\section{Introduction}

To advance the paradigm of autonomous operation for nuclear power plants, a data-driven machine learning approach to control is sought.
Autonomous operation for next-generation reactor designs is anticipated to bolster safety and improve economics.
However, any algorithms that are utilized need to be interpretable, adaptable, and robust.
Interpretable means that one can inspect and understand the underlying relationships between inputs and outputs of the algorithm.
Adaptable refers to the capability of accommodating temporal changes in the underlying system dynamics (e.g.,\@\xspace due to reactivity swings or fouling).
Robustness refers to stability over long horizons under the presence of sensor or process noise.
In this work, we focus on the specific problem of optimal control during autonomous operation.

\subsection{Objectives}

We will demonstrate an interpretable and adaptable data-driven machine learning approach to autonomous control of a molten salt loop.
To address interpretability, we utilize a data-driven algorithm to identify system dynamics in \textit{state-space representation}.
To address adaptability, a control algorithm will be utilized to modify actuator setpoints while enforcing constant, and \textit{time-dependent} constraints.
Robustness is not addressed in this work, and is part of future work.
To demonstrate the approach, we designed a numerical experiment requiring intervention to enforce constraints during a load-follow type transient.

\section{Methods}

In this section we summarize the key algorithms utilized to demonstrate constraint enforcement.
First, the Reference Governor (RG) algorithm is introduced, alongside a formulation of the dynamics model required to utilize it.
Next, the data-driven algorithm utilized to identify system dynamics, Dynamic Mode Decomposition with Control, is introduced.
Finally, the molten salt loop experiment model used to demonstrate constraint enforcement is presented.

\subsection{Reference Governor}

The RG \cite{Bemporad1994} is the constraint enforcement algorithm utilized in this work.
The RG is an \textit{add-on} control scheme that enforces \textit{point-wise in time} constraints.
The terminology \textit{add-on} reflects the nature of its intended use: the algorithm is applied to a system that already has an existing pre-calibrated control logic (e.g.,\@\xspace regulation via PID) for a closed-loop feedback system.
The add-on feature is important as an operator may choose to manually override/disable the RG, while maintaining controllability to maneuver the plant.
The terminology \textit{point-wise in time} reflects the capability to update constraint bounds at each discrete time-step.
This feature allows us to address e.g.,\@\xspace degradation phenomena, where the capabilities of the system vary temporally.
The application of RG is visualized in \cref{fig:RG}.
Previous work demonstrated that the RG framework is applicable to an integrated energy system \cite{Wang2020}.

\begin{figure}
    \includegraphics[width=\linewidth]{RG_Diagram.pdf}
    \caption{Control block diagram visualizing the topology of the Reference Governor (RG).
    Direct access to the underlying closed-loop system is always available.
    }
    \label{fig:RG}
\end{figure}

The RG algorithm is summarized next.
Garone et al.\@\xspace \cite{Garone2017} provide a full review.
First, what the RG algorithm does, with respect to \cref{fig:RG} is described.
To gain some intuition, we first describe the entry-level RG: the Scalar Reference Governor (SRG).
At each time-step $k\in\mathbb{Z}_+$, the SRG receives a reference input, $r_k\in\mathbb{R}^m$, where $m$ is the number of inputs.
Depending on the system's current state, expected evolution, and user defined constraints, an admissible input $v_k\in\mathbb{R}^m$,
\begin{equation}
    v_k = v_{k-1} + \kappa_k \left(r_k - v_{k-1}\right)~,
    \label{eq:rg1}
\end{equation}
is sent to the lower-level controllers.
In \cref{eq:rg1}, $\kappa_k\in[0,1]$ is a \textit{scalar} that governs admissible changes to the inputs, such that a complete rejection, $v_k=v_{k-1}$, acceptance, $v_k=r_k$, or an intermediate change, $v_{k-1}\leq v_k\leq r_k$, is possible.
After determining $\kappa_k$, the output of the RG, $v_k$ is sent to the existing controllers.
For example, a PID controller would receive a component of $v_k$ as a reference set-point for its manipulated variable (MV), and according to its paired MV's current value, issue a control action $u_k$ to the control variable (CV).

Next, how the SRG algorithm determines $\kappa_k$ is summarized.
The RG relies on a state-space representation,
\begin{align}
    x_{k+1} & = Ax_k + Bu_k~, \label{eq:state}\\
    y_k & = Cx_k + Du_k~, \label{eq:outp}
\end{align}
where $x_k\in\mathbb{R}^n$, $u_k\in\mathbb{R}^m$, and $y_k\in\mathbb{R}^p$ are vectors at step $k$ representing states, inputs, and outputs, respectively.
The matrix $A\in\mathbb{R}^{n\times n}$ accounts for the state dynamics.
The matrix $B\in\mathbb{R}^{n\times m}$ is the input matrix.
The matrix $C\in\mathbb{R}^{p\times n}$ is the output matrix.
The matrix $D\in \mathbb{R}^{p\times m}$ is the control pass-through matrix.
Using the system model defined in \cref{eq:state,eq:outp}, constraints are imposed on the output variables, $y_k\in Y$, where $Y$ is defined by a set of linear inequalities.
Constraints may be imposed on $x_k$, or $u_k$.
To ensure constraints are not violated, we define the maximal output admissible set (MOAS).
The MOAS is the set of all $x_k$, and constant input $\Tilde{v}$, such that,
\begin{equation}
    O_\infty = {(x_k, \Tilde{v}) : y_{t+k}\in Y, v_{t+k}=\Tilde{v}, \forall k \in \mathbb{Z}^+}~.
    \label{eq:moas}
\end{equation}
The $k\rightarrow \infty$ assumption of $O_\infty$ is generally relaxed to a suitably large finite horizon, $T$.
Thus, at each time-step $k$, the SRG algorithm determines the admissible $\Tilde{v}$, and therefore $\kappa_k$, such that the system remains in $O_\infty$.
The $O_\infty$ is constructed by evaluating \cref{eq:state,eq:outp} for each $k=0,1...T$ time steps.

A drawback of the SRG formulation is that $\kappa_k$ is a scalar.
Therefore, if $v_k-r_k$ is multi-dimensional and $\kappa_k<1$, input movement in all dimensions is bound.
In our work, we utilize the Command Governor (CG), a variant of the RG which selects $v_k$ by solving the quadratic program,
\begin{align}
v_k = &\argmin_{v_k} ||v_k - r_k ||^2_Q~, \nonumber \\
\mr{s.t.} \hspace{6pt} &(x_k, v_k=\Tilde{v})\in O_\infty~, \label{eq:cg}
\end{align}
where $Q$ is a positive definite matrix signifying the relative importance of each $v_k$ component.
In this work, the quadratic program is solved by the CVXPY convex optimization library \cite{diamond2016cvxpy}.
Using the CG, constraints can be enforced by manipulating $v_k$ arbitrarily in $\mathbb{R}^m$, constrained by $O_\infty$.
This is particularly powerful for systems where $m>1$, i.e.,\@\xspace multiple inputs.

Accurately identifying the system dynamics is a necessary prerequisite for utilizing the RG framework.
In general, and in our particular model, $D=0$.
A system identification method is needed to define the remaining matrices.

\subsection{System Identification}

To identify the system matrices, we utilize the Dynamic Mode Decomposition with Control (DMDc) algorithm \cite{Proctor2016}.
DMDc is an extension of Dynamic Mode Decomposition (DMD), a method that originated in the fluids community to extract spatial-temporal coherent structures from experimental data.
DMDc is a data-driven method relying on snapshots of data, of length $L$,
\begin{equation}
X= \begin{bmatrix}
\vline & \vline & & \vline \\
x_0 & x_1 & \cdots & x_L\\
\vline & \vline & & \vline
\end{bmatrix} ,~~ U = \begin{bmatrix}
\vline & \vline & & \vline \\
u_0 & u_1 & \cdots & u_{L-1} \\
\vline & \vline & & \vline
\end{bmatrix},
\label{eq:snaps}
\end{equation}
corresponding to the states and inputs defined in \cref{eq:state}.
The data in \cref{eq:snaps} is grouped into two matrices: $\Omega=[X_{0...L-1}^\intercal U^\intercal]^\intercal$ and $X'=X_{1...L}$.
Using the singular value decomposition (SVD) of $\Omega$ and $X'$, the reduced order approximations of matrices $A$ and $B$ can be determined  \cite[\S3.3]{Proctor2016}.
Qualitatively, the DMDc methodology combines SVD (dimensionality reduction) with a Fourier transform (finding dominant patterns in time).

