=1}^m su_k b^k,$$ where $b^k$ is the $k$th column of $B$. We can think of $b^k$ as constant vector fields that generate translations. Since $b^k$ all commute to one another, their Lie brackets do not generate terms carrying higher powers of the dispersion parameter $s$. Therefore, the system is not ensemble controllable. Note that this result can also be easily verified by applying the variation of constants formula.\hfill $\Box$

The above example shows that the inability to synthesize higher powers of the dispersion parameter by successive Lie bracketing makes this system not ensemble controllable. The following fact characterizes a necessary controllability condition for a family of single-input linear systems.

\begin{thm}
Consider a family $(A(s),b(s))$ of linear single-input controllable systems
$$\dot{X}(t,s)=A(s)X(t,s)+b(s)u(t),$$
where $s$ takes values from a finite set $D\subset\mathbb{R}$, $A(s)\in\mathbb{R}^{n\times n}$, $b(s)\in\mathbb{R}^{n}$, and $u:[0,T]\rightarrow\mathbb{R}$, $T<\infty$. If the system is controllable, then there are no repeated eigenvalues of $A(s)$ for all $s$ \cite{Li_thesis}.
\end{thm}

\begin{rmk}
For the linear single-input controllable system $(A(s),b(s))$, it is required that $rank\ A(s)\geq n-1$, and the equality holds when $A(s)$ contains at least one eigenvalue equal to $0$ since $\det A(s)=0$. However, the condition that there are no repeated eigenvalues among all $A(s)$ restricts the ensemble to contain at most one singular $A(s)$.
\end{rmk}

These examples motivate the need of developing general controllability conditions. In the following section, we investigate necessary and sufficient controllability conditions for an ensemble of finite dimensional time-varying linear systems. We show that these conditions are associated with the singular value representation of the linear operator characterizing system dynamics.

\section{Linear Operators and Ensemble Controllability}
\label{sec:theory} The main result of this article is to provide the necessary and sufficient controllability conditions for an ensemble of general finite dimensional time-varying linear systems of the form
\begin{eqnarray}
\label{eq:linearTV}
\frac{d}{dt}X(t,s)=A(t,s)X(t,s)+B(t,s)u(t),
\end{eqnarray}
where $A(t,s)$ and $B(t,s)$ are $n\times n$ and $n\times m$ matrices, respectively, whose elements are complex-valued $L_2$ functions defined on a compact set $D=[0,T]\times [s_1,s_2]\subset\mathbb{R}^2$, denoted as $A\in L_2^{\rm n\times n}(D)$ and $B\in L_2^{\rm n\times m}(D)$. In this case, the ensemble controllability conditions are associated with when there exists an open-loop control, $u\in L_2^{\rm m}[0,T]$, which will steer the whole ensemble $X$ between points of interest in the function space $L_2^{\rm n}[s_1,s_2]$.

Let's start with some standard control theoretic analysis for the above system. Consider a fixed finite time $T$, starting from an initial state $X(0,s)$ we have by the variation of constants formula
\begin{eqnarray}
\label{eq:XT}
X(T,s)=\Phi(T,0;s)X(0,s)+\int_0^T \Phi(T,\tau; s)B(\tau,s)u(\tau)d\tau,
\end{eqnarray}
where $\Phi(t,0;s)$ is the transition matrix for $\frac{d}{dt}X(t,s)=A(t,s)X(t,s)$. It is known that for each $s\in[s_1,s_2]$
\begin{align*}
\Phi(t,0;s)=I &+ \int_{0}^{t}A(\sigma_1,s)d\sigma_1+\int_{0}^{t}A(\sigma_1,s) \int_{0}^{\sigma_1}A(\sigma_2,s)d\sigma_2 d\sigma_1\\
&+ \int_{0}^{t}A(\sigma_1,s)\int_{t_0}^{\sigma_1}A(\sigma_2,s)\int_{t_0}^{\sigma_2} A(\sigma_3,s) d\sigma_3 d\sigma_2 d\sigma_1+\ldots,
\end{align*}
is the Peano-Baker series which is uniformly convergent \cite{Brockett}. Given a desired target state $X_F(s)$ and an $\varepsilon>0$, we wish to find a control $u(t)$ such that $\|X(T,s)-X_F(s)\|_2\leq\varepsilon$. A simple manipulation of \eqref{eq:XT} yields
\begin{eqnarray}
\label{eq:xi}
\int_0^T\Phi(0,\tau;s)B(\tau,s)u(\tau)d\tau=\xi(s),
\end{eqnarray}
where $$\xi(s)=\Phi(0,T;s)X_F(s)-X(0,s),$$
and $\xi(s)$ is known as long as the initial and target states are specified. Note that for now we consider $X(T,s)=X_F(s)$. Let $\mathcal{H}_1=L_2^{\rm{m}}[0,T]$ be the set of $m$-tuples, whose elements are complex vector-valued square-integrable measurable functions defined on $0\leq t\leq T$, with an inner product defined by
\begin{eqnarray}
\label{eq:inner1}
\< g,h\> _{\mathcal{H}_1} = \int_0^T g^{\dagger}(t) h(t)dt,
\end{eqnarray}
where $\dagger$ denotes the conjugate transpose. Let $\mathcal{H}_2=L_2^{\rm{n}}[s_1,s_2]$ equipped with an inner product
\begin{eqnarray}
\label{eq:inner2}
\< p,q\> _{\mathcal{H}_2} = \int_{s_1}^{s_2} p^{\dagger}(s) q(s)ds.
\end{eqnarray}
It is clear that, with well-defined addition and scalar multiplication, $\mathcal{H}_1$ and $\mathcal{H}_2$ are separable Hilbert spaces. Now we define $L:\mathcal{H}_1\rightarrow\mathcal{H}_2$ by
$$(Lu)(s)=\int_0^T \Phi(0,\tau;s)B(\tau,s)u(\tau)d\tau,$$
and hence from (\ref{eq:xi})
\begin{eqnarray}
\label{eq:Lu}
(Lu)(s)=\xi(s).
\end{eqnarray}
Denote $\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ as the set of bounded linear operators from  $\mathcal{H}_1$ to $\mathcal{H}_2$. It is then easy to verify that $L\in\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ (see Appendix \ref{apd:Lcompact}). Consequently, $L$ has the adjoint $L^*$ satisfying
$$\left \< f,Lu\right \> _{\mathcal{H}_2}=\left \< L^*f,u \right \> _{\mathcal{H}_1},\quad\forall f\in\mathcal{H}_2, u\in\mathcal{H}_1.$$
This gives, by (\ref{eq:inner1}), (\ref{eq:inner2}), and the Fubini's theorem,
\begin{align*}
\int_{s_1}^{s_2}f^{\dagger}(s)\left(\int_0^{T}\Phi(0,\tau;s)B(\tau,s)u(\tau)d\tau\right)ds &=\int_0^T\left(\int_{s_1}^{s_2}\left[B^{\dagger}(\tau,s)\Phi^{\dagger}(0,\tau;s)f(s)\right]^{\dagger}ds\right)u(\tau)d\tau\\
&=\int_0^T\left(L^* f\right)^{\dagger}u(\tau)d\tau.
\end{align*}
Therefore,
\begin{eqnarray}
\label{eq:L*}
(L^*f)(t)=\int_{s_1}^{s_2}B^{\dagger}(t,s)\Phi^{\dagger}(0,t;s)f(s)ds.
\end{eqnarray}
The study of ensemble controllability for this system boils down to the problem of solving the inverse problem as in (\ref{eq:Lu}). With the above analysis, we show the main result of this paper.

\begin{thm}
\label{thm:main}
Consider a parameterized family of finite dimensional time-varying linear systems
\begin{eqnarray}
\label{eq:maineq}
\frac{d}{dt}X(t,s)=A(t,s)X(t,s)+B(t,s)u(t),
\end{eqnarray}
where $X:D\rightarrow\mathbb{R}^{n}$, $D=[0,T]\times[s_1,s_2]\subset\mathbb{R}^2$, and $u\in L_2^{\rm m}$; $A\in L_2^{\rm n\times n}(D)$, $B\in L_2^{\rm n\times m}(D)$, and time-varying $(A,B)$ are controllable pairs for all $s$. This family is ensemble controllable on the function space $L_2^{\rm n}[s_1,s_2]$ if and only if
\begin{align}
{\rm (i)} & \quad\sum_{n=1}^{\infty}\frac{|\< \xi,\nu_n\>|^2}{\sigma_n^2}<\infty\\
{\rm (ii)} & \quad\xi\in\overline{\mathcal{R}(L)},
\end{align}
where $(\sigma_n,\mu_n,\nu_n)$ is a singular system of $L$. Moreover, the control law
$$u=\sum_{n=1}^{\infty}\frac{1}{\sigma_n}\< \xi,\nu_n\> \mu_n,$$
satisfies $$\<u,u\>\leq\<u_0,u_0\>$$ for all $u_0\in\mathcal{U}$ and $u_0\neq u$, where $\mathcal{U}=\big\{v\,|\,Lv=\xi\ \text{with (i) and (ii)}\big\}$. In addition, $$u_N=\sum_{j=1}^{N(\varepsilon)}\frac{1}{\sigma_j}\< \xi,\nu_j\> \mu_j,$$
is the best approximation of $u$ for a given $\varepsilon>0$, namely, $u_N$ is such that $\|\xi-Lu_m\|\leq\varepsilon$ for all $m\geq N(\varepsilon)$, where $$u_m=\sum_{j=1}^{m}\frac{1}{\sigma_j}\< \xi,\nu_j\> \mu_j.$$ \hfill$\blacksquare$
\end{thm}

Before proving the above theorem, we need the following preliminary tools.

\subsection{Preliminaries}
\begin{prop}
\label{prop:Lcompact}
The operator $L:\mathcal{H}_1\rightarrow\mathcal{H}_2$ defined by
$$(Lu)(s)=\int_0^T \Phi(0,\tau;s)B(\tau,s)u(\tau)d\tau,$$ is compact.
\end{prop}
{\it Proof.} See Appendix \ref{apd:Lcompact}.

\begin{thm}[Spectral Theorem \cite{Porter}]\label{thm:spectral}
Let $X$ be a Hilbert space and $A:X\rightarrow X$ be a compact self adjoint operator. Then there exist a, possibly finite, sequence $\{\mu_n\}$ of nonzero eigenvalues of $K$ and a corresponding orthonormal sequence $\{\phi_n\}$ of eigenvectors such that for each $x\in X$, $Ax=\sum_n\mu_n\<x,\phi_n\>\phi_n$, where the sum is a finite sum if there are only finitely many eigenvalues. Moreover if $\{\mu_n\}$ is an infinite sequence, then it converges to zero.
\end{thm}

\begin{defn}[Singular System \cite{Gohberg}]
\label{def:singular}
Let $X$ and $Y$ be Hilbert spaces and $K:X\rightarrow Y$ be a compact operator. If $(\sigma_n^2,\nu_n)$ is an eigensystem of $KK^*$ and $(\sigma_n^2, \mu_n)$ is an eigensystem of $K^*K$, namely, $KK^* \nu_n=\sigma_n^2 \nu_n$ and $K^* K \mu_n=\sigma_n^2\mu_n$, where $\sigma_n>0$ ($n\geq 1$), and the two systems are related by the equations
\begin{eqnarray}
\label{eq:singular}
K \mu_n=\sigma_n \nu_n,\quad K^* \nu_n=\sigma_n\mu_n,
\end{eqnarray}
we say that $(\sigma_n,\mu_n,\nu_n)$ is a singular system of $K$.
\end{defn}

\begin{rmk}
Since $K$ is compact and then we know $KK^*$ and $K^*K$ are both compact, self-adjoint, and nonnegative operators. Thus by the Spectral theorem, $K^*K$ can be represented in terms of its positive eigenvalues, namely, we have $K^*Kx=\sum_{n}\sigma_n^2\<x,\mu_n\>\mu_n$ for all $x\in X$. Moreover, since $K^*K\mu_n=\sigma_n^2\mu_n$, the relations as in \eqref{eq:singular} follow by taking $\nu_n=\frac{1}{\sigma_n}K\mu_n$. This can be treated as the infinite dimensional analogue of the singular value decomposition of a matrix.
\end{rmk}

