text:
\section{Introduction}
This paper presents a generalization of the minimax state estimation approach to linear Differential-Algebraic Equations (DAE) in the form
\begin{equation}
  \label{eq:state}
  \dfrac {d(Fx)}{dt}=C(t)x(t)+f(t), \quad Fx(t_0)=x_0
\end{equation}
where $F\in\mathbb R^{m\times n}$ is a rectangular $m\times n$-matrix and $t\mapsto C(t)\in\mathbb R^{m\times n}$ is a continuous matrix-valued function. The research presented here may be thought of as a continuation of the paper~\cite{Zhuk2009c}, where the case of discrete time DAEs with time-depending coefficients was investigated.
We stress that the DAE with $F\in\mathbb R^{m\times n}$ is non-causal (the matrix pencil $F-\lambda C(t)$ is singular\cite{Gantmacher1960}) if $m\ne n$. Also the coefficient $C(t)$ depends on time. Therefore the state estimation problem for DAE in the form~\eqref{eq:state} can not be directly addressed by parameter estimation methods (see for instance \cite{Gerdin2007}, \cite{Darouach1997} and citations there), based on the transformation of the regular matrix pencil $F-\lambda C$ ($\mathrm{det}\;(F-\lambda C)\not\equiv0$) to the Weierstrass canonical form \cite{Gantmacher1960}. As it was mentioned in \cite{Gerdin2007}, the latter transformation allows to convert DAE (with regular pencil) into Ordinary Differential Equation (ODE), provided the unknown input $f$ is smooth enough and $C(t)\equiv\,\mathrm{const}$.
On the other hand, in applications $f$ is often modelled as a realization of some random process or as a measurable squared-integrable function with bounded $\mathbb L_2$-norm. One way to go is to take Sobolev-Shvartz derivative of $f$, allowing the state $x(t)$ of DAE to be discontinuous function. If the latter is not acceptable, it is natural to ask if it is possible to derive a state estimation algorithm for DAE (in the form~(\ref{eq:state})) avoiding the differentiation of the unknown input $f$. More generally, is it possible to derive a state estimation algorithm for DAE in the form~(\ref{eq:state}) with measurable $f$ without transforming the pencil $F-\lambda C$ into a canonical form? The same question arises if $C(t)$ is not constant as in this case it may be impossible (see \cite{Campbell1987}) to transform DAE to ODE even if the pencil $F-\lambda C(t)$ is regular for all $t$.
In this paper we give a positive answer to this question for the following state estimation problem: given observations $y(t)$, $t\in[t_0,T]$ of $x(t)$, to reconstruct $Fx(T)$, provided $x$ is a weak solution to \eqref{eq:state}.
We note, that many authors (see \cite{Gerdin2007}, \cite{Darouach1997} and citations there) assume the state vector $x(t)$ of~(\ref{eq:state}) to be a differentiable (in the classical sense) function. In contrast, we only assume that $t\mapsto Fx(t)$ is an absolutely continuous function. In this setting $x(T)$ is not necessary well defined. Hence, it makes sense to estimate $Fx(T)$ only.
In what follows, we assume that $f$ is an unknown squared-integrable function, which belongs to a given bounded set $\mathscr G$. We will also assume that observations $y(t)$ may be incomplete and noisy, that is $y(t)=H(t)x(t)+\eta(t)$, where $\eta$ is a realization of a random process with zero mean and unknown but bounded correlation function.
Following \cite{Nakonechnii1978} we will be looking for the minimax estimate $\widehat{\ell(x)}$ of a linear function
\footnote{Note that in order to reconstruct $Fx(T)$ it is enough to reconstruct a linear function $\ell(x):=\langle\ell,Fx(T)\rangle$ for any $\ell\in\mathbb R^m$.
Having the estimate of $\ell(x)$ for any $\ell\in\mathbb R^m$, one can set $\ell:=e_i=(0,\dots,1,\dots0)^T$ in order to reconstruct $i$-th component of $Fx(T)$.} $\ell(x):=\langle\ell,Fx(T)\rangle$ among all linear functions of observations $u(y)$. Main notions of deterministic minimax state estimation approach \cite{Tempo1985,Chernousko1994}, \cite{Kurzhanski1997} are reachability set, minimax estimate and worst-case error. By definition, reachability set contains all states of the model which are consistent with observed data and uncertainty description. Given a point $P$ within the reachability set one defines a worst-case error as the maximal distance between $P$ and other points of the reachability set. Then the minimax estimate of the state is defined as a point minimizing the worst-case error (a Tchebysheff center of the reachability set). In this paper we deal with random noise in observations. This prevents us from describing the reachability set. Instead, we derive a dynamic mean-squared minimax estimate minimizing mean-squared worst-case error.

The contributions of this paper are a Generalized Kalman Duality (GKD) and sub-optimal minimax estimation algorithm, both for DAE in the form~\eqref{eq:state}. As it was previously noted in \cite{Gerdin2007} the need to differentiate an unknown input
posed a problem of mathematical justification of the filtering framework based on DAE with classical derivatives. In \cite{Gerdin2007} the authors propose a solution, provided $\mathrm{det}(F-\lambda C)\ne 0$ for any $\lambda$ \cite{Gerdin2007}. Here we apply GKD in order to justify the minimax filtering framework for the case of DAEs (\ref{eq:state}) with any rectangular $F$ and time-varying $C(t)$. We do not use the theory of matrix pencils so that the condition of differentiability of the unknown input $f$ in (\ref{eq:state}) is not necessary for our derivations.
Applying GKD we arrive to the Dual Control Problem (DCP) with DAE constraint, which has a unique absolutely continuous solution, provided $\ell$ belongs to the minimax observable subspace $\mathcal L(T)$. Otherwise, the solution of DCP is represented in terms of the impulsive control. In this sense the minimax observable subspace generalizes impulse observability condition (see \cite{Gerdin2007}) to the case of DAE with rectangular time-varying coefficients. The cost function of DCP describes the mean-squared worst-case error and its minimizer represents a minimax estimate. However, Pontryagin Maximum Principle (PMP) can not be applied directly to solve DCP: a straightforward application of the PMP to the dual problem could reduce the minimax observable subspace to the trivial case $\mathcal L(T)=\{0\}$ (see example in Subsection \ref{ss:optcond}). In order to preserve the structure of $\mathcal L(T)$ we apply Tikhonov regularization approach.
As a result (Proposition~\ref{p:3e}) we represent a sub-optimal minimax state estimation algorithm as a unique solution of a well-posed Euler-Lagrange system with a small parameter. This solution converges to the minimax estimate. We represent the sub-optimal estimate in the classical sequential form: as a solution to a Cauchy problem for a linear stochastic ODE, driven by a realization of observations $y(t)$, $t\in[t_0,T]$. We recall that $y(t)$ is perturbed by a ``random noise'', which can be a realization of any random process (not necessary Gaussian as in \cite{Gerdin2007}) with zero mean and unknown but bounded correlation function. \\

This paper is organized as follows. At the beginning of section~\ref{s:obs} we describe the formal problem statement and introduce definitions of the minimax mean-squared estimates and errors. The rest of this section consists of two subsections. Subsection~\ref{s:gkd} presents the GKD (Theorem~\ref{t:1}). In subsection~\ref{ss:optcond} we discuss optimality conditions and derive regularization scheme (Proposition~\ref{p:3e}) along with the representation of the sub-optimal minimax estimate in the sequential form (Corollary~\ref{c:eme}). Also we present an example. Section~\ref{sec:Conclusion} contains conclusion. Appendix contains proofs of technical statements.\\
\emph{Notation}:
$E\eta$ denotes the mean of the random element $\eta$; $\mathrm{int}\, G$ denotes the interior of $G$;
$\mathbb R^n$ denotes the $n$-dimensional Euclidean space;
$\mathbb L_2(t_0,T,\mathbb R^m)$ denotes a space of square-integrable functions with values in $\mathbb R^m$ (in what follows we will often write $\mathbb L_2$ referring $\mathbb L_2(t_0,T,\mathbb R^k)$ where the dimension $k$ will be defined by the context);
$\mathbb H_1(t_0,T,\mathbb R^m)$ denotes a space of absolutely continuous functions with $\mathbb L_2$-derivative and values in $\mathbb R^m$;
the prime $'$ denotes the operation of taking the adjoint: $L'$ denotes adjoint operator, $F'$ denotes the transposed matrix;
$c(G,\cdot)$ denotes the support function of a set $G$;
$\langle\cdot,\cdot\rangle$ denotes the inner product in a Hilbert space $\mathcal H$, $\|x\|^2_{\mathcal H}:=\langle x,x\rangle$, $\|\cdot\|$ denotes norm in $\mathbb R^n$;
$S>0$ means $\langle Sx,x\rangle>0$ for all $x$;
$F^+$ denotes the pseudoinverse matrix; $Q^\frac 12$ denotes the square-root of the symmetric non-negative matrix $Q$, $I_n$ denotes $n\times n$-identity matrix, $0_{n\times m}$ denotes $n\times m$-zero matrix, $I_0:=0$; $\mathrm{tr}$ denotes the trace of the matrix.
\section{Linear minimax estimation for DAE}\label{s:obs}
Consider a pair of system
\begin{align}
      & \dfrac{d(Fx)}{dt}=C(t)x(t)+f(t)\,,\quad Fx(t_0)=x_0\,,\label{eq:dae}\\
      & y(t)=H(t)x(t)+\eta(t)\,,    \label{eq:dae:obs}
\end{align}
where $x(t)\in\mathbb R^n$, $f(t)\in\mathbb R^m$, $y(t)\in\mathbb R^p$,
$\eta(t)\in\mathbb R^p$ represent the state, input, observation and
observation's noise respectively. As above, we assume $F\in\mathbb R^{m\times n}$, $f\in\mathbb L_2(t_0,T,\mathbb R^m)$, and $C(t)$ and $H(t)$ are continuous\footnote{Slightly modifying the proofs we can allow $C(t)$ and $H(t)$ to be just measurable.} matrix-valued functions of $t$ on $[t_0,T]$, $t_0,T\in\mathbb R$. Now let us describe our assumptions on uncertain $x_0,f,\eta$. Let $\eta$ be a realization of a random process such that $E\eta(t)=0$ on $[t_0,T]$ and $\mathcal R_\eta(t,s):=E\Psi(t)\Psi'(s)$ is bounded:
\begin{equation}
  \label{eq:eta_bounds}
\mathcal R_\eta\in W=\{\mathcal R_\eta:\int_{t_0}^T\mathrm{tr}\,( R(t)\mathcal R_\eta(t,t))dt\le 1\}
\end{equation}
We note that the assumption $E\eta(t)=0$ is not restrictive. If  $E\eta(t)=\overline\eta(t)\ne 0$ where $\overline\eta (t)$ is a known measurable function then we can consider new measurements $\overline y(t):=y(t)-\overline\eta(t)$ and new noise $\eta(t)-\overline\eta(t)$. \\
The initial condition $x_0$ and input $f$ are supposed to belong to the following set
\begin{equation}
  \label{eq:G}
  \mathscr G:=\{(x_0,f):\langle Q_0x_0,x_0\rangle+\int_{t_0}^T\langle Q(t)f,f\rangle dt\le 1\},
\end{equation}
where $Q_0,Q(t)\in\mathbb R^{m\times m}$, $Q_0=Q_0'>0$, $Q=Q'>0$, $R(t)\in\mathbb R^{p\times p}$, $R'=R>0$; $Q(t)$, $R(t)$, $R^{-1}(t)$ and $Q^{-1}(t)$ are continuous functions of $t$ on $[t_0,T]$.
\begin{defn}\label{d:1}
Given $T<+\infty$, $u\in\mathbb L_2(t_0,T,\mathbb R^p)$ and $\ell\in\mathbb R^m$ define a mean-squared worst-case estimation error
\begin{equation*}
  \begin{split}
    &\sigma(T,\ell,u):=\sup_{x, (x_0,f)\in \mathscr G, \mathcal R_\eta\in W}
    E[\langle\ell, Fx(T)\rangle-u(y)]^
  \end{split}
\end{equation*}
A function $\hat u(y)=\int_{t_0}^{T}\langle\hat u(t),y(t)\rangle dt$ is called an a priori minimax mean-squared estimate in the direction $\ell$ ($\ell$-estimate) if $\inf_u\sigma(T,\ell,u)=\sigma(T,\ell,\hat u)$. The number
$\hat\sigma(T,\ell)=\sigma(T,\ell,\hat u)$ is called a minimax mean-squared a priori error in the direction $\ell$ at time-instant $T$ ($\ell$-error).
The set $\mathcal L(T)=\{\ell\in\mathbb R^m:\hat\sigma(T,\ell)<+\infty\}$ is called a minimax observable subspace.
\end{defn}
\subsection{Generalized Kalman Duality Principle}\label{s:gkd}
  Definition~\ref{d:1}
  reflects the procedure of deriving the minimax estimation. The first step is, given $\ell$ and $u$ to calculate the worst-case error $\sigma(T,\ell,u)$ by means of the suitable duality concept.
\begin{thm}[Generalized Kalman duality]\label{t:1}
Take $\ell\in\mathbb R^m$. The $\ell$-error $\hat\sigma(T,\ell)$ is finite iff
\begin{equation}
    \label{eq:zul}
    \dfrac{d(F'z)}{dt}=-C'(t)z(t)+H'(t)u(t),\quad F'z(T)=F'\ell
  \end{equation}
for some $z\in\mathbb L_2(0,T,\mathbb R^m)$
and $u\in\mathbb L_2(0,T,\mathbb R^p)$.\\
If $\hat\sigma(T,\ell)<+\infty$ then
\begin{equation}
  \label{eq:umin}
  \begin{split}
\sigma&(T,\ell,u)=
\min_{v,d}\{\| \tilde Q_0^{-\frac12}(z(t_0)-v(t_0))-Q_0^{-\frac12}d\|^2\\
&+\int_{t_0}^T\|Q^{-\frac 12}(z-v)\|^2 dt\}+
\int_{t_0}^T\| R^{-\frac12}u\|^2dt\\
&=\| \tilde Q_0^{-\frac12}(z(t_0)-\tilde v(t_0))-Q_0^{-\frac12}\tilde d\|^2\\
&+\int_{t_0}^T\|Q^{-\frac 12}(z-\tilde v)\|^2 +\| R^{-\frac12}u\|^2dt
  \end{split}
 \end{equation}
where $\tilde Q_0^{-\frac12}=Q_0^{-\frac12}{F'}^+F'$, $\min$ in~\eqref{eq:umin} is taken over all $d$ such that $F'd=0$ and all $v$ verifying \eqref{eq:zul} with $u=0$ and $\ell=0$, and $\min$ is attained at $\tilde v,\tilde d$.
\end{thm}
\begin{rem}\label{r:1}
  An obvious corollary of Theorem~\ref{t:1} is an expression for the minimax observable subspace
  \begin{equation}
    \label{eq:LT}
    \mathcal L(T)=\{\ell\in\mathbb R^m:\text{ \eqref{eq:zul} holds for some }z,\,u\}
  \end{equation}
In the case of stationary $C(t)$ and $H(t)$ the minimax observable subspace may be calculated explicitly, using the canonical Kronecker form \cite{Gantmacher1960}.
\end{rem}
\begin{pf
Take $\ell\in\mathbb R^m$ and $u\in\mathbb L_2(t_0,T,\mathbb R^p)$. Using $E\eta(t)=0$ we compute
\begin{equation}
  \label{eq:error}
  \begin{split}
    E[\ell&(x)-u(y)]^2 = E[\int_{t_0}^T\langle u,\eta\rangle dt]^2(:=\gamma^2)\\
&+\bigl(\langle\ell,Fx(T)\rangle-\int_{t_0}^T\langle H'u,x\rangle dt\bigr)^2(:=\mu^2)
  \end{split}
\end{equation}
Let us transform $\mu^2$. There exists $w\in\mathbb L_2(t_0,T,\mathbb R^m)$ such that: (W) $F'w\in\mathbb H^1(t_0,T,\mathbb R^n)$ and $F'w(T)=F'\ell$. Noting that \cite{Albert1972} $F=FF^+F$ we have
\begin{equation*}
  \langle\ell,Fx(T)\rangle=\langle F'\ell,F^+Fx(T)\rangle =
\langle F'w(T),F^+Fx(T)\rangle\,.
\end{equation*}
The latter equality, an integration-by-parts formula
\begin{equation}
  \label{eq:ibp}
  \begin{split}
\langle F'w(T)&,F^+Fx(T)\rangle-\langle F'w(t_0),F^+Fx(t_0)\rangle=\\
 &= \int_{t_0}^T \langle\dfrac{d(Fx)}{dt},w\rangle+\langle\dfrac{d(F'w)}{dt},x\rangle dt\,.
  \end{split}
\end{equation}
(proved in~\cite{Zhuk2007} for $Fx\in\mathbb H^1(t_0,T,\mathbb R^m)$ and $F'w\in\mathbb H^1(t_0,T,\mathbb R^n)$) and \eqref{eq:dae} gives that
\begin{equation}\label{eq:i1}
  \begin{split}
    \mu=