\subsection{Experiment Model}

This work focuses on demonstrating the applicability of advanced control algorithms for next-generation reactors.
Thus, the System Analysis Module (SAM) code \cite{Hu2017} is utilized.
SAM is developed to provide best-estimate system-level models for Sodium-cooled Fast Reactors, Molten Salt Reactors, Fluoride-cooled High-temperature Reactors, and Lead-cooled Fast Reactors.
A layout of the experiment model and its steady-state characteristics is presented in \cref{fig:loop}.
The model consists of a single loop filled with flibe (\ce{2LiF-BeF_2}), heated through a single channel and cooled with flinak (\ce{LiF-NaF-KF}) via a heat exchanger.
Thermophysical properties of flibe and flinak are embedded within SAM.
In this experiment, there are two fixed boundary conditions: the flinak source temperature ($T_\mr{s,in}$) and the primary-side pump head.

\begin{figure}
        \begin{minipage}[t]{0.5\linewidth}
        \vspace{0pt}
        \centering
        \includegraphics[trim={0 0 0 0.1cm}, clip, width=0.9\linewidth]{Loop.pdf}
        \end{minipage}%
        \begin{minipage}[t]{0.5\linewidth}
        \vspace{0pt}
        \centering
         \begin{tabular}{lc}
             \hline
             \multicolumn{2}{c}{Steady-state Value} \\
             \hline
             \multicolumn{2}{l}{Primary (flibe)} \\
              \hspace{6pt}$\dot{Q}$ [MW] & 18.0 \\
              \hspace{6pt}$\dot{m}_\mr{p}$ [kg/s] & 589 \\
              \hspace{6pt}$T_\mr{p,in}$ [\SI{}{\celsius}] & 585 \\
              \hspace{6pt}$T_\mr{p,out}$ [\SI{}{\celsius}] & 572 \\
              \hspace{6pt}$T_\mr{p,1}$ [\SI{}{\celsius}] &  572 \\
              \hspace{6pt}$T_\mr{p,3}$ [\SI{}{\celsius}] &  585 \\
              \hspace{6pt}$P_\mr{p,out}$ [kPa] & 179 \\
              \hspace{6pt}$P_\mr{p,1}$ [kPa] & 200 \\
            \multicolumn{2}{l}{Secondary (flinak)} \\
              \hspace{6pt}$\dot{m}_\mr{s}$ [kg/s] & 380 \\
              \hspace{6pt}$T_\mr{s,in}$ [\SI{}{\celsius}] & 492 \\
              \hspace{6pt}$T_\mr{s,out}$ [\SI{}{\celsius}] & 517 \\
         \end{tabular}
        \end{minipage}
    \caption{%
        Overview of the experiment model.
        Left: Layout of the model in SAM.
        Right: Corresponding steady-state conditions.
    }
    \label{fig:loop}
\end{figure}

After the model was defined within SAM, a transient to demonstrate the capabilities of the RG was formulated.
A load-follow type transient was chosen.
A pre-defined reference trajectory is assigned for the secondary side mass flow rate ($\dot{m}_\mr{s,ref}$) and the primary-side heat-exchanger inlet temperature ($T_\mr{p,in,ref}$).
A control strategy for the maneuver was then developed, alongside categorization of parameters to identify the system matrices.

\subsection{Control Strategy \& State Selection}

The control strategy encompasses the pairing of CVs to MVs.
Effective pairings minimize cross-talk, which can be quantified via open loop response.
Remaining (uncontrolled) variables are to be categorized into outputs and states.
All outputs are variables that can be directly measured from the plant and that may be subject to constraints.
The states are variables that may not be directly measured from the plant (e.g.,\@\xspace the integral component of the PI controller for secondary-side mass-flow rate, $I_{\dot{m}_\mr{s}}^\mr{PI}$).
The composition of the set of states is determined using a feature-selection algorithm.
A forward sequential feature selection algorithm was utilized.
The objective function to minimize was the mean squared error between the output values from the state-space model, i.e.,\@\xspace \cref{eq:state,eq:outp}, and the SAM model.
The resulting control strategy and classification of variables are summarized in \cref{tab:vars}.
The RG reference inputs, $r_k$ in \cref{eq:rg1}, are defined as the actuators of the entire system.

\begin{table}%
    \begin{tabular}{ccc}%
        \begin{tabular}[t]{l}
            \hline
            Outputs -- $y_k$\\
            \hline
             $T_\mr{p,out}$ \\
             $T_\mr{s,out}$ \\
             $P_\mr{p,out}$ \\
             $P_\mr{p,1}$ \\
        \end{tabular} &
        \begin{tabular}[t]{l}
            \hline
            States -- $x_k$\\
            \hline
             $T_\mr{p,1}$ \\
             $T_\mr{p,3}$ \\
             $I_{\dot{m}_\mr{s}}^\mr{PI}$
        \end{tabular} &
        \begin{tabular}[t]{l}
            \hline
            Inputs -- $r_k$\\
            \hline
            $\dot{m}_\mr{s,ref}$ \\
            $T_\mr{p,in,ref}$
        \end{tabular}
    \end{tabular}
    \begin{tabular}{ll}
        \begin{tabular}[t]{ll}
            \hline
            CVs -- $u_k$ & MVs -- $v_k$\\
            \hline
            $u_\mr{s}$ & $\dot{m}_\mr{s}$ \\
            $\dot{Q}$ & $T_\mr{p,in}$
        \end{tabular} &
        \begin{tabular}[t]{l}
            \hline
            Constrained Variables\\
            \hline
            $T_\mr{p,out}\leq T_\mr{p,out}^\mr{max}$\\
            $T_\mr{s,out}\geq T_\mr{s,out}^\mr{min}$
        \end{tabular}
    \end{tabular}
    \caption{%
    Categorization of variables corresponding to the labels in \cref{fig:RG} and model parameters in \cref{fig:loop}.
    First major row presents the list of outputs, states and inputs.
    Second major row presents the control variables (CV), manipulated variables (MV), and constrained variables.
    }
    \label{tab:vars}
\end{table}

\section{Results}

\raggedbottom
The model implemented in SAM, \cref{fig:loop}, and the control architecture, \cref{fig:RG}, were coupled via python.
A Proportional-Integral (PI) controller was constituted for both MVs tabulated in \cref{tab:vars}, with the CVs as outputs.
For all results presented in this paper, the topology of \cref{fig:RG} is maintained where `Plant' is assumed to be the ground-truth results from SAM and `Existing Controllers' are the PI controllers that were tuned via open-loop response.
A constant $\Delta t \equiv \left(t_{k+1}-t_k\right) = \SI{0.2}{\second}$ was maintained.

\subsection{SAM vs.\@\xspace DMDc}

First, the validity of applying DMDc to identify the SAM model dynamics is discussed.
To simulate a load-follow transient, time-dependent reference trajectories were assigned to $T_\mr{p,out,ref}$ and $\dot{m}_\mr{s,ref}$.
The trajectories are specified such that $T_\mr{p,out}^\mr{max}\equiv \SI{586.85}{\celsius}$ and $T_\mr{s,out}^\mr{min}\equiv \SI{512.85}{\celsius}$ constraints are violated.
These trajectories are presented in the bottom row of \cref{fig:res_dmdc}.
Once the input ($u_k$), state ($x_k$) and output ($y_k$) data is collected, we utilize the DMDc algorithm to identify the matrices in the state-space representation, \cref{eq:state,eq:outp}.
A comparison of the state-space representation to results from SAM are presented in the top row of \cref{fig:res_dmdc}.
By visual inspection, it is apparent that the DMDc algorithm successfully identifies the system dynamics.
Quantitatively, the mean-squared error loss for each state and output is also low.
On additional tests with alternating ramp sequences our algorithms achieved equivalent performance -- we present results for a single sequence for brevity.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{DMDc_comp_red.pdf}
    \caption{Application of DMDc to identify state-space matrices.
    Top rows: Contrasting DMDc model performance vs.\@\xspace SAM.
    Mean-squared error ($\varepsilon_\mr{MSE}$) is presented.
    Bottom row: Reference setpoint trajectory -- the input for both SAM and DMDc.
    }
    \label{fig:res_dmdc}
\end{figure}

\subsection{Constant Constraint Enforcement}

The first result presented utilizes constant constraint values.
Because the constraint values are constant, the admissible region changes only with respect to $x_k$.
The results are presented in \cref{fig:cg}.
The results indicate that the RG algorithm correctly intervenes and restricts $v_k$ such that the constraints assigned to both outputs are enforced.
It is important to note that the results presented in the following figures are from manipulating a SAM plant, rather than the DMDc model.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{CG_sam_const.pdf}
    \caption{Results with application of RG algorithm with constant constraint values.
    Top row displays outputs that are subject to constraints.
    Bottom row displays changes to the reference input trajectory by the RG algorithm.
    }
    \label{fig:cg}
\end{figure}

\subsection{Time-dependent Constraint Enforcement}

The next set of results utilize time-dependent constraints.
In particular the secondary-side outlet temperature is constrained such that,
\begin{equation}
  T_\mr{s,out}^\mr{min}=\begin{cases}
       512.85~\SI{}{\celsius}, &t< \SI{2000}{\second}\\
       512.85\pm\frac{2.5}{1000}(t-2000)~\SI{}{\celsius}, &\SI{2000}{\second}\leq t\leq \SI{2800}{\second}\\
       512.85\pm2.00~\SI{}{\celsius}, &t> \SI{2800}{\second}
    \end{cases}
    \label{eq:constr}
\end{equation}
where the $+~\mr{or}~-$ sign is used consistently for increasing or decreasing $T_\mr{s,out}^\mr{min}$, respectively.
Results for decreasing or increasing the constraint is presented in \cref{fig:cg_tdep}.
First, the scenario where $T_\mr{s,out}^\mr{min}$ decreases is discussed.
As the constraint decreases, the admissible region ($O_\infty$) expands as the admissible setpoint for $T_\mr{p,in,ref}$ can be reduced further to meet the reference value.
The results in \cref{fig:cg_tdep} indicate that the RG algorithm accurately exploits the expansion of $O_\infty$ to minimize the cost in \cref{eq:cg}.
The more challenging scenario, where $T_\mr{s,out}^\mr{min}$ increases is discussed next.
During this scenario, the $O_\infty$ region contracts as the admissible setpoint for $T_\mr{p,in,ref}$ increases in distance from the reference value.
The scenario is more challenging as an incorrect $v_{k+1}$ can immediately violate constraints.
The results in \cref{fig:cg_tdep} indicate that the RG algorithm also optimally responds to the contraction of $O_\infty$.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{CG_comp.pdf}
    \caption{Results with application of RG algorithm with constant $T_\mr{p,out}^\mr{max}$, decreasing $T_\mr{s,out}^\mr{min}$ (top two rows) and increasing $T_\mr{s,out}^\mr{min}$ (bottom two rows).
    }
    \label{fig:cg_tdep}
\end{figure}

\subsection{Admissible Region Evolution}

The admissible region ($O_\infty$) evolution is discussed next.
In \cref{fig:moas}, two separate events are presented.
The evolution of $O_\infty\left( t=550\rightarrow \SI{725}{\second}\right)$ visualizes the geometric objective of the CG, \cref{eq:cg}, a minimization of the distance between $v_k$ and $r_k$ as the bounds of $O_\infty$ evolve.
Note that the evolution of $O_\infty$ in this case occurs solely due to the change in $x_k$, as constraints are constant.
Whereas, the contrast of $O_\infty\left(t=\SI{2750}{\second}\right)$ for the different trajectories of $T_\mr{s,out}^\mr{min}$ visualize the expansion and contraction of $O_\infty$ referred to previously.
In this case, the differences in $O_\infty$ is due to different $x_k$ \textit{and} $T_\mr{s,out}^\mr{min}$ values.

