text:
\section{Introduction}
We consider finite horizon feedback steering of an ensemble of trajectories subject to a controlled stochastic differential equation (SDE) with endpoint joint state probability density function (PDF) constraints -- a topic of growing interest in the systems-control literature. Motivating applications include belief space motion planning for vehicular autonomy, and the steering of robotic or biological swarms via decentralized feedback. While early contributions focused on the covariance control \cite{hotz1987covariance,skelton1989covariance,grigoriadis1997minimum}, more recent papers \cite{chen2015optimal1,chen2015optimal2,chen2016optimal} addressed the optimal feedback synthesis for steering an arbitrary prescribed initial joint state PDF to another prescribed terminal joint state PDF subject to controlled linear dynamics, and revealed the connections between the associated stochastic optimal control problem, the theory of optimal mass transport \cite{villani2003topics}, and the Schr\"{o}dinger bridge \cite{schrodinger1931umkehrung,schrodinger1932theorie}. Follow up works have accounted terminal cost \cite{halder2016finite}, input constraints \cite{bakolas2018finite,okamoto2019input}, output feedback \cite{bakolas2017covariance}, and some nonlinear dynamics \cite{caluya2019finite,caluya2019finiteMulti,caluya2019wasserstein}. As for the state or path constraints, prior work \cite{okamoto2018optimal} incorporated the same in soft probabilistic sense. The contribution of the present paper is to account \emph{hard deterministic path constraints} in the problem of minimum effort finite horizon PDF steering via feedback synthesis. This can be intuitively phrased as the ``hard safety with soft endpoint" problem.

The main idea underlying the ensuing development is to modify the unconstrained It\^{o} SDEs to the ``reflected It\^{o} SDEs" \cite{ikeda1961construction,watanabe1971stochastic,lions1984stochastic,harrison1987multidimensional}, i.e., the controlled sample paths in the state space (in addition to the control-affine deterministic drift) are driven by two stochastic processes: a Wiener process, and a local time stochastic process. The latter enforces the sample paths in the state space to satisfy the deterministic non-strict\footnote{There is no loss of generality in allowing the sample paths to satisfy \emph{non-strict} path containment in given $\mathcal{X} \subset \mathbb{R}^n$ since \emph{strict} containment can be enforced by reflecting them from $\epsilon$-inner boundary layer of $\partial\mathcal{X}$ for $\epsilon$ small enough.} path containment constraints at all times. These considerations engender a Schr\"{o}dinger bridge-like formulation--referred hereafter as the \emph{Reflected Schr\"{o}dinger Bridge Problem (RSBP)}--which unlike its classical counterpart, has extra boundary conditions involving the gradients of the so-called Schr\"{o}dinger factors. We show how recent developments in contraction mapping w.r.t. the Hilbert metric, and the proximal recursion over the Schr\"{o}dinger factors can be harnessed to solve the RSBP.

\section{Reflected Schr\"{o}dinger Bridge Problem}\label{SecRSBP}

\subsection{Formulation}\label{SubsecFormulation}
Consider a connected, smooth\footnote{More precisely, there exists $\xi\in C_{b}^{2}\left(\mathbb{R}^{n}\right)$ such that $\mathcal{X} \equiv \{{\bm{x}}\in\mathbb{R}^{n}\mid \xi({\bm{x}}) > 0\}$ with boundary $\partial\mathcal{X}\equiv\{{\bm{x}}\in\mathbb{R}^{n}\mid \xi({\bm{x}}) = 0\}$.} and bounded domain $\mathcal{X} \subset \mathbb{R}^n$. Let $\overline{\mathcal{X}}:=\mathcal{X}\cup\partial\mathcal{X}$ denote the closure of $\mathcal{X}$. For time $t\in[0,1]$, consider the stochastic control problem
 \begin{subequations} \label{ReflectedOCPM}
\begin{align}
& \underset{{\bm{u}} \in \mathcal{U}} {\text{inf}}
& &  \mathbb{E} \left\{\int_{0}^{1} \frac{1}{2} \| {\bm{u}}(t,{\bm{x}}_{t}^{{\bm{u}}})  \|_{2}^{2}  \: {\rm{d}} t \right\}\label{ReflectedOCPobjM}  \\
& \text{subject to}
&  & {\rm{d}} {\bm{x}}_t^{{\bm{u}}} = {\bm{f}}(t,{\bm{x}}_t^{{\bm{u}}}) \:  {\rm{d}} t + {\bm{u}}(t,{\bm{x}}_t^{{\bm{u}}}) \: {\rm{d}} t \nonumber \\
& &  & \quad \qquad +\sqrt{2\theta} \: {\rm{d}} {\bm{w}}_t +\bm{n}({\bm{x}}_t^{{\bm{u}}}) {\rm{d}} \gamma_t, \label{ReflectedOCPdynM}  \\
& & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\bm{x}}_{0}^{{\bm{u}}} :=  {\bm{x}}_t^{u}\left(t=0\right) \sim  \rho_{0}, \quad {\bm{x}}_1^{{\bm{u}}}:=  {\bm{x}}_t^{u}\left(t=1\right) \sim \rho_{1}, \label{ReflectedOCPbcM}
\end{align}
\end{subequations}
where ${\bm{w}}_t$ is the standard Wiener process in $\mathbb{R}^{n}$, the controlled state ${\bm{x}}_{t}^{{\bm{u}}}\in\overline{\mathcal{X}}$, and the endpoint joint state PDFs $\rho_{0},\rho_{1}$ are prescribed\footnote{The notation ${\bm{x}} \sim \rho$ means that the random vector ${\bm{x}}$ has joint PDF $\rho$.} such that their supports are in $\overline{\mathcal{X}}$, both are everywhere nonnegative, have finite second moments, and $\int\rho_{0}=\int\rho_{1}=1$. The parameter $\theta > 0$ is referred to as the thermodynamic temperature, and the expectation operator $\mathbb{E}\{\cdot\}$ in (\ref{ReflectedOCPobjM}) is w.r.t. the law of the controlled state ${\bm{x}}_{t}^{{\bm{u}}}$. The set $\mathcal{U}$ consists of all admissible feedback policies $\bm{u}(t,{\bm{x}}_t^{u})$, given by $\mathcal{U}:=\{\bm{u}: [0,1]\times\overline{\mathcal{X}} \mapsto \mathbb{R}^{n} \mid \|\bm{u}\|_{2}^{2} < \infty, {\bm{u}}(t,\cdot)\in\text{Lipschitz}\left(\overline{\mathcal{X}}\right)\:\text{for all}\:t\in[0,1]\}$. We assume that the prior drift vector field $\bm{f}$ is bounded Borel measurable in $(t,{\bm{x}}_{t}^{{\bm{u}}})\in[0,1]\times\overline{\mathcal{X}}$, and Lipschitz continuous w.r.t. ${\bm{x}}_{t}^{{\bm{u}}}\in\overline{\mathcal{X}}$. The vector field $\bm{n}$ is set to be the inward unit normal to the boundary $\partial\mathcal{X}$, and gives the direction of reflection. Furthermore, for $t\in[0,1]$, $\gamma_t$ is \emph{minimal local time}: a continuous, non-negative and non-decreasing stochastic process \cite{glynn2018rate,harrison2013brownian,pilipenko2014introduction} that restricts ${\bm{x}}_t^{u}$ to the domain $\mathcal{X}$, with $\gamma_{0}\equiv 0$. Specifically, letting $\mathds{1}_{\{\}}$ denote the indicator function of the subscripted set, we have
\begin{equation}
\gamma_t = \int_{0}^{t}\mathds{1}_{\{{\bm{x}}_s^{{\bm{u}}} \in \partial\mathcal{X}\}}\: {\rm{d}} \gamma_s, \quad \int_{0}^1 \mathds{1}_{\{{\bm{x}}_t^{{\bm{u}}} \notin \partial \mathcal{X}\}}\: {\rm{d}} \gamma_t =0,
\label{gammat}
\end{equation}
which is to say that the process $\gamma_t$ only increases at times $t\in[0,1]$ when ${\bm{x}}_t^{{\bm{u}}}$ hits the boundary, i.e.,  when ${\bm{x}}_t^{{\bm{u}}} \in  \partial \mathcal{X} $. Thus, (\ref{ReflectedOCPdynM}) is a controlled reflected SDE, and the tuple $({\bm{x}}_t^{{\bm{u}}},\gamma_t)$ solves the \emph{Skorokhod problem} \cite{skorokhod1961stochastic,skorokhod1962stochastic,kruk2007explicit}.
We point the readers to \cite{lions1984stochastic} for the proof of existence and uniqueness of solutions to (\ref{ReflectedOCPdynM}) under the stated regularity assumptions.

To formalize the probabilistic setting of the problem at hand, let $\Omega$ be the space of continuous functions $\omega: [0,1] \mapsto \overline{\mathcal{X}}$. We view $\Omega$ as a complete separable metric space endowed with the topology of uniform convergence on compact time intervals. With $\Omega$, we associate the $\sigma$-algebra $\mathscr{F} = \sigma\{\omega(s) \mid 0 \leq s \leq 1\}$. Consider the complete filtered probability space $\left(\Omega,\mathscr{F},\mathbb{P}\right)$ with filtration $\mathscr{F}_{t} = \sigma\{\omega(s)\mid 0\leq s \leq t \leq 1\}$ wherein ``complete" means that $\mathscr{F}_{0}$ contains all $\mathbb{P}$-null sets, and $\mathscr{F}_{t}$ is right continuous. The processes ${\bm{w}}_{t}$, ${\bm{x}}_{t}^{{\bm{u}}}$ (for a given feedback policy ${\bm{u}}$) and $\gamma_{t}$ are $\mathscr{F}_{t}$-adapted (i.e., non-anticipating) for $t\in[0,1]$. In (\ref{ReflectedOCPbcM}), the random vectors ${\bm{x}}_{0}^{{\bm{u}}}$ and ${\bm{x}}_{1}^{{\bm{u}}}$ are respectively $\mathscr{F}_{0}$-measurable and $\mathscr{F}_{1}$-measurable.