\langle (F^+)'F'w(t_0),Fx_0\rangle+\int_{t_0}^T \langle f,w\rangle dt\\
&+\int_{t_0}^T\langle\dfrac{d(F'w)}{dt}+C'w-H'u,x\rangle dt\,.
  \end{split}
\end{equation}
By \eqref{eq:error} and \eqref{eq:i1}:
$\sigma(T,\ell,u)=\sup_{\mathcal R_\eta}\gamma^2+\sup_{f,x_0,x}\mu^2$.
By Cauchy inequality $\gamma^2\le
\int_{t_0}^{T}E\langle R\eta,\eta\rangle dt
\int_{t_0}^{T}\langle R^{-1}u,u\rangle dt$. As $E\langle R\eta,\eta\rangle=\mathrm{tr}\,(R\mathcal R_\eta)$, it follows from~\eqref{eq:eta_bounds}
\begin{equation}
  \label{eq:worstcase_error}
  \sigma(T,\ell,u)=
\int_{t_0}^{T}\langle R^{-1}u,u\rangle dt + \sup_{f,x_0,x}\mu^
\end{equation}
Assume $\hat\sigma(T,\ell)<+\infty$. Then $\sigma(T,\ell,u)<\infty$ for at least one $u$ so that $\sup_{f,x_0,x}\mu^2<\infty$. The term $\int_{t_0}^T \langle f,w\rangle dt$ in the first line of~\eqref{eq:i1} is bounded due to~\eqref{eq:G}. Thus
\begin{equation}
  \label{eq:bound_on_w}
  \begin{split}
    \sup_{x} &\{
\langle F'w(t_0),F^+Fx(t_0)\rangle\\
&+\int_{t_0}^T\langle\dfrac{d(F'w)}{dt}+C'w-H'u,x\rangle dt\}<\infty
  \end{split}
\end{equation}
where $\sup$ is taken over all $x$ solving~\eqref{eq:dae} with $(x_0,f)\in\mathscr G$. \eqref{eq:bound_on_w} allows us to prove that there exists $z$ such that~\eqref{eq:zul} holds for the given $\ell$ and $u$. To do so we apply a general duality result\footnote{The proof of \eqref{eq:LfiG} for bounded $L$ and Banach spaces $\mathcal H_{1,2}$ can be found in \cite{Ioffe1974}} from~\cite{Zhuk2009d}:
\begin{equation}
  \label{eq:LfiG}
  \sup_{x\in\mathscr D(L)}\{\langle\mathcal F, x\rangle, Lx\in G\}=
\inf_{b\in\mathscr D(L')}\{c(G,b),L'b=\mathcal F\}
\end{equation}
provided (A1) $L:\mathscr D(L)\subset\mathcal H_1\to\mathcal H_2$ is a closed dense-defined linear mapping, (A2) $G\subset H_2$ is a closed bounded convex set and $\mathcal H_{1,2}$ are abstract Hilbert spaces. Define
\begin{equation}
  \label{eq:Lx}
  \begin{split}
    &(Lx)(t)=(\dfrac{d(Fx)}{dt}-C(t)x(t),Fx(t_0)),\,x\in\mathscr D(L) \\
    &\mathscr D(L):=\{x:Fx\in\mathbb H_1(t_0,T,\mathbb R^n)\}
  \end{split}
\end{equation}
Then $L$ is a closed dense defined linear mapping \cite{Zhuk2007} and
\begin{equation}
  \label{eq:sLz}
  \begin{split}
  (L'&b)(t)=-\dfrac{d(F'z)}{dt}-C'z,b\in\mathscr D(L'),\\
  \mathscr D&(L'):=\{b=(z,z_0):F'z\in\mathbb H_1(t_0,T,\mathbb R^m),\\
  &F'z(T)=0, z_0={F^+}'F'z(t_0)+d, F'd=0 \}
  \end{split}
\end{equation}
Setting $\mathcal F:=((F^+)'F'w(t_0),\dfrac{d(F'w)}{dt}+C'w-H'u)$ we see from~\eqref{eq:bound_on_w} that the right-hand part of~\eqref{eq:LfiG} is finite. Hence, there exists at least one $b\in\mathscr D(L')$ such that $L'b=\mathcal F$ or (using~\eqref{eq:sLz})
$-\dfrac{d(F'z)}{dt}-C'z=\dfrac{d(F'w)}{dt}+C'w-H'u$.
Setting $\tilde z:=(w+z)$ we obtain $
\dfrac{d(F'\tilde z)}{dt}+C'\tilde z-H'u=0$ and $F'\tilde z=F'\ell$.
This proves~\eqref{eq:zul} has a solution. \\
On the contrary, let $z$ verify~\eqref{eq:zul} for the given $\ell$ and $u$.
Then $z$ verifies conditions (W), therefore we can plug $z$ into \eqref{eq:i1} instead of $w$.
Define\footnote{$G_1$ is a set of all $x_0,f$ such that $(x_0,f)\in\mathscr G$ and ~\eqref{eq:dae} has a solution $x$.}   $G_1:=\mathscr G\cap \mathcal R(L)$, where $
\mathcal R(L)$ is the range of the linear mapping $L$ defined by~\eqref{eq:Lx} and set $
S:=\sup_{(x_0,f)\in G_1}\langle F'z(t_0),F^+x_0\rangle+
\int_{t_0}^T\langle z,f\rangle dt$. Now, using \eqref{eq:worstcase_error}
one derives easily
\begin{equation}
  \label{eq:gerr}
\sigma(T,\ell,u)=\int_{t_0}^{T}\langle R^{-1}u,u\rangle dt+S^2
\end{equation}
Since $G_1$ is bounded, it follows that $\sigma(T,\ell,u)$ is finite. Let us prove \eqref{eq:umin}. Note that $S$ is a value of the support function of the set $G_1=\mathscr G\cap \mathcal R(L)$ on $({F'}^+F'z(t_0),z)$. To compute $S$ we note that $L$, $\mathscr G$ verify (A1), (A2) and $\mathrm{ int}\,G_1\ne\varnothing$. Thus (see \cite{Zhuk2009d}):
\begin{equation}
  \label{eq:supGRL}