\section{Concluding Remarks}

To advance the paradigm of autonomous operation for nuclear power plants, a data-driven machine learning approach to control is sought.
This work demonstrates an interpretable and adaptable approach to constraint enforcement for a molten salt loop, with multiple actuators and time-dependent constraints.
The major contribution of this works are: developing a framework to interface DMDc and RG algorithms with SAM; demonstrating that DMDc provides accurate state-space representations for load-follow type transients for flibe-flinak loops; and demonstrating successful actuator setpoint intervention to enforce constant and time-dependent constraints using the RG framework.
The next steps of our work include: introduction of noise to gauge robustness of the proposed framework, and application to a SAM plant with nuclear dynamics.

\begin{figure}
    \centering
    \includegraphics[width=\linewidth]{CG_moas.pdf}
    \caption{Visualizations of $O_\infty$.
    Top: $O_\infty\left( t=550\rightarrow \SI{725}{\second}\right)$ with constant constraints.
    Bottom: Contrast of $O_\infty\left(t=\SI{2750}{\second}\right)$ with constant (const.), decreased (dec.), and increased (inc.) values of $T_\mr{s,out}^\mr{min}$, according to \cref{eq:constr}.
    In both figures, the axes represent input values relative to steady-state, \cref{fig:loop}.
    }
    \label{fig:moas}
\end{figure}

\section{Acknowledgments}
This work is supported by U.S. Department of Energy Nuclear Energy Enabling Technologies Project 20-19321, and
U.S. Department of Energy Advanced Research Projects Agency–Energy Project 2174-1556.

\bibliographystyle{ans}


\section{Short introduction}

This preprint's aim is to present the statement and the proof of the theorem \ref{mainth}. The text is a continuation of \cite{Rem}, which is available for free from arXiv. In \cite{Rem} one can find wider introduction to the field, necessary background and the bibliography, as well as comments and examples to the abstract theorem proven.

\begin{definition} A quasi-Feynman formula is a representation of a function in a form which includes multiple integrals of an infinitely increasing multiplicity. \end{definition}

Formulas (\ref{supadupaF1}) and (\ref{supadupaF2}) from theorem \ref{mainth} are examples
of quasi-Feynman formulas if at least one of the families $(S_k(t))_{t\geq 0}$ consists of integral operators. The obtained formulas give the exact solution to the Cauchy problem for the Schr\"{o}dinger equation

\begin{equation}\label{SE}
\left\{ \begin{array}{ll}
 i\psi'_t(t,x)=\mathcal{H}\psi(t,x);\quad t\in\mathbb{R}, x\in Q\\
 \psi(0,x)=\psi_0(x);\quad x\in Q\\
  \end{array} \right.
\end{equation}

where $\mathcal{H}$ is the Hamiltonian of the Schr\"odinger equation (\ref{SE}). The operator $\mathcal{H}$ is a self-adjoint operator in Hilbert space $\mathcal{F},$ and $\mathcal{F}$ is a space of wave functions, i.e. for each $t\in\mathbb{R}$ we have $\psi(t,\cdot)\in\mathcal{F}.$ Usually $\mathcal{F}$ is $L_2(Q),$ where $Q$ is a configuration space of the system described by (\ref{SE}). The vector $\psi_0\in\mathcal{F}$ is regarded as initial state of the system, and vector $\psi(t)=e^{-it\mathcal{H}}\psi_0$ is a state of the system at the time $t$.

In this paper I propose a method of evaluating $e^{-it\mathcal{H}}\psi_0$ for all $\psi_0\in\mathcal{F}$. It uses the decomposition
$$-\mathcal{H}=L=a_1L_1+\dots+a_mL_m$$

and the existence of functions $S_k$ that are Chernoff-tangent to $L_k$.

\section{Short preliminaries}

Here only several definitions and basic theorems are listed, and only those that are used in the proof of the main theorem. Wider discussion and citations see in \cite{Rem}.
\vskip0.3cm

This paper is about $C_0$-groups. The definition of a $C_0$-group is classical and is supposed to be known, but we state it here for the sake of good order.

\begin{definition}Let $\mathcal{F}$ be a Banach space over the field $\mathbb{C}$. Let $\mathcal{L}(\mathcal{F})$ be a set of all bounded linear operators in $\mathcal{F}$. Suppose we have a mapping $V\colon [0,+\infty)\to \mathcal{L}(\mathcal{F}),$ i.e. $V(t)$ is a bounded linear operator $V(t)\colon \mathcal{F}\to \mathcal{F}$ for each $t\geq 0.$ The mapping $V$ is called a \textit{$C_0$-semigroup}, or \textit{a strongly continuous one-parameter semigroup} if it satisfies the following conditions:

1) $V(0)$ is the identity operator $I$, i.e. $\forall \varphi\in \mathcal{F}: V(0)\varphi=\varphi;$

2) $V$ maps the addition of numbers in $[0,+\infty)$ into the composition of operators in $\mathcal{L}(\mathcal{F})$, i.e. $\forall t\geq 0,\forall s\geq 0: V(t+s)=V(t)\circ V(s),$ where for each $\varphi\in\mathcal{F}$ the notation $(A\circ B)(\varphi)=A(B(\varphi))$ is used;

3) $V$ is continuous with respect to the strong operator topology in $\mathcal{L}(\mathcal{F})$, i.e. $\forall \varphi\in \mathcal{F}$ function $t\longmapsto V(t)\varphi$ is continuous as a mapping $[0,+\infty)\to \mathcal{F}.$

The definition of a \textit{$C_0$-group} is obtained by the substitution of $[0,+\infty)$ by $\mathbb{R}$ in the paragraph above.
\end{definition}

\begin{remark}If $(V(t))_{t\geq 0}$ is a $C_0$-semigroup in Banach space $\mathcal{F}$, then the set $$\left\{\varphi\in \mathcal{F}: \exists \lim_{t\to +0}\frac{V(t)\varphi-\varphi}{t}\right\}\stackrel{denote}{=}Dom(L)$$ is dense in $\mathcal{F}$. The operator $L$ defined on the domain $Dom(L)$ by the equality $$L\varphi=\lim_{t\to +0}\frac{V(t)\varphi-\varphi}{t}$$ is called an infinitesimal generator (or just generator to make it shorter) of the $C_0$-semigroup $(V(t))_{t\geq 0}$. The generator is a closed linear operator that defines the $C_0$-semigroup uniquely, and the notation $V(t)=e^{tL}$ is used. If $L$ is a bounded operator and $Dom(L)=\mathcal{F}$ then $e^{tL}$ is indeed the exponent defined by the power series $e^{tL}=\sum_{k=0}^\infty\frac{t^kL^k}{k!}$ converging with respect to the norm topology in $\mathcal{L}(\mathcal{F})$. In most interesting cases the generator is an unbounded differential operator such as Laplacian $\Delta$.
\end{remark}

The Stone theorem implies that $e^{-it\mathcal{H}}$ exists for each self-adjoint $\mathcal{H}$ and each $t\in\mathbb{R}$:

\begin{theorem} (\textsc{M.\,H.~Stone \cite{Stone}, 1932}) There is a one-to-one correspondence between the linear self-adjoint operators $H$ in Hilbert space $\mathcal{F}$ and the unitary strongly continuous groups $(W_t)_{t\in \mathbb{R}}$ of linear bounded operators in $\mathcal{F}$. This correspondence is the following: $iH$ is the generator of $(W_t)_{t\in \mathbb{R}}$, which is denoted as $W_t=e^{itH}.$
\end{theorem}

\begin{corollary}\label{cor} If $A$ is a linear self-adjoint operator in Hilbert space, then $\left\|e^{iA}\right\|=1.$
\end{corollary}

The following definition of Chernoff tangency (introduced in \cite{Rem}) plays the main role in what follows:

\begin{definition} Let $\mathcal{F}$ be a Banach space,
and $\mathcal{L}(\mathcal{F})$ be the space of all linear bounded operators in $\mathcal{F}$ endowed with the operator norm. Let $L\colon Dom(L)\to \mathcal{F}$ be a linear operator defined on $Dom(L)\subset\mathcal{F}$.

A function $G$ is said to be \textit{Chernoff-tangent} to the operator $L$ and the symbols $G\stackrel{Ch}{\asymp} L$ are written iff:

(CT1). $G$ is defined on $[0,+\infty)$, takes values in $\mathcal{L}(\mathcal{F})$ and $t\longmapsto G(t)f$ is continuous for every vector $f\in\mathcal{F}$.