Denote the Euclidean gradient operator as $\nabla$, the inner product as $\langle\cdot,\cdot\rangle$, and the Laplacian as $\Delta$. Letting
\[\mathcal{L} := \theta\Delta + \langle {\bm{f}} + {\bm{u}},\nabla\rangle,\]
the law of the sample path of (\ref{ReflectedOCPdynM}) can be characterized \cite{stroock1971diffusion} as follows: for each ${\bm{x}}\in\overline{\mathcal{X}}$, there is a \emph{unique} probability measure $\mathbb{P}_{{\bm{x}}}^{\mu}$ on $\Omega$ such that (i) $\mathbb{P}_{{\bm{x}}}^{\mu}\left({\bm{x}}_t^{{\bm{u}}}(t=0) = {\bm{x}}\right) = 1$, (ii) for any $\phi\in C_{c}^{1,2}\left([0,1];\overline{\mathcal{X}}\right)$ whose inner normal derivative on $\partial\mathcal{X}$ is nonnegative,
\[\phi(t,{\bm{x}}_{t}^{{\bm{u}}}) - \displaystyle\int_{0}^{t}\left(\dfrac{\partial\phi}{\partial s} + \mathcal{L}\phi\right)\left(s,{\bm{x}}_{s}^{{\bm{u}}}\right)\:{\rm{d}} s\]
is $\mathbb{P}^{\mu}_{{\bm{x}}}$-submartingale, and (iii) there is a continuous, nonnegative, nondecreasing stochastic process $\gamma_{t}$ satisfying (\ref{gammat}). As a consequence \cite[p. 196]{stroock1971diffusion} of this characterization it follows that the process ${\bm{x}}_{t}^{{\bm{u}}}$ is Feller continuous and strongly Markov. In particular, the measure-valued trajectory $\mathbb{P}_{{\bm{x}}_{t}^{{\bm{u}}}}^{\mu(t)}$ comprises of absolutely continuous measures w.r.t. Lebesgue measure.

The objective in problem (\ref{ReflectedOCPM}) is to perform the minimum control effort steering of the given initial state PDF $\rho_{0}$ at $t=0$ to the given terminal state PDF $\rho_{1}$ at $t=1$ subject to the controlled sample path dynamics (\ref{ReflectedOCPdynM}). In other words, the data of the problem consists of the domain $\overline{\mathcal{X}}$, the prior dynamics data $\bm{f},\theta$, and the two endpoint PDFs $\rho_{0},\rho_{1}$. \par
Formally, we can transcribe (\ref{ReflectedOCPM}) into the following variational problem \cite{benamou2000computational}:
\begin{subequations}  \label{SBproblem}
\begin{align}
& \underset{(\rho,{\bm{u}})\in \mathcal{P}_2(\overline{\mathcal{X}}) \times \mathcal{U}} {\text{inf}}
& & \int_{0}^{1}\!\!\!\int_{\overline{\mathcal{X}}}\frac{1}{2} \| {\bm{u}}(t,{\bm{x}}_{t}^{u}) \|_{2}^{2}\: \rho(t,{\bm{x}}_{t}^{u})  \: {\rm{d}}{\bm{x}}_{t}^{u}{\rm{d}} t \label{SBproblemConstraint1}\\
& \text{subject to}
& & \frac{\partial \rho }{\partial t} +\nabla \cdot \left(\rho ({\bm{u}}+{\bm{f}}) \right) = \theta \Delta \rho ,  \label{SBproblemConstraint2} \\
& & &\langle -({\bm{u}}+{\bm{f}}) \rho + \theta \nabla \rho,{\bm{n}}\rangle \big |_{\partial \mathcal{X}}=0, \label{SBproblemConstraint3}\\
& & & \rho(0,{\bm{x}}_{t}^{u}) = \rho_{0}, \quad \rho(1,{\bm{x}}_{t}^{u}) = \rho_{1},  \label{SBproblemConstraint4}
\end{align}
\end{subequations}
where a PDF-valued curve $\rho(t,\cdot)\in\mathcal{P}_2(\overline{\mathcal{X}})$ if for each $t\in [0,1]$, the PDF $\rho$ is supported on $\overline{\mathcal{X}}$, and has finite second moment. In this paper, we will not focus on the rather technical direction of establishing the existence of minimizer for (\ref{SBproblem}), which can be pursued along the lines of \cite[p. 243--245]{villani2003topics}. Instead, we will formally derive the conditions of optimality, convert them to the so-called Schr\"{o}dinger system, and argue the existence-uniqueness of solutions for the same.

\subsection{Necessary Conditions of Optimality}\label{SubsecCondOptimality}
The following result summarizes how the optimal pair $(\rho^{\rm{opt}},\bm{u}^{\rm{opt}})$ for problem (\ref{SBproblem}) can be obtained.
\begin{theorem} [\textbf{Optimal control and optimal state PDF}]
A pair $(\rho^{\rm{opt}},\bm{u}^{\rm{opt}})$ solving the variational problem (\ref{SBproblem}) must satisfy the system of coupled nonlinear PDEs:
\begin{subequations} \label{FPKHJB}
\begin{align}
&\frac{\partial \rho^{\rm{opt}} }{\partial t} +\nabla \cdot\left(\rho^{\rm{opt}} (\nabla \psi+{\bm{f}}) \right) = \theta\Delta \rho^{\rm{opt}}, \label{FPK}\\
& \frac{\partial \psi}{\partial t} + \frac{1}{2}\lVert  \nabla \psi \rVert_{2}^2 + \langle \nabla \psi ,{\bm{f}} \rangle = -\theta\Delta \psi,\label{HJB}
\end{align}
\end{subequations}
where
\begin{align} \label{optcntrl}
\bm{u}^{\rm{opt}}(t,\cdot) = \nabla \psi(t,\cdot),
\end{align}
subject to the boundary conditions
\begin{subequations}
\begin{align}
    &\langle \nabla \psi, {\bm{n}} \rangle \big|_{\partial \mathcal{X}}= 0, \quad \text{for all}\quad t\in[0,1], \label{HJBneumann} \\
    &\rho^{\rm{opt}}(0,\cdot) = \rho_0, \quad \rho^{\rm{opt}}(1,\cdot)= \rho_1, \label{BndryCond} \\
    &\langle \rho^{\rm{opt}}(\nabla \psi + {\bm{f}}) -\theta \nabla \rho^{\rm{opt}},{\bm{n}} \rangle \big |_{\partial \mathcal{X}}\!=0, \quad\text{for all}\quad t \in [0,1]. \label{ZeroFluxBndryCond}
\end{align}
\end{subequations}
\end{theorem}
\vspace*{0.1in}
The PDE (\ref{FPK}) is a controlled Fokker-Planck-Kolmogorov (FPK) equation, and (\ref{HJB}) is a Hamilton-Jacobi-Bellman (HJB) equation. Because the equations (\ref{FPK})-(\ref{HJB}) have one way coupling, and the boundary conditions (\ref{HJBneumann})-(\ref{ZeroFluxBndryCond}) are atypical, solving (\ref{FPKHJB}) is a challenging task in general. In the following, we show that it is possible to transform the \emph{coupled nonlinear} system (\ref{FPKHJB}) into a boundary coupled \emph{linear} system of PDEs which we refer to as the \emph{Schr\"{o}dinger system}. We will see that the resulting system paves way to a computational pipeline for solving the density steering problem with path constraints.

\subsection{Schr\"{o}dinger System}\label{SubsecSchrodingerSystem}
We now apply the Hopf-Cole transform \cite{cole1951quasi,hopf1950partial} to the system of nonlinear PDEs  (\ref{FPKHJB}).
\begin{theorem}[\textbf{Schr\"{o}dinger system}]\label{SchrodingerSystemTheorem}
Given the data $\overline{\mathcal{X}},{\bm{f}},\theta,\rho_0,\rho_1$ for problem (\ref{SBproblem}), consider the Hopf-Cole transform $(\rho^{\rm{opt}},\psi) \mapsto (\varphi,\hat{\varphi})$ given by
\begin{subequations}
\begin{align}
    \varphi(t,\cdot) &:= \exp\left(\psi(t,\cdot)/2\theta \right), \label{HCback}\\
    \hat{\varphi}(t,\cdot) &:= \rho^{\rm{opt}}(t,\cdot) \exp \left(-\psi(t,\cdot)/2\theta \right), \label{HCforw}
\end{align}
\label{DefHC}
\end{subequations}
applied to (\ref{FPKHJB}) where $t\in[0,1]$. For $k\in\{0,1\}$, introduce the notation $\varphi_{k}:=\varphi(k,\cdot)$, $\hat{\varphi}_{k}:=\hat{\varphi}(k,\cdot)$. Then the pair $(\varphi,\hat{\varphi})$ satisfies the system of linear PDEs
\begin{subequations} \label{ForwardBackwardSystem}
\begin{align}
    \frac{\partial \varphi}{\partial t} &=-\langle \nabla \varphi,{\bm{f}} \rangle -\theta \Delta \varphi,  \label{BackwardEqn}\\
    \frac{\partial \hat{\varphi}}{\partial t} & = -\nabla \cdot( {\bm{f}} \hat{\varphi})+\theta \Delta \hat{\varphi},\label{ForwardEqn}
\end{align}
\end{subequations}
subject to the boundary conditions
\begin{subequations}
\begin{align}
&\varphi_{0} \hat{\varphi}_{0} = \rho_0, \quad \varphi_{1} \hat{\varphi}_{1} = \rho_1,\label{HCbndrycond1}\\
 &\langle \nabla \varphi, {\bm{n}} \rangle \big|_{\partial \mathcal{X}} =
 \langle {\bm{f}} \hat{\varphi} - \theta \nabla \hat{\varphi},{\bm{n}}\rangle \big|_{\partial \mathcal{X}} = 0.\label{HCbndrycond2}
 \end{align}
 \label{HCbndrycond}
\end{subequations}
For all $t\in[0,1]$, the pair $(\rho^{\rm{opt}},\bm{u}^{\rm{opt}})$ can be recovered as
\begin{align}
\!\rho^{\rm{opt}}(t,\cdot) = \varphi(t,\cdot)\hat{\varphi}(t,\cdot), \!\!\quad \!\!\bm{u}^{\rm{opt}}(t,\cdot) = 2\theta\nabla\log\varphi(t,\cdot).
\label{recoverroptuopt}
\end{align}
\end{theorem}

\begin{remark}
From (\ref{DefHC}), both $\varphi,\hat{\varphi}$ are nonnegative by definition, and strictly positive if $\psi$ is bounded and $\rho^{\rm{opt}}$ is positive.
\end{remark}