S=\min_{(z_0,z)\in N(L')}\{c(G,{F'}^+F'z(t_0)-z_0,z-v)\}
\end{equation}
and the $\min$ in~\eqref{eq:supGRL} is attained on some $(\tilde z_0,\tilde z)\in N(L')$.
Recalling the definition of $L'$ (formula~\eqref{eq:sLz}) and noting that $
c^2(G,z-b)=\| Q_0^{-\frac12}({F'}^+F'(z(t_0)-v(t_0))-d)\|^2+
\int_{t_0}^T\|Q^{-\frac 12}(z-v)\|^2 dt
$ we derive \eqref{eq:umin} from \eqref{eq:gerr}-\eqref{eq:supGRL}. This completes the proof.
\end{pf}
\subsection{Optimality conditions}
\label{ss:optcond}
Assume $\ell\in\mathcal L(T)$. By definition~\ref{d:1} and due to Generalized Kalman Duality (GKD) principle (see Theorem \ref{t:1}) the $\ell$-estimate $\hat u$ is a solution of the Dual Control Problem (DCP), that is the optimal control problem with cost~\eqref{eq:umin} and DAE constraint~\eqref{eq:zul} for any constant
$F\in\mathbb R^{m\times n}$ and continuous $t\mapsto C(t)\in\mathbb R^{m\times n}$, $t\in[t_0,T]$. If $F=I_{n\times n}$ then $\hat u=RHp$ where $p$ may be found from the following optimality conditions (Euler-Lagrange System in the Hamilton form \cite{Ioffe1974}):
\begin{equation}
    \label{eq:dae_EL}
    \begin{split}
      &\dfrac{dFp}{dt}=Cp+Q^{-1}z,\,Fp(t_0)=\tilde Q_0z(t_0),\\
      &\dfrac{dF'z}{dt}=-C'z+H'RHp,\,F'z(T)=F'\ell\,.
    \end{split}
  \end{equation}
with $\tilde Q_0 = Q_0^{-1}{F'}^+F'$. In the general case $F\in\mathbb R^{m\times n}$, let us assume that (AS) the system~\eqref{eq:dae_EL} is solvable. One can prove using direct variational method (see \cite{Ioffe1974})) that $\hat u=RHp$ solves the DCP with cost~\eqref{eq:umin} and DAE constraint~\eqref{eq:zul}. Although the assumption (AS) allows one to solve the optimal control problem with DAE constraints,
 it may be very restrictive for state estimation problems.
To illustrate this, let us consider an example. Define
\begin{equation}
  \label{eq:exmpl_FCH}
  F'=\left[
\begin{smallmatrix}
  1&&0\\
  0&&1\\
  0&&0\\
  0&&0\\
\end{smallmatrix}\right],
 C'(t)=\left[
\begin{smallmatrix}
0&&-1\\
0&&0\\
1&&0\\
0&&-1
\end{smallmatrix}\right],
H'(t)=\left[\begin{smallmatrix}
1&&0&&0\\
0&&0&&1\\
0&&0&&0\\
0&&1&&0
\end{smallmatrix}\right]
\end{equation} and take $Q_0=Q(t)=I_{2\times 2}$, $R(t)=I_{4\times 4}$.
In this case~\eqref{eq:dae_EL} reads as:
\begin{equation}
  \label{eq:bvppr}
  \begin{split}
    &\dfrac{dz_1}{dt}=z_2+p_1, z_1(T)=\ell_1, -z_1=0, \\
    &\dfrac{dz_2}{dt}=p_2,z_2(T)=\ell_2, z_2+p_4=0,\\
    &\dfrac{dp_1}{dt}=p_3+z_1, p_1(t_0)=z_1(t_0),\\
    &\dfrac{dp_2}{dt}=-p_1-p_4+z_2,p_2(t_0)=z_2(t_0)\,.
  \end{split}
\end{equation}
We claim that~\eqref{eq:bvppr} has a solution iff $\ell_1=\ell_2=0$. Really, $z_1(t)\equiv0$ implies $z_1(T)=\ell_1=0$, $-z_2=p_1=p_4$ and $\frac d{dt}p_1=p_3$. According to this we rewrite~\eqref{eq:bvppr} as follows:
\begin{equation}
  \label{eq:bvpr2}
  \begin{split}
    &\dfrac{dp_1}{dt}=-p_2,p_1(t_0)=0,p_1(T)=\ell_2,\\
    &\dfrac {dp_2}{dt}=-3p_1,p_2(t_0)=0\,.
  \end{split}
\end{equation}
It is clear that~\eqref{eq:bvpr2} has a solution iff $\ell_2=0$. Thus, the assumption (AS) leads to the trivial minimax observability subspace: $\mathcal L(T)=\{0\}\times\{0\}$. However, $\mathcal L(T)=\{0\}\times\mathbb R$. To see this, take $u_3\in\mathbb L_2$, $\ell_2\in\mathbb R$ and $\ell_1=0$, and define $z_1=0$, $u_{1,2}=-z_2$, $z_2=\ell_2-\int_t^Tu_3(s)ds$. By direct substitution one checks that $z_{1,2}$ and $u_{1,2,3}$ solve~\eqref{eq:zul}.
Therefore $\mathcal L(T)=\{0\}\times\mathbb R$ due to~\eqref{eq:LT}. We see that classical optimality condition (Euler-Lagrange system in the form~\eqref{eq:dae_EL}) may be inefficient for solving the minimax state estimation problems for DAEs.
In the next proposition we prove that optimal control problem with cost~\eqref{eq:umin} and DAE constraint~\eqref{eq:zul} has a unique solution $\hat u$, $\hat z$, provided $\ell\in\mathcal L(T)$, and we present one possible approximation of $\hat u$, $\hat z$ based on the Tikhonov regularization method \cite{Tikhonov1977}.
\begin{prop}[optimality conditions]
\label{p:3e}
Let $\varepsilon>0$.
The DAE boundary-value problem
\begin{equation}
\label{eq:daebvpe}
\begin{split}
&\frac{d(F'z)}{dt}=-C'z+H'\hat u+\hat p,\\
&\frac{d(Fp)}{dt}=Cp+\varepsilon Q^{-1}z,\,\varepsilon\hat u = RHp,\\
&F'z(T)+F^+Fp(T)=F'\ell,\,F'd=0\,,\\
&\frac 1\varepsilon Fp(t_0)= Q^{-1}_0({F'}^+F'z(t_0)-d)\,.
\end{split}
\end{equation}
has a unique solution $\hat u_\varepsilon$, $\hat p_\varepsilon$, $\hat z_\varepsilon$, $\hat d_\varepsilon$. If $
\ell\in\mathcal L(T)$ then there exists $\hat d,\hat u$ and $\hat z$ such that 1) $\hat d_\varepsilon\to\hat d$ in $\mathbb R^n$ and $\hat u_\varepsilon\to\hat u$, $\hat z_\varepsilon\to\hat z$ in $\mathbb L_2$, 2) $\hat u$ and $\hat z$ verify \eqref{eq:zul} and 3) $\hat u$ is the $\ell$-estimate and $\ell$-error is given by
\begin{equation}
  \label{eq:mnmx_err}
  \begin{split}
    \hat\sigma(T,&\ell)=
    \sigma(T,\ell,\hat u) = \Omega(\hat u,\hat z,\hat d):=\|R^{-\frac 12}\hat u\|_{\mathbb L_2}^2\\
    &+\| Q_0^{-\frac12}({F'}^+F'\hat z(t_0)-\hat d)\|^2+\|Q^{-\frac 12}\hat z\|_{\mathbb L_2}^2\,.
  \end{split}
\end{equation}
 \end{prop}
 \begin{pf}
Define $r:=\mathrm{rang} F$ and $D=\mathrm{diag}(\lambda_1\dots\lambda_r)$ where $\lambda_i$, $i=\overline{1,r}$ are positive eigen values of $FF'$. If $r=0$ then~(\ref{eq:daebvpe}) is obviously uniquely solvable. Assume $r>0$.
It is easy to see, applying the SVD decomposition \cite{Albert1972} to $F$, that
$ F=U'S V$, where $UU'=I_m$, $V'V=I_{n}$ and $S=\left
  [\begin{smallmatrix}
    D^\frac12&&0_{r\times n-r}\\
    0_{m-r\times r}&&0_{m-r\times n-r}
  \end{smallmatrix}\right]$.
Thus multiplying the first equation of~\eqref{eq:daebvpe} by $U$, the second -- by $V$, and changing variables
one can reduce the general case to the case of DAE~\eqref{eq:daebvpe} with
$F=\left[
\begin{smallmatrix}
  I_{r}&&0_{r\times n-r}\\0_{m-r\times r}&&0_{m-r\times n-r}
\end{smallmatrix}
\right]$. In what follows, therefore, we can focus on this case only.
Having in mind the above 4-block representation for $F$
we split the coefficients of~\eqref{eq:daebvpe} as follows:
$C(t)=\left[
\begin{smallmatrix}
C_1&&C_2\\
C_3&&C_4
\end{smallmatrix}
\right]$, $
Q=\left[
\begin{smallmatrix}
Q_1&&Q_2\\
Q'_2&&Q_4
\end{smallmatrix}
\right]
$, $Q_0=\left[
\begin{smallmatrix}
Q_1^0&& Q^0_2\\
(Q^0_2)'&& Q^0_4
\end{smallmatrix}\right]
$, $
H'RH=\left[
\begin{smallmatrix}
S_1&&S_2\\
S_2'&&S_4
\end{smallmatrix}
\right]
$, $\ell=\bigl(
\begin{smallmatrix}
  \ell_1\\
  \ell_2
\end{smallmatrix}\bigr)$, $d=\bigl(
\begin{smallmatrix}d_1\\d_2\end{smallmatrix}\bigr)$. If 1) $n-r=0$ and $m-r>0$ we set $
C_2:=0_{r\times 1}$, $C_4:=0_{m-r\times 1}$ and $S_2:=0_{r\times 1}$, $S_4:=0$; if 2) $n-r>0$ and $m-r=0$ we set $C_3:=0_{1\times r}$, $C_4:=0_{1\times n-r}$ and $Q^0_2,Q_2:=0_{m\times 1}$, $Q^0_4,Q_4:=0$; if 3) $n=m=r$ we set $C_4:=0$, $C_2:=0_{r\times 1}$, $C_3:=0_{1\times r}$ and let $S_i$, $Q^0_i,Q_i$ be defined as in 1) and 2) respectively, $i\in\{2,4\}$.
According to this \eqref{eq:daebvpe} splits into
\begin{equation*}
  \begin{split}
    &\dfrac{dp_1}{dt}=C_1p_1+C_2p_2+\varepsilon (Q_1z_1+Q_2z_2),\\
    &\dfrac{dz_1}{dt}=-C_1'z_1-C_3'z_2+p_1+\frac 1\varepsilon (S_1p_1+S_2p_2),\\
    &z_1(T)+p_1(T)=\ell_1, p_1(t_0)=\varepsilon(Q^0_1z_1(t_0)-Q^0_2d_2)\\
  \end{split}
\end{equation*}
and algebraic part: $Q^0_4d_2=(Q^0_2)'z_1(t_0)$,
\begin{equation*}
  \begin{split}
&C_3p_1+C_4p_2+\varepsilon (Q_2'z_1+Q_4z_2)=0,\\% Q^0_4d_2=(Q^0_2)'z_1(t_0),\\
&-C_2'z_1-C'_4z_2+\frac 1\varepsilon S_2'p_1+(I+\frac 1\varepsilon S_4)p_2=0.
  \end{split}
\end{equation*}
Define $\widetilde Q_4:=Q^0_1-Q^0_2(Q^0_4)^{-1}(Q^0_2)'$ and set
\begin{equation*}
  \begin{split}
&W(t,\varepsilon)=\varepsilon I_{n-r}+S_4+C_4'Q_4^{-1}C_4\,, M(t,\varepsilon)=W^{-1}(t,\varepsilon)\,,\\
&A(t)=(C_3'Q_4^{-1}C_4+S_2),\,B(t)=(C'_2-C_4'Q_4^{-1}Q_2')\,, \\
&C_\varepsilon(t)=-C_1'+C_3'Q_4^{-1}Q_2'+A(t)M(t,\varepsilon)B(t)\,,\\
&Q_\varepsilon(t)=-\frac 1\varepsilon A(t)M(t,\varepsilon)A'(t)+I_{r}+\frac 1\varepsilon[S_1+C_3'Q_4^{-1}C_3]\,,\\
&S_\varepsilon(t)=\varepsilon (Q_1 -Q_2Q_4^{-1}Q_2'
+ B'(t)M(t,\varepsilon)B(t))\,.\\
  \end{split}
\end{equation*}
Solving the algebraic equations for $z_2,p_2,d_2$ we find:
\begin{equation}\label{eq:p2z2}
    \begin{split}
      &Q_4z_2= (-Q_2'-C_4MB)z_1+\frac 1\varepsilon (C_4MA'-C_3)p_1,\\
      &p_2=\varepsilon MBz_1- MA'p_1, d_2=(Q^0_4)^{-1}(Q^0_2)'z_1(t_0).
    \end{split}
  \end{equation}
Substituting~\eqref{eq:p2z2}
into differential equations for $p_1,z_1$ we obtain
\begin{equation}
\label{eq:ebvp_reduced}
  \begin{split}
    &\dfrac{dz_1}{dt}=C_\varepsilon z_1+Q_\varepsilon p_1,z_1(T)+p_1(T)=\ell_1,\\
    &\dfrac{dp_1}{dt}=-C'_\varepsilon p_1+S_\varepsilon z_1, p_1(t_0)=
\varepsilon\widetilde Q_4z_1(t_0).
  \end{split}
\end{equation}
We claim that \eqref{eq:ebvp_reduced} has a unique solution for any $\ell_1\in\mathbb R^r$ and $\varepsilon>0$. Let us prove uniqueness. Note that $Q_1 -Q_2Q_4^{-1}Q_2'>0$ as $Q(t)>0$ (see \cite{Albert1972} for details) and thus $S_\varepsilon(t)>0$ for $\varepsilon>0$ (as $B'MB\ge0$). Applying simple matrix manipulations one can prove (see, for instance, \cite{Kurina1986g}) that $Q_\varepsilon(t)\ge0$ for $\varepsilon>0$. Assume $z_1,p_1$ solve \eqref{eq:ebvp_reduced} for $\ell_1=0$. Then, integrating by parts and using \eqref{eq:ebvp_reduced} we obtain
\begin{equation*}
  \begin{split}
    -\langle z_1(T),z_1(T)\rangle&-\langle z_1(t_0),\varepsilon\widetilde Q_4z_1(t_0)\rangle\\
    &=\int_{t_0}^T\langle Q_\varepsilon p_1,p_1\rangle+\langle S_\varepsilon z_1,z_1\rangle dt
  \end{split}
\end{equation*}
This equality is possible only if $p_1=0$, $z_1=0$ as $S_\varepsilon(t),Q_\varepsilon(t)\ge0$. As \eqref{eq:ebvp_reduced} is a Noether Boundary-Value Problem (BVP), which has a unique solution for $\ell_1=0$, it follows from the general theory of linear BVP \cite{Boichuk2004} that \eqref{eq:ebvp_reduced} has a unique solution for any $\ell_1$. Thus, we proved unique solvability of~\eqref{eq:daebvpe}.
Let us introduce the following definitions. Take $u\in\mathbb L_2$ and $z\in\mathbb L_2$ such that $F'z\in\mathbb H_1$, and assign to $u,z$ a number $\delta(u,z)$:
$$
\delta(u,z):=\|F'z(T)-F'\ell\|^2+\|\dfrac{dF'z}{dt}+C'z-H'u\|^2_{\mathbb L_2}
$$
It was proved in \cite{Zhuk2007} that $\delta$ is convex and weakly low semi-continuous\footnote{that is $\delta(u,z)\le\lim\delta(u_n,z_n)$, provided $u_n,z_n$ converges weakly to $(u,z)$ or equally $\int_{t_0}^T\langle u_n,p\rangle+\langle z_n,q\rangle dt\to\int_{t_0}^T\langle u,p\rangle+\langle z,q\rangle dt$ for any $(p,q)$ in $\mathbb L_2\times\mathbb L_2$.
} (w.l.s.c). Define a Tikhonov function $\mathcal T_\varepsilon(u,z,d):=\delta(u,z)+\varepsilon\Omega(u,z,d)$ with
$\Omega(u,z,d)$ defined by~\eqref{eq:mnmx_err}.
We claim (see appendix for the details) that
\begin{equation}
  \label{eq:infTikhonovFunc}
  \mathcal T_\varepsilon(\hat u_\varepsilon,\hat z_\varepsilon,\hat d_\varepsilon)=
\inf_{u,z,d}\mathcal T_\varepsilon(u,z,d):=\mathcal T^*_\varepsilon, \quad \forall\varepsilon>0
\end{equation}
Take any $\ell\in\mathcal L(T)$. By~\eqref{eq:LT} there exists
$u$ and $z$ such that $\inf\delta=\delta(u,z)=0$. Using \eqref{eq:infTikhonovFunc} we obtain
\begin{equation}
  \label{eq:TikhonovInequalities}
  \begin{split}