(CT2). $G(0)=I,$ i.e. for all $f\in\mathcal{F}$ we have $G(0)f=f.$

(CT3). There exists a dense subspace $\mathcal{D}\subset \mathcal{F}$ such that for every $f\in \mathcal{D}$ there exists
a limit $G'(0)f=\lim_{t\to 0}(G(t)f-f)/t$.

(CT4). The operator  $(G'(0),\mathcal{D})$ has the closure $(L,Dom(L)).$

\end{definition}

The following Chernoff theorem allows to construct the $C_0$-semigroup in $\mathcal{F}$ having suitable family of linear bounded operators in $\mathcal{F}$. This family usually does not have a semigroup composition property but is pretty close to a $C_0$-semigroup in the sense described in the theorem below. For many $C_0$-semigroups such families $G$ are known or have been constructed in the past 15 years by the group of O.G.Smolyanov. We provide the wording of the Chernoff theorem based on the concept of the Chernoff tangency.

\begin{theorem}\label{FormulaChernova}(\textsc{P.\,R.~Chernoff \cite{Chernoff}, 1968}) Let $\mathcal{F}$ be a Banach space,
and $\mathcal{L}(\mathcal{F})$ be the space of all linear bounded operators in $\mathcal{F}$ endowed with the operator norm. Let $L\colon Dom(L)\to \mathcal{F}$ be a linear operator defined on $Dom(L)\subset\mathcal{F}$.

\textbf{Suppose} there is a function $G$ such that:

(E). There exists a strongly continuous semigroup $(e^{tL})_{t\geq 0}$ and its generator is $(L,Dom(L))$.

(CT). $G$ is Chernoff-tangent to $L.$

(N). There exists $\omega\in\mathbb{R}$ such that $\|G(t)\|\leq e^{\omega t}$ for all $t\geq 0$.

\textbf{Then} for every $f\in \mathcal{F}$ we have $(G(t/n))^nf\to e^{tL}f$ as $n\to \infty$, and the limit
is uniform with respect to $t$ from every segment $[0,t_0]$ for every fixed $t_0>0$.
\end{theorem}

From the Chernoff theorem one can infer this simple fact that will be used to obtain quasi-Feynman formulas:
\begin{corollary}\label{corboundop} If $\mathcal{F}$ is a Banach space, and $A\colon \mathcal{F}\to \mathcal{F}$ is a linear bounded operator, then
$$e^A=\sum_{k=0}^\infty\frac{A^k}{k!}=\lim_{k\to\infty}\left(I+\frac{A}{k}\right)^k.$$
\end{corollary}

\begin{definition}\label{Cheq}Let $\mathcal{F}$ and $\mathcal{L}(\mathcal{F})$ be as before. Let us call two $\mathcal{L}(\mathcal{F})$-valued mappings $G_1$ and $G_2$ defined both on $[0,+\infty)$ (respectively, both on $\mathbb{R}$)  \textit{Chernoff-equivalent} (denoting this as $G_1\stackrel{Ch}{\sim} G_2$) if and only if  $G_1(0)=G_2(0)=I$ and for each $f\in\mathcal{F}$ and each~$T>0$
$$\lim_{n\to\infty}\sup_{\scriptsize\begin{array}{cc}
t\in [0,T]\\
(resp.\ t\in [-T,T]) \end{array}}\left\|\left(G_1\left(\frac{t}{n}\right)\right)^nf-\left(G_2\left(\frac{t}{n}\right)\right)^nf\right\|=0.$$
\end{definition}

\begin{remark} There are several slightly different definitions of the Chernoff equivalence, I will just follow \cite{OSS} not going into details. The only thing I need from this definition is that if $G_1$ and $L$ satisfy all the conditions of the Chernoff theorem, then the mapping $G_1$ is Chernoff-equivalent to the mapping $G_2(t)=e^{tL}$. In other words, the limit of $(G_1(t/n))^n$ yields the $C_0$-semigroup $(e^{tL})_{t\geq 0}$ as $n$ tends to infinity.
\end{remark}

\begin{remark} With these definitions, the Chernoff-equivalence of $G$ to $e^{tL}$ follows via Chernoff theorem from the existence (E) of the $C_0$-(semi)group plus Chernoff-tangency (CT) plus the growth of the norm  bound (N). \end{remark}

\begin{remark} Denoting the Chernoff equivalence by $\stackrel{Ch}{\sim}$ and the Chernoff tangency by $\stackrel{Ch}{\asymp}$ one can compare the approximation theorems:
\end{remark}

\begin{center}
  \begin{tabular}{ | c | c  | c | c | }
    \hline

		\hline
\multicolumn{4}{ |c| }{How to obtain $\exp(tL)$} \\
\hline

    Case & We have & Then & History \\ \hline

		$L=A+B$ & $e^{tA}$, $e^{tB}$ & $e^{tL}=\lim\limits_{n\to\infty} \Big(e^{\frac{t}{n}A}\circ e^{\frac{t}{n}B}\Big)^n$  & Trotter \cite{Trotter}, 1958 \\ \hline

    General & $S(t)\stackrel{Ch}{\sim}e^{tL}$ & $e^{tL}=\lim\limits_{n\to\infty} \left(S\left(\frac{t}{n}\right)\right)^n$ & Chernoff \cite{Chernoff}, 1968 \\ \hline

$L=L_1+\dots+L_m$ &  $\forall k=1,2,\dots,m$ &  & Butko-Schilling\\

where $m\in\mathbb{N}$ is fixed  &$S_k(t)\stackrel{Ch}{\sim}e^{tL_k}$ & $e^{tL}=\lim\limits_{n\to\infty} \Big(S_1\left(\frac{t}{n}\right)\circ \dots\circ S_m\left(\frac{t}{n}\right)\Big)^n$ &-Smolyanov \cite{ButSS}, 2012\\ \hline

$L=iaH$      & $S(t)\stackrel{Ch}{\asymp}H$  &  $e^{tL}=\lim\limits_{n\to\infty} \left(R\left(\frac{t}{n}\right)\right)^n$ & Remizov \cite{Rem}, 2014 \\

where $H=H^*$, $a\in\mathbb{R}$ &  $S(t)^*=S(t)$                     & where $R(t)=\exp\big[ia(S(t)-I)\big]$ &\\ \hline

$L=iH,$ $H=H^*$  &  & & \\

$H=a_1H_1+\dots+a_m H_m$ & $\forall k=1,2,\dots,m$ &$e^{tL}=\lim\limits_{n\to\infty} \left(R\left(\frac{t}{n}\right)\right) ^n$ where  & Remizov, 2015 \\

where $m\in\mathbb{N}$ is fixed and & $S_k(t)\stackrel{Ch}{\asymp}H_k$,  & $R(t)=\exp\Big[i\sum_{k=1}^ma_k(S_k(t)-I)\Big]$  & (theorem \ref{mainth} below) \\

$a_k\in\mathbb{R}$ $\forall k=1,2,\dots,m$ &$S_k(t)^*=S_k(t)$ &  & \\ \hline

  \end{tabular}
\end{center}

\section{Main theorem}

\begin{theorem}\label{mainth}
Let $\mathcal{F}$ be a complex Hilbert space, and let $(L_k,Dom(L_k))_{k=1}^m$ be a finite set of linear (bounded or unbounded) operators $L_k\colon Dom(L_k)\to \mathcal{F},$ where $Dom(L_k)\subset\mathcal{F}$. Let $a_1,\dots,a_m$ be a finite set of non-zero (real or complex) numbers.

\textbf{Suppose} that

1) For all $k=\overline{1,m}$ the function $S_k$ is Chernoff-tangent to the operator $L_k$.

2) In item 1) above one can take the same linear dense subspace $\mathcal{D}\subset\mathcal{F}$ in (CT3) and (CT4) for all $k=\overline{1,m}$.

3) The closure $(L,Dom(L))$ of the operator $(a_1L_1+\dots+a_mL_m,\mathcal{D})$ is self-adjoint.

4) For each $t\geq 0$ the operator $a_1S_1(t)+\dots+a_mS_m(t)\stackrel{denote}{=}S(t)$ is self-adjoint.

5) The number $a=\sum_{k=1}^ma_k$ is real.

\textbf{Then}

1. There exists a $C_0$-group $e^{itL}$ with the generator $iL$.

2. The following family of bounded linear operators is well-defined $$R(t)=\exp\left[i\sum_{k=1}^ma_k(S_k(|t|)-I)\, \mathrm{sign}(t)\right]=\exp\Big[i(S(|t|)-aI)\, \mathrm{sign}(t)\Big].$$

3. The family $(R(t))_{t\in\mathbb{R}}$ is Chernoff-equivalent to the group $\left(e^{itL}\right)_{t\in\mathbb{R}}.$ It means that for all $t\in\mathbb{R}$

\begin{equation}\label{supadupaF}e^{itL}f=\lim_{n\to\infty}(R(t/n))^nf\end{equation}

4. Denote $a=a_1+\dots+a_m,$ $a_{m+1}=-a$ and  $S_{m+1}(t)=I$ for all $t\geq 0$ for simpicity of the formulas below. Then   for each $f\in\mathcal{F}$ and each $t\in\mathbb{R}$ the following formulas hold:

\begin{equation} \label{supadupaF1}e^{itL}f=\lim_{n\to\infty}\lim_{j\to\infty}\sum_{p=0}^j \frac{(in\, \mathrm{sign}(t))^p}{p!}
\sum_{k_1=1}^{m+1}\dots\sum_{k_p=1}^{m+1} a_{k_1}\dots a_{k_p} S_{k_1}(|t|/n)\circ \dots \circ S_{k_p}(|t|/n)f \end{equation}

\begin{equation}\label{supadupaF2}e^{itL}f=\lim_{n\to\infty}\lim_{p\to\infty} p!\,\sum_{d=0}^p \frac{(in\, \mathrm{sign}(t))^{d}}{p^dd!(p-d)!} \sum_{k_1=1}^{m+1}\dots\sum_{k_d=1}^{m+1} a_{k_1}\dots a_{k_d} S_{k_1}(|t|/n)\circ \dots \circ S_{k_d}(|t|/n)f\end{equation}

\end{theorem}
\begin{remark}
The conditions 4) and 5) of theorem \ref{mainth} trivially follow from the conditions 4') and 5') that are simpler in checking:

4') For each $t\geq 0$ and each $k=\overline{1,m}$ the operator $S_k(t)$ is self-adjoint.