\begin{remark} Under the regularity assumptions on ${\bm{f}}$ and $\overline{\mathcal{X}}$ stated in Section \ref{SubsecFormulation}, the process ${\bm{x}}_t$ satisfying the \emph{uncontrolled} reflected It\^{o} SDE
\begin{align} \label{uncontrolledSDE}
\!\!{\rm{d}}{\bm{x}}_t = {\bm{f}}(t,{\bm{x}}_t) \:  {\rm{d}} t +\sqrt{2\theta} \: {\rm{d}} {\bm{w}}_t +\bm{n}({\bm{x}}_t) \: {\rm{d}} \gamma_t, \: t\in[0,1],
\end{align}
is a Feller continuous strongly Markov process. Therefore, the theory of semigroups applies and the transition density of (\ref{uncontrolledSDE}) satisfies Kolmogorov's equations. Notice that the transition density or Green's function will depend on the domain $\overline{\mathcal{X}}$. In particular, we point out that (\ref{BackwardEqn}) is the \emph{backward Kolmogorov equation} in
unkonwn $\varphi$ with the corresponding \emph{Neumann boundary condition} $\langle\nabla\varphi,{\bm{n}}\rangle\big |_{\partial\mathcal{X}}=0$ in (\ref{HCbndrycond2}). On the other hand, (\ref{ForwardEqn}) is the \emph{forward Kolmogorov equation} in unkonwn $\hat{\varphi}$ with the corresponding \emph{Robin boundary condition} $\langle {\bm{f}} \hat{\varphi} - \theta \nabla \hat{\varphi},{\bm{n}}\rangle \big|_{\partial \mathcal{X}} = 0$ in (\ref{HCbndrycond2}). These ``backward Kolmogorov with Neumann" and ``forward Kolmogorov with Robin" system of PDE boundary value problems are coupled via the atypical boundary conditions (\ref{HCbndrycond1}).
\end{remark}

\begin{figure}[t]
\centering
    \includegraphics[width=0.95\linewidth]{CommutativeDiagramSBP.pdf}
     \caption{{\small{Schematic of the fixed point recursion for the Schr\"{o}dinger system (\ref{ForwardBackwardSystem})-(\ref{HCbndrycond}). The abbreviation ``b.c." stands for boundary condition, the symbol $\oslash$ denotes the Hadamard division.}}}
    \label{FigCommutativeDiagm}
    \vspace*{-0.23in}
\end{figure}

Theorem \ref{SchrodingerSystemTheorem} reduces finding the optimal pair $\left(\rho^{\rm{opt}},\bm{u}^{\rm{opt}}\right)$ for the RSBP to that of finding the pair\footnote{We refer to $\varphi(t,{\bm{x}}_t),\hat{\varphi}(t,{\bm{x}}_t)$ as the \emph{Schr\"{o}dinger factors}.} $(\varphi(t,{\bm{x}}_t),\hat{\varphi}(t,{\bm{x}}_t))$ associated with the uncontrolled SDE (\ref{uncontrolledSDE}). To do so, we need to compute the terminal-initial condition pair $(\varphi_1,\hat{\varphi}_0)$, which can be obtained by first making an initial guess for $(\varphi_1,\hat{\varphi}_0)$ and then performing time update by integrating the system (\ref{ForwardBackwardSystem})-(\ref{HCbndrycond2}). Using (\ref{HCbndrycond1}), this then sets up a fixed point recursion over the pair $(\varphi_1,\hat{\varphi}_0)$ (see Fig. \ref{FigCommutativeDiagm}). If this recursion converges to a unique pair, then the converged pair $(\varphi_1,\hat{\varphi}_0)$ can be used to compute the transient factors $(\varphi(t,{\bm{x}}_t),\hat{\varphi}(t,{\bm{x}}_t))$, and we can recover $(\rho^{\rm{opt}},\bm{u}^{\rm{opt}})$ via (\ref{recoverroptuopt}). This computational pipeline will be pursued in this paper.

Since the PDEs in (\ref{ForwardBackwardSystem}) are linear, and the boundary couplings in (\ref{HCbndrycond1}) are in product form, the nonnegative function pair $(\varphi_1,\hat{\varphi}_0)$ can only be unique in the projective sense, i.e., if $(\varphi_1,\hat{\varphi}_0)$ is a solution then so is $(\alpha\varphi_1,\hat{\varphi}_0/\alpha)$ for any $\alpha>0$. In \cite{chen2016entropic}, it was shown that the aforesaid fixed point recursion is in fact \emph{contractive} on a suitable cone in Hilbert's projective metric, and hence guaranteed to converge to a unique pair $(\varphi_1,\hat{\varphi}_0)$, provided that the transition density for (\ref{uncontrolledSDE}) is positive and continuous\footnote{Under the regularity assumptions on ${\bm{f}}$ and $\overline{\mathcal{X}}$ stated in Section \ref{SubsecFormulation}, the transition density for (\ref{uncontrolledSDE}) indeed satisfies
these conditions.} on $\overline{\mathcal{X}}\times\overline{\mathcal{X}}$ for all $t\in[0,1]$, and $\rho_{0},\rho_{1}$ are supported on compact subsets of $\overline{\mathcal{X}}$.

\begin{figure}[t]
\centering
    \includegraphics[width=0.9\linewidth]{PathPlots.pdf}
     \caption{{\small{For $t\in[0,1]$, the \emph{solid line} shows a sample path $x_{t}$ for (\ref{TwosidedRBM}) with $[a,b]\equiv[-1,1]$, $\theta=0.5$. The \emph{dotted line} shows the corresponding unconstrained sample path $x_{t}^{\text{unconstrained}}$, computed using the two-sided Skorokhod map \cite{kruk2007explicit}.}}}
    \label{FigPathPlot}
    \vspace*{-0.23in}
\end{figure}

\section{Case Study: RSBP in 1D without Prior Drift} \label{RSBPInterval}
To illustrate the ideas presented thus far, we now consider a simple instance of problem (\ref{SBproblem}) over the state space $\overline{\mathcal{X}} =[a,b] \subset \mathbb{R}$, and with the prior drift $f\equiv 0$. That is to say, we consider the finite horizon density steering subject to the controlled two-sided reflected Brownian motion. Using some properties of the associated Markov kernel, we will show that the Schr\"{o}dinger system (\ref{ForwardBackwardSystem})-(\ref{HCbndrycond}) corresponding to this particular RSBP has a unique solution which can be obtained by the kind of fixed point recursion mentioned toward the end of Section \ref{SubsecSchrodingerSystem}.

In this case, the Schr\"{o}dinger system (\ref{ForwardBackwardSystem})-(\ref{HCbndrycond}) reduces to
\begin{subequations} \label{ForwardBackwardHeat}
\begin{align}
&\frac{\partial \varphi}{\partial t} = -\theta \frac{\partial^2 \varphi}{\partial x^2}, \label{BackwardHeatPDE}\\
 &\frac{\partial \hat{\varphi}}{\partial t} = \theta \frac{\partial^2 \hat{\varphi}}{\partial x^2}, \label{ForwardHeatPDE}\\
 &\varphi_0\hat{\varphi}_0 = \rho_0, \quad \varphi_1\hat{\varphi}_1 = \rho_1, \label{ForwardBackwardHeatbc2} \\
&\frac{\partial \varphi}{\partial x}\bigg|_{x=a,b} = \frac{\partial \hat{\varphi}}{\partial x}\bigg|_{x=a,b} =0. \label{gradBCforwardbackwardHeat}
\end{align}
\end{subequations}
Notice that (\ref{BackwardHeatPDE})-(\ref{ForwardHeatPDE}) are the backward and forward heat PDEs, respectively, which subject to (\ref{gradBCforwardbackwardHeat}), have solutions
\begin{subequations} \label{FormalSolution}
\begin{align}
    \varphi(x,t) &=\int_{[a ,b]} K_{\theta}(x,y,1-t) \varphi_1(y)\: {\rm{d}} y  ,\quad t\leq 1, \\
    \hat{\varphi}(x,t) &=\int_{[a,b]} K_{\theta}(y,x,t) \hat{\varphi}_0(y)\: {\rm{d}} y  , \qquad \: \: \: t \geq 0,
\end{align}
\end{subequations}
where
\begin{align} \label{TransitionDensity}
    K_{\theta}(x,y,t)& :=\frac{1}{b-a} + \frac{2}{b-a}\sum_{m=1}^{\infty} \exp\left(-\frac{\theta \pi^2 m^2 }{(b-a)^2} t \right)  \nonumber \\
    & \times \cos\left(\frac{m \pi (x-a) }{b-a} \right)\cos\left(\frac{m \pi (y-a)}{b-a} \right)
\end{align}
is the Markov kernel or transition density \cite[Sec. 4.1]{linetsky2005transition},\cite[p. 410-411]{bhattacharya2009stochastic} associated with the uncontrolled reflected SDE
\begin{align}
    {\rm{d}} x_t = \sqrt{2\theta}\: {\rm{d}} w_t + {\rm{d}} L_t - {\rm{d}} U_t, \quad t\in[0,1].
    \label{TwosidedRBM}
\end{align}
In (\ref{TwosidedRBM}), $L_t,U_t$ are the two local time stochastic processes \cite{glynn2018rate,harrison2013brownian} at the lower and upper boundaries respectively, which restrict $x_t$ to the interval $[a,b]$; see Fig. \ref{FigPathPlot}.