0\le
\mathcal T^*_\varepsilon\le \mathcal T_\varepsilon(u,z,d)
\le \delta(\hat u_\varepsilon,\hat z_\varepsilon)+\varepsilon \Omega(u,z,d)
  \end{split}
\end{equation}
for any $d:F'd=0$. Due to \eqref{eq:TikhonovInequalities}:
$\mathcal T^*_\varepsilon=
\delta(\hat u_\varepsilon,\hat z_\varepsilon)+
\varepsilon\Omega(\hat u_\varepsilon,\hat z_\varepsilon,\hat d_\varepsilon)\le
\delta(\hat u_\varepsilon,\hat z_\varepsilon)+\varepsilon \Omega(u,z,d)
$ so that
\begin{equation}
  \label{eq:uezeMinimizesOmegaOverAdjDAE}
 \Omega(\hat u_\varepsilon,\hat z_\varepsilon,\hat d_\varepsilon)\le \Omega(u,z,d),\quad \delta (u,z)=0,F'd=0
\end{equation}
\eqref{eq:uezeMinimizesOmegaOverAdjDAE} proves that $\{\hat u_\varepsilon,\hat z_\varepsilon,\hat d_\varepsilon\}$ is a bounded sequence in the Hilbert space $\mathbb L_2\times\mathbb L_2\times\mathbb R^n$. Thus
it contains a sub-sequence $\{\hat u_{\varepsilon_k},\hat z_{\varepsilon_k},\hat d_{\varepsilon_k}\}$ which converges weakly to some element $\hat u,\hat z,\hat d$. By~\eqref{eq:TikhonovInequalities}
$0=\delta(u,z)\le \delta(\hat u_{\varepsilon_k},\hat z_{\varepsilon_k})\le\mathcal T_\varepsilon^*\le
  \delta(u,z)+\varepsilon\Omega(u,z,d)$
so that by w.l.s.c. of $\delta$:
\begin{equation}
  \label{eq:TuepstoTopt}
\delta(\hat u,\hat z)\le
\lim_{k\to\infty}\delta(\hat u_{\varepsilon_k},\hat z_{\varepsilon_k})=\delta(u,z)  =\inf_{u,z} \delta=0
\end{equation}
We claim that (see appendix for technical details)
\begin{equation}
  \label{eq:OmegaLowCont}
\Omega(\hat u,\hat z,\hat d)
\le\lim_{\varepsilon\to0}\Omega(\hat u_{\varepsilon},\hat z_{\varepsilon},\hat d_\varepsilon)
\end{equation}
By \eqref{eq:OmegaLowCont} and \eqref{eq:uezeMinimizesOmegaOverAdjDAE} we get:
\begin{equation}
  \label{eq:Omegainf}
  \Omega(\hat u,\hat z,\hat d)=\Omega^*:=\inf_{\{(u,z):\delta(u,z)=0\},F'd=0}\Omega(u,z,d)
\end{equation}
Note that $\Omega$ is strictly convex, therefore $\Omega$ has a unique minimizer $w^*$, which coincides with $(\hat u,\hat z,\hat d)$ by~\eqref{eq:Omegainf}. This proves that $w^*$ is a unique weak limiting point for the bounded sequence $\{\hat u_\varepsilon,\hat z_\varepsilon,\hat d_\varepsilon\}$. Thus, $\{\hat d_\varepsilon\}$ converges to $\hat d$ in $\mathbb R^n$ as in $\mathbb R^n$ the weak convergence is equivalent to the strong convergence. Moreover,~\eqref{eq:OmegaLowCont} and \eqref{eq:uezeMinimizesOmegaOverAdjDAE} implies $\lim\Omega(\hat u_{\varepsilon},\hat z_{\varepsilon},\hat d_\varepsilon)=\Omega(\hat u,\hat z,\hat d)$. The latter proves 1) as $\{\hat u_\varepsilon,\hat z_\varepsilon\}$ converges to $(\hat u,\hat z)$ in $\mathbb L_2$ if and only if $\{\hat u_\varepsilon,\hat z_\varepsilon\}$ converges to $(\hat u,\hat z)$ weakly and $\lim\|\hat u_\varepsilon\|^2_{\mathbb L_2}+\|\hat z_\varepsilon\|^2_{\mathbb L_2}=
\|\hat u\|^2_{\mathbb L_2}+\|\hat z\|^2_{\mathbb L_2}$ (see \cite{Ioffe1974} for details). 2) also holds as $\delta(\hat u,\hat z)=0$ by~\eqref{eq:TuepstoTopt}.
Let us prove 3).
Take any $u,z$ verifying~(\ref{eq:zul}) and $\tilde v,\tilde d$ defined by \eqref{eq:umin}.
Define $\tilde z:=z-\tilde v$.
Using the definition of $\tilde v$ (see \eqref{eq:umin} and notes after it) we
find that $\tilde z$ also solves~(\ref{eq:zul}). Thus, $\sigma(T,\ell,u)=
\Omega(u,\tilde z,\tilde d)\ge\Omega(\hat u,\hat z,\hat d)$ by \eqref{eq:Omegainf}.
Hence, $\hat\sigma(T,\ell)\ge\Omega(\hat u,\hat z,\hat d)$.
On the other hand, we get by 1):
\begin{equation*}
  \begin{split}
    &\hat\sigma(T,\ell)\le\sigma(T,\ell,\hat u)
    \le\min_{v}\Omega(\hat u,\hat z-v,\hat d)\\

    &\l
    \min_{v}\{\lim_{\varepsilon\to0}\| Q_0^{-\frac12}({F^+}'F'\hat z_\varepsilon(t_0)-{F^+}'F'v(t_0)-\hat d_\varepsilon)\|^2\\
    &+\lim_{\varepsilon\to0}\|Q^{-\frac 12}(\hat z_\varepsilon-v)\|^2\}+
    \| R^{-\frac12}\hat u\|_{\mathbb L_2}^2\\
    &=\lim_{\varepsilon\to0}\Omega(\hat u,\hat z_\varepsilon,\hat d_\varepsilon)+
    \min_{v}\{\|Q_0^{-\frac12}{F^+}'F' v(t_0)\|^2+\|Q^{-\frac 12}v\|_{\mathbb L_2}^2\}\\
    &=\Omega(\hat u,\hat z,\hat d)
  \end{split}
\end{equation*}
where we obtained the 4th line noting that $\|Q^{-\frac 12}(\hat z_\varepsilon-v)\|_{\mathbb L_2}^2 =
    \|Q^{-\frac 12}\hat z_\varepsilon\|_{\mathbb L_2}^2 +
    \|Q^{-\frac 12}v\|_{\mathbb L_2}^2 -
    2\int_{t_0}^T\langle Q^{-1}\hat z_\varepsilon,v\rangle dt$ and
   \begin{equation*}
  \begin{split}
&\int_{t_0}^T\langle Q^{-1}\hat z_\varepsilon,v\rangle dt
    =\frac 1\varepsilon \int_{t_0}^T\langle\dfrac{dF\hat p_\varepsilon}{dt}-C\hat p_\varepsilon,v\rangle dt\\
&=-\langle Q_0^{-1}({F^+}'F'\hat z_\varepsilon(t_0)-\hat d_\varepsilon),{F^+}'F'v(t_0)\rangle
\end{split}
\end{equation*}
where the latter equality follows from~(\ref{eq:ibp}), definition of $v$ (see notes after~\eqref{eq:umin}) and~(\ref{eq:daebvpe}).
Thus $\hat\sigma(T,\ell)\le\sigma(T,\ell,\hat u)\le\mu_\varepsilon(\hat u)=\Omega(\hat u,\hat z,\hat d)$ implying~\eqref{eq:mnmx_err}. \eqref{eq:mnmx_err}
implies, in turn, $\hat u$ is the $\ell$-estimate by definition. This completes the proof.
 \end{pf}
We will refer $\hat u_\varepsilon$ as a sub-optimal $\ell$-estimate.
Let us represent $\hat u_\varepsilon(y)$ in the form of the minimax filter.
Recalling definitions of $M,A,B$ introduced at the beginning of the proof of Proposition~\ref{p:3e}, and splittings for $\ell,Q,R,H,C$ we define $\Phi(t,\varepsilon)= \left[
\begin{smallmatrix}
  K(t,\varepsilon)\\
M(t,\varepsilon)[\varepsilon B(t)-A'(t)K(t,\varepsilon)]
\end{smallmatrix}\right]$, where $K(t,\varepsilon)$ solves $$
\frac {dK}{dt}=-KC_\varepsilon-C'_\varepsilon K-
   KQ_\varepsilon K+S_\varepsilon,\,K(0)=\varepsilon \tilde Q_4
$$ Define $z_1(T)=(I_r+K(T,\varepsilon))^{-1}\ell_1$ and let $z_1$ solve
\begin{equation}
  \label{eq:zK}
  \begin{split}
    &  \frac{dz_1}{dt}=C_\varepsilon z_1+Q_\varepsilon K(t,\varepsilon)z_1,\\
 \end{split}
\end{equation}
Define $\hat x_\varepsilon(t_0)=0$
and let $\hat x_\varepsilon$ solve the following linear stochastic differential equation:
\begin{equation}\label{eq:xKz}
  \begin{split}
   &d\hat x_\varepsilon = [-C'_\varepsilon-K(\varepsilon,t)Q_\varepsilon]
   \hat x_\varepsilon dt + \Phi H'R dy,\\%\hat x_\varepsilon(0)=0\\