5') Numbers $a_1, \dots, a_m$ real.
\end{remark}
\textbf{Proof of the theorem \ref{mainth}.} Item 1 follows from 3) and the Stone theorem. Item 2 follows from the fact that in the power index of the exponent for every fixed $t$ one finds linear bounded operators only thanks to 1), so the exponent can be defined as the power series  $e^A=\sum_{j=1}^\infty A^j/j!$ for $A= i\sum_{k=1}^ma_k(S_k(|t|)-I)\, \mathrm{sign}(t)$.

To prove item 3 let us check the conditions of the Chernoff theorem for $R^+(t)=\exp\left[i\sum_{k=1}^ma_k(S_k(t)-I)\right]$ and $L$. The condition (E) follows from the Stone theorem: self-adjoint operator $(L,Dom(L))$ is the infinitesimal generator of the $C_0$-group $(e^{itL})_{t\in\mathbb{R}}$ and of the $C_0$-semigroup $(e^{itL})_{t\geq 0}$.

Now let us check that $R^+$ is Chernoff-tangent to $(L,Dom(L))$.

CT1). The continuity of $t\longmapsto R^+(t)$ in the strong operator topology follows from the continuity of $t\longmapsto \sum_{k=1}^ma_k(S_k(t)-I)$ in the strong operator topology (recall (C1) for $S_k$) and the continuity of the exponent in the norm topology. So (CT1) for $R^+$ is completed.

CT2). $R^+(0)=\exp\left[i\sum_{k=1}^ma_k(S_k(0)-I)\right]=\exp\left[i\sum_{k=1}^ma_k(I-I)\right]=e^0=I.$

CT3) i) Remember that (CT1) for $S_k$ says that for every $f\in \mathcal{F}$ the function $M_f\colon [0,+\infty)\ni t\longmapsto S_k(t)f\in\mathcal{F}$ is continuous. So by the Weierstrass extreme value theorem the set $M_f([0,1])\subset \mathcal{F}$ is compact and hence bounded for each $f\in\mathcal{F}$. This means that for each $f\in\mathcal{F}$ there exists a number $C_k(f)>0$ such that $\|S_k(t)f\|\leq C_k(f)$ for all $t\in[0,1]$. Next, by the Banach-Steinhaus uniform boundedness principle the family of linear bounded operators $(S_k(t))_{t\in[0,1]}$ is bounded collectively, i.e. there exists a number $C_k>0$ such that $\|S_k(t)\|<C_k$ for all $t\in[0,1]$. Finally, for all $t\in[0,1]$ and all $k=\overline{1,m}$ we have
$$\|S_k(t)f\|\leq C=\max(C_1,\dots,C_n).$$

ii) Suppose that linear operator $A\colon \mathcal{F}\to\mathcal{F}$ is bounded. Then $e^A=I+A+A^2\frac{1}{2!}+A^3\frac{1}{3!}+\dots=I+A+ \sum_{n=0}^\infty\frac{A^n}{(n+2)!}A^2\stackrel{denote}{=}I+A+ \Psi(A)A^2.$  One can see that $$\|\Psi(A)\|=\left\|\sum_{n=0}^\infty\frac{A^n}{(n+2)!}\right\|\leq\sum_{n=0}^\infty\frac{\|A\|^n}{(n+2)!}\leq\sum_{n=0}^\infty\frac{\|A\|^n}{n!}=e^{\|A\|}.$$

iii) Set $A=i\sum_{k=1}^ma_k(S_k(t)-I)$. Then for all $t\in[0,1]$ the estimates $\|A\|=\|i\sum_{k=1}^ma_k(S_k(t)-I)\|\leq \sum_{k=1}^m|a_k|(C+1)$ and $\Psi(i\sum_{k-1}^ma_k(S_k(t)-I))\leq \exp[\sum_{k=1}^m|a_k|(C+1)]\stackrel{denote}{=}M$ hold. So for all $t\in(0,1]$ we have
\begin{equation}\label{rpluseq}\frac{R^+(t)f-f}{t}= i\sum_{k=1}^ma_k\frac{S_k(t)f-f}{t} + h(t,f)\end{equation}
where
$$\|h(t,f)\| =\left\|i^2 \Psi\left(\sum_{k=1}^ma_k(S_k(t)-I)\right) \left(\sum_{j=1}^ma_j(S_j(t)-I)\right) \sum_{k=1}^ma_k\frac{S_k(t)f-f}{t}  \right\|\leq$$
$$\leq M \sum_{j=1}^m\sum_{k=1}^m|a_j||a_k|\left\|(S_j(t)-I) \frac{S_k(t)f-f}{t} \right\|\leq$$
/(CT3) for $S_k$ says that $\frac{S_k(t)f-f}{t}=L_kf + \alpha_k(t,f),$ and $\lim\limits_{t\to 0}\|\alpha_k(t,f)\|=0$/
$$\leq M \sum_{j=1}^m\sum_{k=1}^m|a_j|\cdot|a_k|\cdot\Big(\|(S_j(t)-I)(L_kf)\| + \|(S_j(t)-I)\alpha_k(t,f))\|\Big).$$
Now let $t\to 0.$ From (CT1) and (CT2) for $S_j$ we infer $\|(S_j(t)-I)(L_kf)\|\to~0.$ Relying on the fact that $\lim\limits_{t\to 0}\|\alpha_k(t,f)\|=0$ and on the estimate $\|(S_j(t)-I)\|\leq C+1$ we infer that $\|(S_j(t)-I)\alpha_k(t,f))\|\leq \|(S_j(t)-I)\|\cdot \|\alpha_k(t,f))\|\to 0.$ So one can proceed in (\ref{rpluseq}) to the limit as $t\to0$ and obtain
$$\lim\limits_{t\to 0}\frac{R^+(t)f-f}{t}= iLf.$$

CT4) for $R^+$ is a part of the condition 3), which completes the proof of the fact that $R^+$ is Chernoff-tangent to $(L,Dom(L))$.

The (N) condition for $R^+$ with $\omega=0$ follows from the conditions 4), 5) and the corollary \ref{cor}. Indeed, the operator $S(t)-aI$ is self-adjoint so $\left\|\exp[i(S(t)-aI)]\right\|=1.$

All the conditions of the Chernoff theorem for $R^+$ are fulfilled, which implies that for all $t\geq 0$ and all $f\in\mathcal{F}$ the following formula holds: $$e^{itL}f=\lim_{n\to\infty}(R^+(t/n))^nf.$$ To prove (\ref{supadupaF}) substitute $t$ by $-t$, $a_k$ by $-a_k$ and apply the generation theorem for the groups from \cite{EN1} at p.79.

Now let me prove (\ref{supadupaF1}) and (\ref{supadupaF2}). Formula (\ref{supadupaF}) says that $e^{itL}=\lim_{n\to\infty}\exp\big[in\sum_{k=1}^ma_k(S_k(|t/n|)-I)\, \mathrm{sign}(t)\big].$ To pass from this formula to (\ref{supadupaF1}) and (\ref{supadupaF2}) let me first calculate $\big[\sum_{k=1}^ma_k(S_k(|t/n|)-I)\big]^p$ for every $p\in\mathbb{N}$. Denote $a=a_1+\dots+a_m,$ $a_{m+1}=-a$ and  $S_{m+1}(t)=I$ for all $t\geq 0$ to get more regular form of the expression studied. Then $$\sum_{k=1}^ma_k(S_k(|t/n|)-I)=\sum_{k=1}^{m+1}a_kS_k(|t/n|)$$
and
\begin{equation}\label{helpful}\left(\sum_{k=1}^{m+1}a_kS_k(|t/n|)\right)^p=\sum_{k_1=1}^{m+1}\dots\sum_{k_p=1}^{m+1} a_{k_1}\dots a_{k_p} S_{k_1}(|t|/n)\circ \dots \circ S_{k_p}(|t|/n).
\end{equation}

Now set $A=in\sum_{k=1}^ma_k(S_k(|t/n|)-I)\, \mathrm{sign}(t)$ in formula $e^{A}=\sum_{p=1}^\infty A^p/p!$ and use (\ref{helpful}), which leads to (\ref{supadupaF1}).