Combining (\ref{FormalSolution}) and (\ref{ForwardBackwardHeatbc2}), we get a system of coupled nonlinear integral equations in unknowns $(\varphi_1,\hat{\varphi}_0)$, given by
\begin{subequations} \label{IntegralEqns}
\begin{align}
\rho_0(x) &= \hat{\varphi}_0(x) \int_{[a,b]} K_{\theta}(x,y,1) \varphi_1(y) \: {\rm{d}} y, \\
\rho_1(x) &= \varphi_1(x) \int_{[a,b]} K_{\theta}(y,x,1) \hat{\varphi}_0(y) \: {\rm{d}} y.
\end{align}
\end{subequations}
Clearly, solving (\ref{IntegralEqns}) is equivalent to solving (\ref{ForwardBackwardHeat}). The pair $(\varphi_1,\hat{\varphi}_0)$ can be solved from (\ref{IntegralEqns}) iteratively as a fixed point recursion with guaranteed convergence established through contraction mapping in Hilbert's projective metric;  see \cite{chen2016entropic}. The Lemma \ref{lemma} stated next will be used in the Proposition \ref{PropRBMcontraction} that follows, showing the existence and uniqueness of the pair $(\varphi_1,\hat{\varphi}_0)$ in (\ref{IntegralEqns}) as well as the fact that the aforesaid fixed point recursion is guaranteed to converge to that pair.

 \begin{lemma}
For $0<\theta$, $a<b$, consider the transition probability density $K_{\theta}(x,y,t)$ in (\ref{TransitionDensity}). Then,
\begin{enumerate}[i.]
    \item[(i)] $K_{\theta}(x,y,t=1)$ is continuous on the set $[a,b] \times [a,b]$.
    \item[(ii)] $K_{\theta}(x,y,t=1)>0$ for all $(x,y) \in [a,b] \times [a,b]$.
\end{enumerate}
\label{lemma}
\end{lemma}
\begin{proposition} Given $0<\theta$, $a<b$, and the endpoint PDFs $\rho_0,\rho_1$ having compact supports $\subseteq [a,b]$. There exists a unique pair $(\varphi_1,\hat{\varphi}_0)$ that solves (\ref{IntegralEqns}), and equivalently (\ref{ForwardBackwardHeat}). Moreover, this unique pair can be computed by the fixed point recursion shown in Fig. \ref{FigCommutativeDiagm}.
\label{PropRBMcontraction}
\end{proposition}

To illustrate how the above results can be used for practical computation, consider solving the RSBP (\ref{ReflectedOCPM}) with $f\equiv 0$, $\theta=0.5$, $\overline{\mathcal{X}} = [a,b]\equiv [-4,4]$, and $\rho_{0},\rho_{1}$ as (see Fig. \ref{FigEndpointPDFs1d})
\begin{subequations}\label{EndpointPDFs1Dexample}
\begin{align}
\rho_0(x) &\propto 1 + (x^2 - 16)^{2}\exp(-x/2), \label{1dexamplerho0}\\
\rho_1(x) &\propto 1.2 -\cos(\pi (x+4)/2),\label{1dexamplerho1}
\end{align}
\end{subequations}
where the supports of (\ref{EndpointPDFs1Dexample}) are restricted to $[-4,4]$, and the proportionality constants are determined accordingly. For state feedback synthesis enabling this unimodal to bimodal steering over $t\in[0,1]$, we performed the fixed point recursion over the pair $\left(\varphi_{1},\hat{\varphi}_{0}\right)$ using (\ref{IntegralEqns}) with $\rho_{0},\rho_{1}$ as in (\ref{EndpointPDFs1Dexample}), and $K_{\theta}$ given by (\ref{TransitionDensity}). For numerical implementation, we truncated the infinite sum in (\ref{TransitionDensity}) after the first 100 terms. Fig. \ref{FigDistanceContraction} shows the convergence of this fixed point recursion w.r.t. Hilbert's projective metric. The converged pair $\left(\varphi_{1},\hat{\varphi}_{0}\right)$ is used to compute the transient Schr\"{o}dinger factors $(\varphi(t,{\bm{x}}_t),\hat{\varphi}(t,{\bm{x}}_t))$ via (\ref{FormalSolution}), and then the pair $(\rho^{\rm{opt}}(t,{\bm{x}}_t^u),\bm{u}^{\rm{opt}}(t,{\bm{x}}_t^u))$ via (\ref{recoverroptuopt}). Fig. \ref{FigRBM} depicts the evolution of the optimal controlled transient joint state PDFs $\rho^{\rm{opt}}(t,x_{t}^{u})$ as well as 100 sample paths $x_{t}^{u}$ of the optimal closed-loop reflected SDE. These sample paths were computed by applying the Euler-Maruyama scheme with time-step size $10^{-3}$. Notice from Fig. \ref{FigRBM} that (i) the closed-loop sample paths satisfy $-4\leq x_{t}^{u} \leq 4$ for all $t\in[0,1]$, and (ii) in the absence of feedback, the terminal constraint $\rho(1,x_{1}^{u})=\rho_{1}$ (given by (\ref{1dexamplerho1})) cannot be satisfied.
\begin{figure}[t]
    \centering
    \includegraphics[width=0.9\linewidth]{EndpointPDFs1Dexample.png}
     \caption{{\small{The endpoint PDFs $\rho_{0},\rho_{1}$ shown above are supported on $[-4,4]$, and are given by (\ref{EndpointPDFs1Dexample}).}}}
     \vspace*{-0.23in}
    \label{FigEndpointPDFs1d}
\end{figure}

\begin{figure}[htpb]
    \centering
    \includegraphics[width=0.95\linewidth]{dHilbert.pdf}
     \caption{{\small{Convergence of the fixed point recursion over $(\varphi_{1},\hat{\varphi}_{0})$ in Hilbert's projective metric $d_{\text{Hilbert}}$.}}}
     \vspace*{-0.23in}
    \label{FigDistanceContraction}
\end{figure}

\begin{figure}[h]
    \centering
    \includegraphics[width=0.95\linewidth]{ReflectedBrownianMotionControlled.png}
    \caption{{\small{Shown as the black curves are the optimal controlled transient joint state PDFs $\rho^{\rm{opt}}(t,x_{t}^{u})$ for steering the two-sided reflecting Brownian motion with endpoint PDFs $\rho_0,\rho_1$ as in Fig. \ref{FigEndpointPDFs1d}. The red curve $\rho_{1}^{\rm{unc}}$ is the uncontrolled state PDF at $t=1$, i.e., obtained by setting $u\equiv 0$. Also depicted are the 100 sample paths of the optimally controlled (i.e., closed-loop) reflected SDE. This simulation corresponds to the RSBP (\ref{ReflectedOCPM}) with problem data $f\equiv 0, [a,b]=[-4,4], \theta = 0.5$, and $\rho_0,\rho_1$ given by (\ref{EndpointPDFs1Dexample}).}}}
    \vspace*{-0.2in}
    \label{FigRBM}
\end{figure}

\section{RSBP with Prior Drift}\label{RSBPprox}
For generic ${\bm{f}}$, $\overline{\mathcal{X}}$, there is no closed-form expression of the Markov Kernel associated with (\ref{ForwardBackwardSystem})-(\ref{HCbndrycond2}). Hence, unlike the situation in Section \ref{RSBPInterval}, we cannot explicitly set up coupled integral equations of the form (\ref{IntegralEqns}), thus preventing the numerical implementation of the fixed point recursion (Fig. \ref{FigCommutativeDiagm}) via direct matrix-vector recursion. In this Section, we will show that if ${\bm{f}}$ is gradient of a potential, then we can reformulate (\ref{ForwardBackwardSystem})-(\ref{HCbndrycond}) in a way that leads to a variational recursion which in turn enables us to implement the fixed point recursion (Fig. \ref{FigCommutativeDiagm}) in an implicit manner.

\subsection{Reformulation of the Schr\"{o}dinger System}
Let ${\bm{f}}$ be a  gradient vector field, i.e., ${\bm{f}}=-\nabla V$ for some potential $V\in C^2(\overline{\mathcal{X}})$. The associated Schr\"{o}dinger system (\ref{ForwardBackwardSystem})-(\ref{HCbndrycond}) becomes
\begin{subequations}\label{ForwardBackwardGrad}
\begin{align}
    &\frac{\partial \varphi}{\partial t} =\langle \nabla \varphi,\nabla V \rangle -\theta \Delta \varphi, \label{BackwardEqnGrad} \\
    &\frac{\partial \hat{\varphi}}{\partial t} = \nabla \cdot( \nabla V \hat{\varphi})+\theta \Delta \hat{\varphi}, \label{ForwardEqnGrad} \\
    &\varphi_0\hat{\varphi}_0 = \rho_0, \quad \varphi_1\hat{\varphi}_1 = \rho_1, \\
 &\langle \nabla \varphi, {\bm{n}} \rangle \big|_{\partial \mathcal{X}}=\langle \nabla V\hat{\varphi}+ \theta \nabla \hat{\varphi},{\bm{n}}\rangle \big|_{\partial \mathcal{X}} = 0.
 \end{align}
\end{subequations}
The idea now is to exploit the structural nonlinearities in (\ref{ForwardBackwardGrad}) to design an algorithm that allows computing the Schr\"{o}dinger factors $(\varphi,\hat{\varphi})$. To that end, the following is a crucial step.
\begin{theorem}\label{ThmReformulateSS}
Given $V\in C^2(\overline{\mathcal{X}}),\theta>0$, and $t\in[0,1]$, consider $\varphi(t,{\bm{x}}_t)$ in (\ref{ForwardBackwardGrad}). Let $s:=1-t$, and define the mappings $\varphi \mapsto q \mapsto p$ given by $q(s,{\bm{x}}_s):= \varphi(t,{\bm{x}}_t)=\varphi(1-s,{\bm{x}}_{1-s})$, $p(s,{\bm{x}}_s):=q(s,{\bm{x}}_s)\exp(-V({\bm{x}}_s)/\theta)$. Then $p(s,{\bm{x}}_s)$ solves the PDE initial boundary value problem:
\begin{subequations}
\begin{align}
    &\frac{\partial p}{\partial s} = \nabla \cdot (p \nabla V) + \theta \Delta p, \label{pPDE}\\
    &p(0,{\bm{x}})= \varphi_1({\bm{x}})\exp(-V({\bm{x}})/\theta), \label{pInitialCondition}\\
    &\langle \nabla V p + \theta\nabla p,{\bm{n}} \rangle \big|_{\partial \mathcal{X}} =0.\label{pBoundaryCondition}
\end{align}
\end{subequations}
\end{theorem}
Thanks to Theorem \ref{ThmReformulateSS}, solving (\ref{ForwardBackwardGrad}) is equivalent to solving
\begin{subequations}\label{ForwardBackwardGradREform}
\begin{align}
    &\frac{\partial p}{\partial s} = \nabla \cdot (p \nabla V) + \theta \Delta p, \label{BackwardEqnGradReform} \\
    &\frac{\partial \hat{\varphi}}{\partial t} = \nabla \cdot( \nabla V \hat{\varphi})+\theta \Delta \hat{\varphi},\label{ForwardEqnGradReform} \\
    &p(s=1,{\bm{x}})\exp(V({\bm{x}})/\theta) \hat{\varphi}_0({\bm{x}}) = \rho_0,\nonumber\\
    &p(s=0,{\bm{x}}) \exp(V({\bm{x}})/\theta)\hat{\varphi}_1({\bm{x}}) = \rho_1, \\
 &\langle \nabla V p+\theta\nabla p,{\bm{n}} \rangle \big|_{\partial \mathcal{X}}=\langle \nabla V\hat{\varphi}+ \theta \nabla \hat{\varphi},{\bm{n}}\rangle \big|_{\partial \mathcal{X}} = 0. \label{RobinBC}
 \end{align}
\end{subequations}
From (\ref{BackwardEqnGradReform})-(\ref{ForwardEqnGradReform}), $\varphi$ and $p$ satisfy the exact same FPK PDE with different initial conditions and integrated in different time coordinates $t$ and $s$. From (\ref{RobinBC}), $\varphi$ and $p$ satisfy the same Robin boundary condition. Therefore, a single FPK initial boundary value problem solver can be used to set up the fixed point recursion to solve for $(p_1,\hat{\varphi}_0)$, and hence $(p(s,{\bm{x}}_s),\hat{\varphi}(t,{\bm{x}}_t))$. From $p$, we can recover $\varphi$ as
\[\varphi(t,{\bm{x}}_t)= \varphi(1-s,{\bm{x}}_{1-s}) = p(s,{\bm{x}}_s) \exp(-V({\bm{x}}_s)/\theta).\]