    \end{split}
  \end{equation}
\begin{cor
\label{c:eme}
Assume $F=\bigl[
\begin{smallmatrix}
  I_{r}&&0_{r\times n-r}\\
0_{m-r,r}&&0_{m-r\times m-r}
\end{smallmatrix}
\bigr]$.
Then
$$
\hat u_\varepsilon(y)=\int_{t_0}^T\langle\hat u_\varepsilon{},y\rangle{dt}=\langle\varepsilon^{-1}(I_r+K(T,\varepsilon))^{-1}\hat x_\varepsilon(T),\ell_1\rangle
$$ The sub-optimal $\ell$-error is given by
$$
\hat\sigma^\varepsilon(T,\ell):=\frac1\varepsilon\bigr[
\langle z_1(T),z_1(T)
\rangle
-\int_{t_0}^T\|\Phi(t,\varepsilon)z_1\|^2dt\bigl]
$$
\end{cor}
 \begin{pf}
Assume $\hat p_\varepsilon$ solves \eqref{eq:daebvpe}. We split $p_\varepsilon=\bigl(
\begin{smallmatrix}
  p_1\\p_2
\end{smallmatrix}\bigr)$ where $p_1$ solves \eqref{eq:ebvp_reduced} and $p_2$ is defined by \eqref{eq:p2z2}. It can be checked by direct calculation that $p_1(t)=K(t,\varepsilon)z_1(t)$ where
$z_1$ is defined by~\eqref{eq:zK}. Using this and \eqref{eq:p2z2} we deduce $\hat p_\varepsilon
= \Phi(t,\varepsilon)z_1$. \eqref{eq:daebvpe} implies $\hat u_\varepsilon=\frac 1\varepsilon RH\hat p_\varepsilon$. Finally, using the obtained representations for $\hat p_\varepsilon$, $\hat u_\varepsilon$ and~\eqref{eq:xKz} we obtain integrating by parts that
\begin{equation*}
  \begin{split}
    \hat u_\varepsilon&(y)=\int_{t_0}^T\langle y,\frac 1\varepsilon RH\hat p_\varepsilon\rangle dt = \int_{t_0}^T\langle \frac 1\varepsilon \Phi' H'R y, z_1\rangle dt\\
&=\langle \ell_1,\varepsilon^{-1}(I_r+K(T,\varepsilon))^{-1}\hat x_\varepsilon(T)\rangle\to
\int_{t_0}^T\langle \hat u,y\rangle dt=\hat u(y)
  \end{split}
\end{equation*}
By~\eqref{eq:uezeMinimizesOmegaOverAdjDAE}-\eqref{eq:OmegaLowCont}
$\Omega(\hat u_\varepsilon,\hat z_\varepsilon,\hat d_\varepsilon)\to\hat\sigma(T,\ell)$. It is easy to compute using~\eqref{eq:daebvpe} that $\Omega(\hat u_\varepsilon,\hat z_\varepsilon,\hat d_\varepsilon)=\varepsilon^{-1}(\langle \ell,F\hat p_\varepsilon\rangle+\|\hat p_\varepsilon\|_{\mathbb L_2}^2)$. To conclude it is sufficient to substitute $\hat p_\varepsilon= \Phi(t,\varepsilon)z_1$ into the latter formula.
\end{pf}
\textbf{Example.} In order to demonstrate main benefits of Proposition~\ref{p:3e} we will apply it to the example presented above: assume that the bounding set, state equation and observation operator are defined by~\eqref{eq:exmpl_FCH}. Note that ${F'}^+F'=I_4$. Thus $F'd=0$ implies $d=0$. According to Theorem~\ref{t:1} the exact $\ell$-estimate $\hat u = \bigl(
\begin{smallmatrix}
  \hat u_1\\
  \hat u_2\\
  \hat u_3
\end{smallmatrix}\bigr)$ may be obtained minimizing  $
\sigma(T,\ell,u)=\sum_{i=1}^2\|z_i(t_0)\|^2+\|z_i\|_{\mathbb L_2}^2+
\sum_{j=1}^3\|u_j\|^2_{\mathbb L_2}
$ over solutions of the DAE
\begin{equation}
  \label{eq:w12}
  \begin{split}
    &\dfrac {dz_1}{dt}-z_2-u_1=0, z_1(T)=\ell_1, -z_1=0\\
    &\dfrac{dz_2}{dt}-u_3=0,z_2(T)=\ell_2,-z_2-u_2=0
  \end{split}
\end{equation}
Assume $\ell_1=0$ so that $\ell=\bigl(
\begin{smallmatrix}
  \ell_1\\
  \ell_2
\end{smallmatrix}\bigr)\in\mathcal L(T)$. If $\hat u_{1,2}$ solves~\eqref{eq:w12} then $\hat u_{1,2}=-z_2$. Hence, $\hat u_3$ may be found minimizing $
\sigma(T,\ell,u)=\|z_2(t_0)\|^2+\int_{t_0}^T 3z_2^2+u_3^2dt
$ over $\dfrac{dz_2}{dt}=u_3$, $z_2(T)=\ell_2$. The optimality condition takes the following form: $\hat u_3=p$, $\frac{dz_2}{dt}=p$, $z_2(T)=\ell_2$, $\frac{dp}{dt}=3z_2$, $p(t_0)=z_2(t_0)$. Let us represent the estimate in the form of the minimax filter. Introducing $k$ as a solution of the Riccati equation $\frac{dk}{dt}=k^2-3$, $k(0)=1$ we find that $\hat u_2=kz_2$ where $z_2$ solves the following Cauchy problem: $\frac {dz_2}{dt} = kz_2$, $z_2(T)=\ell_2$. Let $\hat x$ be a solution to $\frac{d\hat x}{dt}=-k\hat x-y_1-y_2+ky_3$, $\hat x(t_0)=0$. Then it is easy to see that $\hat u(y)=\int_{t_0}^T\langle \hat u, y\rangle dt = \ell_2\hat x(T)$, where $y=\bigl(
\begin{smallmatrix}
  y_1\\y_2\\y_3
\end{smallmatrix}\bigr)$ denotes a realization of the random process representing observed data. Let us compute the sub-optimal $\ell$-estimate. \eqref{eq:xKz} reads as
\begin{equation*}
  \begin{split}
    \dfrac d{dt}&
    \bigl(
    \begin{smallmatrix}
      x_1\\x_2
    \end{smallmatrix}
    \bigr)
    =\left[\begin{smallmatrix}
       -(1+\frac 1\varepsilon )k_1&&-(1+\frac 1\varepsilon )k_2\\
       -1-(1+\frac 1\varepsilon )k_2&&-(1+\frac 1\varepsilon )k_4
      \end{smallmatrix}\right]
    \bigl(
    \begin{smallmatrix}
      x_1\\x_2
    \end{smallmatrix}
    \bigr)\\
&+\left[\begin{smallmatrix}
        k_1&&k_2&&1&&0\\
        k_2&&k_4&&0&&\frac{-\varepsilon}{1+\varepsilon}
      \end{smallmatrix}\right]
    \left[\begin{smallmatrix}
        1&&0&&0\\
        0&&0&&1\\
        0&&0&&0\\
        0&&1&&0
      \end{smallmatrix}\right]
    \bigl(
    \begin{smallmatrix}
      y_1\\y_2\\y_3
    \end{smallmatrix}
    \bigr),\bigl(
    \begin{smallmatrix}
      x_1(t_0)\\x_2(t_0)
    \end{smallmatrix}
    \bigr)=\bigl(
    \begin{smallmatrix}
      0\\0
    \end{smallmatrix}
\bigr)
\\
\dfrac d{dt}&\left[\begin{smallmatrix}
        k_1&&k_2\\
        k_2&&k_4
      \end{smallmatrix}\right]=
    \left[\begin{smallmatrix}
        0&&0\\
        -1&&0
      \end{smallmatrix}\right]
    \left[\begin{smallmatrix}
        k_1&&k_2\\
        k_2&&k_4
      \end{smallmatrix}\right]+
\left[\begin{smallmatrix}
        k_1&&k_2\\
        k_2&&k_4
      \end{smallmatrix}\right]
\left[\begin{smallmatrix}
        0&&-1\\
        0&&0
      \end{smallmatrix}\right]\\
&\left[\begin{smallmatrix}
        1+\varepsilon&&0\\
        0&&\varepsilon+\frac\varepsilon{1+\varepsilon}
      \end{smallmatrix}\right]-
(1+\frac 1\varepsilon)
\left[\begin{smallmatrix}
        k_1&&k_2\\
        k_2&&k_4
      \end{smallmatrix}\right]^2
,\left[\begin{smallmatrix}
        k_1(t_0)&&k_2(t_0)\\
        k_2(t_0)&&k_4(t_0)
      \end{smallmatrix}\right]=\left[\begin{smallmatrix}
        1&&0\\
        0&&1
      \end{smallmatrix}\right].
  \end{split}
\end{equation*}
Define $\bigl(
    \begin{smallmatrix}
      \tilde \ell_1\\\tilde \ell_2
    \end{smallmatrix}
    \bigr)=\left[\begin{smallmatrix}
        1+k_1(T)&&k_2(T)\\
        k_2(T)&&1+k_4(T)
      \end{smallmatrix}
\right]^{-1}\bigl(
    \begin{smallmatrix}
      \ell_1\\\ell_2
    \end{smallmatrix}
    \bigr)$.
Due to Corollary~\ref{c:eme} the sub-optimal $\ell$-estimate may be represented as $\hat u_\varepsilon(y)=\int_{t_0}^T\langle \hat u_\varepsilon, y\rangle dt = \frac 1\varepsilon
\langle \bigl(
    \begin{smallmatrix}
      \tilde \ell_1\\\tilde \ell_2
    \end{smallmatrix}
    \bigr),\bigl(
    \begin{smallmatrix}
      x_1(T)\\x_2(T)
    \end{smallmatrix}
    \bigr)\rangle$. If $\ell_1=0$ then the sub-optimal $\ell$-error is given by
    $\hat\sigma^\varepsilon(T,\ell)=\frac 1\varepsilon
\langle\left[\begin{smallmatrix}
        k_1(T)&&k_2(T)\\
        k_2(T)&&k_4(T)
      \end{smallmatrix}
\right]
\bigl(
    \begin{smallmatrix}
      \tilde \ell_1\\\tilde \ell_2
    \end{smallmatrix}
    \bigr),
\bigl(
    \begin{smallmatrix}
      \tilde \ell_1\\\tilde \ell_2
    \end{smallmatrix}
    \bigr)\rangle$.
 \begin{figure}[t]
   \centering
\includegraphics[width=.4\textheight]{x2plot.pdf}
   \caption{
Optimal $\ell$-estimate $\hat u(y)$ (Dashed), sub-optimal $\ell$-estimate $\hat u_\varepsilon(y)$ (Dotted) and error $\hat\sigma^\varepsilon(t,\ell)$ (DotDashed) and simulated $x_2(t)$ (Solid), $\ell=(0,1)^T$, $t\in[0,1]$, $\varepsilon=\exp(-30)$.
}
   \label{fig:2}
 \end{figure}
Take $t_0=0$, $T=1$ and assume that $F$ and $C$ are defined by~\eqref{eq:exmpl_FCH}. In the corresponding DAE $x_{3,4}$ are free components. For simulations we choose $x_3=\cos(t)$ and $x_4=\sin(t)$, $x_1(0)=0.1$, $x_2(0)=-0.1$, $f_1=f_2=0$. In order to generate artificial observations $y$ we take $\eta(t)=\bigl(
\begin{smallmatrix}
  -0.1\\-0.2\\0.3
\end{smallmatrix}
\bigr)$. In Figure~\ref{fig:2} the optimal $\ell$-estimate, sub-optimal $\ell$-estimate and sub-optimal $\ell$-error are presented, provided $\ell_1=0$, $\ell_2=1$. As $\mathcal L(t)\equiv\{0\}\times\mathbb R$ we see that $x_1$ is not observable in the minimax sense. This can be explained as follows. The derivative $x_3$ of $x_1$ may be any element of $\mathbb L_2$. As we apply integration by parts formula in order to compute $\sigma(T,\ell,u)$ (see~\eqref{eq:ibp}), the expression for $\sigma(T,\ell,u)$ contains $\int_{t_0}^Tx_3z_1dt$. Thus, $\sigma(T,\ell,u)<+\infty$ implies $z_1(t)\equiv 0$ for any $t\in[t_0,T]$, in particular $z_1(T)=\ell_1=0$. In this case $\hat u\in\mathbb L_2$ and $\hat u_\varepsilon\to\hat u$ in $\mathbb L_2$. If $\ell_1\ne0$ then the only candidate for $\hat u_1$ is the impulse control $\delta(T-t)\ell_1$ switching $z_1$ from $0$ to $\ell_1$ at time-instant $T$. However, in this case
the numerical sub-optimal $\ell$-error computed by the algorithm of Corollary~\ref{c:eme} increases: $\hat\sigma^\varepsilon(1,\ell)\approx 3\times 10^6$, provided $\ell_1=1$, $\ell_2=0$ and $\varepsilon=\exp(-30)$.
Let us illustrate this. Assume $ C'(t)=\left[
\begin{smallmatrix}
0&&-1\\
0&&0\\
c(t)&&0\\
0&&-1
\end{smallmatrix}\right]$, where $c(t)=0$ for $t_1\le t\le T$ and $c(t)>0$ for $t_0\le t<t_1$.  Then~\eqref{eq:zul} is solvable for any $\ell$: the solution is given by $u_1:=\frac {dz_1}{dt}-z_2$, $u_2=-z_2$, $z_1(t)=0$, $t_0\le t\le t_1$ and $z_1(t)=\frac{t-t_1}{T-t_1}\ell$ for $t_1\le t\le T$. As $\hat u_1=\frac {dz_1}{dt}-z_2$, it follows that $\|\hat u_1\|_{\mathbb L_2}$ goes to infinity if $t_1\to T$. We stress that the limiting case $t_1=T$ with $c(t)\equiv 1$ corresponds to $C'(t)$ which is being considered in our example (see~\eqref{eq:exmpl_FCH}).

\section{Conclusion}
 \label{sec:Conclusion}
The paper presents one way to generalize the minimax state estimation approach for linear time-varying DAE~\eqref{eq:dae}. The only restriction we impose here is that $F$ does not depend on time. But our approach can be generalized to the class of time-varying $F(t)$ with constant (or piece-wise constant) rank by means of Lyapunov-Floke theorem. The main idea behind the generalization is the Generalized Kalman Duality (GKD) principle. GKD allows to formulate a Dual Control Problem (DCP) which gives an expression for the Worst-Case Error (WCE). Due to GKD the WCE is finite if and only if
Also GKD gives necessary and sufficient conditions (in the form of the minimax observable subspace) for the WCE finiteness. In order to compute $\ell$-estimate one needs to solve the DCP, that is a linear-quadratic control problem with DAE constraints. Application of the classical optimality conditions (Euler-Lagrange equations) imposes additional constraints onto the minimax observability subspace $\mathcal L(T)$. To avoid this we apply a Tikhonov regularization approach allowing to construct sub-optimal solutions of DCP or, sub-optimal estimates. If $\ell\in\mathcal L(T)$ then the sequence of sub-optimal estimates converges to the $\ell$-estimate which belongs to $\mathbb L_2$. Otherwise sub-optimal estimates weakly converge to the linear combination of delta-functions. The $\mathbb L_2$-norms of the sub-optimal $\ell$-estimates grow infinitely in this case.