The above definition immediately gives rise to the following results.

\begin{prop}
\label{prop:singular}
Let $X$ and $Y$ be Hilbert spaces and $K:X\rightarrow Y$ be a compact operator. If $(\sigma_n,\mu_n,\nu_n)$ is a singular system of $K$, then
\begin{enumerate}
\item[(i)] $\{\mu_n\}$ is an orthonormal basis of $\overline{\mathcal{R}(K^*)}$,
\item[(ii)] $\{\nu_n\}$ is an orthonormal basis of $\overline{\mathcal{R}(K)}$.
\end{enumerate}
\end{prop}
{\it Proof.} (i) Since $\mu_n=\frac{1}{\sigma_n^2}K^*K\mu_n\in\mathcal{R}(K^*K)$ and $K^*K$ is compact and self adjoint, the Spectral theorem, $K^*K x=\sum_j\sigma_j^2\<x,\mu_j\>\mu_j$, for all $x\in X$, implies that $span\{\mu_n\}$ is dense in $\mathcal{R}(K^*K)$. It follows that $\overline{span}\{\mu_n\}=\overline{\mathcal{R}(K^*K)}=\overline{\mathcal{R}(K^*)}$. (ii) can be proved similarly. \hfill$\Box$

\begin{thm}[Singular value expansion \cite{Gohberg}]\label{thm:singular}
Let $X$ and $Y$ be Hilbert spaces, $K:X\rightarrow Y$ be a compact operator and $\{(\sigma,\mu_n,\nu_n)\ |\ n\in\Delta\}$ be a singular system for $K$. Then
$$Kx=\sum_{n\in\Delta}\sigma_n\<x,\mu_n\>\nu_n,\quad K^* y=\sum_{n\in\Delta}\sigma_n\<y,\nu_n\>\mu_n,$$
for all $x\in X$, $y\in Y$. In particular, if
$$K_nx=\sum_{j=1}^{n}\sigma_j\<x,\mu_j\>\nu_j,\quad x\in X,$$
and $K$ is of infinite rank, namely, $\Delta=\mathbb{N}$, then
$$\|K-K_n\|\leq\sup_{j>n}\sigma_j\rightarrow 0 \quad as\quad n\rightarrow\infty.$$
\end{thm}
{\it Proof.} Since, by Proposition \ref{prop:singular}, $\{\nu_n\}$ is an orthonormal basis of $\overline{\mathcal{R}(K)}$, the Fourier expansion gives for all $x\in X$
$$Kx=\sum_{j}\<Kx,\nu_j\>\nu_j=\sum_j\<x,K^*\nu_j\>\nu_j=\sum_j\sigma_j\<x,\mu_j\>\nu_j.$$
The other part can be shown similarly. Also, we have
\begin{align*}
\|(K-K_n)\,x\|^2=\|\sum_{j>n}\sigma_j\<x,\mu_j\>\nu_j\|^2=\sum_{j>n}|\sigma_j|^2|\<x,\mu_j\>|^2
\leq\sup_{j>n}\ \sigma_j^2\ \|x\|^2.
\end{align*}
Therefore, $$\|K-K_n\|\leq\sup_{j>n}\ \sigma_j\rightarrow 0\quad as\quad n\rightarrow\infty,$$
since $\sigma_n^2$ is an eigenvalue of $K^* K$ and $\sigma_n^2\rightarrow 0$ as $n\rightarrow\infty$. \hfill$\Box$

\begin{thm}[Riesz-Fischer Theorem \cite{Gohberg}]\label{thm:Riesz}
Let $\{u_1,u_2,\ldots\}$ be an orthonormal set in a Hilbert space $X$ and let $\{\alpha_n\}$ be a sequence of scalars. Then
$$\sum_{n=1}^{\infty}|\alpha_n|^2 \quad\text{converges if and only if} \quad\sum_{n=1}^{\infty}\alpha_n u_n \quad\text{converges},$$
and, in that case,
$$\alpha_n=\< x,u_n\> \quad\forall\, n\in\mathbb{N}, \quad\text{where}\quad x=\sum_{n=1}^{\infty}\alpha_n u_n.$$
\end{thm}

\begin{thm}[Minimum Norm \cite{Luenberger}]
\label{thm:minnorm} Let $G$ and $H$ be Hilbert spaces and let $A\in\mathcal{B}(G, H)$ with range closed in $H$. Then, the vector $x$ of minimum norm satisfying $Ax=y$ is given by $x=A^{*}z$ where $z$ is any solution of $AA^{*}z=y$ and $A^{*}$ is the adjoint operator of $A$.
\end{thm}

{\it Proof of Theorem \ref{thm:main}.} {\it Necessity:} Since $L$ is compact by Proposition \ref{prop:Lcompact}, we then let $(\sigma_n,\mu_n,\nu_n)$ be a singular system of $L$. Now suppose that $u(t)$ is a solution to (\ref{eq:Lu}) and $u\in\mathcal{H}_1$. Then
$$\< \xi,\nu_n\> =\< Lu,\nu_n\> =\< u,L^* \nu_n\> =\sigma_n\< u,\mu_n\> .$$
Hence, $\< u,\mu_n\> =\frac{1}{\sigma_n}\< \xi,\nu_n\> .$ By the application of Bessel's inequality to $u$ and the orthonormal system $\{\mu_n\}$, we have
$$\sum_{n=1}^{\infty}\frac{|\< \xi,\nu_n\> |^2}{\sigma_n^2}=\sum_{n=1}^{\infty}|\< u,\mu_n\> |^2\leq\|u\|^2<\infty.$$
Furthermore, for any $\eta\in\mathcal{N}(L^*)$, i.e., $L^*\eta=0$ and $\eta\in\mathcal{H}_2$, we have
$$\< \xi,\eta\> =\< Lu,\eta\> =\< u,L^*\eta\> =0.$$
Therefore, $\xi\in\mathcal{N}(L^*)^{\perp}=\overline{\mathcal{R}(L)}.$

{\it Sufficiency:} Conversely, we suppose that both conditions (i) and (ii) are satisfied. Let $\alpha_n=\frac{1}{\sigma_n}{\<\xi,\nu_n\>}$, hence $\sum_{n=1}^{\infty}|\alpha_n|^2$ converges according to the condition (i). By the Riesz-Fischer theorem, there exists a $u\in\mathcal{H}_1$ so that
\begin{eqnarray}
\label{eq:u_RF}
u=\sum_{n=1}^{\infty}\alpha_n \mu_n,
\end{eqnarray}
and then
\begin{eqnarray}
\label{eq:alphan}
\alpha_n=\<u,\mu_n\>=\frac{1}{\sigma_n}\<\xi,\nu_n\>.
\end{eqnarray}
Note that $u\in\mathcal{N}(L)^{\perp}\subset\mathcal{H}_1$ since $\{\mu_n\}$ spans $\overline{\mathcal{R}(L^*)}$. Hence, from Theorem \ref{thm:singular} and (\ref{eq:alphan}), we obtain
\begin{eqnarray}
\label{eq:Lu1}
Lu=\sum_{n=1}^{\infty}\sigma_n\<u,\mu_n\>\nu_n=\sum_{n=1}^{\infty}\<\xi,\nu_n\>\nu_n.
\end{eqnarray}
Since $\xi\in\overline{\mathcal{R}(L)}$ by the condition (ii) and $\{\nu_n\}$ spans $\overline{\mathcal{R}(L)}$ by Proposition \ref{prop:singular}, $\xi$ can be expressed by the Fourier expansion, $$\xi=\sum_{n=1}^{\infty}\<\xi,\nu_n\>\nu_n.$$
Combining this with \eqref{eq:Lu1} and \eqref{eq:u_RF}, we conclude that
\begin{eqnarray}
\label{eq:control}
u=\sum_{n=1}^{\infty}\frac{1}{\sigma_n}\<\xi,\nu_n\>\mu_n,
\end{eqnarray}
$u\in\mathcal{N}(L)^{\perp}\subset\mathcal{H}_1$, is a solution of (\ref{eq:Lu}).
We now put
$$u_N=\sum_{j=1}^{N}\frac{1}{\sigma_j}\<\xi,\nu_j\>\mu_j,$$
where $N\in\mathbb{N}$. By the fact that $\{\mu_n\}$ is an orthonormal sequence, we have
$$\|u-u_N\|^2=\sum_{j=N+1}^{\infty}\frac{1}{\sigma_j^2}\,|\<\xi,\nu_j\>|^2\rightarrow 0 \quad as \quad N\rightarrow\infty,$$
and then, by Theorem \ref{thm:singular},
$$\|Lu-Lu_N\|^2=\sum_{j=N+1}^{\infty}\sigma_j^2\,|\<u,\mu_j\>|^2\rightarrow 0 \quad as \quad N\rightarrow\infty.$$
Therefore, given any $\varepsilon>0$, we can find $u_N$ such that $\|\xi-Lu_N\|\leq\varepsilon$ for an appropriate choice of $N=N(\varepsilon)$. Moreover, since $u\in\mathcal{N}(L)^{\perp}=\overline{\mathcal{R}(L^*)}$, according to Theorem \ref{thm:minnorm}, $u$ is of minimum norm satisfying (\ref{eq:Lu}), that is,
$$\<u,u\>\leq\<u_0,u_0\>$$
for all $u_0\in\mathcal{U}$ and $u_0\neq u$, where $\mathcal{U}=\{v\,|\,Lv=\xi\ \text{with (i) and (ii)}\}$.  $\hfill\Box$

\begin{rmk}
The controllability condition (i) implies that an ensemble control law exists if and only if the Fourier coefficients $\<\xi,\nu_n\>$ with respect to the singular functions $\nu_n$ decay fast enough relative to the singular values $\sigma_n$. Note that $(\sigma_n^2,\nu_n)$ is an eigensystem of the Hermitian operator $LL^*:\mathcal{H}_2\rightarrow\mathcal{H}_2$ defined by
$$(LL^* z)(s)=\int_{s_1}^{s_2}\int_0^T\Phi(0,\tau;s)B(\tau,s)B^{\dagger}(\tau,\sigma)\Phi^{\dagger}(0,\tau;\sigma)z(\sigma) \,d\tau d\sigma,$$
an analogy of the classical controllability Gramian. The ensemble controllability condition (i) coincides with the so called Picard criterion in the literature of integral equations.

\end{rmk}

\begin{cor}
For any given initial state $X_0(s)=X(0,s)$, the ensemble control law $u$ as in (\ref{eq:control}) does not depend continuously on the target state $X_F(s)$.
\end{cor}
{\it Proof.} Suppose that $\delta_n$ is a perturbation of $\xi$ with $\delta_n\rightarrow 0$ as $n\rightarrow\infty$ and that $\tilde{\xi}_n=\xi+\delta_n$. Let $u$ and $\tilde{u}_n$ are solutions to the integral equations $Lu=\xi$ and $L\tilde{u}_n=\tilde{\xi}_n$, respectively. We now consider a qualified perturbation $\delta_n=a\sqrt{\sigma_n}\nu_n$, where $a\in\mathbb{R}$. It is clear that $\delta_n\rightarrow 0$ as $n\rightarrow\infty$ because $\{\nu_n\}$ is an orthonormal basis and $\sigma_n\rightarrow 0$ as $n\rightarrow\infty$. Thus, we have
$$\|\tilde{\xi}_n-\xi\|=|a|\sqrt{\sigma_n}\rightarrow 0 \quad {\it as} \quad n\rightarrow\infty.$$
However,
$$\tilde{u}_n=\sum_{n=1}^{\infty}\frac{1}{\sigma_n}\<\tilde{\xi}_n,\nu_n\>\mu_n=u+\frac{a}{\sqrt{\sigma_n}}\ \mu_n,$$
and hence
$$\|\tilde{u}_n-u\|=\frac{|a|}{\sqrt{\sigma_n}}\rightarrow\infty\quad {\it as} \quad n\rightarrow\infty.$$
Therefore, the control $u$ doesn't depend continuously on $\xi(s)$ and thus neither on the target state $X_F(s)=\Phi(T,0;s)\left[\xi(s)+X_0(s)\right]$. $\hfill\Box$

\section{Optimal Control of An Ensemble of Harmonic Oscillators}
\label{sec:optimal}
In this section, we study in detail the ensemble control of a family of harmonic oscillators that demonstrates our main results in Section \ref{sec:theory}. We show that this system is ensemble controllable and derive an analytical optimal control law. The analysis of this ensemble control system is related to the study of time-frequency limited signals and prolate spheroidal wave functions. Alternatively, this problem can be viewed as control of a harmonic oscillator with parameter uncertainty, where the frequency is unknown but only its range is provided.