\subsection{Computation via Wasserstein Proximal Recursion} \label{proxrecur}
Building on our previous works \cite{caluya2019proximal,caluya2019gradient,caluya2019wasserstein}, we propose proximal recursions to numerically time march the solutions of the PDE initial boundary value problems (\ref{ForwardBackwardGradREform}) by exploiting certain infinite dimensional gradient descent structure. This enables us to perform the computation associated with the horizontal arrows in Fig. \ref{FigCommutativeDiagm}, and hence the fixed point recursions to solve for $(p,\hat{\varphi})$, and consequently for $(\varphi,\hat{\varphi})$. We give here a brief outline of the ideas behind these proximal recursions.

It is well-known \cite{jordan1998variational,santambrogio2017euclidean} that the flows generated by (\ref{BackwardEqnGradReform}),(\ref{ForwardEqnGradReform}),(\ref{RobinBC}) can be viewed as the gradient descent of the Lyapunov functional
\begin{align}
F(\varrho):= \int_{\overline{\mathcal{X}}} V({\bm{x}})\varrho({\bm{x}}) \:{\rm{d}} {\bm{x}}\: + \theta\int_{\overline{\mathcal{X}}} \varrho({\bm{x}}) \log \varrho({\bm{x}}) \:{\rm{d}} {\bm{x}}
\end{align}
w.r.t. the distance metric $W$ referred to as the (quadratic) Wassertein metric \cite{villani2003topics} on $\mathcal{P}_2(\overline{\mathcal{X}})$.
For chosen time-steps $\tau,\sigma$, this allows us to set up a variational recursion over the discrete time pair $(t_{k-1},s_{k-1}):=((k-1)\tau,(k-1)\sigma)$ as
\begin{align}\label{ProxRecursions}
\begin{pmatrix}
    \hat{\phi}_{t_{k}} \\
    \varpi_{s_{k}}
\end{pmatrix} =
\begin{pmatrix}
{\rm{prox}}_{\tau F}^{W^2}(\hat{\phi}_{t_{k-1}})\\
{\rm{prox}}_{\sigma F}^{W^2}(\varpi_{s_{k-1}})
\end{pmatrix}, \quad  k \in \mathbb{N},
\end{align}
wherein the Wasserstein proximal operator
\begin{align}
{\rm{prox}}_{h F}^{W^2}(\cdot):= \underset{\varrho \in \mathcal{P}_{2}(\overline{\mathcal{X}})}{\arg\inf} \frac{1}{2}W^2(\cdot,\varrho) + h F(\varrho), \quad h>0.
\end{align}
The sequence of functions generated by the proximal recursions (\ref{ProxRecursions}) approximate the flows $(p(s,{\bm{x}}_s),\hat{\varphi}(t,{\bm{x}}_t))$ for (\ref{BackwardEqnGradReform}),(\ref{ForwardEqnGradReform}),(\ref{RobinBC}) in the small time step limit, i.e.,
\begin{align*}
   \hat{\phi}_{t_{k-1}} \rightarrow \hat{\varphi}(t=(k-1)\tau,{\bm{x}}_{t}) \quad &\text{in $L^{1}(\overline{\mathcal{X}})$ as $\tau \downarrow 0$}, \\
   \varpi_{s_{k-1}} \rightarrow p(s=(k-1)\sigma,{\bm{x}}_s) \quad &\text{in $L^{1}(\overline{\mathcal{X}})$ as $\sigma \downarrow 0$}.
\end{align*}
In the numerical example provided next, we solved (\ref{ProxRecursions}) using the algorithm developed in  \cite{caluya2019gradient}.

\subsection{Numerical Example}
 We consider an instance of the RSBP with $\overline{\mathcal{X}}=[-4,4]^{2}$, ${\bm{f}} = -\nabla V$, $V(x_1,x_2):=(x_1^2+x_2^3)/5$. For
 \begin{subequations}\label{EndpointPDFs2Dexample}
\begin{align}
\rho_0(x_1,x_2) &\propto \prod_{i=1,2} \left(1 + (x_i^2 - 16)^{2}\exp(-x_i/2)\right), \label{2dexamplerho0}\\
\rho_1(x_1,x_2) &\propto \prod_{i=1,2} \left(1.2 -\cos(\pi (x_i+4)/2)\right),\label{2dexamplerho1}
\end{align}
\end{subequations}
the optimal controlled joint state PDFs $\rho^{\rm{opt}}(t,{\bm{x}}_t^{{\bm{u}}})$ are shown in Fig. \ref{rho_con}. The corresponding uncontrolled joint state PDFs $\rho^{\text{unc}}(t,{\bm{x}}_t)$ are shown in Fig. \ref{rho_unc}. These results were obtained by solving (\ref{ProxRecursions}) via \cite[Sec. III.B]{caluya2019gradient} with $\tau=\sigma=10^{-3}$ to perform the fixed point recursion (Fig. \ref{FigCommutativeDiagm}) applied to (\ref{ForwardBackwardGradREform}).

\begin{figure*}[t]
\hspace{.1cm}
    \centering
    \includegraphics[width=\linewidth]{rho_con.png}
     \caption{For the RSBP in Section \ref{proxrecur}, shown here are the contour plots of the optimal controlled joint state PDFs $\rho^{\rm{opt}}(t,{\bm{x}}_t^{{\bm{u}}})$ over $\overline{\mathcal{X}}=[-4,4]^2$. Each subplot corresponds to a different snapshot of $\rho^{\rm{opt}}$ in time. The color denotes the joint PDF value; see colorbar (dark hue = high, light hue = low).}
     \vspace*{-0.1in}
    \label{rho_con}
\end{figure*}

\begin{figure*}[t]
\centering
\hspace{.2cm}
    \includegraphics[width=\linewidth]{rho_uncon.png}
     \caption{For the RSBP in Section \ref{proxrecur}, shown here are the contour plots of the uncontrolled joint state PDFs $\rho^{{\rm{unc}}}(t,{\bm{x}}_t)$ over $\overline{\mathcal{X}}=[-4,4]^2$ starting from (\ref{2dexamplerho0}). Each subplot corresponds to a different snapshot of $\rho^{{\rm{unc}}}$ in time. The color denotes the joint PDF value; see colorbar (dark hue = high, light hue = low).}
     \vspace*{-0.15in}
    \label{rho_unc}
\end{figure*}

\section{Conclusions}
In this paper, we introduced the Reflected Schr\"{o}dinger Bridge Problem (RSBP) -- a stochastic optimal control problem for minimum energy feedback steering of a given joint PDF to another over finite horizon subject to reflecting boundary conditions on the controlled state trajectories. Combining our prior work on Wasserstein proximal recursions with some recent results on contraction mapping associated with the Schr\"{o}dinger system, we provide a computational pipeline for optimal feedback synthesis. Numerical examples are given to highlight the proposed framework.

redpajama_set_name: RedPajamaArXiv


text:
<?php

class MenuModel extends Model{
    /**
     * get modules
     * @return array
     */
    public function getModules(){
        $amodules = $this->table('permission')
                        ->where('showInMenu=1 and pId=0 and type="backend_menu"')
                        ->order('level')
                        ->getField('id,title,URL,icon');
        return $amodules;
    }

    /**
     * get menu match with current url
     * @param $modelName
     * @return array
     */
    public function getMenuByModelName($modelName){
        $amenu = $this->table('permission')
            ->where('showInMenu=1 and type="backend_menu" and `level` like (SELECT CONCAT(p2.`level`,",","%") as `level` FROM permission p1 LEFT JOIN permission p2 ON p1.pId=p2.id WHERE p1.`name`="'.$modelName.'")')
            ->order('`level`')
            ->getField('id,pId,name,title,URL,icon,level');
        //dump($this->getlastsql());
        return $amenu;
    }
}
?>

redpajama_set_name: RedPajamaGithub


Distinguishing Characteristics: White male. Blond hair, blue eyes. Christopher is of Swedish, German and Spanish descent. His hair may darken to brown or brownish-blond as he grows older. His father and brothers are all of average to above average height and Christopher may be also.
In the early morning hours of July 15, 1986, Christopher's parents couldn't locate their son. They had put him to bed at midnight and found his crib empty in the morning. Their garage door opener was missing and a basement window was open. Christopher has never been heard from again. At 6:30 a.m., his parents alerted the police. Arriving officers couldn't find any signs of forced entry, but the front door was unlocked on the night of his disappearance.
Members of Christopher's family initially seemed like potential suspects in his case. His parents had separated prior to his disappearance but were reconciling, and the night of Christopher's disappearance was the first night his father had spent in the home since the separation. Christopher's mother failed two polygraph exams, which she attributed to a withdrawal from tranquilizers. Authorities haven't cleared either of his parents of involvement in the case or charged them in connection with it. Three weeks after he vanished, they drained Quail Lake, three blocks from the family's home, to search for his body. They recovered nothing.
Christopher's family member refuse to consider him deceased. They believe he and his guardians don't know he's an abducted child. Christopher's loved ones operate a site featuring information about his case. Texas and New Mexico top the list of his possible locations. According to the website, the family had been receiving hang-up phone calls for about six months before Christopher disappeared. The calls stopped when he vanished, then resumed again several months later. His sister believes the abductor was Emma Bradshaw, a woman who'd had an affair with Christopher's father and had a history of break-ins. Bradshaw maintains her innocence in the baby's disappearance and the police have not named her as a suspect. In 2015, Bradshaw won a civil judgement for $150,000 against Christopher's sister for interfering with her employment.