\textbf{Appendix.}

redpajama_set_name: RedPajamaArXiv


text:
What Many PR Users Ignore
Simply that the behaviors of their most important outside audiences rank pretty low on their list of things to worry about. And this despite the reality that, properly cared for, those behaviors can affect whether or not those managers achieve their managerial objectives.
Unfortunately, many business, non-profit and association public relations budgets are used pretty much to produce newspaper and radio mentions, or to fund somebody's favorite special event. And this at a time when they should be driving an action plan that persuades those key external stakeholders to the PR user's way of thinking, then moves those audiences to take actions that help departments, divisions or subsidiaries succeed.
After all, since that's public relations' strongest suit, shouldn't you be getting that first, THEN incremental publicity exposure?
Run this idea by the public relations team assigned to your unit: people act on their own perception of the facts before them, which leads to predictable behaviors about which something can be done. When we create, change or reinforce that opinion by reaching, persuading and moving-to-desired-action the very people whose behaviors affect the organization the most, the public relations mission is accomplished.
If you get agreement, you'll share a simple blueprint that gets everyone working towards the same external audience behaviors insuring that your public relations effort stays focused.
And there's no end to the possible benefits: capital givers or specifying sources beginning to look your way; prospects starting to do business with you; membership applications on the rise; customers making repeat purchases; community leaders beginning to seek you out; welcome bounces in show room visits; fresh proposals for strategic alliances and joint ventures; higher employee retention rates, and even politicians and legislators starting to view you as a key member of the business, non-profit or association communities.
But first, you need to find out who among your important outside audiences is behaving in ways that help or hinder the achievement of your objectives. And then, list them according to how severely their behaviors affect your organization.
Are you really certain as to HOW most members of that key outside audience perceive your organization? Since there's a good chance you don't have the budget to accommodate expensive professional survey work, you and your PR colleagues (they should be quite familiar with perception and behavior matters) must monitor those perceptions yourself.
Sit down with members of that outside audience and ask questions like "Are you familiar with our services or products?" "Have you ever had contact with anyone from our organization? Was it a satisfactory experience?" Stay alert to negative statements, especially evasive or hesitant replies. Watch carefully for false assumptions, untruths, misconceptions, inaccuracies and potentially damaging rumors. Any of which will need to be corrected, because experience shows they usually lead to negative behaviors.
Now, because the obvious objective here is to correct those same untruths, inaccuracies, misconceptions and false assumptions, you must select the specific perception to be altered which, in turn, becomes your public relations goal.
Unfortunately, a PR goal without a strategy to show you how to get there, is like Shrimp Lo Mein without the noodles. That's why you must select one of three strategies especially designed to create perception or opinion where there may be none, or change existing perception, or reinforce it. The challenge here (albeit small) is to insure that the goal and its strategy match each other. You wouldn't want to select "change existing perception" when current perception is just right suggesting a "reinforce" strategy.
Writing ability comes to the fore here as you create a compelling message carefully designed to alter your key target audience's perception, if called for by your public relations goal.
Here's a tip. Combining your corrective message with another news announcement or presentation may lend more credibility by downplaying the need for such a correction.
Be very clear about what perception needs clarification or correction, and why. Your facts must be truthful and your position must be logically explained and believable if it is to hold the attention of members of that target audience, and actually move perception in your direction. In other words, your message must be compelling.
You could call the communications tactics you will use to move your message to the attention of that key external audience, "beasts of burden" because they must carry your persuasive new thoughts to the eyes and ears of those important outside people.
Luckily, the list of tactics is extensive. It includes letters- to-the-editor, brochures, press releases and speeches. Or, you might select radio and newspaper interviews, personal contacts, facility tours or customer briefings. There are dozens in waiting and the only selection requirement is that the communications tactics you choose have a record of reaching people just like the members of your key target audience.
By the way, things can always be moved along at a faster clip by adding more communications tactics, AND by increasing their frequencies.
It won't be long before those around you will be asking if any progress is being made. By which time you already will be hard at work remonitoring perceptions among your target audience members. Using questions similar to those used during your earlier monitoring session, you'll now be on the lookout for indications that audience perceptions are beginning to move the way you want them to move.
The best way to satisfy your associates' curiosity is with the results you will receive when you undertake this aggressive public relations plan. In other words, targeting the kind of key stakeholder behavior change that leads directly to achieving your department, division or subsidiary objectives.
Please feel free to publish this article and resource box in your ezine, newsletter, offline publication or website. A copy would be appreciated at mailto:bobkelly@TNI.net. Robert A. Kelly � 2004.
Bob Kelly counsels, writes and speaks to business, non-profit and association managers about using the fundamental premise of public relations to achieve their operating objectives. He has been DPR, Pepsi-Cola Co.; AGM-PR, Texaco Inc.; VP-PR, Olin Corp.; VP-PR, Newport News Shipbuilding & Drydock Co.; director of communi- cations, U.S. Department of the Interior, and deputy assistant press secretary, The White House. He holds a bachelor of science degree from Columbia University, major in public relations. mailto:bobkelly@TNI.net. Visit: http://www.prcommentary.com
Promoting Your Website Through a Press Release
Have you ever gotten one of those letters from your localproperty tax appraiser, informing you that your tax bill isgoing up about 20 percent?
Oprah! How to Appear on The Oprah Winfrey Show
Do you dream of being on Oprah Winfrey's television show? Lots of people do. An appearance on Oprah is considered by many to be the pinnacle of success. Authors dream of having bestselling books as a result of an Oprah appearance. Even David Letterman staged a long-running bit on his show where he openly campaigned to be invited on Oprah's show.
Tough Times, Tough Tactics
When times are tough, it's no time to ignore those external audiences whose behaviors matter so much to your organization.
Managers and PR: One Thing Is Clear
As a business, non-profit or association manager, you have a clear choice when you set up your public relations. Arrange your resources to generate a variety of product and service plugs on radio, and in newspapers and in magazines. Or, use a broader, more comprehensive and workable public relations blueprint to alter key external audience perceptions that lead to changed behaviors ? behaviors you will need to reach your managerial objectives.
Publicity From Thin Air
In an ideal world, your business would be overflowing withnewsworthy stories, and the media would be waiting with batedbreath for your next press release, ready to give you front pagecoverage.
Media Training: How To Speak During a Media Interview
Media Contact Lists and How to Build a Fantastic One
I got the latest issue of Internet Works in the post yesterday and was disappointed to find out that it's going to be the last. As well as having the good taste to run a feature on me last year (!) Internet Works has been a great source of information and ideas for me for the last couple of years and it's demise will leave a gap.
When Tactics Are Not Enough
Your public relations people are busy. The buzz is all about hits on a radio show or mentions in a newspaper column. Or, which to do first, the trade show exhibit or the video clip. All useful tactics, but hardly the detailed planning needed to REALLY do something about the behaviors of those outside audiences that impact you the most.
3 Essential Elements For Turning Your PR Pitch Into Publicity Exposure
1) Establish Rapport, then get the editor/producer excited.
Driving Near School Buses in Company Vehicles
Public Relations: Understanding Educated Gambling
As an entry level position to PR, I found myself typing up a forecast by a major Public Relation's firm for a major pharmaceutical company of what life would be like in the year 2000. Market research predictions included telephones with monitors that could help you see people while you talked, fax machines that could transmit information over telephone wires, microwave ovens for reducing food defrosting time from hours to minutes and other devices that have certainly come to pass. In the lifestyle area, predictions proved less valid. Not only would Americans be enjoying longer lives, it foretold, but they would have shorter work weeks, more vacations and overall, a more leisurely lifestyle. An iota of truth, but mostly wishful thinking when we read 2005 front pages.
What? You Havent Got a Capability Statement?
What's a Capability Statement?
Right PR Empowers a Manager
Business, non-profit and association managers are in a stronger position to succeed when they use their public relations resources in a way that alters individual perception leading to changed external stakeholder behavior.
How To Make Time For Public Relations
"Don't say you don't have enough time. You have exactly the same number of hours per day that were given to Helen Keller, Pasteur, Michalangelo, Mother Theresa, Leonardo da Vinci, Thomas Jefferson, and Albert Einstein."
Possibilities Of The Blogosphere For The PR Industry In Spanish-Speaking Countries
Only two media in Spanish speaking countries offer RSS: the Spanish newspaper El Mundo and the Argentine Clar�n. Although the blogs are becoming more visible in the media and are becoming a research topic, still they are something of small "evangelist groups", who promote its use? but this can change in little time.
When Should You Outsource Your PR Efforts?
1) Do you NEED solid, consistant media exposure...week after week, or are you satisfied with "occasional" exposure? Now, this question alone is important... but not enough. The main component of this question is the IMPORTANCE of PR.
Prep for a Successful Trade Show
Well, autumn is upon us and with the onset of this season comes cleaner air and colourful outdoor scenery and, it is also prime season for trade shows. Sure, trade shows happen all throughout the year but, with many areas recognizing small business month/week, there is a greater opportunity for entrepreneurs to showcase their products or services to their target markets.
Media Training 201: The Reporters Have Done Their Homework. Have You Done Yours?
Just about anyone who has been in the public eye has a story of the media interview that went south. "I talked to that reporter for an hour and all they used was a ten-second sound bite!" or, "He said he wanted to ask me about X when that was just a way to get in the door so he could talk about Y." Chances are, the reporter came armed with questions and if he really did his homework, knew what answers to expect. You should be just as prepared. Media training can't make the tough questions go away, but it can give you the tools to control the interview. Here are some tips:

redpajama_set_name: RedPajamaCommonCrawl


My oldest 2 sons go through flip flops like crazy. I don't know if it's because they're grown or very active, or both but their slip-on shoes just don't seem to last long. I've picked up a new pair of Cool Men Flip Flops British Style from Top Tier Style and I'm hoping they will do the trick of lasting longer than others have before. I have never tried the brand or the website, but check them out if you get a chance. There are a TON of great gift ideas there, and ideas for you too. It is primarily fashion items for men and women.
I'm just glad to have a new brand of shoes to try. They're legit from the UK, so we'll see if the British Style flips stand up on my busy boys. I surely hope so. And does it make me want to take a trip to the UK (totally unrelated but I can't mention that area with saying 'vacation' at the same time)? You bet your shoes it does.
Learn more about these shoes and the many other great items available for sale at Top Tier Style by visiting their website. You can also follow along on Facebook.
Comments are turned off for this post. Please feel free to share on social media.


text:
Сенко́ За́нкович Го́стський (? — після 1440/1456) — руський (український) боярин з роду Кирдійовичів, гербу Кирдій. Зем'янин у Луцькому повіті Волинської землі Великого князівства Литовського . Родове гніздо — Гоща.

Відомості
Сенко Занкович Гостський — син Занка Кирдійовича, одного з нащадків пана Кирдія і є найдавнішим з відомих носіїв прізвища — Гостський. Водночас від його патроніма "Занкович", протопластом чи репрезентантом роду Гостських можна уважати Занка Кирдійовича. Діяльність Сенка Гостського припадає на другу третину XV ст. Вперше він фігурує у привілеї князя Свидригайла Ольгердовича від 1436 року, який було надано Петрашку Ланевичу-Мильському на село Сенне поблизу Гощі. Згодом, близько сер. XV ст., Сенко Занкович Гостський фігурує свідком у листі Єска Івановича Чаплича, яким той записує село Курозвани своїй доньці Настасії та зятю Патрикію Некрашовичу. Єдиним з відомих нащадків Сенка Гостського є Богдан Сенкович Гостський, чашник у 1483 році, луцький ключник у 1483 - 1506 роках. Він же прямий продовжувач гілки роду Гостських.

Джерела

 Собчук В.Д. «Від коріння до корони». Кременець 2014 р. – 506 ст. Ст. 260, 261.
 Яковенко Н.М. «Українська шляхта з кін. XIV - до сер. XVII ст. Волинь і Центральна Україна» Київ. 2008 р. –  409 ст. Ст. 148.