\subsection{Unconstrained Optimal Ensemble Control}
We first look at a fixed end-point optimal ensemble control problem without constraints on the control signals.

\begin{problem}
\label{prob:harmonic1} \rm Consider an ensemble of harmonic oscillators with a variation in their natural frequencies
\begin{equation}
\label{eq:harmonic}
\frac{d}{dt}\left[\begin{array}{c}x(t,\omega)\\y(t,\omega)\end{array}\right] =\left[\begin{array}{cc}0&-\omega\\ \omega&0\end{array}\right] \left[\begin{array}{c}x(t,\omega)\\y(t,\omega)\end{array}\right]
+\left[\begin{array}{c}u(t) \\ v(t)\end{array}\right],
\end{equation}
where $\omega\in D=[\omega_1,\omega_2]\subset\mathbb{R}$, $X(\cdot,\omega)=(x(\cdot,\omega),y(\cdot,\omega))^T \in L_2^{\rm 2}(D)$, and $U=(u,v)^T \in L_2^{\rm 2}[0,T]$. Find controls $u(t)$ and $v(t)$ that steer this continuum of systems from an initial state $X_0=(x(0,\omega),y(0,\omega))^{T}$ to within a ball of radius $\varepsilon$ around the final state $X_F=(x_F(\omega), y_F(\omega))^{T}$ at time $T<\infty$, and minimize the cost functional
\begin{eqnarray}
\label{eq:J} J=\int_0^T \left[u(t)^2+v(t)^2\right]dt.
\end{eqnarray}
\end{problem}

We first observe that each element of the ensemble in (\ref{eq:harmonic}) with a frequency $\omega\in D$ is controllable, because the Gramian matrix is of full rank, i.e.,
$$rank \Big[B\Big|AB\Big]=2,$$
where $$A=\left[\begin{array}{cc}0&-\omega\\ \omega&0\end{array}\right],\quad B=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right].$$
\begin{thm}
\label{thm:smallcontrol} An ensemble of harmonic oscillators modeled as in (\ref{eq:harmonic}) is ensemble controllable on $L_2^2(D)$.
\end{thm}
{\it Proof.} Without loss of generality, we consider the frequency distributes in a symmetric domain $D_s=[-\beta,\beta]$ since the system (\ref{eq:harmonic}) with $\omega\in[\omega_1,\omega_2]$ can be readily transformed to a frame with $\omega\in D_s$ by a simple change of coordinate. Let's rewrite (\ref{eq:harmonic}) as $\dot{X}=\omega\Omega X+BU$, where
$$\Omega=\left[\begin{array}{cc}0&-1\\ 1&0\end{array}\right].$$
Let $\tilde{X}(t)=\exp(-\tilde{\omega}\Omega t)X$, then we obtain the differential equation in the new coordinate, $\dot{\tilde{X}}=(\omega-\tilde{\omega})\Omega\tilde{X}+\exp(-\tilde{\omega}\Omega t)BU$, with frequencies $\nu=\omega-\tilde{\omega}\in[\omega_1-\tilde{\omega},\omega_2-\tilde{\omega}]$. Taking $\tilde{\omega}=(\omega_1+\omega_2)/2$, then we have
$$\frac{d}{dt}\tilde{X}(t,\nu)=\left[\begin{array}{cc}0&-\nu\\ \nu&0\end{array}\right]\tilde{X}(t,\nu) +\left[\begin{array}{c}\tilde{u}(t) \\ \tilde{v}(t)\end{array}\right],$$
where $\tilde{u}(t)=u(t)\cos(\tilde{\omega}t)+v(t)\sin(\tilde{\omega}t)$, $\tilde{v}(t)=-u(t)\sin(\tilde{\omega}t)+v(t)\cos(\tilde{\omega}t)$, $\nu\in[-\beta,\beta]$, and $\beta=(\omega_2-\omega_1)/2$.

Let
\begin{align*}
p(t,\omega) &= x(t,\omega)+iy(t,\omega),\\
\alpha(t) &= u(t)+iv(t),
\end{align*}
where $i=\sqrt{-1}$. The system (\ref{eq:harmonic}) can then be written as
\begin{eqnarray*}
\label{eq:maineqn} \dot{p}(t,\omega)=i\omega p(t,\omega)+\alpha(t),
\end{eqnarray*}
with $p(0,\omega)=x(0,\omega)+iy(0,\omega)$. By the variation of constants formula, we have at time $T$
\begin{equation}
\label{eq:pomega} p(T,\omega)=e^{i\omega T}p(0,\omega) +\int_{0}^{T}e^{i\omega(T-\tau)}\alpha(\tau)d\tau,
\end{equation}
for all $\omega\in D_s$. This gives
\begin{equation}
\label{eq:varnat}
\int_{0}^{T}e^{-i\omega\tau}\alpha(\tau)d\tau=e^{-i\omega T}p(T,\omega)-p(0,\omega)\doteq\xi(\omega).
\end{equation}
Let $\mathcal{H}_1=L_{2}[0, T]$ and $\mathcal{H}_2=L_{2}[-\beta,\beta]$ be Hilbert spaces over $\mathbb{C}$. Defining the linear operator $L: \mathcal{H}_1\rightarrow \mathcal{H}_2$
by
\begin{equation}
\label{eq:L}
(L\alpha)(\omega)=\int_{0}^{T}e^{-i\omega\tau}\alpha(\tau)d\tau\doteq\int_0^T k(\omega,\tau)\alpha(\tau)d\tau,
\end{equation}
we then have from (\ref{eq:varnat}) and (\ref{eq:L}) that
\begin{equation}
\label{eq:Lalpha} (L\alpha)(\omega)=\xi(\omega).
\end{equation}
Observe that $L$ is bounded since for every $f\in\mathcal{H}_1$, by the Cauchy-Schwartz inequality,
\begin{eqnarray*}
\|Lf\|_{\mathcal{H}_2}\leq\|(\sqrt{T}\ \|f\|_{\mathcal{H}_1})\|_{\mathcal{H}_2}=\sqrt{2BT}\|f\|_{\mathcal{H}_1}.
\end{eqnarray*}
Therefore, $L$ has the adjoint $L^*$ defined by
\begin{eqnarray}
\label{eq:L*}
(L^* g)(t)=\int_{-\beta}^{\beta}k(\omega,t)^{\dagger}g(\omega)d\omega
=\int_{-\beta}^{\beta}e^{i\omega t}g(\omega)d\omega.
\end{eqnarray}
Moreover, since $L: \mathcal{H}_1\rightarrow \mathcal{H}_2$ and $k(\omega,\tau)\in L_2([-\beta,\beta]\times [0,T])$, $L$ is a Hilbert-Schmidt operator on $\mathcal{H}_1$ and hence $L$ is compact (the compactness can also be shown following the proof of Proposition \ref{prop:Lcompact}). According to Theorem \ref{thm:minnorm}, the function $\alpha$ of minimum norm satisfying (\ref{eq:Lalpha}) is given by
\begin{equation}
\label{eq:controllaw} \alpha(t)=L^{*}z(\omega),
\end{equation}
where $z$ satisfies
\begin{equation}
\label{eq:zomega} (Wz)(\omega)=\xi(\omega),
\end{equation}
and the operator $W:\mathcal{H}_2\rightarrow\mathcal{H}_2$ is defined by
\begin{align}
(Wz)(\omega) &= (LL^{*}z)(\omega)=\int_{-\beta}^{\beta}\frac{e^{i(\omega^{\prime}-\omega)T}-1}
{i(\omega^{\prime}-\omega)}z(\omega^{\prime})d\omega^{\prime}\nonumber\\
\label{eq:LLstar}&= \int_{-\beta}^{\beta}2\pi e^{i\frac{T}{2}(\omega^{\prime}-\omega)} \Big[\frac{\sin\big(\frac{T}{2}(\omega-\omega^{\prime})\big)}
{\pi(\omega-\omega^{\prime})}\Big]z(\omega^{\prime})d\omega^{\prime}
\end{align}
for $\omega, \omega^{\prime}\in [-\beta,\beta]$. A simple change of variables converts (\ref{eq:LLstar}) into
\begin{eqnarray}
\label{eq:W2} (Wz)(\Omega)=\int_{-1}^{1}2\pi e^{ic(\Omega^{\prime}-\Omega)} \left[\frac{\sin[c(\Omega-\Omega^{\prime})]}{\pi(\Omega-\Omega^{\prime})}\right]z(\Omega^{\prime})d\Omega^{\prime},
\end{eqnarray}
in which $c=\frac{\beta T}{2}$, $\Omega=\frac{\omega}{\beta}$, $\Omega^{\prime}=\frac{\omega^{\prime}}{\beta}$, and $\Omega,\Omega^{\prime}\in [-1, 1]$. Notice that the term inside the bracket in (\ref{eq:W2}) is the kernel of the following integral equation:
\begin{eqnarray}
\label{eq:pswf}\int_{-1}^{1}\frac{\sin[c(\Omega-\Omega^{\prime})]}{\pi(\Omega-\Omega^{\prime})} \,\psi_{n}(\Omega^{\prime},c)\,d\Omega^{\prime}=\kappa_{n}(c)\, \psi_{n}(\Omega,c),
\end{eqnarray}
where the $n^{th}$ eigenfunction $\psi_{n}(\Omega,c)$ is the well-known \emph{prolate spheroidal wave function} (\emph{pswf}), and $\kappa_{n}(c)$ is the associated eigenvalue \cite{Percival, Flammer, Slepian1, Slepian2, Slepian3}, where $\kappa_n>0$ and $\kappa_n\rightarrow 0$ as $n\rightarrow\infty$. Consequently, the $n^{th}$ eigenfunction and the corresponding eigenvalue for $W$ as in (\ref{eq:W2}) can be easily represented in terms of $\psi_{n}$ and $\kappa_{n}$ by $\phi_{n}=e^{-i\omega\frac{T}{2}}\psi_{n}$, and $\lambda_{n}=2\pi\kappa_{n}$. Note that $\psi_{n}$'s are orthogonal and complete on $L_{2}[-1,1]$ \cite{Percival}. Since $L$ is compact, $W=LL^*$ is compact. It can then be spectral decomposed by the orthonormal basis $\{\tilde{\phi}_n\}$ applied to (\ref{eq:zomega})
\begin{eqnarray}
\label{eq:Wz} Wz=\sum_{n=1}^{\infty}\lambda_{n}\<z,\tilde{\phi}_{n}\>\tilde{\phi}_{n}=\xi,\quad \tilde{\phi}_{n}=e^{-i\omega\frac{T}{2}}\frac{\psi_{n}}{\|\psi_{n}\|},
\end{eqnarray}
and this sequence is being uniformly convergent by the spectral theorem \cite{Gohberg}. It is also clear that $\{\tilde{\phi}_{n}\}$ is an orthonormal basis of $W$. The solution of (\ref{eq:Wz}) takes the form
\begin{eqnarray*}
z=\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}}\<\xi,\tilde{\phi}_{n}\>\tilde{\phi}_{n}.
\end{eqnarray*}
Finally, we show that the above series $z(\omega)$ can be truncated to $z_{N}(\omega)$ so that $\|Wz-Wz_N\|\rightarrow 0$ as $N\rightarrow\infty$. Then, we obtain the best approximation of the minimum energy control law $\alpha_N=L^* z_N$ by (\ref{eq:controllaw}).