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Definition at line 90 of file detect-engine-prefilter.c.
References PrefilterEngine_::alproto, PrefilterEngine_::cb, Packet_::flow, PrefilterEngine_::gid, PrefilterEngine_::is_last, likely, PrefilterEngine_::local_id, next, PACKET_PROFILING_DETECT_END, PACKET_PROFILING_DETECT_START, PrefilterEngine_::pectx, DetectEngineThreadCtx_::pmq, DetectTransaction_::prefilter_flags, PREFILTER_PROFILING_END, PREFILTER_PROFILING_START, PrefilterEngine_::PrefilterTx, PROF_DETECT_PF_SORT1, PrefilterRuleStore_::rule_id_array, PrefilterRuleStore_::rule_id_array_cnt, SCLogDebug, SigGroupHead_::tx_engines, DetectTransaction_::tx_id, PrefilterEngine_::tx_min_progress, DetectTransaction_::tx_progress, and DetectTransaction_::tx_ptr.
Definition at line 139 of file detect-engine-prefilter.c.
References PrefilterEngine_::cb, Packet_::flags, PrefilterEngine_::gid, PrefilterEngine_::is_last, likely, PACKET_PROFILING_DETECT_END, PACKET_PROFILING_DETECT_START, SigGroupHead_::payload_engines, Packet_::payload_len, PrefilterEngine_::pectx, PKT_DETECT_HAS_STREAMDATA, SigGroupHead_::pkt_engines, PKT_NOPAYLOAD_INSPECTION, DetectEngineThreadCtx_::pmq, PrefilterEngine_::Prefilter, PREFILTER_PROFILING_END, PREFILTER_PROFILING_START, PROF_DETECT_PF_PAYLOAD, PROF_DETECT_PF_PKT, PROF_DETECT_PF_SORT1, PrefilterRuleStore_::rule_id_array, PrefilterRuleStore_::rule_id_array_cnt, SCEnter, and SCReturn.
Definition at line 189 of file detect-engine-prefilter.c.
References CLS, PrefilterEngineList_::Free, PrefilterEngineList_::gid, PrefilterEngineList_::id, SigGroupHead_::init, PrefilterEngineList_::name, PrefilterEngineList_::next, PrefilterEngineList_::pectx, SigGroupHeadInitData_::pkt_engines, PrefilterEngineList_::Prefilter, and SCMallocAligned.
Definition at line 223 of file detect-engine-prefilter.c.
References CLS, PrefilterEngineList_::Free, PrefilterEngineList_::gid, PrefilterEngineList_::id, SigGroupHead_::init, PrefilterEngineList_::name, PrefilterEngineList_::next, SigGroupHeadInitData_::payload_engines, PrefilterEngineList_::pectx, PrefilterEngineList_::Prefilter, and SCMallocAligned.
Referenced by PrefilterPktPayloadRegister(), and PrefilterPktStreamRegister().
Definition at line 257 of file detect-engine-prefilter.c.
References PrefilterEngineList_::alproto, CLS, PrefilterEngineList_::Free, PrefilterEngineList_::gid, PrefilterEngineList_::id, SigGroupHead_::init, PrefilterEngineList_::name, PrefilterEngineList_::next, PrefilterEngineList_::pectx, PrefilterEngineList_::PrefilterTx, SCFreeAligned, SCMallocAligned, SigGroupHeadInitData_::tx_engines, and PrefilterEngineList_::tx_min_progress.
Referenced by DetectHttpResponseLineRegister(), PrefilterGenericMpmRegister(), and PrefilterMpmFiledataRegister().
Definition at line 332 of file detect-engine-prefilter.c.
References SigGroupHead_::payload_engines, SigGroupHead_::pkt_engines, and SigGroupHead_::tx_engines.
Definition at line 480 of file detect-engine-prefilter.c.
Definition at line 304 of file detect-engine-prefilter.c.
References PrefilterStore_::FreeFunc, PrefilterEngine_::gid, PrefilterEngine_::is_last, next, PrefilterEngineList_::next, PrefilterEngine_::pectx, and SCFreeAligned.
Definition at line 608 of file detect-engine-prefilter.c.
References DetectMpmAppLayerRegistery_::alproto, PrefilterMpmCtx::GetData, DetectMpmAppLayerRegistery_::GetData, PrefilterMpmCtx::list_id, PrefilterMpmCtx::mpm_ctx, DetectMpmAppLayerRegistery_::pname, PrefilterAppendTxEngine(), SCCalloc, SCEnter, SCFree, PrefilterMpmCtx::transforms, DetectMpmAppLayerRegistery_::transforms, DetectMpmAppLayerRegistery_::tx_min_progress, and DetectMpmAppLayerRegistery_::v2.
Referenced by DetectAppLayerMpmRegister2(), DetectFiledataRegister(), DetectHttpClientBodyRegister(), DetectHttpCookieRegister(), DetectHttpHHRegister(), DetectHttpMethodRegister(), DetectHttpRequestLineRegister(), DetectHttpStatCodeRegister(), DetectHttpStatMsgRegister(), DetectHttpUARegister(), DetectHttpUriRegister(), DetectSmbNamedPipeRegister(), DetectSshProtocolRegister(), DetectSshSoftwareRegister(), DetectTemplateBufferRegister(), DetectTlsFingerprintRegister(), DetectTlsIssuerRegister(), DetectTlsJa3HashRegister(), DetectTlsJa3StringRegister(), DetectTlsSerialRegister(), DetectTlsSniRegister(), and DetectTlsSubjectRegister().
Definition at line 487 of file detect-engine-prefilter.c.
References BUG_ON, PrefilterStore_::FreeFunc, HashListTableAdd(), HashListTableGetListData, HashListTableGetListHead(), HashListTableGetListNext, HashListTableInit(), HashListTableLookup(), PrefilterStore_::id, PrefilterStore_::name, DetectEngineCtx_::prefilter_hash_table, DetectEngineCtx_::prefilter_id, SCCalloc, SCFree, and SCLogDebug.
Definition at line 348 of file detect-engine-prefilter.c.
References PrefilterEngineList_::alproto, PrefilterEngine_::alproto, HashListTable_::array_size, BUG_ON, PrefilterEngine_::cb, CLS, DETECT_PREFILTER_AUTO, DETECT_TBLSIZE, PrefilterEngineList_::gid, PrefilterEngine_::gid, PrefilterEngineList_::id, SigGroupHead_::init, PrefilterEngine_::is_last, PrefilterEngine_::local_id, MAX, PrefilterStore_::name, PrefilterEngineList_::next, PatternMatchPrepareGroup(), SigGroupHeadInitData_::payload_engines, SigGroupHead_::payload_engines, PrefilterEngineList_::pectx, PrefilterEngine_::pectx, SigGroupHeadInitData_::pkt_engines, SigGroupHead_::pkt_engines, PrefilterEngineList_::Prefilter, PrefilterEngine_::Prefilter, DetectEngineCtx_::prefilter_maxid, DetectEngineCtx_::prefilter_setting, PrefilterEngineList_::PrefilterTx, PrefilterEngine_::PrefilterTx, SCFree, SCLogDebug, SCMallocAligned, SigTableElmt_::SetupPrefilter, sigmatch_table, DetectEngineCtx_::sm_types_prefilter, TRUE, SigGroupHeadInitData_::tx_engines, SigGroupHead_::tx_engines, PrefilterEngineList_::tx_min_progress, and PrefilterEngine_::tx_min_progress.
Definition at line 552 of file detect-engine-prefilter.c.


text:
KAR Flexing Tech Muscle
Saturday, October 21, 2017 03:56 PM EDT Updated: Monday, October 23, 2017 10:18 AM EDT
By Andy Ober
An executive with Carmel-based KAR Auction Services Inc. (NYSE: KAR) says the company's acquisition of TradeRev, marks a "transformative moment in the industry." Becca Polak, who leads TradeRev, says the platform allows dealers to launch auctions remotely, potentially doubling the industry's footprint from 10 million to 20 million vehicles per year. Chief Executive Officer Jim Hallett says the move strengthens KAR's position as "a technology company that happens to sell cars."
Polak says the platform allows dealers to launch auctions remotely using a picture, rather than having to physically bring a car to multiple sellers. She says KAR plans to continue bringing the technology to new markets, including California in 2018, growing the company's reach coast-to-coast.
Earlier this month, KAR announced the acquisition of the remaining half of Nth Gen Software Inc., a dealer-to-dealer auction technology platform. The deal is valued at up to $125 million. KAR purchased the first half of the business in 2014 for $30 million. At the time, Hallett said the deal will help KAR maintain a leadership position now that "the digital revolution in remarketing has begun."
The acquisition came on the heels of September's announcement that KAR would invest $80 million to expand its headquarters in a move expected to create 400 jobs by 2020. The company says the project will create a state-of-the-art campus to house 1,500 workers, and give it room for future growth. Groundbreaking is set for early next year. KAR plans to move in during the second half of 2019.
KAR has around 300 locations around the world and about 17,500 employees. Hallett says the company sold more than 5 million vehicles last year in 110 countries.

redpajama_set_name: RedPajamaCommonCrawl


In order for marketing to be effective, it must work closely with research teams and development teams, as well as finance, operations, and human resource departments to ensure the right message is getting out to the right people at the right time. Without a marketing strategy that actually works, many companies are forced to close prematurely due to tanking sales and unhappy customers or no customers at all. Since much of the success of your company revolves around marketing, it is imperative that you gain insights into the latest trends. These marketing events will reveal trends that drive customer experience, the importance of data, creating effective email marketing campaigns, and achieving your marketing goals through video.
MozCon is a full three day event that focus mostly on one overarching theme - Marketing.
Sponsored by HubSpot, INBOUND's three-day event will be filled with guest speakers who will narrate their success stories, launch new products, and create networking opportunities so that you will have a round idea on how you can improve your professional life.
The Festival of Marketing is an experience like no other where marketers can discover, learn, celebrate, and shape the future.
The B2B Marketing Forum hosted by MarketingProfs is a conference where leaders, creators, and innovators get together to learn the latest in B2B marketing.