Примітки
Українська шляхта
Кирдійовичі
Гойські

Гоща
Луцький повіт

redpajama_set_name: RedPajamaWikipedia


text:
Iran dismisses U.S. talks -- for now 'Mistrust' must end before Khatami will speak directly to U.S.
Mark MatthewsTHE BALTIMORE SUN
WASHINGTON -- Iran's president rejected anew last night the possibility of an official dialogue with Washington anytime soon. But he called for the peoples of the two countries to expand their understanding of each other through academic and cultural exchanges.
In what was billed as a message to the American people, President Mohammad Khatami said in an unusual interview with CNN that before the two nations could end 19 years of antagonism, there must first be "a crack in the wall of mistrust." Khatami blamed "hostile" U.S. policies for causing that mistrust.
"When I speak of dialogue, I intend dialogue between civilizations and culture," said Khatami, a moderate cleric who since his election last spring has sought to ease tensions with the West without angering hard-line Iranians who despise the United States. "Such dialogue should be centered around thinkers and intellectuals."
"I believe that all doors should now be opened for such dialogue and understanding and possibilities for contact even between American and Iranian citizens should become available," said Khatami, who served as culture minister after the 1979 Islamic Revolution.
Khatami's offer was greeted with disappointment from the State Department. In a statement released moments after the taped televised interview ended, the department's chief spokesman, James P. Rubin, said: "We welcome the fact that he wants a dialogue with the American people. But we continue to believe that the way to address the issues between us is for our two governments to talk directly."
Khatami had delivered a similar message of rapprochement with the American people at a news conference Dec. 14 during a Muslim conference in Tehran.
What was new in his remarks last night was his specific suggestion of academic and cultural exchanges. Travel to Iran is not now officially barred, and American journalists travel there periodically. In addition, there are limited academic exchanges between the two nations. But Washington warns U.S. citizens that it is unsafe to travel there.
In an Iranian government controlled by Islamic clerics, Khatami is decidedly moderate. Since he was elected president on a platform of openness in domestic policy, Western and officials and analysts have looked to him to help diminish Iran's role as a threat to Western nations and Israel.
U.S. officials say Iran subsidizes acts of terrorism against Western interests, is trying to develop an atomic bomb and other weapons of mass destruction and foments violent opposition to the Middle East peace process between Israel and its Arab neighbors. Khatami denied these charges last night.
The United States severed ties with Iran in 1979 after Islamic militants loyal to the revolutionary government of Ayatollah Ruhollah Khomeini stormed the U.S. Embassy in Tehran and held 52 Americans hostage for 444 days.
Khatami said he regretted that American feelings had been hurt.
"In the heat of the revolutionary fervor, things happen which cannot be fullly contained or judged according to usual norms," he said. Today, Americans would be safer, he indicated.
The Khatami interview, conducted by CNN's Christiane Amanpour, was eagerly awaited by the Clinton administration, which has grown weary of its unsuccessful effort to isolate the Iranian regime internationally.
Earlier yesterday, Rubin said: "The president is going to be watching carefully what President Khatami has to say. Secretary [of State Madeleine K.] Albright will be watching carefully what President Khatami has to say."
Khatami, however, suggested that for a dialogue to open, the United States would have to moderate its own policies, which include economic sanctions against U.S. and foreign companies that invest in Iran's oil industry.
"Unfortunately, the behavior of American government[s] in the past, up to this date, has always exacerbated the climate of mistrust, and we do not detect any sign of change of behavior. We are looking for a world in which misunderstanding can be overcome, nations can understand one another, and mutual respect and logic govern relations among states."
One U.S. analyst, Michael Eisenstadt of the Washington Institute for Near East Policy, said last night that Khatami had effectively ruled out any dialogue between the two governments.
In the interview, Khatami rejected the main U.S. accusations against his government. He said terrorism was incompatible with Islam and that Iran has no intention of building weapons of mass destruction. He also said Iranian leaders oppose actions -- such as burning the American flag -- that might offend U.S. citizens.
Khatami spoke of his admiration for the American people and their history. But he criticized Washington for a "flawed policy of domination" he said was rooted in a Cold War mentality.
During the Khatami interview, conducted Tuesday, Amanpour wore a dark blue cloak covering her hair in deference to Iran's Muslim customs.
While denying U.S. charges that Iran has tried to develop nuclear weapons or commit acts of terror, Khatami spoke of Iranian opposition to the Middle East peace process. He denounced Israel as a "racist" government.
The administration has tried to pursue a policy of "dual containment" against Iran and its neighbor and adversary, Iraq. But the policy has come under increasing stress. European countries deeply resent U.S. threats to punish them for trading with Iran.
But the administration's freedom of action is tightly limited by Congress. The administration is considering whether to impose sanctions against French oil companies that are investing $2 billion in Iranian oil development.
In a sign of frustration, the State Department announced yesterday that it was reviewing ways to make sanctions a more effective tool. Among the guiding principles of the review is the idea that sanctions in which other countries also participate, even if weaker than Americans would prefer, "may actually prove more effective" than tough punishment by the United States alone.
The administration is also weighing sanctions against Russian companies that have helped Iran develop a new missile capable of hitting Israel and American soldiers in the Persian Gulf.
For more information about topics covered in this article, go to The Sun's Web site, SunSpot, at www.sunspot.net/news/
Pub Date: 1/08/98


By Sarah Ravani Updated 7:19 am, Wednesday, June 13, 2018 Photo: Mark J. Rebilas / Reuters Image 1of/1 CaptionClose Image 1 of 1 Apr 2, 2014; Glendale, AZ, USA; USA forward Chris Wondolowski (17) celebrates after scoring a goal in the first half against Mexico during a friendly match at University of Phoenix Stadium. On Wednesday, June 13, 2018, U.S. Soccer officials announed that North America would host the 2026 World Cup. (Mark J. Rebilas-USA TODAY Sports) less Apr 2, 2014; Glendale, AZ, USA; USA forward Chris Wondolowski (17) celebrates after scoring a goal in the first half against Mexico during a friendly match at University of Phoenix Stadium. On Wednesday, June ... more Photo: Mark J. Rebilas / Reuters North America wins bid for World Cup 2026 — could Bay Area host games? 1 / 1 Back to Gallery … [Read more...] about North America wins bid for World Cup 2026 — could Bay Area host games?


In the present work, the authors describe the synthesis of some mixed ligand complexes of thorium(IV) derived from 4[N-(2'-hydroxy-1'-naphthalidene)amino]antipyrine semicarbazone (HNAAPS) or 4[N-(cinnamalidene)amino]antipyrinesemicarbazone (CAAPS) as primary ligand and diphenyl sulfoxide (DPSO) as secondary ligand with the general composition ThX4.n(L).DPSO (n = 1, X = Cl, Br, NCS or NO3; n = 2, X = I or ClO4, L = HNAAPS or CAAPS). All the compounds were characterized through elemental analysis, molar conductance, molecular weight, infrared data and thermogravimetric analysis. The infrared studies reveal that the semicarbazones behave as neutral tridentate (N,N,O) while DPSO coordinates through its oxygen atom. The nitrates are bicovalently bonded, while thiocyanates are N-coordinated in these compounds. In conclusion, the coordination number of the central metal ion displays coordination number 7, 8, 9 or 12 depending on the nature of the anionic ligands.


If you're going out to space, at least have some fun.
WWW.SKYGLOWPROJECT.COM presents POLI'AHU, a timelapse trip above the clouds to Mauna Kea in Hawaii, an inactive volcano 14,000 feet above sea level. Much of the mountain is under water; when measured from its oceanic base, Mauna Kea is over 10,000 m (33,000 ft) tall. In Hawaiian mythology, Poliʻahu is one of the four goddesses of snow, thought to reside on Mauna Kea. Today, the mountain is known as one of the most important land-based astronomical research centers in the world. Our light pollution project WWW.SKYGLOWPROJECT.COM has taken us all across the continent to some of the most incredible dark sky locations, but the night sky quality here was absolutely stunning. We really lucked out with a crystal clear night and one of the best milky ways we've ever seen. You can even see the faint glow of the Halemaʻumaʻu Crater from the active Kīlauea Volcano at 2:41 (bottom right) and at 3:02. Very special thanks to everyone at NASA's Infrared Telescope Facility for their incredible support and access, allowing us to get the really cool interior perspective at 2:06. From University of Hawaii: "Mauna Kea is unique as an astronomical observing site. The atmosphere above the mountain is extremely dry -- which is important in measuring infrared and submillimeter radiation from celestial sources - and cloud-free, so that the proportion of clear nights is among the highest in the world. The exceptional stability of the atmosphere above Mauna Kea permits more detailed studies than are possible elsewhere, while its distance from city lights and a strong island-wide lighting ordinance ensure an extremely dark sky, allowing observation of the faintest galaxies that lie at the very edge of the observable Universe." Shot by Gavin Heffernan (www.SunchaserPictures.com) and Harun Mehmedinovic (www.Bloodhoney.com). SKYGLOW is endorsed by the International Dark Sky Association www.darksky.org/. Download Stills Here: www.bit.ly/2p8Piwn Music: "Genesis One" by David O'Brien / Chris Egan and "E Ala E" by Pualani Kanahele. Special Thanks: Alan Tokunaga, Lars Bergknut, Dawn Pamarang, Brian McOuat, Nasa Infrared Telescope Facility, University of Hawaii, Mauna Kea Observatories, Hawaii Film Office, International Dark Sky Association. ------- Timelapse artists and filmmakers Gavin Heffernan and Harun Mehmedinović are proud to introduce WWW.SKYGLOWPROJECT.COM, a 192-page hardcover photobook and timelapse video series exploring North America's remaining magnificent night skies and the increasing impact of light pollution on our highly fragile environment. A blend of images, stories, essays, and anecdotal captions, SKYGLOW explores the history and mythology of celestial observation and the proliferation of electrical outdoor lighting that spurred the rise of the phenomena known as "light pollution," a grave threat not only to our incredible starscapes but also to the very ecosystem itself. After a highly publicized Kickstarter campaign that ended as the fourth-most earning Photobook campaign ever, Harun and Gavin traveled over 150,000 miles and logged more than 3,000,000 photos on their grueling three-year quest. From incredible locations like the active Kīlauea volcano in Hawaii to Alberta's majestic Northern Lights, SKYGLOW takes viewers on a visual journey through time, exploring our civilization's evolving relationship with light and the night sky through the ages. See how the ancient Puebloan archaeoastronomy sites of our native elders have now been replaced with the blinding "artificial day" of over-lit modern metropolises, and learn about the "Dark Sky Movement" fighting to reclaim the pristine darkness the Earth had enjoyed for billions of years. The importance of America's threatened National Parks is also highlighted in a section of stunning landscapes from numerous parks, including Shenandoah, Yosemite, Acadia, Death Valley, Yellowstone and many more. Completed in collaboration with the International Dark Sky Association (IDA), SKYGLOW also explores the numerous towns and sites that IDA has identified as official "Dark-Sky" Communities, Cities, Parks, Reserves and Sanctuaries. Light pollution (aka "skyglow") affects human health, animal migratory patterns, obstructs astronomy research and leads to over two billion dollars of lost energy every year in the USA. The book includes original essays on the subject from key science voices like BAD ASTRONOMY blogger / author Phil Plait, Eric Betz of ASTRONOMY MAGAZINE, and night sky crusader / poster artist Tyler Nordgren. Though this is the first time Heffernan and Mehmedinović have put their night sky photos in print, their SKYGLOW timelapse collaborations have been seen by over 50 million viewers, and featured in a vast array of media, like The Rolling Stones 2015 ZIP CODE stadium tour and their 2016 DESERT TRIP shows, the Roger Waters DESERT TRIP shows and his 2016 Mexico concerts.
HAWAI'I VOLCANOES 8K is the culmination of several weeks spent filming in the rugged volcanic landscapes of Hawai'i Volcanoes National Park. Situated on the island of Hawaii in the heart of the remote south pacific, Hawai'i Volcanoes National Park boasts an awe-inspiring array of massive mountains, rare wildlife, and fiery volcanoes. Journey with More Than Just Parks as we discover an ever-growing land shaped by fire. This is Hawai'i Volcanoes. Filmed primarily in 8K. To see more National Park films or learn more about More Than Just Parks, visit our website: morethanjustparks.com/ Licensing: morethanjustparks.com/licensing Prints: morethanjustparks.smugmug.com/MoreThanJustParks/ Social: FB facebook.com/morethanjustparks, Twitter: twitter.com/MTJParks, Insta: instagram.com/morethanjustparks, Email: ask@morethanjustparks.com Sony a7R II Canon 5D Mark III Blackmagic URSA Mini 4.6K Timelapse Gear: Pulse & Radian by Alpine Labs alpinelaboratories.com/voyageurs Stage One Motion Slider Plus with NMX by Dynamic Perception dynamicperception.com/ Music: La Mar "Still Wild" 100% of your donations go toward making more captivating short films like the ones we've already made.