\begin{lem}
Given any $\varepsilon>0$, there exists a finite series $z_N$,
\begin{eqnarray}
\label{eq:zN}
z_N=\sum_{n=1}^{N}\frac{1}{\lambda_{n}}\<\xi,\tilde{\phi}_{n}\>\tilde{\phi}_{n},
\end{eqnarray}
such that
$$\|Wz-Wz_N\|\rightarrow 0 \quad {\it as} \quad N\rightarrow\infty,$$
where $N=N(\varepsilon)$ depends on the choice of $\varepsilon$.
\end{lem}
{\it Proof.} By the orthonormality of $\{\tilde{\phi}_{n}\}$, we get
$$Wz_N=\sum_{n=1}^{N}\<\xi,\tilde{\phi}_{n}\>\tilde{\phi}_{n}.$$
Let $a_{n}=\<\xi,\tilde{\phi}_{n}\>$, and then we have
\begin{eqnarray}
\label{eq:error1}
\|Wz-Wz_{N}\|^{2}=\sum_{N+1}^{\infty}|a_{n}|^{2}.
\end{eqnarray}
Since, by the Bessel's inequality, $$\sum_{n=1}^{\infty}|a_{n}|^{2}<\|\xi\|^2<\infty,$$
the error in (\ref{eq:error1}) can be made in response to the desired $\varepsilon$ by the selection of $N=N(\varepsilon)$. \hfill $\blacksquare$

It follows from (\ref{eq:L*}), (\ref{eq:controllaw}) and (\ref{eq:zN}) that
\begin{eqnarray}
\label{eq:alphaN}\alpha_{N}(t)=\int_{-\beta}^{\beta}e^{i\omega t}\sum_{n=1}^{N(\varepsilon)} \frac{1} {\lambda_{n}}\<\xi,\tilde{\phi}_{n}\>\tilde{\phi}_{n}\,d\omega,
\end{eqnarray}
which will steer the system (\ref{eq:harmonic}) from $p(0,\omega)$ to within the ball $\mathbf{B_\varepsilon}\big(p(T,\omega)\big)$ at time $T$, where $\mathbf{B_\varepsilon}\big(p(T,\omega)\big)=\{h\in\mathcal{H}_2\,:\,\|p(T,\omega)-h(\omega)\|\leq\varepsilon\}$. In addition, $\alpha_N$ is the best approximation, for the given $\varepsilon>0$, of the control law $\hat{\alpha}=u+iv$ that minimizes the cost functional $J$ as in (\ref{eq:J}). \hfill$\Box$

We now have ensemble controllability for the system (\ref{eq:harmonic}), however, this result fails when either $u(t)$ or $v(t)$ is not available.

\begin{cor}
\label{rmk:harmonic} An ensemble of systems as in (\ref{eq:harmonic}) is not ensemble controllable if either $u(t)\equiv 0$ or $v(t)\equiv 0$.
\end{cor}
{\it Proof.} Without loss of generality, we suppose now that $v(t)=0$ and that the initial state $(x(0,\omega),y(0,\omega))=(0,0)$ for all $\omega\in D_s$. Note that each element of the ensemble is still controllable in this case. Let
\begin{eqnarray}
\tilde{X}(t,\omega)&=&x(t,\omega)-x(t,-\omega),\nonumber\\
\tilde{Y}(t,\omega)&=&y(t,\omega)+y(t,-\omega).\nonumber
\end{eqnarray}
The system described in (\ref{eq:harmonic}) can then be transformed to
\begin{equation}
\label{eq:harmonic1}
\frac{d}{dt}\left[\begin{array}{c}\tilde{X}\\ \tilde{Y}\end{array}\right] =\left[\begin{array}{cc}0&-\omega\\ \omega&0\end{array}\right]\left[\begin{array}{c}\tilde{X}\\ \tilde{Y}\end{array}\right]; \,\,\left[\begin{array}{c}\tilde{X}(0,\omega)\\ \tilde{Y}(0,\omega)\end{array}\right]
=\left[\begin{array}{c}0\\0\end{array}\right].\nonumber
\end{equation}
Since the above system is autonomous, it stays at the origin for all $t$, i.e., $(\tilde{X}(t,\omega),\tilde{Y}(t,\omega))\equiv (0,0)$. Thus, the system is not ensemble controllable. \hfill $\Box$

\begin{figure}[t]
\centering
\begin{tabular}{cc}
{\small (a)}\includegraphics[scale=0.53]{fig2a.eps}&
{\small (b)}\includegraphics[scale=0.53]{fig2b.eps}
\end{tabular}
\caption{\small The simulation results of Problem \ref{prob:harmonic1} for $N=1001$, $T=1$, and $\beta=10$. The initial state $X_0=(1,0)$ and the target state $X_F=(0,0)$. (a) The optimal control law $(u(t),v(t))$ for $t\in[0,1]$, and the final states for all systems $\omega\in[-10,10]$. (b) The trajectories for $\omega=-10$, $\omega=0$ and $\omega=5$ following $(u(t),v(t))$.}\label{fig:case1}
\end{figure}

\begin{figure}[h]
\centering
\begin{tabular}{cc}
{\small (a)}\includegraphics[scale=0.54]{fig3a.eps}&\,
{\small (b)}\includegraphics[scale=0.5]{fig3b.eps}\\
{\small (c)}\includegraphics[scale=0.5]{fig3c.eps}&\,
{\small (d)}\includegraphics[scale=0.5]{fig3d.eps}
\end{tabular}
\caption{\small The simulation results of Problem \ref{prob:harmonic1} for $N=1001$, $T=1$, and $\beta=10$. The initial state $X_0=(1,2)$ and the target state $X_F=(0,0)$. (a) The optimal control law $(u(t),v(t))$ for $t\in[0,1]$, and the final states for all systems $\omega\in[-10,10]$. (b) The trajectory for $\omega=-10$ following $(u(t),v(t))$. (c) The trajectory for $\omega=0$. (d) The trajectory for $\omega=5$.}\label{fig:case2}
\end{figure}

\subsection{Simulations}
\label{sec:simulation}
Here, we provide numerical solutions for $\alpha_N(t)$ since it is not of closed form. As shown in (\ref{eq:alphaN}), the ensemble control law $\alpha_N(t)$ is synthesized by the set of eigenfunctions $\{\tilde{\phi}_{n}\}$ and the corresponding eigenvalues $\{\lambda_{n}\}$ associated with the pswf's. These functions can be approximated by the \emph{discrete prolate spheroidal sequences (dpss's)}, denoted as $\{v_{t, k}(N, W)\}$, which are defined via the solution to the following equation \cite{Percival, Slepian4}
\begin{eqnarray}
\sum_{t^{\prime}=0}^{N-1}\frac{\sin\big[2\pi W(t-t^{\prime})\big]}{\pi(t-t^{\prime})}\, v_{t^{\prime}, k}(N,
W)=\lambda_{k}(N, W)v_{t, k}(N, W),\nonumber
\end{eqnarray}
where $0<W<\frac{1}{2}$ and $t = 0, 1, \ldots, N-1$. It is equivalent to saying that $\lambda_{k}(N, W)$ are the eigenvalues of the $N\times N$ matrix $A$ whose $(t, t^{\prime})$th element is
\begin{eqnarray}
\label{eq:dpssmatrix}(A)_{t,t^{\prime}}=\frac{\sin\big[2\pi W(t-t^{\prime})\big]}{\pi(t-t^{\prime})},\quad t,
t^{\prime}=0,1,\ldots,N-1,
\end{eqnarray}
and that the $N$ elements of the corresponding eigenvectors for this matrix are in fact subsequences of length $N$ of the \emph{dpss}'s. Note that $\lambda_{k}(N, W)$ are distinct, real, and ordered non-zero eigenvalues such that
\begin{eqnarray}
1>\lambda_{0}(N, W)>\lambda_{1}(N, W)>\ldots >\lambda_{N-1}(N,
W)>0,\nonumber
\end{eqnarray}
and the $dpss$'s are real-valued. Now, we show how to compute $\alpha_{N}(t)$. We present two cases with different initial states for $\beta=10$ and $T=1$:
\begin{enumerate}
\item[(1)] Consider $X(0,\omega)=(1,0)$ and $X(1,\omega)=(0,0)$. Then we have $p(0,\omega)=1$, $p(1,\omega)=0$, and hence, by (\ref{eq:varnat}), $\xi(\omega)=-1$ is a constant function.
\item[(2)] Consider $X(0,\omega)=(1,2)$ and $X(1,\omega)=(0,0)$. Then we have $p(0,\omega)=1+2i$, $p(1,\omega)=0$, and hence $\xi(\omega)=-1-2i$.
\end{enumerate}
According to the analysis above, the ``sinc'' kernel in (\ref{eq:pswf}) is replaced by the symmetric matrix $A$ as in (\ref{eq:dpssmatrix}), where $W=\frac{T\beta}{2\pi(N-1)}$. Note that the number of harmonic oscillators $N$ must be large enough to satisfy $W<\frac{1}{2}$ \cite{Percival}. Here we consider $N = 1001$ and the frequencies are uniformly sampled within $[-10,10]$. The simulation results are shown in Figure \ref{fig:case1} and Figure \ref{fig:case2}. It can be seen that following the resulting optimal control laws, the final states of all systems converge to a neighborhood of the desired target state, the origin. The trajectories for $\omega=-10$, $\omega=0$, and $\omega=5$ are displayed. Observe that in both cases a strong impulse is implemented initially, $|\alpha_N(0)|=191.7$ and $428.6$, respectively, as shown in Figure \ref{fig:amplitude}. Practical applications make it desirable to design a control with a limited amplitude. This leads to the following problem.

\begin{figure}[t]
\centering
\begin{tabular}{cc}
{\small (a)}\includegraphics[scale=0.52]{fig4a.eps}&\,
{\small (b)}\includegraphics[scale=0.52]{fig4b.eps}
\end{tabular}
\caption{\small (a) The control amplitude for the case (1). (b) The control amplitude for the case (2).}
\label{fig:amplitude}
\end{figure}

\subsection{Constrained Convex Optimization Problem}
In practice, the problem of interest is when the control amplitude is limited.

\begin{problem}
\label{prob:bounded}\rm Given a fixed time $T$, find bounded controls $u(t)$ and $v(t)$ satisfying the constraint $\sqrt{u^{2}(t)+v^{2}(t)}\leq A_{\rm{max}}\in\mathbb{R}^{+}$ for all $t\in[0,T]$, which will steer an ensemble of systems in (\ref{eq:harmonic}) from an initial state $X(0,\omega)=(1,0)$ as close as possible, in the $L_{2}$ sense, to the origin, $X(T,\omega)=(0,0)$, at time $T$.
\end{problem}