Formula (\ref{supadupaF2}) is inferred from (\ref{supadupaF}) and the formula $e^{A}=\lim_{p\to\infty}(I+A/p)^p$ for the same operator $A$. First mention that
$$\left(I+\frac{A}{p}\right)^p= \left(\frac{
in\, \mathrm{sign}(t)}{p}\right)^p \left(\frac{p}{in\, \mathrm{sign}(t)}I + \sum_{k=1}^{m+1}a_kS_k(|t|/n)  \right)^p$$
then recall that $I$ commutes with all operators and apply Newton's binomial formula
$$\left(\frac{p}{in\, \mathrm{sign}(t)}I + \sum_{k=1}^{m+1}a_kS_k(|t|/n)  \right)^p=
\sum_{d=0}^p \frac{p!}{(p-d)!d!}\left(\frac{p}{in\, \mathrm{sign}(t)}I\right)^{p-d}\left(\sum_{k=1}^{m+1}a_kS_k(|t|/n)\right)^d.$$
Now mention that $$\left(\frac{in\, \mathrm{sign}(t)}{p}\right)^p\frac{p!}{(p-d)!d!} \left(\frac{p}{in\, \mathrm{sign}(t)}\right)^{p-d}=\frac{(in\, \mathrm{sign}(t))^{d}p!}{p^dd!(p-d)!}$$
then substitute $p$ by $d$ in (\ref{helpful}) and obtain
$$\left(I+\frac{A}{p}\right)^p=  p!\,\sum_{d=0}^p \frac{(in\, \mathrm{sign}(t))^{d}}{p^dd!(p-d)!}    \left(\sum_{k=1}^{m+1}a_kS_k(|t|/n)  \right)^d \quad \Box$$


\section{Introduction and Preliminary Remarks}

\subsection{General, Type Independent Theories for PDEs}

It is widely believed that there is no general, type independent
theory concerning existence and regularity of solutions to
arbitrary nonlinear PDEs.  Indeed, the book `Lecture notes on
PDEs' by I. V. Arnold \cite{Arnold} starts with the following
remark:
\begin{eqnarray}
&&\mbox{``In contrast to ordinary differential equations, there is
no unified}\nonumber\\
&&\mbox{theory of partial differential equations.  Some equations
have their}\nonumber\\
&&\mbox{own theories, while others have no theory at all.  The
reason for}\nonumber\\
&&\mbox{this complexity is a more complicated
geometry...''}\nonumber
\end{eqnarray}
Within the confines of functional analysis, and more broadly
speaking, general topology, this was until very recently the case.
However, within the last fifteen years, two such general, type
independent theories for the solutions PDEs appeared.

In 1994 there appeared a type independent theory of existence and
regularity of solutions to nonlinear systems of PDEs based on
order completion of sets of functions, see \cite{Obergugenberger
and Rosinger}.  This theory yields the existence, and uniqueness,
of solutions to arbitrary continuous and nonlinear systems of
PDEs, with the solutions assimilated with Hausdorff continuous
functions, see \cite{AnguelovRosinger}.

Also, within the confines of functional analysis, there recently
emerged a general and type independent theory of PDEs, see
\cite{Neuberger 1} through \cite{Neuberger 4}. This theory,
developed by Neuberger, is based on approximations within Hilbert
space obtained by a generalized method of Steepest Descent.  This
method has yielded exceptional numerical results.

Both methods are in fact able to solve equations that are far more
general than PDEs.  This is precisely the feature which makes them
so powerful and type independent when applied to the particular
case of PDEs.

\subsection{A Brief Overview of the Order Completion Method}

Let us recall the general construction for solving a PDE of the
form
\begin{equation}\label{PDE1}
T\left(x,D\right)u\left(x\right)=f\left(x\right)\mbox{,
}x\in\Omega
\end{equation}
 by topological methods.  Here $\Omega\subseteq\mathbb{R}^{n}$ is supposed to be nonempty
and open.  The right hand term $f$ is assumed continuous, and the
unknown function satisfies, initially at least,
$u\in\mathcal{C}^{m}\left(\Omega\right)$.  The operator
$T\left(x,D\right)$ is supposed to be defined by some continuous
function $F:\Omega\times\mathbb{R}^{M}\rightarrow \mathbb{R}$
through
\begin{equation}\label{PDE2}
T\left(x,D\right)u\left(x\right)=F\left(x,u\left(x\right),...,D^{\alpha}u\left(x\right),...\right)\mbox{,
}x\in\Omega\mbox{ and }|\alpha|\leq m
\end{equation}
The general construction for finding generalized solutions to
(\ref{PDE1}) through (\ref{PDE2}), within the context of topology,
and specifically functional analysis, is summarized as follows:
\begin{eqnarray}
&\mbox{(1)  }&\mbox{Start with a PDE-operator $T$ mapping some
initial space of clas-}\nonumber\\
&&\mbox{sical functions $X$ into some space of functions $Y$ ,
with the right-}\nonumber\\
&&\mbox{hand term $f\in Y$.}\nonumber\\
&\mbox{(2)  }&\mbox{Define some structure, e.g. a uniformity, on
$Y$.}\nonumber\\
&\mbox{(3)  }&\mbox{Define a structure of the same sort on $X$
through pullback of the}\nonumber\\
&&\mbox{operator $T$ to obtain a space $X_{T}$ so that $T$ is
compatible with}\nonumber\\
&&\mbox{the customary structures on $X$ and $Y$.}\nonumber\\
&\mbox{(4)  }&\mbox{Construct the completions of $X_{T}$ and $Y$,
say
$X_{T}^{\sharp}$ and $Y^{\sharp}$, and extend}\nonumber\\
&&\mbox{$T$ to a mapping $T^{\sharp}:X_{T}^{\sharp}\rightarrow
Y^{\sharp}$}\nonumber\\
&\mbox{(5)  }&\mbox{If $f\in
T^{\sharp}\left(X_{T}^{\sharp}\right)$, one obtains a generalized
solution to the equation}\nonumber
\end{eqnarray}

The order completion method operates in a similar fashion, see
\cite{Obergugenberger and Rosinger}.  It is based on the
fundamental approximation result that, under a condition which is
necessary for the existence of a classical solution at
$x\in\Omega$, every continuous righthand term $f$ in (\ref{PDE1})
can be approximated from below (or from above) by functions in
$\mathcal{C}_{nd}^{m}\left(\Omega\right)$ where
\begin{equation}\label{CNDDef}
f:\Omega\rightarrow\mathbb{R}\in\mathcal{C}_{nd}^{m}\left(\Omega\right)\Leftrightarrow\left(\begin{array}{l}
  \exists\mbox{ }\Gamma_{f}\subset\Omega\mbox{ closed nowhere dense :} \\
  f\in\mathcal{C}^{m}\left(\Omega\setminus\Gamma_{f}\right) \\
\end{array}\right)
\end{equation}
In this regard, for any $f\in\mathcal{C}^{0}\left(\Omega\right)$
we have
\begin{eqnarray}\label{Approx1}
&&\forall\mbox{ }\epsilon
> 0\mbox{ :}\nonumber\\
&&\exists\mbox{ }\Gamma_{\epsilon}\subset\Omega\mbox{ closed
nowhere
dense, }U_{\epsilon}\in\mathcal{C}^{m}\left(\Omega\setminus\Gamma_{\epsilon}\right)\mbox{ :}\label{OCApprox}\\
&& f\left(x\right)-\epsilon\leq
T\left(x,D\right)U_{\epsilon}\left(x\right)\leq
f\left(x\right),x\in\Omega\setminus\Gamma_{\epsilon}\nonumber
\end{eqnarray}
Essentially, what this means is that the image of
$\mathcal{C}_{nd}^{m}\left(\Omega\right)$ under
$T\left(x,D\right)$ is order dense in
$\mathcal{C}_{nd}^{0}\left(\Omega\right)$.

One now proceeds as follows:  On the set
$\mathcal{C}^{0}_{nd}\left(\Omega\right)$ one considers the
equivalence relation
\begin{equation}\label{CNDEq}
f\sim g\Leftrightarrow\left(\begin{array}{l}
  \exists\mbox{ }\Gamma_{f,g}\subset\Omega\mbox{ closed, nowhere
dense}\mbox{ :} \\
  x\in\Omega\setminus\Gamma_{f,g}\Rightarrow f\left(x\right)=g\left(x\right) \\
\end{array}\right)
\end{equation}
On the quotient space
$\mathcal{M}^{0}\left(\Omega\right)=\mathcal{C}^{0}_{nd}\left(\Omega\right)/\sim$,
one introduces the partial order
\begin{equation}\label{CNDOrder}
F\leq G\Leftrightarrow\left(%
\begin{array}{l}
   \exists\mbox{ }f\in F\mbox{, }g\in G\mbox{, }\Gamma\subset\Omega\mbox{, closed, nowhere dense}\mbox{ :} \\
   (1)\mbox{  }f,g\in\mathcal{C}^{0}\left(\Omega\setminus\Gamma\right) \\
   (2)\mbox{  }f\leq g\mbox{ on }\Omega\setminus\Gamma \\
\end{array}%
\right)
\end{equation}
By the continuity of the operator $T\left(x,D\right)$ it follows
that if $u\in\mathcal{C}^{m}_{nd}\left(\Omega\right)$, and
$u\in\mathcal{C}^{m}\left(\Omega\setminus\Gamma\right)$, then
$T\left(x,D\right)u\in\mathcal{C}^{0}\left(\Omega\setminus\Gamma\right)$
so that one has the well defined mapping
\begin{equation}\label{ExtPDE1}
T\left(x,D\right):\mathcal{C}^{m}_{nd}\left(\Omega\right)\rightarrow\mathcal{C}^{0}_{nd}\left(\Omega\right)
\end{equation}