Just got back from shooting our Fall/Winter Lookbook. Here's a tiny behind-the-scenes taste!
Stay tuned to our Facebook for more sneak peeks coming tomorrow!
I recently scored a sick Guy Laroche knock-off bag – a black leather, bucket-shaped, mini-backpack with gold grommet details – and it's got me falling hard for early 90's fashion!
My current 90's style muse? Marisa Tomei's Oscar-winning character, Mona Lisa Vito, from "My Cousin Vinny." Everyday seems like the perfect day to rock her giant sunglasses, over-sized shoulder pads, and big hair-sprayed hair, with the tiniest of mini skirts and barely-there bandage dresses!
Black black black and leather everything.
Totally loving the strong-shouldered silhouette and all-over stud detail!
Coveting her peek-a-boo blonde streak!
Awesome: deep v-neck with delicate gold pendant and enormous black shades!
And who could forget this classic clip, featuring the most uh-mazing floral print catsuit? Pure 90's perfection!
Are you hooked yet? We here at Calico strongly suggest you Netflix "My Cousin Vinny" immediately. And don't forget to visit our blog again soon for more vintage style inspiration!
Our resident window-dresser and merchandiser-extraordinaire, Cody Nowell, did a bang-up job designing and installing this display. We love the spooky Halloween feel it has at night!
You can check out this window at Calico (173 Union St, downtown New Bedford, MA) until mid-November!


In part 2 of our prognosis for 2010 we look at gold stocks and other.
For gold stocks (XAU index) our cycles point to another flat to down year. We look for a peak around June, with bottoms around March and in September-October.
Euro – US$: Look for a stronger US dollar in the first half of the year, next an appreciating Euro after July.
Oil prices: Chinese cycles indicate rising oil prices for this year and also next. So we want to keep a good portion of investments in oil and related stocks.
Other commodities: Cycles for most other commodities like grains and metals point to downward pressure on prices this year.
Bonds and interest rates: Look for some recovery in bond prices , then another decline in the second half of the year. Interest rates should end the year flat to higher.
Bottom line: keep a good portion in energy related investments, and keep enough cash to take advantage of great buying opportunities we are likely to see within the next two years.
The first two years of a new decade (which are always Metal years) typically generate some great buying opportunities that last for the entire decade. Computers and internet dominated the 90s, and during the last decade metals and commodities have done extremely well.
For this decade I expect the best performance to come from certain new technologies again. Main candidates: biotech, nanotech, robotics, and space travel and exploration. So that's what I want to buy when the market falls in a pessimistic mood this year or next.
Will gradually become defunct in the longer term, so better avoid: newspapers, banks, insurance, manufacturing sector, large "dinosaur" companies.
Today we start giving our outlook for the new year, and first we will take a look at the general picture for stock markets.
In last month's analysis for Nasdaq we mentioned a possible jump to the 2300 level, and that's where we are there right now. So, what next?
As you can see the chart is mostly red, so we can expect to get renewed downside action this year. January is still slightly positive, but next February we will enter the year of Yang Metal Tiger.
Tiger is considered Wood, so the main elements this year will be Metal and Wood. This is very much a mixed bag, as Metal is usually negative for stocks while Wood has a positive influence. Metal controls and wounds Wood, so the growth(Wood) will be subdued, it is like a wounded Tiger, or say wounded growth.
Looking at the monthly variations: after an OK start in January new clouds will appear on the economic sky and investors will realize that things are not nearly as rosy as we have been told to believe recently. At best markets will (try to) hold up till March, because the FPF index stays close to neutral till March, but April-May I will be looking for some noticeable downside action. July-August and December offer the best chances to produce rallies this year. Expect the moves to be of the fast-and-unexpected variety (Tiger style), sudden shocks. Thus more volatile, contrary to the more steady moves we got in Ox year.
Given the zigzag nature of the expected pattern it is more difficult to say where the Nasdaq is likely to end the year. A breakeven year is possible, but I think it is more likely we will get another down year with the Nasdaq Composite Index ending somewhere below 2000 .
So for the moment I would stay out of most stocks and just wait for better entry levels later this year. Metal years have rarely been easy sailing.
look for important news and rare breakthroughs in science, technology and medicine, also in next year 2011 – Metal Rabbit.
a new wave of protectionism, trade wars, and more economic controls as politicians continue to increase their grip on the economy (which will fail as it always has).
a sudden currency or bond market panic.
major changes for Cuba, could be the lifting of the US embargo or the passing away of Fidel Castro.
the green movement and green stocks plagued by scandals.
separation movements become more vigorous.
actions or violence by single individuals make the news.
Check back for our prognosis for gold and other markets in the coming days.
Today we look back on the predictions we made last year, and within the next couple of days we will present our prognosis for 2010.
For the stock market our cycles called for an up year, with economies stabilizing and volatility in the markets decreasing. We put a target of at least 2000 and possibly 2300-2400 for the Nasdaq index.
This worked out very nicely as the Nasdaq just ended the year at 2269. Remember that Nasdaq started the year at 1600 and economies looked very bleak, so a 40% rise was not so obvious.
The VIX volatility index reached its lowest level since the financial crisis started more than a year ago.
While our cycles did not indicate the stock market bottom in March, they were correct in showing a strong up year.
For gold stocks and Euro-US$ rate our prediction was not on the mark.
Gold stocks did quite a bit better than we expected and the XAU index ended well above the range we indicated.
The dollar kept weakening throughout the year and only started showing some strength in the last months.
Our commodity picks were coffee, wheat and silver.
Coffee prices rose gradually throughout the year and ended up about 30%.
Wheat prices stayed largely flat.
Silver went up strongly: from $11/ounce at the beginning of the year silver went as high as $19 and ended the year near $17, up more than 60%.
We also recommended to use price weakness to buy crude oil and sugar for the long term.
Crude oil traded as low as $40/barrel till late February, then went up and now sells near $80.
Sugar could be bought at $12 till April, then went up and ended the year at $27 , that's a gain of 125%.
Overall most of our predictions did well and it will not be easy to match this for 2010.
But of course we will try.
So check back for our 2010 prognosis in the next few days.