This problem can be formulated as the following minimization problem
\begin{eqnarray}
\label{eq:obj}
&& \min_{\alpha}\quad\int_{-\beta}^{\beta}\parallel p(T,\omega)-\mathbf{0}\parallel^2 d\omega, \\
\label{eq:cons} && {\rm s.t.}\quad\,\,\, u^2(t)+v^2(t)\leq A_{\rm{max}}^2,
\end{eqnarray}
where $p(T,\omega)$ defined in (\ref{eq:pomega}) depends on the control $\alpha$. It follows from (\ref{eq:pomega}) and (\ref{eq:obj}) that the problem can be further simplified as to minimize the following cost functional
\begin{eqnarray}
J=\int_{-\beta}^{\beta}\|\int_{0}^{T}e^{-i\omega\tau}\alpha(\tau)d\tau+1\|^2 d\omega,\nonumber
\end{eqnarray}
subject to the constraint (\ref{eq:cons}). By first integrating over $\omega$, we get
\begin{align}
J&=\int_{0}^{T}\int_{0}^{T}\frac{2\sin[\beta(\tau-\sigma)]}{\tau-\sigma}\,\alpha(\tau)\alpha^{\dag}(\sigma)d\tau
d\sigma\nonumber\\
&+\int_{0}^{T}\frac{2\sin(\beta\tau)}{\tau}\,\alpha(\tau)d\tau+\int_{0}^{T}\frac{2\sin(\beta\sigma)}{\sigma} \alpha^{\dag}(\sigma)d\sigma+2\beta.\nonumber
\end{align}
Observe that the imaginary part of the double integration in the expression for $J$ vanishes by antisymmetry, i.e.
\begin{eqnarray}
\int_{0}^{T}\int_{0}^{T}\frac{2\sin[\beta(\tau-\sigma)]}{\tau-\sigma} \Big[v(\tau)u(\sigma)-u(\tau)v(\sigma)\Big]d\tau d\sigma=0.\nonumber
\end{eqnarray}
Moreover, the sinc kernel is positive definite since
\begin{eqnarray}
\label{eq:PD}
\int_{0}^{T}\int_{0}^{T}\frac{2\sin[\beta(\tau-\sigma)]}{\tau-\sigma}\,
v(\tau)v(\sigma)d\tau\,d\sigma\nonumber\\
=\int_{-\beta}^{\beta}\parallel\int_{0}^{T}e^{-i\omega\tau}v(\tau)d\tau\parallel^2\,
d\omega > 0.
\end{eqnarray}
According to these observations, we can always minimize $J$ by the appropriate choice of $u(t)$ disregarding $v(t)$, because
\begin{align}
J & \geq\int_{0}^{T}\int_{0}^{T}\frac{2\sin[\beta(\tau-\sigma)]}{\tau-\sigma}u(\tau)u(\sigma)d\tau d\sigma\nonumber\\
& +\int_{0}^{T}\frac{4\sin(\beta\tau)}{\tau}u(\tau)d\tau+2\beta\nonumber.
\end{align}
Without loss of generality, we assume $A_{\rm{max}}=1$ and $\beta=1$ such that the ratio $\frac{A_{\max}}{\beta}=1$. The original problem described in (\ref{eq:obj}) and (\ref{eq:cons}) can now be recapitulated as follows:
\begin{align}
&\min_{u}\,\,\int_{0}^{T}\left[\int_{0}^{T}\frac{\sin(\tau-\sigma)}{\tau-\sigma}u(\tau)u(\sigma) d\sigma+\frac{2\sin(\tau)}{\tau}u(\tau)\right]d\tau\nonumber\\
\label{eq:consnew} & \ {\rm s.t.}\quad\,\, u^2(t)\leq 1.
\end{align}

\begin{prop}\label{prop:convex}
Let $\mathcal{S}=:\Big\{(u,v)\,\Big\vert\, u^2(t)+v^2(t)\leq A_{\rm{max}}^2\Big\}$. A local minimum of the cost functional $J$ over $\mathcal{S}$ is the global minimum.
\end{prop}
{\it Proof.} First, observe that $\mathcal{S}$ is a convex set. Furthermore, the cost function $J$ is quadratic in $\alpha$ with positive definite Hessian (see (\ref{eq:PD})). \hfill $\Box$

\begin{figure}[t]
\centering
\begin{tabular}{cc}
{\small (a)}\includegraphics[scale=0.44]{fig5a.eps}&
{\small (b)}\includegraphics[scale=0.44]{fig5b.eps}
\end{tabular}
\caption[The simulation results using Iterative Algorithm.]{(a) The optimal control laws for $T=1$, $T=\pi$, $T=5\pi$, and $T=10\pi$. (b) The distance between the final state and the origin, $|p(T,\omega)-{\bf 0}|$, of 51 harmonic oscillators for $T=1$, $T=\pi$, $T=5\pi$, and $T=10\pi$.}\label{fig:constrained}
\end{figure}

As a result, the model described in (\ref{eq:consnew}) is a convex optimization problem with a unique global minimum. We solve this problem numerically as a discrete quadratic optimization problem of the form
\begin{eqnarray}
\label{eq:objdis}
&&\min_{X}\quad X^{t}HX+2X^{t}Q \\
\label{eq:consdis} && {\rm s.t.}\quad\, |x_{i}|\leq 1, \quad i=1, 2,
\ldots, n,
\end{eqnarray}
where $X=(x_{1},x_{2}\ldots,x_{n})^{T}$, $t_{1}=0$, $t_{n}=T$, and
\begin{eqnarray}
H=\left[\begin{array}{cccc}\frac{\sin(t_{1}-t_{1})}{t_{1}-t_{1}} & \frac{\sin(t_{1}-t_{2})}{t_{1}-t_{2}} & \ldots & \frac{\sin(t_{1}-t_{n})}{t_{1}-t_{n}}\\ \frac{\sin(t_{2}-t_{1})}{t_{2}-t_{1}} & \frac{\sin(t_{2}-t_{2})}{t_{2}-t_{2}} & \ldots&\frac{\sin(t_{2}-t_{n})}{t_{2}-t_{n}}
\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\sin(t_{n}-t_{1})}{t_{n}-t_{1}} & \frac{\sin(t_{n}-t_{2})}{t_{n}-t_{2}} & \ldots & \frac{\sin(t_{n}-t_{n})}{t_{n}-t_{n}}\end{array}\right],\,\,
Q=\left[\begin{array}{c}\frac{\sin(t_{1})}{t_{1}}\\ \frac{\sin(t_{2})}{t_{2}}\\ \vdots\\
\frac{\sin(t_{n})}{t_{n}}\end{array}\right], \nonumber
\end{eqnarray}
Some simulation results of Problem \ref{prob:bounded} for various values of $T$ are shown in Figure \ref{fig:constrained}, where we assume $A_{\rm{max}}=1$ and $\beta=1$. We consider 51 harmonic oscillators ($n=51$) with their frequencies uniformly distributed in $[-1,1]$. The optimal control laws with square wave forms are illustrated in Figure \ref{fig:constrained}(a). Figure \ref{fig:constrained}(b) shows the $L_2$ distance between the final state and the origin of each harmonic oscillator following the corresponding designed control laws.

\section{Conclusion}
In this paper, we studied ensemble control of general time-varying linear systems and derived the necessary and sufficient controllability conditions. The key idea of understanding controllability relies on investigating the solvability of the integral equation associated with the system dynamics, and it is of Fredholm equations of the first kind. We highlighted the role of singular systems and spectral theorem in designing ensemble control laws, and an analytical optimal control is provided. The work on computing singular values and eigenfunctions of operators will be pursued. Another interesting topic is to characterize the researchable set for the system \eqref{eq:linearTV} under the constrained controls, i.e., $u\in U\subset\mathbb{R}^{\rm m}$. We plan to continue working on extending and generalizing our current results towards the goal of developing a theory of ensemble control. We believe the study of ensemble control problems will foster further developments in control and systems theory with broader applications such as systems with parameter uncertainties as well as system identification.

\appendices
\section{
\label{apd:Lcompact} We prove Proposition 1.

{\it Proposition 1:} The operator $L:\mathcal{H}_1\rightarrow\mathcal{H}_2$ defined by $$(Lu)(s)=\int_0^T\Phi(0,\tau;s)B(\tau,s)u(\tau)d\tau,$$ is compact, where $\Phi(t,0;s)$ satisfies for all $s\in[s_1,s_2]\subset\mathbb{R}$
$$\frac{d}{dt}\Phi(t,0;s)=A(t,s)\Phi(t,0;s);\quad\Phi(0,0;s)=I.$$
{\it Proof.} To prove this, we need the following tools.

\begin{defn}
Let $\mathcal{H}_0=L_2^{\rm n\times m}([s_1,s_2]\times [0,T])$ be the vector space of all those matrix valued functions $f$ whose elements $f_{ij}(s,t)$, $i=1,\ldots,n$, $j=1,\ldots,m$, are Lebesgue measurable on $[s_1,s_2]\times [0,T]$ and for which
$$\int_{s_1}^{s_2}\int_0^T\|f(s,t)\|^2 dt\,ds<\infty.$$
With the inner product for $f,g\in\mathcal{H}_0$ defined by
$$\< f,g\> =tr\int_{s_1}^{s_2}\int_0^T f(s,t)g(s,t)^{\dagger}\,dt ds,$$
$\mathcal{H}_0$ is a Hilbert space.
\end{defn}

Let $h(s,t)=\Phi(0,t;s)B(t,s)$, and then we have $$(Lu)(s)=\int_0^T h(s,t)u(t)dt.$$
We first show that $L$ is a bounded operator.

\begin{lem}
If $h(s,t)\in\mathcal{H}_0$, then $L\in\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$.
\end{lem}
{\it Proof.} By Schwarz's inequality,
\begin{align*}
\|Lu\|^2 &= \int_{s_1}^{s_2}\left(\int_0^T h(s,t)u(t)dt\right)^{\dagger}\left(\int_0^T h(s,t)u(t)dt\right)ds\\
& \leq\|u\|^2\int_{s_1}^{s_2}\int_0^T h(s,t)^{\dagger}h(s,t)dtds\\
& \leq\|u\|^2\|h\|^2.
\end{align*}
Thus,
\begin{equation}
\label{eq:Lbdd}
\|L\|\leq\|h\|<\infty.\qquad\qquad\qquad\qquad\qquad\hfill\Box
\end{equation}

\begin{lem}
\label{lem:basis}
Suppose that $\{\phi_1,\phi_2,\ldots\}$ and $\{\psi_1,\psi_2,\ldots\}$ are orthonormal bases for $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, and
$$\Psi_{ij}(s,t)=\psi_i(s)\phi_j(t)^{\dagger}$$
for $(s,t)\in[s_1,s_2]\times [0,T], (i,j)\in\mathbb{N}\times\mathbb{N}$. Then, $\{\Psi_{ij}\}$ is an orthonormal basis of $\mathcal{H}_0$.
\end{lem}
{\it Proof.} For $j,k,m,n\in\mathbb{N}$,
\begin{align*}
\< \Psi_{jk},\Psi_{mn}\> &= tr\int_{s_1}^{s_2}\int_0^T\left[\psi_j(s)\phi_k(t)^{\dagger}\right] \left[\psi_m(s)\phi_n(t)^{\dagger}\right]^{\dagger} dtds\\
&= tr\int_{s_1}^{s_2}\psi_j(s)\left[\int_0^T\phi_k(t)^{\dagger}\phi_n(t)dt\right]\psi_m(s)^{\dagger}ds\\
&= \delta_{kn}\left[tr\int_{s_1}^{s_2}\psi_j(s)\psi_m(s)^{\dagger}ds\right]\\
&= \delta_{kn}\left[\int_{s_1}^{s_2}\psi_m(s)^{\dagger}\psi_j(s)ds\right]\\
&= \delta_{kn}\delta_{mj},
\end{align*}
so that $\{\Psi_{ij}\}$ is an orthonormal set in $\mathcal{H}_0$.