On the set $\mathcal{C}^{m}_{nd}\left(\Omega\right)$ one considers
the equivalence relation induced by the PDE operator
$T\left(x,D\right)$
\begin{eqnarray}\nonumber
u\sim_{T}v\Leftrightarrow T\left(x,D\right)u\sim
T\left(x,D\right)v.\nonumber
\end{eqnarray}
With the mapping (\ref{ExtPDE1}) one can now associate in a
canonical way an injective mapping
\begin{eqnarray}\nonumber
T:\mathcal{M}^{m}_{T}\left(\Omega\right)\rightarrow\mathcal{M}^{0}\left(\Omega\right),\nonumber
\end{eqnarray}
where
$\mathcal{M}^{m}_{T}\left(\Omega\right)=\mathcal{C}^{m}_{nd}\left(\Omega\right)/\sim_{T}$.
 Next one introduces a partial order on
$\mathcal{M}^{m}_{T}\left(\Omega\right)$ by
\begin{equation}\label{PulbackOrder}
U\leq_{T}V\Leftrightarrow TU\leq TV\mbox{ in
}\mathcal{M}^{0}\left(\Omega\right).
\end{equation}
This construction results in an order isomorphic embedding
\begin{eqnarray}\nonumber
T:\mathcal{M}^{m}_{T}\left(\Omega\right)\hookrightarrow
\mathcal{M}^{0}\left(\Omega\right).\nonumber
\end{eqnarray}
By extension of $T$ to the Dedekind completion
$\mathcal{M}_{T}^{m}\left(\Omega\right)^{\sharp}$ one obtains the
following commutative diagram

\begin{math}
\setlength{\unitlength}{1cm} \thicklines
\begin{picture}(13,6)
\put(2.4,5){$\mathcal{M}_{T}^{m}\left(\Omega\right)$}
\put(3.8,5.1){\vector(1,0){6.5}}
\put(10.5,5){$\mathcal{M}^{0}\left(\Omega\right)$}
\put(6.6,5.4){$T$} \put(3.0,4.8){\vector(0,-1){3.3}}
\put(2.4,1){$\mathcal{M}_{T}^{m}\left(\Omega\right)^{\sharp}$}
\put(4.0,1.1){\vector(1,0){6.2}} \put(6.6,1.3){$T^{\sharp}$}
\put(10.5,1){$\mathcal{M}^{0}\left(\Omega\right)^{\sharp}$}
\put(11.2,4.8){\vector(0,-1){3.3}}
\end{picture}
\end{math}
\\
\\
with $T^{\sharp}$ an order isomorphism.  Hence, for every
$f\in\mathcal{C}^{0}\left(\Omega\right)$, the generalized equation
\begin{eqnarray}
T^{\sharp}U=f\nonumber
\end{eqnarray}
has a unique solution in
$\mathcal{M}^{m}_{T}\left(\Omega\right)^{\sharp}$.

\subsection{Extended Concepts of Topology}

The aim of this paper is to cast the Order Completion Method of
Oberguggenberger and Rosinger within the framework of the
traditional topological and function analytical method.  In order
to achieve this aim it is necessary to consider topological type
structures that are more general than the usual
Hausdorff-Kuratowski-Bourbaki concept of topology.  Note that this
need for generalized concepts of topology is not unique to the
problem considered here, but appears frequently in analysis- even
in the relatively simple setting of locally convex linear spaces.
Indeed, recall that there is no topology on the topological dual
$E^{*}$ of a locally convex space $E$ such that the simple
evaluation mapping
\begin{eqnarray}
ev:E\times E^{*}\rightarrow\mathbb{R}\nonumber
\end{eqnarray}
defined through
\begin{eqnarray}
ev:\left(x,x^{*}\right)\mapsto\langle x^{*},x\rangle\nonumber
\end{eqnarray}
is jointly continuous, unless $E$ is a normable space.  Other
examples include some natural notions of convergence in measure
theory and in the theory of ordered spaces, which can not be
induced by a topology. For a more detailed motivation of the need
for generalized concepts of topology we refer the reader to
\cite{Beattie} and \cite{RosingervdWalt}.

The most general such topological type structures so far
considered in the literature was introduced by Rosinger, see
\cite{Rosinger2} through \cite{Rosinger4}, and more recently
\cite{RosingervdWalt}. A highly particular case of these
topological type structures are the celebrated convergence
structures and convergence spaces, see \cite{Beattie}, which are
defined in terms of convergent filters.  The collection of
convergence structures on a given set $X$ contains the collection
of all topologies on $X$ so that Hausdorff-Kuratowsky-Bourbaki
topology is a special case of convergence structures.

The theory of uniform spaces is a particular but highly important
case of general topology, among others due to the issue of
completeness and completion.  The generalization of uniform spaces
and uniformities within the setting of convergence spaces are
uniform convergence structures and uniform convergence spaces.
Whereas the uniform spaces are rather particular amongst
topological spaces, almost all convergence spaces are uniform
convergence space. Indeed, every Hausdorff convergence space is a
uniform convergence space, and there are even some uniform
convergence spaces that are not Hausdorff!  As is the case for
uniform spaces, every Hausdorff uniform convergence space admits a
Hausdorff completion which is unique up to homeomorphism.  For
details on uniform convergence spaces we refer the reader to
\cite{Beattie}, \cite{Gahler 1}, \cite{Gahler 2} or Appendix A.

\subsection{Order Convergence and the Order Convergence Structure}

As mentioned, there are many natural and useful modes of
convergence that can be defined in terms of a partial order $\leq$
on a set $X$, see for instance \cite{Birkhoff}, \cite{RieszI} and
\cite{Peressini}.  One of the better known such modes of
convergence is order convergence of sequences, which can be
defined on an arbitrary poset $X$ through
\begin{eqnarray}
\left(x_{n}\right)\mbox{ order converges to }x\in X
\Leftrightarrow\left(\begin{array}{l}
  \exists\mbox{ }\left(\lambda_{n}\right)\mbox{,
  }\left(\mu_{n}\right)\subset X\mbox{ :}\\
  \left(\lambda_{n}\right)\mbox{ increases to }x\mbox{ :} \\
  \left(\mu_{n}\right)\mbox{ decreases to }x\mbox{ :} \\
  n\in\mathbb{N}\Rightarrow \lambda_{n}\leq x_{n} \leq\mu_{n} \\
\end{array}\right)\label{OrderConvDef}
\end{eqnarray}
It is well know that order convergence of sequences can, in
general, not be induced by a topology, that is, for an arbitrary
poset $X$ there is no topology $\tau$ on $X$ such that the
convergent sequences with respect to $\tau$ are exactly those
sequences that order converge, see for instance \cite{AngvdWalt}.
However, recently it was shown that for a $\sigma$-distributive
lattice $X$ there is a convergence structure $\lambda$ on $X$ that
induces order convergence of sequences, see \cite{AngvdWalt} and
\cite{vdWalt1}.  One such a convergence structure, the Order
Convergence Structure $\lambda_{o}$, was defined as
\begin{eqnarray}
\mathcal{F}\in\lambda_{o}\left(x\right)\Leftrightarrow
\left(\begin{array}{l}
  \exists\mbox{ }\left(\lambda_{n}\right)\mbox{, } \left(\mu_{n}\right)\mbox{ on }X\mbox{ :}\\
  \left(\lambda_{n}\right)\mbox{ increases to }x\mbox{ :} \\
  \left(\mu_{n}\right)\mbox{ decreases to }x\mbox{ :} \\
  \{[\lambda_{n},\mu_{n}]\mbox{ : }n\in\mathbb{N}\} \subseteq\mathcal{F} \\
\end{array}\right)\label{OConStrucDef}
\end{eqnarray}
Note that this result applies amongst others to all vector
lattices.  Moreover, if a vector lattice $X$ is also Archimedean,
then the Order Convergence Structure is a vector space convergence
structure in the sense of \cite{Beattie}.  An important result
obtained in \cite{vdWalt1} is that the completion of the uniform
convergence structure induced on $X$ by $\lambda_{o}$ and the
linear structure is the Dedekind $\sigma$-completion of $X$
equipped with the Order Convergence Structure.  In case $X$ is
order separable \cite{RieszI}, the completion is in fact the
Dedekind completion of $X$.

Since the space $\mathcal{M}^{0}\left(\Omega\right)$ that appears
in the Order Completion Method can be viewed as an order separable
Archimedean vector lattice, the completion result described above
appears to provide a way of casting the Order Completion Method in
the framework of uniform convergence spaces.  However, one of the
main advantages of the Order Completion Method is that it makes no
distinction between linear and nonlinear equations, as is usually
the case for topological methods.  The completion result for the
Order Convergence Structure on an Archimedean vector lattice, on
the other hand, is a distinctly linear result.   However, in this
regard it was recently shown \cite{vdWalt2} that the Order
Convergence Structure on the order isomorphic copy
$\mathcal{ML}^{0}\left(\Omega\right)$ of
$\mathcal{M}^{0}\left(\Omega\right)$ is induced by a nontrivial
uniform convergence structure, and the resulting completion is
again the Dedekind completion $\mathbb{H}_{nf}\left(\Omega\right)$
of $\mathcal{ML}^{0}\left(\Omega\right)$.  Here
$\mathbb{H}_{nf}\left(\Omega\right)$ denotes the set of all nearly
finite Hausdorff continuous functions, \cite{Anguelov} and
\cite{Sendov}.