Organizing the perfect slumber party for your kid, tween or teen? Or maybe you're a teen yourself and you're looking for some inspiration to really impress your friends! Well, there are some awesome ideas out there on how to throw a trendy and unforgettable slumber party!
Kids can have fun adding flavors to their popcorn and enjoy watching their favorite animated movie in matching pajamas! Tweens can enjoy making friendship bracelets in their individual bohemian sleeping tents. Meanwhile, teens can enjoy taking selfies while drinking mocktails and playing Truth or Dare.
Check out some of these ideas to inspire your next slumber party!
All best friends need matching pajamas! Supply your child and their friends with matching pajamas that you can either include with the invitation or have ready for when they arrive!
Organize the coolest movie night around! Pillows everywhere for ultimate comfort, premade popcorn in sweet or salty flavors, teddy bears to snuggle up with and fruity punch! Your kids will have so much fun sitting on their bed of pillows and watching their favorite animated movie!
Every amazing slumber party includes a movie night and what's movie night without popcorn!? This rustic popcorn bar is going to be the hit of the party with delicious toppings and lots of options. Let's just hope your little ones can get to sleep after this special treat!
Put out toothbrush party favours to make sure that all the kiddos brush their teeth before falling asleep. The kids might not find it that great but the parents will love you for it!
Throw an indoor camping slumber party for your kids with the ultimate planning. S'mores in a cup, a fake fire pit, sleeping tents, wildlife stuffs for them to snuggle up with and a fun activity like painting birdhouses!
Create a comfortable Princess Castle Fort for your little girls to snuggle up in, play make-believe and tell stories. A princess fort is a wonderful idea for a girly slumber party. You can put out pretty nail polishes, decorate with pink frilly materials and set up a crown making activity!
Provide cute fuzzy housecoats for your kids and their guests to really make them feel relaxed and comfortable. They will love running around in these puffy and warm coats as they do crafts and watch movies.
Got any Star Wars fans in the house? You can put together a totally cool Star Wars themed slumber party at your house! Look for decorations, bring out the Darth Vader puzzles, and print out some fun coloring sheets!
A princess tea party for your little sleeping beauties. Delicate cookies, elegant teacups and of course some golden crowns to wear with their matching dresses! After all the tea party chit-chat, kiss your sleepy beauties to sleep.
Is there anybody that doesn't love breakfast for dinner? I know that I do, and the kids will definitely love it too! This is a fabulous sleepover theme that lets your kids be a little spoiled for the night with delicious pancakes as they play in their pajamas and make friendship bracelets!
This is SO cool and sure to be the hit of your sleepover! Try this fun DIY out for the craft of the night and get your kiddos having fun squishing slime between their little fingers.
Little bohemian tents for all the tweens to sleep in for the night. Everyone gets their own individual tent with sleepmasks, headbands, and party favors. Maybe this way they will actually want to go to sleep? We can only hope!
Seal the friendship with a colorful, crazy, and cute friendship bracelet! This is a wonderful way to keep the tweens busy for the night having fun until it's time to shut their eyes.
Your teens will love making their own sleepover eye masks and these unicorn ones are the coolest! Give them a hand by making the first one and then let them take it from there! Careful using hot glue guns!
Oh my! Your tweens will absolutely adore these fabulous fuzzy slipper cookies! Or even better yet, for an evening activity, you can make the cookies with everybody lending a hand and decorating their own slippers!
Every tween sleepover needs a good old fashioned pillow fight! So get your contestants ready with their soft and cushiony customized pillow fight pillows!
Unicorn popcorn is the new must-have at any sleepover party! Kids, tweens or teens will adore this special addition to their late night movie fun!
This is an amazing slumber party activity! It's a great way to get all the tweens fed while still having fun! Each tween gets to personalize their own pizza to their own taste and you get to make sure everyone eats!
Every amazing party needs games! Game night at your house will be a night they won't forget; complete with snacks, drinks and all the board games you can imagine! Check out your local thrift store to find cheap board games but make sure you check that they have all the pieces.
Design the most perfect invitation to your teens totally cool slumber party! Charm your guests with funky eye mask invitations with all the details of what to bring, how to get there and what time to show up.
Relax your young teenagers with a pampering spa night before they tuck into bed with a romantic chick flick! Soaking their feet while drinking virgin Mimosas and painting their nails… What a hard life they have!
Building a blanket fort is one of those things that is fun to do at any age! Even as an adult! It's a great place to get away with friends and drink mocktails. So set up your teens at their next slumber party and help them build the ultimate blanket fort!
Keep their fingers warm with a hot cup of cocoa. The best part? They get to make their hot cocoa exactly how they'd like it! Don't forget to put out the marshmallows!
Decorations at your tweens slumber bash is super important! That's because everybody will be snapping selfies and the background has got to be spot on!
Every slumber party needs a photo booth and every photo booth needs props! Get silly with your teens and take hilarious pictures with big moustaches and funny glasses.
Give your teenagers a glamping experience right in your backyard. A place where they can gossip, talk about their favorite new movie, and play fun games!
This is an all girls slumber bash! A place where you can get together with your besties, paint your nails, talk about that cute boy, and take selfies! Fun and bright neon colors, big signs and of course a lot of balloons!
Truth or Dare will always be fun! It's just one of those games that make you laugh, scream and be surprised! It's a great way for everybody to get to know each other better and to have a good time.
Print out a list of your favorite chick flicks and then get all the party guests to vote on a number! It's an easy way to make up your mind on which Netflix film you're going to watch.
When you're staying up late with your besties gossiping about the cute new guy in school there's gonna need to be a midnight snack!
So grab your scissors and glue guns and start making up those slumber party invitations! Having all the kids over to your house will be such a blast when you've got the whole thing planned out!
Start with some fun crafts and activities to get the party going. DIY eye masks, friendship bracelets and unicorn slime are fun ones! Afterward, play some board games or make your own pizza dinner. When it gets a bit later, get comfy in the living room and laugh out loud while watching that new animated movie for kids or that hilarious chick flick for the teens! Of course, when it's time to go to sleep and all the kids crawl into their own sleeping tents you can finally relax.
As you can see, there are lots of ways to host a totally awesome slumber party no matter how old your kids are! Fun and colorful, trendy and stylish or themed to their current interest. Get creative and have a wonderful night with everyone.


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Positioner can be set up in three minutes
A venerable British engineering company specialising in fluid controls has developed a smart valve positioner which it claims puts it at leading edge of 21st century technology. Wirral-based Heap & Partners, which was founded in 1866, says that its positioner can be set up in just three minutes compared to about an hour needed for a conventional installation.
Heap has joined forces with Hoerbiger-Origa, which is supplying a low-current piezo-pneumatic valve technology which allows the positioner - called Storm - to be powered from a 4-20mA loop signal with a voltage drop of 5V.
A direct-drive potentiometer with a torque-limiting clutch allows the positioner to be mounted and calibrated quickly, without needing time-consuming alignment. The device sets up the valve travel, control parameters and auto-tunes a PID algorithm to give optimum performance for the specific valve and actuator package.
The positioner is housed in an enclosure submersible to IP67 and has been certified intrinsically safe to EExia IIC T5 for use in hazardous areas - making it the only Zone 0 certified digital positioner on the market, according to Heap.


Motive Me Monday – Be a GOAL GETTER!
Need motivation? I find there's no better way to get motivated than to set a goal and GET IT! Accordingly, Love Fit Life is doing a 5K!
Ok, but what is a 5K exactly?
Sunday, June 24, 2012 @ 8:30 a.m.
6. Start walking/running because Sunday, June 24, 2012 will be here before you know it!
8. Invite all your family and friends to join too – the more the merrier and you definitely don't want to miss this party!
O.I.: No problemo! Check out these websites for more races and train along with us anyway!
O.I.: Don't worry, I didn't either when I did my first 5K. I started by running at least a little bit 3-4 times a week. At first, at least a little bit meant just around the block. As I progressed, I ran based on time and slowly increased my time until I was able to run for 25-30 minutes straight. On race day, the crowd and my adrenaline did a lot more of the work than I thought! I finished in 30 minutes and could not have been happier!
O.I.: Talk to your family, friends, and co-workers to sponsor you! One way to pitch it is to have people sponsor you per mile – "Hey, I'm walking/running in a 5K to prioritize a fit life for myself and support children with cancer and their families – what do you say to sponsoring my 1st mile with a $5-$10 donation?" We are running to support a great cause and since it's only $25 you can easily get 1 or more people to help you cover the cost.
O.I. Part 1: What doesn't kill you makes you stronger!
O.I. Part 2: Get over it.
O.I.: Great news! You are the best possible role model for your kids and family! Lead by example and show them that living a healthy and fit life is crucial for so many reasons, AND that it can be done as a family. Not only that, but EVERYONE IS WELCOME! Got a bun in a oven? You should definitely be walking so that you're ready for delivery day. Got a little one running around? Pop them in the stroller and walk/run while pushing or if they're old enough to walk, walk as a group! Got a significant other/spouse that's not on board? Tell them that you're prioritizing yourself and your health and they can make their own breakfast that morning!
O.I.: No prob Bob… as we've already mentioned, you can WALK! We already have walkers confirmed too so you won't be alone!
O.I. Part 1: Just replace walking/running for some of the time you spend everyday being unhealthy, unhappy, stressed, tired, and worrying about how your life will be shorter because you don't take care of yourself. Oh, did I say that out loud? Yes I did. Oh, you didn't want to hear that? The truth hurts.
O.I. Part 2: You will have walked or run your way to a completed 5K by 9:30 a.m. on Sunday morning – You'll probably call all your friends to brag about your accomplishment and they won't even be up yet! You'll probably have a meeting or event to go to that day and you'll have tons of time to kill before it even starts! You'll probably get home before your family or roommates even wake up!
O.I.: We want a fit life community… that means being together, being happy in positive company and not leaving anyone behind. Let's brainstorm a way to get you across the finish line! We want everyone there, no matter what.
O.I.: Don't worry, I was there too once! Trust me and everyone else who has ever broken that cycle – YOU WILL NOT REGRET IT ONE BIT…EVER!


Model # GDS26L5-1818WG Elegantly scrolled pilasters are the hallmark detail of the Samuel Mantel Electric Fireplace.
Elegantly scrolled pilasters are the hallmark detail of the Samuel Mantel Electric Fireplace. Strong, geometric carvings anchor the header while a mitered picture frame surround elegantly frames the firebox. A Weathered Grey finish enhances the characteristic knots of the rustic pine wood, giving the Samuel a final note of comfort and charm.


text:
The Matrix Revolutions (2003)
Watch The Matrix Revolutions (2003) Online
The human city of Zion defends itself against the massive invasion of the machines as Neo fights to end the war at another front while also opposing the rogue Agent Smith.
Genre: Action, Science Fiction
Director: Lana Wachowski, Lilly Wachowski
Actors: Carrie-Anne Moss, Helmut Bakaitis, Hugo Weaving, Keanu Reeves, Lambert Wilson, Laurence Fishburne, Mary Alice
Star Trek VI: The Undiscovered Country (1991)
On the eve of retirement, Kirk and McCoy are charged with assassinating the Klingon High Chancellor and imprisoned. The Enterprise crew must help them escape to thwart a conspiracy aimed…
Teenage Mutant Ninja Turtles II: The Secret of the Ooze (1991)
The Turtles and the Shredder battle once again, this time for the last cannister of the ooze that created the Turtles, which Shredder wants to create an army of new…
Heist (2015)
A father is without the means to pay for his daughter's medical treatment. As a last resort, he partners with a greedy co-worker to rob a casino. When things go…
Wonderful Days
Civilization has been destroyed by war and pollution, but the survivors have built the last city of Ecoban. As most natural resources have been exhausted, Ecoban is powered by pollution….
Genre: Action, Adventure, Animation, Fantasy, Science Fiction
Trancers 3: Deth Lives (1992)
A time traveling cop, Jack Deth, from the future is taken back to the past to be given the task of destroying the Trancer program before it has a chance…
In a gritty and alternate 1985 the glory days of costumed vigilantes have been brought to a close by a government crackdown, but after one of the masked veterans is…
Genre: Action, Drama, Mystery, Science Fiction
The Amazing Spider-Man (2012)
Peter Parker is an outcast high schooler abandoned by his parents as a boy, leaving him to be raised by his Uncle Ben and Aunt May. Like most teenagers, Peter…
Indiana Jones and the Kingdom of the Crystal Skull (2008)
Set during the Cold War, the Soviets – led by sword-wielding Irina Spalko – are in search of a crystal skull which has supernatural powers related to a mystical Lost…
The Last Stand (2013)
Ray Owens is sheriff of the quiet US border town of Sommerton Junction after leaving the LAPD following a bungled operation. Following his escape from the FBI, a notorious drug…
The Shadow (1994)
Based on the 1930's comic strip, puts the hero up against his arch enemy, Shiwan Khan, who plans to take over the world by holding a city to ransom using…
Genre: Action, Adventure, Crime, Fantasy, Mystery, Thriller
Vigilante (1982)
New York City factory worker Eddie Marino is a solid citizen and regular guy, until the day a sadistic street gang brutally assaults his wife and murders his child. When…
Trailer: The Matrix Revolutions (2003)


Are you worried about following USAU's uniform requirements? Steering clear of trademark infringements? Complying with your school's logo licensing?
Our Custom Crew is super knowledgeable about all things related to ultimate uniforms. We took that expertise and put together some good rules of thumb to make sure you don't violate any rules.
Or start an order and let us worry about following the rules with your team gear for you!
With custom jerseys, you can print 99% of pretty much anything you can imagine. But let's take a second to look at that 1% you can't, or at least shouldn't, print.
Did you know that nationally competitive teams are required to have numbers on both the front AND back of their jerseys?
Ordering jerseys for the USAU college and club series is a little different than ordering for Potlatch or Poultry Days. Most sports organizations, including USAU, reserve the right to alter jerseys should they deem them to be inappropriate for a sanctioned program or event.
If you don't follow these rules, your team may be disqualified. Here's how to avoid that.
While these rules are really only enforced at the higher competitive echelons of the game (for now), all teams competing in USAU sanctioned tournaments are encouraged to follow suit.
You can read the FAQ's here, as well as the uniform requirements for both the Triple Crown Tour and the College Series.
A lot of teams take jersey inspiration from movies, television, and other pop culture references. While these designs are made in good faith, they can be susceptible to trademark and copyright restrictions.
Legally speaking, if you copy-paste the Coca-Cola logo from the internet and print it on your jersey, Coke could come in and ask that you either stop wearing their logo on your jerseys or force you to pay them for use of their trademark.
No one wants to deal with legal action from a big company. Ultimate was born in the counterculture after all, and our sport has a deep-seated distrust of authority.
Luckily for us, parody is thoroughly protected under our 1st Amendment right to free speech. Even better: the protection of parody extends all the way to something as silly and awesome as the logo on an ultimate jersey.
Parody is the deliberate and exaggerated imitation of a design for comic effect.
That means that if you want to use a major brand's logo as an influence, you have to alter it in a way that makes it clear that you're doing it for comic effect.
When you think about it, the rules around trademarks and parody are actually kind of a benefit for us ultimate players.
It's not very cool to copy-paste a big company's logo from the internet and paste it on your chest.
What IS cool is taking a big company's logo, making silly pun that reinforces the culture of fun and awesomeness in ultimate, and making a logo out of that!
We love clever and wacky spinoffs of well-known pop culture references. We also love not getting sued. If you're worried your design isn't parody enough, our artists can help you get there!
Some schools require their club sports teams to use the official school logo on all team apparel in order to qualify for funding.
Of course, colleges and universities ALSO have very specific and restrictive guidelines for licensing and using their officials logos, which just makes it more difficult to order gear.
If you're in this group, and you want to go with and ultimate apparel company, make sure to ask ahead of time if they are willing and able to help you license your school's logo.
Luckily, we're experts in working with schools to license their logos! We've worked with Pitt, Michigan, Florida, and a bunch of other schools.
Simply let us know up front that you need (or want!) to license your school's official logo. We'll take the work off of your hands as much as possible, and make sure that your jerseys comply with your school's official guidelines.
Did we miss anything? Let us know below! Or if you're unsure if your team design is riding the line between original and copyright infringement, just ask us by emailing team@fiveultimate.com!