Now, suppose that $k\in\mathcal{H}_0$ be such that $\< k,\Psi_{ij}\> =0$ for all $i,j\in\mathbb{N}$. Corresponding to this $k$, let $K:\mathcal{H}_1\rightarrow\mathcal{H}_2$ be defined by
$$(Ku)(s)=\int_0^T k(s,t)u(t)dt,\quad u\in\mathcal{H}_1.$$
Then, we have for all $i,j\in\mathbb{N}$
\begin{align*}
0=\< k,\Psi_{ij}\> &= tr\int_{s_1}^{s_2}\int_0^T k(s,t)\left[\psi_i(s)\phi_j(t)^{\dagger}\right]^{\dagger}dtds\\
&= tr\int_{s_1}^{s_2}\left[\int_0^T k(s,t)\phi_j(t)dt\right]\psi_i(s)^{\dagger}ds\\
&= tr\int_{s_1}^{s_2}(K\phi_j)\psi_i(s)^{\dagger}ds\\
&= \int_{s_1}^{s_2}\psi_i(s)^{\dagger}(K\phi_j)ds\\
&= \< \psi_i,K\phi_j\>_{\mathcal{H}_2}.
\end{align*}
Since $\psi_i$ is an orthonormal basis of $\mathcal{H}_2$ and $\phi_j$ is an orthonormal basis of $\mathcal{H}_1$, it follows that $$Ku=0 \quad\forall\, u\in\mathcal{H}_1.$$ Consequently, $k=0$ a.e. and hence $\{\Psi_{ij}\}$ is an orthonormal basis for $H_0$. $\hfill\Box$
\begin{thm}
\label{thm:compactseq}
Suppose $\{K_n\}$ is a sequence of compact operators in $\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ and $\|K_n-K\|\rightarrow 0$, where $K$ is in $\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$. Then $K$ is compact.
\end{thm}
{\it Proof.} See \cite{Gohberg}. $\hfill\Box$

According to Lemma \ref{lem:basis}, $h(s,t)$ can be represented as $h=\sum_{i,j=1}^{\infty}\< h,\Psi_{ij}\> \Psi_{ij}$. Define
$$h_n(s,t)=\sum_{i,j=1}^{n}\< h,\Psi_{ij}\> \Psi_{ij}(s,t).$$
Then,
\begin{eqnarray}
\label{eq:h-hn}
\|h-h_n\|\rightarrow 0.
\end{eqnarray}
Let $L_n$ be the integral operator defined on $\mathcal{H}_1$ by
$$(L_n u)(s)=\int_0^T h_n(s,t)u(t)dt.$$
Now $L_n$ is a bounded linear operator of finite rank since $\mathcal{I}m L_n\subset\rm{sp}\{\psi_1,\ldots,\psi_n\}$, and hence $L_n$ is compact. By (\ref{eq:Lbdd}) and (\ref{eq:h-hn}) applied to $L-L_n$,
$$\|L-L_n\|\leq\|h-h_n\|\rightarrow 0.$$
This follows $L$ is compact by Theorem \ref{thm:compactseq}. Note that the compactness of $L$ can also be shown by using the properties of Hilbert-Schmidt operators \cite{Gohberg}. $\hfill\Box$

redpajama_set_name: RedPajamaArXiv


If you love the taste of a velvety coconut cream pie, you're in luck! This lovely, Coconut Cream Pie latte recipe is surprisingly light, undeniably tasty, and captures the essence of one of our favorite summer desserts. Best of all, it can be made in less than five minutes!
Step 2: Add 8 ounces coffee to your mug and mix well. If you enjoy flavored coffee, try using Coconut Cream Pie coffee.
Step 3: Add 1/2 cup coconut milk to your mug.* If you are heating the milk, heat it until it is warm to the touch. You can also froth the milk before adding it to the coffee to create additional texture.
Step 4: Top with milk foam and / or whipped cream and toasted coconut flakes.
*We used 1/2 cup of coconut milk and enjoyed it but the quantity of milk added is really up to personal taste! Regular milk or cream can be used in place of coconut milk.


text:
News, Students
Students serve in Houston following Harvey aftermath
By: Rachel Thoms, Strategic Communications January 11, 2018
It has been nearly five months since the costliest tropical cyclone on record made landfall in southeast Texas and the surrounding area, inundating hundreds of thousands of homes and displacing tens of thousands of people.
Victims of Hurricane Harvey's path are still recovering, and last week, students from CUAA were able to play a part in the recovery.
"The devastation was heartbreaking, but I saw Jesus working in and through the brokenness our students encountered." said Randy Duncan, director of campus ministry. "They rose up and were the hands and feet of Christ to the people of Houston."
From January 2-6, 2018, Duncan traveled to Houston, Texas with 16 students from Concordia University Ann Arbor to aid in the ongoing hurricane relief, a majority of their time spent insulating and installing dry wall in homes that were completely gutted after being submerged in over four feet of sewage water.
Concordia's team partnered with Trinity Klein Lutheran Church in Houston, Texas and Salem Lutheran Church in Tomball, Texas who are both members of Heart for Texas, an organization made up of a coalition of Lutheran organizations that formed in the wake of Hurricane Harvey, serving as boots on the ground to provide disaster relief throughout southeast Texas.
"The houses we were working in had nothing; they were completely empty and didn't even have the bottom few feet of their walls," said Ashtyn Hare, junior at CUAA. "It made us all realize how much these families' lives have been impacted and how grateful I am for what I have."
Hare went on to share that one of her favorite parts of the experience was being able to interact with the family that owned the home they were repairing.
"Our new friend Barry, who owned the house we were working on, even ordered pizza for us one day for lunch," said Hare. "You could tell they were so appreciative, when really, we were the ones who felt inspired by them."
Each year, Concordia's campus ministry department hosts a number of mission trips for students to be a part of. Some trips are reoccurring, like the annual trip to New Orleans over spring break, while other mission teams are formed on an as needed basis, such as this trip to Texas. CUAA will also be sending 11 students to Guatemala in March.
"This was the first mission trip for two-thirds of the students who went on this trip, and I was blessed to witness the deepening love for each other and the Lord by these students during tour time in Houston." said Duncan.
Hare shared that this was her first time doing a mission trip, and she hopes to join CUAA's trip to Haiti next year if she is able to raise the funds.
"It was amazing to see the houses begin to look like homes again by the end of the week," said Hare. "After spending the days working and the evenings recovering, hanging out, and worshipping together, I really grew in my faith. This is a trip I will never forget."
Honor Band Days Culminate With Joint Concert
Popular Faculty Music Recital Set For February 19

redpajama_set_name: RedPajamaCommonCrawl


There Are Angels Hovering Round is a song that has come to the MMC network through multiple channels. St. Gregory of Nyssa congregants have been singing this song at funerals for many years. Peter and Mary Alice Amidon have a lovely arrangement of it on their website amidonmusic.com, where you can listen to an mp3 and purchase sheet music.
This prayer song was written by Carol Webb and would be especially effective during prayers for healing and wholeness. She writes about the inspiration for the song in a beautiful post.
Sheet music can be found in Singing In Community, our latest songbook published by Augsburg Fortress. Find a preview copy of the score here.


text:
https://artoyz.com/shop/en/state-of-the-art/7020-wave-s-mandrill.html 7020 100TAUR - Wave's Mandrill https://media3.artoyz.net/shop/18775-nosto/wave-s-mandrill.jpg 59.99 79.99 EUR OutOfStock /Home/ART & BOOKS/ART/Silkscreen /Home/Winter Sales /Home/ART & BOOKS
Painting by 100TAUR created for the opening of his show "Monsters in my Closet" @ Artoyz Shop + Galerie.October 2014.

Silkscreen print on paper - limited to 50 pieces.

Between innocence and darkness, 100Taur explores the allure of difference and imperfection, by creating a world inhabited by fantastical creatures, half human-half animal hybrids, deformed, tarnished, where every detail is a tribute to the etching's caption « the sleep of the reason produces monsters », made by the spanish painter Francisco de Goya in 1799.

His work might be affiliated to « Low Brow », and there's in it an assumed fascination for mythologies and monsters, even a claimed sympathy. Its aim is to dispute the idea of standardization, and to reveal that any standard is monstrous itself.

« Through my childish eyes, I invent nightmare creatures in order to make them my imaginary confidents. I love to pay a particular attention to the tiny details of the backgrounds, offering to my creatures a rich and colorful universe, inspired by mythological descriptions in literature, japanese icons...We must not forget that «monstrum», means, in Latin, a miracle, an incredible thing, also a show. Wonderful, horrible or disgusting. It's what it's necessary to see and to show ».

Behind every drawing, painting, sculpture or installation, a story is waiting to be discovered.

He uses acrylic paint, ink, walnut stain, spray painting, egg tempera technique, in superimposed layers.

He tirelessly works for the creation of a dark enchanted world.

100Taur was born in 1982 in the south of France. He has always cultivated a deep curiosity for nature and he finds his inspiration in «classical » paintings -Jerome Bosch, Bruegel...-, surrealistic pop and Low Brow, mythology, religious icons, taxidermy, fantasy literature – Lovecraft, Edgar Allan Poe –, comics and B movies. His eyes are wide open all the time.
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100TAUR - Wave's Mandrill
Painting by 100TAUR created for the opening of his show "Monsters in my Closet" @ Artoyz Shop + Galerie.October 2014.
Silkscreen print on paper - limited to 50 pieces.
Between innocence and darkness, 100Taur explores the allure of difference and imperfection, by creating a world inhabited by fantastical creatures, half human-half animal hybrids, deformed, tarnished, where every detail is a tribute to the etching's caption « the sleep of the reason produces monsters », made by the spanish painter Francisco de Goya in 1799.
His work might be affiliated to « Low Brow », and there's in it an assumed fascination for mythologies and monsters, even a claimed sympathy. Its aim is to dispute the idea of standardization, and to reveal that any standard is monstrous itself.
« Through my childish eyes, I invent nightmare creatures in order to make them my imaginary confidents. I love to pay a particular attention to the tiny details of the backgrounds, offering to my creatures a rich and colorful universe, inspired by mythological descriptions in literature, japanese icons...We must not forget that «monstrum», means, in Latin, a miracle, an incredible thing, also a show. Wonderful, horrible or disgusting. It's what it's necessary to see and to show ».
Behind every drawing, painting, sculpture or installation, a story is waiting to be discovered.
He uses acrylic paint, ink, walnut stain, spray painting, egg tempera technique, in superimposed layers.
He tirelessly works for the creation of a dark enchanted world.
100Taur was born in 1982 in the south of France. He has always cultivated a deep curiosity for nature and he finds his inspiration in «classical » paintings -Jerome Bosch, Bruegel...-, surrealistic pop and Low Brow, mythology, religious icons, taxidermy, fantasy literature – Lovecraft, Edgar Allan Poe –, comics and B movies. His eyes are wide open all the time.
Width : 69.50 cm
Designer : 100TAUR (France)
Limited Edition : Limited to 50 pieces
Technique : Silkscreen print on paper.


text:
McLaren 765LT Spider unveiled: Bloody fast
This is the fastest convertible that McLaren has ever produced
Behold! This is the fastest convertible that McLaren has ever produced. The 765LT Spider does the 0-100-kph and 0-200-kph dash in 2.8 seconds and 7.2 seconds respectively. When you want to go fast, you have to be lighter on the scale, too. This one's 80 kg lighter than the 720S Spider and only 49 kg heavier than the 765LT Coupe.
Powering the McLaren 765LT Spider is a 4.0-litre twin-turbo V8 that delivers 754 bhp and 60.16 kgm of torque. Tipping the scale at 1388 kg, the 765LT Spider is the fastest convertible in the McLaren lineup. Since this is a convertible, the extra weight over the Coupe comes from the one-piece carbon roof. Up to 50-kph speeds, the roof can be retracted, and it will take only 11 seconds. Interested in only hearing the dramatic sound coming from the quad pipes, are you? Then, the rear window can be independently lowered. Safety is obviously a matter of concern, but McLaren says that it uses the 'MonoCage II' carbon fibre structure that makes the 765LT Spider robust and doesn't require any additional bracing.
Now, the 765LT Coupe is an extraordinary car in itself, but McLaren's retuned the dampers and active aero on the Spider for that added fun and drama. Having said that, the fun and drama don't come cheap. The prices for these beauts start at £310,500 (Rs 3.2 crore) excluding taxes. Well, these are going to be limited, and just like the 765 examples of the 765LT Coupe, it'll not take much time for the Spider to be sold out as well. Desirable much?