text:
Mary McKenzie1
F, #44851, b. circa 1896, d. 1948
Pedigree to immigrant Edward Ball of Newark NJ
Pedigree to immigrant Susannah Ball of New Haven CT
Mary McKenzie was born circa 1896.1 She married Francis Willard Van Antwerp, son of Russell Willard Van Antwerp and Lulu Nellie Allen, circa 1925.1 Mary McKenzie and Francis Willard Van Antwerp were divorced in 1945.1 Mary McKenzie died in 1948 at Boston, Suffolk Co., MA.1
[S524] Michel Metford-Platt and others, The Big gEDcom - Edward Ball of Newark New Jersey (GEDCOM), South Africa, copyright 1996-2003 by The Descendants of Edward Ball of New Jersey Group (used with permission granted 1 May 2005).
Irmine Ericson1
Wilma Marie Van Antwerp1
F, #44853, b. 14 August 1902, d. 31 December 1989
Wilma Marie Van Antwerp was born on 14 August 1902 at Dowagiac, Cass Co., MI.1 She was the daughter of Russell Willard Van Antwerp and Lulu Nellie Allen.1 Wilma Marie Van Antwerp married Walter D. Allen on 14 June 1924 at Dowagiac, Cass Co., MI.1 Wilma Marie Van Antwerp died on 31 December 1989 at Kalamazoo, Kalamazoo Co., MI, at age 87.1
a bookkeeper1
Walter D. Allen1
M, #44854, b. 10 December 1895, d. 1972
Walter D. Allen was born on 10 December 1895 at Dowagiac, Cass Co., MI.1 He married Wilma Marie Van Antwerp, daughter of Russell Willard Van Antwerp and Lulu Nellie Allen, on 14 June 1924 at Dowagiac, Cass Co., MI.1 Walter D. Allen died in 1972 at Kalamazoo, Kalamazoo Co., MI.1
Helen Beryl Van Antwerp1
F, #44855, b. 28 August 1912, d. 18 September 1963
Helen Beryl Van Antwerp was born on 28 August 1912 at Dowagiac, Cass Co., MI.1 She was the daughter of Russell Willard Van Antwerp and Lulu Nellie Allen.1 Helen Beryl Van Antwerp died on 18 September 1963 at Kalamazoo, Kalamazoo Co., MI, at age 51.1
Fred Garland1
Francis Oscar Van Antwerp1
M, #44857, b. 12 September 1827, d. 29 November 1888
Francis Oscar Van Antwerp was born on 12 September 1827 at Oakland Co., MI.1 He married Hester Ball, daughter of Stephen Ball and Susan Berry, on 4 April 1850 at Cassopolis, Cass Co., MI.1 Francis Oscar Van Antwerp died on 29 November 1888 at Dowagiac, Cass Co., MI, at age 61.1 He was buried at Cassopolis, Cass Co., MI.1
He owned and operated a blacksmith and wagon shop for over 25 years.1
27 August 1850 LaGrange, Cass Co., MI "Francis" age 22, b. MI, carpenter, with wife Hester and brother George2
5 June 1860 Dowagiac, Cass Co., MI "F. O." age 34, b. MI, wagon maker, with wife Hester and sons Ritner, Daniel, Frank, and Charles3
Children of Francis Oscar Van Antwerp and Hester Ball
Adelia M. Van Antwerp1 b. 19 Jan 1851, d. 3 Sep 1852
Ritner L. Van Antwerp+1 b. 15 Aug 1853, d. 31 Oct 1924
Daniel Winnie Van Antwerp+1 b. 23 Dec 1855, d. 11 Apr 1927
Frank Wilbur Van Antwerp+1 b. Jan 1858
Charles E. Van Antwerp1 b. 23 Nov 1859, d. 20 Sep 1879
Alice M. Van Antwerp1 b. 9 Dec 1861, d. 8 Feb 1935
Eva M. Van Antwerp+1 b. 19 Apr 1864
Leslie Van Antwerp1 b. 25 Dec 1866, d. 18 Aug 1867
Lester Van Antwerp1 b. 25 Dec 1866, d. 21 Aug 1867
George B. Van Antwerp1 b. 1 Jun 1869, d. 12 Mar 1872
Dora B. Van Antwerp+1 b. 24 Jun 1871, d. 7 May 1933
[S7] 1850 (June 1) U.S. Census, LaGrange, Cass Co., MI, p. 625 (stamped 314).
[S8] 1860 (June 1) U.S. Census, Dowagiac, Cass Co., MI, p. 15.
Hester Ball1
F, #44858, b. 19 September 1829, d. 24 April 1900
Hester Ball was born on 19 September 1829 at Butler Co., OH.1 She was the daughter of Stephen Ball and Susan Berry.1 Hester Ball married Francis Oscar Van Antwerp on 4 April 1850 at Cassopolis, Cass Co., MI.1 Hester Ball died on 24 April 1900 at Dowagiac, Cass Co., MI, at age 70.1 She was buried on 26 April 1900 at Cassopolis, Cass Co., MI.1
27 August 1850 LaGrange, Cass Co., MI "Hester" age 20, b. OH, with husband2
5 June 1860 Dowagiac, Cass Co., MI "Hester" age 32, b. OH, with husband, etc.3
Children of Hester Ball and Francis Oscar Van Antwerp
Adelia M. Van Antwerp1
F, #44859, b. 19 January 1851, d. 3 September 1852
Adelia M. Van Antwerp was born on 19 January 1851 at Cassopolis, Cass Co., MI.1 She was the daughter of Francis Oscar Van Antwerp and Hester Ball.1 Adelia M. Van Antwerp died on 3 September 1852 at Cassopolis, Cass Co., MI, at age 1.1 She was buried at Cassopolis, Cass Co., MI.1
Ritner L. Van Antwerp1
M, #44860, b. 15 August 1853, d. 31 October 1924
Ritner L. Van Antwerp was born on 15 August 1853 at Cassopolis, Cass Co., MI.1 He was the son of Francis Oscar Van Antwerp and Hester Ball.1 Ritner L. Van Antwerp married Dora A. Wells, daughter of Willard Wells, on 28 October 1873 at Dowagiac, Cass Co., MI.1 Ritner L. Van Antwerp died on 31 October 1924 at Dowagiac, Cass Co., MI, at age 71.1 He was buried on 2 November 1924 at Dowagiac, Cass Co., MI.1
5 June 1860 Dowagiac, Cass Co., MI "R. L." age 6, b. MI, with parents, etc.2
Children of Ritner L. Van Antwerp and Dora A. Wells
Russell Willard Van Antwerp+1 b. 13 May 1874, d. 27 Nov 1949
Beryl Nina Van Antwerp1 b. Jun 1883, d. 1956
Dora A. Wells1
F, #44861, b. 29 September 1853, d. 6 April 1915
Dora A. Wells was born on 29 September 1853 at Onondaga Co., NY.1 She was the daughter of Willard Wells.1 Dora A. Wells married Ritner L. Van Antwerp, son of Francis Oscar Van Antwerp and Hester Ball, on 28 October 1873 at Dowagiac, Cass Co., MI.1 Dora A. Wells died on 6 April 1915 at Dowagiac, Cass Co., MI, at age 61.1 She was buried on 8 April 1915 at Dowagiac, Cass Co., MI.1
Children of Dora A. Wells and Ritner L. Van Antwerp
Daniel Winnie Van Antwerp1
M, #44862, b. 23 December 1855, d. 11 April 1927
Daniel Winnie Van Antwerp was born on 23 December 1855 at Cassopolis, Cass Co., MI.1 He was the son of Francis Oscar Van Antwerp and Hester Ball.1 Daniel Winnie Van Antwerp married Ada Hardy in 1882 at Cass Co., MI.1 Daniel Winnie Van Antwerp married Cora A. Howard on 18 November 1886 at Cass Co., MI.1 Daniel Winnie Van Antwerp married Pearl McStay Lake on 2 July 1919 at Cass Co., MI.1 Daniel Winnie Van Antwerp died on 11 April 1927 at Dowagiac, Cass Co., MI, at age 71.1
a bookkeeper, a traveling salesman for the Beckwith Company, and a real estate agent. Just prior to his death, he was re-elected as Justice of the Peace for a second term1
5 June 1860 Dowagiac, Cass Co., MI "D. W." age 4, b. MI, with parents, etc.2
Child of Daniel Winnie Van Antwerp and Ada Hardy
Ada M. Van Antwerp1 b. 21 Feb 1883
Ada Hardy1
F, #44863, b. 15 February 1860, d. 21 February 1883
Ada Hardy was born on 15 February 1860 at MI.1 She married Daniel Winnie Van Antwerp, son of Francis Oscar Van Antwerp and Hester Ball, in 1882 at Cass Co., MI.1 Ada Hardy died on 21 February 1883 at Cass Co., MI, at age 23.1
Child of Ada Hardy and Daniel Winnie Van Antwerp
Cora A. Howard1
F, #44864, b. June 1861, d. 20 June 1911
Cora A. Howard was born in June 1861 at MI.1 She married Daniel Winnie Van Antwerp, son of Francis Oscar Van Antwerp and Hester Ball, on 18 November 1886 at Cass Co., MI.1 Cora A. Howard died on 20 June 1911 at Cass Co., MI.1
Frank Wilbur Van Antwerp1
M, #44865, b. January 1858
Frank Wilbur Van Antwerp was born in January 1858 at Cassopolis, Cass Co., MI.1 He was the son of Francis Oscar Van Antwerp and Hester Ball.1 Frank Wilbur Van Antwerp married Ada B. Warner on 2 August 1882 at Dowagiac, Cass Co., MI.1
After the death of his father in 1888, Wilbur took over the operation of the wagon shop in Dowagiac.1
5 June 1860 Dowagiac, Cass Co., MI "F. W." age 2, b. MI, with parents, etc.2
Children of Frank Wilbur Van Antwerp and Ada B. Warner
Nellie Van Antwerp1 b. May 1883, d. 26 Jan 1903
Leigh Van Antwerp1 b. Jun 1888
Ada B. Warner1
F, #44866, b. October 1861
Ada B. Warner was born in October 1861 at MI.1 She married Frank Wilbur Van Antwerp, son of Francis Oscar Van Antwerp and Hester Ball, on 2 August 1882 at Dowagiac, Cass Co., MI.1
Children of Ada B. Warner and Frank Wilbur Van Antwerp
Charles E. Van Antwerp1
M, #44867, b. 23 November 1859, d. 20 September 1879
Charles E. Van Antwerp was born on 23 November 1859 at Dowagiac, Cass Co., MI.1 He was the son of Francis Oscar Van Antwerp and Hester Ball.1 Charles E. Van Antwerp died on 20 September 1879 at Dowagiac, Cass Co., MI, at age 19.1 He was buried at Cassopolis, Cass Co., MI.1
5 June 1860 Dowagiac, Cass Co., MI "C. E." age 6/12, b. MI, with parents, etc.2
Alice M. Van Antwerp1
F, #44868, b. 9 December 1861, d. 8 February 1935
Alice M. Van Antwerp was born on 9 December 1861 at Dowagiac, Cass Co., MI.1 She was the daughter of Francis Oscar Van Antwerp and Hester Ball.1 Alice M. Van Antwerp died on 8 February 1935 at Dowagiac, Cass Co., MI, at age 73.1 She was buried at Dowagiac, Cass Co., MI.1
Eva M. Van Antwerp1
F, #44869, b. 19 April 1864
Eva M. Van Antwerp was born on 19 April 1864 at Dowagiac, Cass Co., MI.1 She was the daughter of Francis Oscar Van Antwerp and Hester Ball.1 Eva M. Van Antwerp married Edgar Pollard in 1885 at Cass Co., MI.1
1935 LaGrange, Cass Co., MI 1
Children of Eva M. Van Antwerp and Edgar Pollard
Arlynn Pollard1 b. Jul 1886
Mabel Pollard1 b. Jun 1895
Edgar Pollard1
M, #44870, b. July 1860
Edgar Pollard was born in July 1860 at NY.1 He married Eva M. Van Antwerp, daughter of Francis Oscar Van Antwerp and Hester Ball, in 1885 at Cass Co., MI.1
Edgar Pollard owned and operated a restaurant at Dowagiac, Cass Co., MI.1
Children of Edgar Pollard and Eva M. Van Antwerp
Leslie Van Antwerp1
M, #44871, b. 25 December 1866, d. 18 August 1867
Leslie Van Antwerp was born on 25 December 1866 at Dowagiac, Cass Co., MI.1 He was the son of Francis Oscar Van Antwerp and Hester Ball.1 Leslie Van Antwerp died on 18 August 1867 at Dowagiac, Cass Co., MI.1 He was buried at Cassopolis, Cass Co., MI.1
Lester Van Antwerp1
Lester Van Antwerp was born on 25 December 1866 at Dowagiac, Cass Co., MI.1 He was the son of Francis Oscar Van Antwerp and Hester Ball.1 Lester Van Antwerp died on 21 August 1867 at Dowagiac, Cass Co., MI.1 He was buried at Cassopolis, Cass Co., MI.1
George B. Van Antwerp1
M, #44873, b. 1 June 1869, d. 12 March 1872
George B. Van Antwerp was born on 1 June 1869 at Dowagiac, Cass Co., MI.1 He was the son of Francis Oscar Van Antwerp and Hester Ball.1 George B. Van Antwerp died on 12 March 1872 at Dowagiac, Cass Co., MI, at age 2.1 He was buried at Cassopolis, Cass Co., MI.1
Dora B. Van Antwerp1
F, #44874, b. 24 June 1871, d. 7 May 1933
Dora B. Van Antwerp was born on 24 June 1871 at Dowagiac, Cass Co., MI.1 She was the daughter of Francis Oscar Van Antwerp and Hester Ball.1 Dora B. Van Antwerp married Cyrus Glenn on 18 July 1906 at Dowagiac, Cass Co., MI.1 Dora B. Van Antwerp married Frank Wheaton after 1910 at Cass Co., MI.1 Dora B. Van Antwerp died on 7 May 1933 at Dowagiac, Cass Co., MI, at age 61.1
a school teacher1
Children of Dora B. Van Antwerp and Cyrus Glenn
Francis D. Glenn1 b. c 1907
Hester Glenn1 b. c 1910, d. b 1920
Cyrus Glenn1
M, #44875, b. circa 1879, d. after 1910
Cyrus Glenn was born circa 1879.1 He married Dora B. Van Antwerp, daughter of Francis Oscar Van Antwerp and Hester Ball, on 18 July 1906 at Dowagiac, Cass Co., MI.1 Cyrus Glenn died after 1910 at Dowagiac, Cass Co., MI.1
Children of Cyrus Glenn and Dora B. Van Antwerp


Chrissy Marie dresses in a Winter Fetish catsuit, a tight satin corset, and some tall vinyl boots. We stuff her mouth and fit her with a heavy leather muzzle gag, and strap it up to the steel canopy. An armbinder is added next, and then we strap her midsection very tightly, welding her arms in place behind her. Her legs are banded together, the armbinder "D" ring is anchored to the strap at her knees, and her boots are belted to the bottom of the frame. She is now standing helpless and completely immobilized. She struggles, twists, and pleads with those big blue eyes..