Electronic waste is the fastest growing waste stream globally, to deal with the scourge of illegal dumping of this harmful waste measures have to be put in place to divert these materials from landfills. Qobo Computer Recycling has been at the forefront of assisting numerous organisations in Gauteng with the proper disposal of their electronic waste.
Copyright © Qobo Recycling CC 2013 2011-2012. All Rights Reserved.


text:
Home » Groups » Security Section » 2017 Section Election Information
Vacant Position
Chair-Elect: Three-year term - first year Chair-elect, second year as Chair, and third year as immediate past Chair
The main responsibilities of the vacant position are providing leadership & direction for the Security Section. This includes, but is not limited to, planning the annual meeting, holding additional meetings as needed to evaluate activities/direction of section, sending relevant announcements to the section via the listserv, and updating the section's blog and microsite with news and resources.
Candidates for Chair-Elect
James Havron
James "Jim" Havron C.A., has worked in government, private, and academic archives, as well as public library special collections. His graduate degrees, an M.A. in public history (archival management), and an M.S. in computer information systems (security and assurance concentration), both come from Middle Tennessee State University. He has experience with records, manuscripts, and other media, and is a member of The Academy of Certified Archivists, a holder of the Digital Archivist Certification of the Society of American Archivists, and holds the CompTIA Security+ cybersecurity certification.
He held a position as Archivist/Coordinator and Oral History Projects Coordinator for the Albert Gore Research Center at Middle Tennessee State University for a number of years, working particularly with electronic records, developing access and security for databases, and doing some work managing aspects of the center's network security. Prior to that position, he was archivist for the Tennessee Conference of the United Methodist Church and an archivist with the Special Collections Division of the Nashville Public Library. He has also worked for Dell Computers and does cybersecurity consulting. He has been with SAA for a number of years. He is particularly involved with emergency preparedness/disaster recovery, security threat and risk assessment, and understanding cybersecurity for archivists.
Statement of Interest
I have been a professional archivist for a number of years, but I entered the field with the specific plan of also becoming an electronic information professional. I felt, even over a decade and a half ago, that the ability to preserve and understand the records being generated by most entities depends on the ability to preserve and understand electronic records. While most archivists will learn to do this on some level as their carriers progress, I felt then and now that there was a need for people who had the core professional training in each field, thereby being able to work where the professions meet. In the process I developed an understanding that professionals often need a different viewpoint when they work outside the areas they normally work, one coming from professionals in that new area.
I think of security holistically. It is part of all we do. Because so much information, both static and transactional, is in an electronic format, cybersecurity must be part of the "whole." Prior to obtaining my information systems education, I was already specializing to some degree in security. I have always felt that providing a secure environment for records and people, as well as preparing for disaster recovery, was an essential component to what we do as archivists. This was reinforced when I entered the cybersecurity/InfoSec (information security) world as the primary targets for maintaining security were identified as CIA. "C" for confidentiality- records/information should be kept securely confidential when appropriate. "I" for integrity of data- all that is possible is done to verity the records and other information that might provide context for interpreting them. "A" for availability/accessibility- records should be accessible to the appropriate people at all times. CIA is also a good summing up of most of what we do as archives professionals.
I have discovered that, at least in cybersecurity/InfoSec, security slips from the mind as a priority unless the mind is specifically trained to examine security as second nature. This requires access to the proper resources, as well as an understanding of the source of danger and a willingness to do the training. As members of SAA's Security Section, we have resources we can share, as we do on the list and blog, and as professionals we can find ways to share these resources of skill and knowledge with others. The rate development and implementation of electronic information creation, use, and storage, and threats to these practices, grows too quickly to handle without reaching out. I advocate finding as many platforms as possible to reach out to ourselves and others, providing possible solutions to dealing with rapidly growing and changing issues in security. I would like to see more professional discussion on different types of threats, either via online forums, Webinars, or personal meetings. I would like us to advocate for regional and state organizations to provide workshops and forums on things such as general security, disaster planning, and cybersecurity. I feel that the section does well in its discussion forums and news dissemination as it is, but should I be elected I hope to have the chance to help us become more proactive in taking these things to others.
Blake Relle
Blake Relle received his Master's Degree in Library and Information Science with a concentration in archive management from LSU in 2013. He obtained his bachelor's degree in accounting from Our Lady of Holy Cross College in New Orleans in 2007. Currently, he serves as an Archives Specialist at the Louisiana State Archives. In this position he fulfills request for materials made by the public as well as state agencies. Also, he processes and creates a DACS compliant finding aid for each collection transferred to the Archives. From 2013 to 2015, he served as a digitization intern at the National World War II Museum in New Orleans. In this role, he scanned items from the Museum's archives including photographs, photographic negatives, propaganda leaflets, journals, flight books, and letters.
Professionally, Relle has presented on "How to get new archival professionals involved in archival organizations?", and "Should archival professionals do continuing education?" At the SAA meeting in August of 2016, Relle was on a panel that discussed how archives and museums can provide access to their collections to people with disabilities. Relle serves on the Steering Committee of the Government Records Section (SAA) as well as on the Louisiana Archives and Manuscript Association (LAMA) Scholarship Committee.
By serving on the Leadership Team of the Security Section, I will have an opportunity to help develop solutions and best practices that will help keep archival collections secured. Threats to our collections can come in many forms as well as come from internal or external forces. By exchanging ideas and coming up with solutions, we can work towards keeping our collections secured for the enjoyment of future generations. We must be proactive in keeping our repository's collections secured.
Standing Rules Referendum
The Leadership Team of the Security Section is proposing a Standing Rules change, augmenting the organization of the Officers. Currently, the Security Section Officers is a group of 3 individuals: a Secretary (1-year term), a Junior Co-Chair (first year of a 2-year term), and a Senior Co-Chair (second year of a 2-year term). The proposed change will maintain a 3-person leadership team, but will do away with the Secretary role in favor a single, annually-elected position with a three-year term (year one as Chair-elect, year two as Chair, and year three as immediate past-Chair).
Below is the text of the proposed Standing Rules change with the deletions struck out and the additions underlined:
IV. Governance
A. Officers. The officers of the roundtable shall be a Senior Co-chair, a Junior Co-chair, and a Secretary. The Junior Co-chair shall be elected annually for a two-year term, serving in year one as Junior Co-chair and in year two as Senior Co-chair. The Secretary is elected annually and may not serve more than three successive years.
The three-person team of officers of the Section shall be a Chair, a Chair-elect, and an immediate past Chair. The Chair shall be elected annually for a three-year term, serving in year one as the Chair-elect, serving in year two as the Chair, and serving in year three as the immediate past Chair.
B. Nominations.The Chair shall issue a call for nominations, including self-nominations, for the positions of Chair-elect Junior Co-chair and Secretary every June to all Section roundtable members via the Section's roundtable official email discussion list and website. A slate of candidates shall be established by the officers and announced to Section roundtable members no later than June 15.


Debbie S. Miller has written one of the best, if not the best, descriptive books on the Arctic National Wildlife Refuge, and it is certainly the most enjoyable that I have read. Debbie has real world years of feet on the ground experience in ANWR and her book will impart to you her passion for a place of moving beauty. I'll let her introduce herself in her own words (http://www.debbiemilleralaska.com/). Midnight Wilderness is a book to read if you want to know what the fuss is about with regard to drilling ANWR. http://www.debbiemilleralaska.com/MidnightWilderness.htm. It is available on Amazon of course.
Just got back from Denali NP and Denail State Park so I have a lot of catching up to do. I have a ton of photos to go through; I will put some up asap.
This entry was posted on Monday, June 1st, 2009 at 2:22 pm	and is filed under Anything But... and tagged with Alaska, ANWR, Arctic, fall river photography, Midnight Wilderness. You can follow any responses to this entry through the RSS 2.0 feed.


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Honorary degree for Mugabe shames system, says rector
THE rector of Edinburgh University has called on the
institution to review its policy of honouring foreign politicians, claiming it faced embarrassment over its award to Robert Mugabe, the president of Zimbabwe.
Robin Harper said the university had been hasty in 1984 in handing Mr Mugabe - whom he described as a ''nasty tyrant'' - an honorary doctorate.
Mr Harper, the Green MSP, has been a firm critic of Mr Mugabe but until now has limited his comments to welcoming discussion of rescinding the honorary degree.
His term as rector ends in March and he has indicated he will not run for re-election.
The criticism comes as arguments continue over the England cricket team's decision to fulfil its world cup fixture with Zimbabwe in Harare next month. The government has recommended the team boycotts the country in protest at the regime.
A wave of violence has swept the country as Mr Mugabe pursues his land-grab policy, confiscating farms from white Zimbabweans.
More than six million of his people now face starvation in a famine stretching across Southern Africa, and aid agencies have accused supporters of the ruling Zanu-PF party of denying food to opponents.
However, two decades ago Mr Mugabe was honoured by Edinburgh University for his contribution to education in Zimbabwe and his role in bringing the country to independence. Students have been lobbying the university to repeal his honorary doctorate, although there is no mechanism for revoking awards.
Mr Harper said: ''Perhaps the answer is not to award honorary degrees to politicians, particularly those in foreign countries, until they have run their course. Maybe the end of a presidency is the time for an honour to be bestowed and not simply when a person is elected. There is always an element of caution because you don't know how people will use their power.
''I believe Edinburgh University was precipitous in honouring him so early in his tenure.''
However, Mr Harper said that a mechanism for withdrawing honours, once awarded, would not be the answer.
''That would be mean-
minded and it could be open to all sorts of abuse. It would be difficult to draw up a list of criteria that would apply,'' he said.
''Perhaps a motion by students is enough in itself as a mark of severe criticism of a regime. It's just as much as can be done.''
He said he would be against retrospective university legislation which would strip people of degrees originally awarded in good faith.
''In this case, that faith has been thoroughly broken,'' said Mr Harper, who added: ''The policy of honorary degrees for foreign politicians deserves a revisit, looking at the criteria and the point at which an honour should be bestowed.''
Jonathan Meenagh, students' association president, said the Mugabe case highlighted the case for reviewing the policy. He said: ''Mr Harper has a point. You don't know what an honorary graduate is going to do - not just politicians, scientist could carry out controversial and unethical research.''
Mr Meenagh said there was a motion coming before the students' representative council requesting that university heads look at the whole concept of handing out honorary degrees.
''In this case, it's done and dusted, but I hope the university will consider the issue.''
The university, which said it supported the UK government's line on the Mugabe issue, said yesterday that the degree was awarded ''on the merits of the case at the time''.
A spokesman added: ''The University of Edinburgh has a concern that the withdrawal of such a degree might rebound on any in the country associated with the university - our graduates in Zimbabwe and also the Edinburgh graduates of many nationalities who work all over the world, including Zimbabwe.''
He added that it had no procedures for withdrawing such degrees. He said: ''We do not feel that embarking on a time-consuming, lengthy process of change to provide us with a mechanism for this is the best use of resources.''
controversial AWARDS
Harry Truman: The US president who ordered the use of nuclear weapons against Japan faced a furious protest in 1957 when he was honoured by Oxford University.
Sir Peter Swinnerton-Dyer, a former chairman of the University Grants Committee, in 1991 was conferred with one of Aberdeen University's most controversial honorary awards. Some held him responsible, through UGC swingeing cuts, for the loss of more than 130 jobs there.
George Younger, former Scottish secretary, faced a walkout by a group of Glasgow University students at a ceremony in 1992 to award
an honorary degree of doctor of laws. The problem appeared to be that he was a ''high
Tory''.
Kenneth Clarke, while Tory chancellor, faced action in 1996 by lecturers when Nottingham Trent University awarded him an honorary degree. They blamed him for education cuts.


File Explorer 6.0.2.3 Apk Download by NextApp, Inc.
This app uses the Device Administrator permission. This permission is used only to lock the device's screen automatically when you press the "lock" button in the image viewer (to allow you to show a single picture on your device to another person without them swiping through your images).
SMB v2. Major UI updates. New themes. New dual window mode. 6.0.2: More UI improvements, bugfixes.


Worried you've got Yellow Candy Fever? There's no need to get all worked up over a Google Diagnosis. We have all of the symptoms laid out here. Are you constantly seeking out a bright golden hue? Are your daydreams filled up with vivid memories of silky milk chocolate? Do you turn around in a hurry whenever you hear a sound similar to the lip-smacking crunch of a malt ball? If you answered yes to any of these questions, then you are most likely in need of a sweet cure. Look no further than these yellow milk chocolate covered malt balls. They're therapeutically tasty. Note: Yellow milk chocolate balls can also be beneficial for regular sugar cravings or celebrations. This remedy will not treat any other fever.
Bag contains 5 pounds of Yellow Milk Chocolate Covered Malt Balls.


Heliotropium arborescens for sale – Easy plant to grow, mostly grown for the ornamental flowers and for the leaves, planting season in spring to autumn, better to buy plant or another option to start from seeds yet more challenging.
Growing information: perennial plant, growing hardiness zone: 10+, water needed – small to average amount, light conditions – full sun to partial shade, height: 40-120cm, 16-98 inches.
Blooming in the spring to autumn in star shaped flowers that appear in pink, white, red or blue color.


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SJB est un sigle qui peut désigner :
 Saint-Jean-Baptiste et par extension la fête nationale des Canadiens français et des Québécois éponyme ;
 Salaire journalier de base, salaire déterminé par les services de la sécurité sociale afin de chiffrer les indemnités journalières ;
Simon Jolin-Barrette, avocat et homme politique québécois.

redpajama_set_name: RedPajamaWikipedia


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NSS Position Paper: A Public/Private COTS-Type Program to Develop Space Solar Power
by National Space Society | Feb 5, 2020 | National Space Society, Space Development, Space Policy, Space Solar Power | 3 comments
The National Space Society (NSS) proposes a public/private partnership approach to develop and demonstrate space solar power at a sufficiently high level of technical maturity that commercial energy providers can subsequently build initial operational systems for high-energy-cost environments such as remote locations (islands, mining facilities, military encampments, etc.), emergency settings (e.g., evacuation camps) and others. The outlines of the partnership are based on the successful Commercial Orbital Transportation Services program that helped develop the SpaceX Falcon 9 launcher/Dragon capsule and the Orbital Sciences Corp. Antares launcher and Cygnus spacecraft. The Space Solar Power Demo development described here features minimal NASA oversight, milestone-driven fixed-price pay-outs, minimal exit criteria, substantial commercial partner funding commitments, non-traditional contracts (e.g., Space Act Agreements with NASA), commercial partner choice of energy market and energy consumer, enabling system development (e.g., space robotics), and at least two winners.
READ FULL PAPER
Robert Sugg on February 6, 2020 at 8:08 am
Gerard O'Neill's group sponsored a study in the mid-1980s which found that 99% of the bulk mass of a solar power satellite can come from the Moon, saving 97% in fabrication costs vs. Earth-launched materials. The key resources are lunar metals and water, which was discovered at the lunar poles in 2009.
One of the most significant benefits of the Artemis lunar mission can be the kickstart of the space solar power industry. SSP can extend coal, oil, & gas reserves and permanently solve the climate/energy problem on the ground.
If SpaceX Starships fly, Artemis can inaugurate simple lunar ISRU for the paving of concrete landing pads at the lunar south pole and at an equatorial location for these ships, which can transport cargo, house workers, and rotate crews.
HR 5666 should be adjusted for an emphasis on lunar prospecting and development with competitive commercial participation.
Eventually, a multinational SSP consortium led by the United States may be the smart way to go. Nations who participate in SSP can be the chief energy exporters ahead, and the dominant space and terrestrial powers of the 21st century.
David on February 6, 2020 at 11:39 pm
Terrible idea. We should spend money on advanced nuclear liquid fuel reactors instead.
Bob Brodbeck on February 9, 2020 at 9:25 pm
So who/which entity or office might pull this together and get the ball rolling? Yes, seems to make sense to have "expanded Government representation" with NASA, DOD, DOE,and Commerce. But what office today might undertake the early organizing for a possible program? AFRL?
Also, the expanded representation would seem to increase likelihood of "At least two commercial partners (CPs)", and not just Northrup-Grumman. However, if a CP's ability to put their "own skin in the game", perhaps to the tune a couple hundred million dollars, is a criteria for acceptance in the program, that would seem to possibly eliminate smaller entities with potentially very valid designs, perhaps such as International Electric Company's CASSIOPeiA design. Hopefully there would be a role in this for smaller firms too.


Image Title: Vanity Duvet Covers Dormify White Twin Xl Cover Windigoturbines Throughout Plan 13. Filename: vanity-duvet-covers-dormify-white-twin-xl-cover-windigoturbines-throughout-plan-13.jpg. Image Dimension: 480 x 480 pixels. Images Format: jpg/jpeg. Publisher/Author: Nikolas Nolan. Uploaded Date: Monday - May 14th. 2018 02:52:16 AM. Category: Architecture. Image Source: digsdigs.com.
Tap The Thumbnail Bellow to See Related Gallery of "Vanity Duvet Covers Dormify White Twin Xl Cover Windigoturbines Throughout Plan 13"


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RebelMouse CEO Paul Berry Talks Publishing Platforms And Content Startups
Anthony Ha	@anthonyha / 4 years
RebelMouse founder and CEO Paul Berry joined us at Disrupt NY to talk about how the company has evolved from its early days as a social media aggregator.
"The product was always headed where we are now," Berry said. He added that the company has built "an enterprise class CMS that's solving for media companies — like startups that are venture-backed and large media companies — how you build your direct relationship with consumers, with an awesome site and apps that use all the latest viral trends and beautiful editing tools. And we're doing the same for brands as well."
Berry also talked about some of the sites built using RebelMouse, including cute animal website The Dodo, as well as Fatherly, which he described as "a babysitter for dads." RebelMouse likes to work closely with publishers to build their sites and apps, and he said more and more publishers are using the platform because they'd otherwise need "a huge army of developers" to maintain their site and apps.


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Empal gepuk (Sundanese) or sometimes simply known just as empal or gepuk is an Indonesian sweet and spicy fried beef dish. This dish is commonly popular in Java island, but can trace its origin to Sundanese cuisine of West Java, Indonesia.

Ingredients
The beef preferably used in this dish is shank part. First the beef is boiled until medium well, then being cut quite thick along the muscle fiber into half-palm size. Then  using stone pestle and mortar, the beef pieces are beaten mildly to loosen the meat fibers and spread its size a little bit. The spices used in this dish are ground shallot, garlic, chili pepper, coriander, palm sugar and salt, mixed with bruised lemongrass, galangal, daun salam (Indonesian bay leaf), a little coconut milk and turmeric water. The meat pieces are cooked with the spices well until the spice are absorbed into the meat and the stock evaporate. Then the meat pieces are fried in coconut oil until the color darken and the meat is done. Empal gepuk is sprinkled with bawang goreng (fried shallot) and served with steamed rice. This fried beef dish tastes succulent with mild sweetness acquired from palm sugar and a hint of spiciness.

Gallery

See also

 Empal gentong
 Rendang

References

External links
 Empal gepuk recipe

Sundanese cuisine
Beef dishes
Fried foods


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We know that startup ventures are often working on shoestring budgets but they still need competent legal assistance. This is one of the reasons we offer a flat rate option as well as a subscription service for some of our clients.
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Whether you need help deciding on the business structure that best meets your needs, or you are uncertain about how to go about protecting your trade name, you can count on us to provide you with the guidance and assistance you need. As your business grows, you will have an advocate by your side to provide litigation services to protect your intellectual property, resolve employment disputes, and more.
We understand the importance of feeling like your attorney is not only advocating on your behalf but also works closely with you to understand your goals. Contact Jayaraman Law if you are starting a technology-oriented business in the greater Chicago area and need help with your corporate governance documents.


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Brooklyn Chamber brings food, fun times to State Capitol
March 16, 2016 By Paula Katinas Brooklyn Daily Eagle
Chamber President Carlo Scissura welcomes members of the Brooklyn delegation to the Brooklyn Night celebration. Eagle photos by Paula Katinas
Who would have guessed that the best way to show off Brooklyn's culinary status to Democrats and Republicans in the state Legislature was by presenting chocolate treats shaped like donkeys and elephants?
The eatable political party symbols, which came in both milk and dark chocolate, were part of Brooklyn Night, an event sponsored by the Brooklyn Chamber of Commerce in Albany that served as a highlight to the Chamber's two-day visit to the state Capitol this week.
Brooklyn Night took place at 1 Empire State Plaza on the grounds of the state Capitol Monday night and attracted hundreds of people, including Lt. Gov. Kathy Hochul, State Comptroller Thomas DiNapoli, state Senate Majority Leader John Flanagan (R-Suffolk County), Assembly Speaker Carl Heastie (D-Bronx) and nearly every member of the Brooklyn delegation to the state Legislature.
There was mouth watering food everywhere.
More than 50 vendors participated in Brooklyn Night, offering free samples of their products to hungry visitors.
"Tonight is an amazing night," Chamber President and CEO Carlo Scissura told the crowd.
The donkey and elephant candies came courtesy of Li-Lac Chocolates. Brooklyn Bangers served sausages and mashed potatoes. There was also beer, courtesy of Coney Island Brewery.
Carmen Rodriguez, owner of Brooklyn Cupcake, served her delicious cupcakes and told visitors a touching story of how the Chamber helped her get back on her feet and start a business after she had lost her job.
"We've all got stories. Someone inspired us," said Rodriguez, who started her company in 2011 and now sells her products in Barclays Center.
It was a night when the best of Brooklyn was celebrated.
"There's only two kinds of people. Those from Brooklyn and those who want to be from Brooklyn!" said Assemblymember Joseph Lentol (D-North Brooklyn), who is the leader of the Assembly's Brooklyn delegation.
It was also a night to celebrate the diversity, drive and achievements of the borough's small business community. "Small businesses are the lifeblood of our communities," Heastie told the crowd.
The majority of the members of the Brooklyn Chamber of Commerce are small business owners.
Chamber brings Brooklyn business concerns to Albany
March 16 | By Paula Katinas


Bots, or Internet robots, are also known as spiders, crawlers, and web bots. While they may be utilized to perform repetitive jobs, such as indexing a search engine, they often come in the form of malware. Malware bots are used to gain total control over a computer.
One of the typical "good" bot uses is to gather information. Bots in such guises are called web crawlers. Another "good" use is automatic interaction with instant messaging, instant relay chat, or assorted other web interfaces. Dynamic interaction with websites is yet another way bots are used for positive purposes.
Bots are usually used to infect large numbers of computers. These computers form a "botnet," or a bot network.
One of the many unfortunate things about malicious bots is the fact that they can easily go unnoticed. They hide in "the shadows" of a computer, and many times have file names and processes similar if not identical to regular system files/processes.
The download method is one of the ways in which bots infect your computer. Malware is delivered in download format via social media or email messages that advise clicking a link. The link is often in picture or video form, with either containing viruses and other malware.
Another typical bot attack comes in the form of "scare tactics." Many times a bot will appear as a warning saying that if you do not click on the associated link, your computer will get a virus. Clicking the link subsequently infects your computer with a virus.
Internet access is slow for no apparent reason.
The computer crashes for no apparent reason.
The fan goes into overdrive when the device is idle.
The computer takes a long time to shut down, or fails to shut down correctly.
Pop-up windows and advertisements appear even when you aren't using a web browser.
Friends and family receive email messages you did not send.
Computer programs are running slowly.
Settings have changed, and there's no way to reverse them.
The browser features components you didn't download.
How Do I Protect My Computer From Bots?
Install firewalls to block malicious attacks and never turn them off.
Use a long and complicated password that contains numbers and symbols.
Never use the same password for multiple programs.
Install quality anti-malware software such as Norton Security to protect your device.
Ensure software is up to date, and never ignore system updates.
Refrain from using flash drives, or thumb drives, in an infected computer.
How Do I Clean Up My Infected Computer?
If your computer is already infected by bots, protecting your data is the most important thing. Disconnect the computer from the network as soon as possible, which will halt the theft of sensitive information. It also prevents your computer from being used to attack other networks. The next step is moving all important/personal data to another computer or external hard drive—just make certain they're malware-free first! Once this is complete, you'll need to clean your computer using assorted security tools, or by having a professional work on the device.
Remember, prevention is the best medicine in regard to bots and all other malware. Stay up to date with your software, never click on anything suspicious, and utilize anti-malware techniques to the fullest extent.