\section{Spaces of Functions}

This section is devoted to a short introduction of the spaces of
functions, and spaces of generalized functions on which the PDEs,
and their extensions, are to act.  As with the Order Completion
Method, we start with the space
$\mathcal{C}^{m}_{nd}\left(\Omega\right)$.  On this space we can
consider the Lower- and Upper Baire Operators
\begin{eqnarray}\label{IDef}
I\left(f\right)\left(x\right)=\sup\{\inf\{z\in
f\left(y\right)\mbox{ : }y\in V\}\mbox{ :
}V\in\mathcal{V}_{x}\}\mbox{, }x\in X
\end{eqnarray}
\begin{eqnarray}\label{SDef}
S\left(f\right)\left(x\right)=\inf\{\sup\{z\in
f\left(y\right)\mbox{ : }y\in V\}\mbox{ :
}V\in\mathcal{V}_{x}\}\mbox{, }x\in X
\end{eqnarray}
where $\mathcal{V}_{x}=\{V\subseteq X|V\mbox{ a neighbourhood of
}x\}$ is the neighborhood filter at $x\in\Omega$.  These operators
were first introduced by Baire \cite{Baire} for real valued
functions of a real variable, and later generalized by Sendov
\cite{Sendov} and Anguelov \cite{Anguelov}.  Note that the natural
setting of these operators, within the context of point valued
functions, is the set $\mathcal{A}\left(\Omega\right)$ consisting
of all functions $f$ that act on $\Omega$ and take values in the
extended real line
$\overline{\mathbb{R}}=\mathbb{R}\cup\{\pm\infty\}$.  There the
operators $I$ and $S$ characterize semi-continuity.  Indeed,
\begin{eqnarray}\label{LSCChar}
f\in\mathcal{A}\left(\Omega\right)\mbox{ is lower semi-continuous
}\Leftrightarrow I\left(f\right)=f
\end{eqnarray}
\begin{eqnarray}\label{USCChar}
f\in\mathcal{A}\left(\Omega\right)\mbox{ is upper semi-continuous
}\Leftrightarrow S\left(f\right)=f
\end{eqnarray}
Moreover, compositions of these operators characterize the normal
semi-continuous functions, see \cite{Dilworth} and
\cite{Anguelov}:
\begin{eqnarray}\label{NLSCChar}
f\in\mathcal{A}\left(\Omega\right)\mbox{ is normal lower
semi-continuous }\Leftrightarrow I\left(S\left(f\right)\right)=f
\end{eqnarray}
\begin{eqnarray}\label{NUSCChar}
f\in\mathcal{A}\left(\Omega\right)\mbox{ is normal upper
semi-continuous }\Leftrightarrow
S\left(I\left(f\right)\right)\left(x\right)=f
\end{eqnarray}
The normal lower and upper semi-continuous functions were
introduced in \cite{Dilworth} in connection with his attempts to
find the Dedekind order completion of spaces of continuous
functions.  This problem was finally solved by Anguelov, see
\cite{Anguelov}, where he characterized the Dedekind completion of
$\mathcal{C}\left(X\right)$ and some of its subspaces as spaces of
Hausdorff continuous functions.

We now consider the space
\begin{eqnarray}
\mathcal{ML}^{m}\left(\Omega\right)=\{I\left(S\left(f\right)\right)\mbox{
: }f\in\mathcal{C}^{m}_{nd}\left(\Omega\right)\}.\label{MLDef}
\end{eqnarray}
That is, $\mathcal{ML}^{m}\left(\Omega\right)$ is the image of
$\mathcal{C}^{m}_{nd}\left(\Omega\right)$ under the operator
$I\circ S$.  Since $\left(I\circ
S\right)\left(f\right)\left(x\right)$ is equal to
$f\left(x\right)$ whenever $f$ is continuous at $x$, it is $m$
times continuously differentiable everywhere except on a closed
nowhere dense subset of $\Omega$. Hence it follows that
$\mathcal{ML}^{m}\left(\Omega\right)\subset
\mathcal{C}^{m}_{nd}\left(\Omega\right)$.  By the idempotence of
the composite operator $I\circ S$, \cite{Anguelov}, it follows by
(\ref{NLSCChar}) and (\ref{CNDDef}) that
$\mathcal{ML}^{m}\left(\Omega\right)$ consists of all normal lower
semi-continuous functions which are $m$-times continuously
differentiable everywhere except on some closed nowhere dense
subset of $\Omega$. This space was introduced and studied in
\cite{vdWalt2} and \cite{vdWalt3}, and it was shown that
$\mathcal{ML}^{m}\left(\Omega\right)$ is an order isomorphic copy
of $\mathcal{M}^{m}\left(\Omega\right)$. Recall that
$\mathcal{M}^{m}\left(\Omega\right)=
\mathcal{C}^{m}_{nd}\left(\Omega\right)/\sim$, where $\sim$ is the
equivalence relation (\ref{CNDEq}).

We now consider the family $\mathcal{J}_{o}$ of filters on the
Cartesian product $\mathcal{ML}^{0}\left(\Omega\right)
\times\mathcal{ML}^{0}\left(\Omega\right)$.
\begin{definition}
A filter $\mathcal{U}$ on
$\mathcal{ML}^{0}\left(\Omega\right)\times\mathcal{ML}^{0}\left(\Omega\right)$
belongs to the family $\mathcal{J}_{o}$ whenever there exists
$k\in\mathbb{N}$ such that
\begin{eqnarray}\label{UOC}
&&\forall\mbox{ }i=1,...,k\mbox{ :}\nonumber\\
&&\exists\mbox{ }\left(I_{n}^{i}\right)\mbox{ nonempty order
intervals :}\nonumber\\
&&1)\mbox{  }I_{n}^{i}\supseteq I_{n+1}^{i}\mbox{ for every
}n\in\mathbb{N}\nonumber\\
&&2)\mbox{  }\forall\mbox{ }V\subseteq\Omega\mbox{ open
}\cap_{n\in\mathbb{N}}I_{n|V}^{i}=\emptyset\mbox{ or
there}\nonumber\\
&&\mbox{is }f_{i}\in \mathcal{ML}^{0}\left(V\right)\mbox{ such
that }\cap_{n\in\mathbb{N}}I_{n|V}^{i}=\{f_{i}\}\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\left(\mathcal{I}^{1}\times\mathcal{I}^{1}\right)\cap...
\cap\left(\mathcal{I}^{k}\times\mathcal{I}^{k}\right)
\subset\mathcal{U}\mbox{ where }\mathcal{I}^{i}=[\{I^{i}_{n}\mbox{
: }n\in\mathbb{N}\}]\nonumber
\end{eqnarray}
\end{definition}
The family $\mathcal{J}_{o}$ is in fact a uniform convergence
structure on $\mathcal{ML}^{0}\left(\Omega\right)$, which we call
the Uniform Order Convergence Structure.  It induces the Order
Convergence Structure on $\mathcal{ML}^{0}\left(\Omega\right)$,
see \cite{vdWalt2} and \cite{vdWalt3}, and hence the following
properties are immediate:
\begin{eqnarray}
&\bullet&\mbox{A filter $\mathcal{F}$ on
$\mathcal{ML}^{0}\left(\Omega\right)$ converges with respect to
the induced convergence}\nonumber\\
&&\mbox{structure if and only if $\mathcal{F}$ converges in the
Order Convergence Structure}\nonumber\\
&&\mbox{on
$\mathcal{ML}^{0}\left(\Omega\right)$.}\label{UOCSP1}\\
&\bullet&\mbox{A sequence $\left(f_{n}\right)$ on
$\mathcal{ML}^{0}\left(\Omega\right)$ converges to
$f\in\mathcal{ML}^{0}\left(\Omega\right)$ if and only if
$\left(f_{n}\right)$}\nonumber\\
&&\mbox{order converges to $f$.}\label{UOCSP2}\\
&\bullet&\mbox{The uniform convergence space
$\mathcal{ML}^{0}\left(\Omega\right)$ is uniformly
Hausdorff.}\label{UOCSP3}
\end{eqnarray}

We now consider a system of $K$ nonlinear PDEs, each of order at
most $m$, in $K$ unknown functions $u_{1},...,u_{K}$
\begin{equation}\label{PDEDef}
\textbf{T}\left(x,D\right)\textbf{u}\left(x\right)=\textbf{f}\left(x\right)\mbox{,
}x\in\Omega\subseteq\mathbb{R}^{n}\mbox{ an open set.}
\end{equation}
Here $\textbf{u}:\Omega\rightarrow\mathbb{R}^{K}$ is a
$K$-dimensional vector valued function with components
$u_{1},...,u_{K}\in\mathcal{C}^{m}\left(\Omega\right)$.  The
righthand term $\textbf{f}:\Omega\rightarrow\mathbb{R}^{K}$ is
supposed to have continuous components
$f_{1},...,f_{K}:\Omega\rightarrow\mathbb{R}$.  The PDE-operator
$\textbf{T}\left(x,D\right)$ is defined through
\begin{equation}\label{PDEOperator}
\textbf{T}\left(x,D\right):\textbf{u}\left(x\right)\mapsto\textbf{F}\left(x,
...,u_{i}\left(x\right),...,D^{\alpha}u_{i},...\right)\mbox{ with
}i=1,...,K\mbox{ ,}|\alpha|\leq m
\end{equation}
where
$\textbf{F}:\Omega\times\mathbb{R}^{M}\rightarrow\mathbb{R}^{K}$
is a jointly continuous function in all of its arguments with
components
$F_{1},...,F_{K}:\Omega\times\mathbb{R}^{M}\rightarrow\mathbb{R}$.
One can now write equation (\ref{PDEDef}) in the form
\begin{eqnarray}\nonumber
\begin{array}{ccc}
  T_{1}\left(x,D\right)\textbf{u}\left(x\right) & = & f_{1}\left(x\right) \\
  \vdots & \vdots & \vdots \\
  T_{K}\left(x,D\right)\textbf{u}\left(x\right) & = & f_{k}\left(x\right) \\
\end{array}
\mbox{, }x\in\Omega.\nonumber
\end{eqnarray}
where each $T_{i}\left(x,D\right)$ is defined through
\begin{eqnarray}\nonumber
T_{i}\left(x,D\right):\textbf{u}\left(x\right)\mapsto
F_{i}\left(x,...,u_{j}\left(x\right),...,D^{\alpha}u_{j},...\right)\mbox{
with }j=1,...,K\nonumber
\end{eqnarray}

With the operator $T\left(x,D\right)$ we associate a mapping
\begin{eqnarray}
\widetilde{\textbf{T}}:\mathcal{ML}^{m}\left(\Omega\right)^{K}\rightarrow
\mathcal{ML}^{0}\left(\Omega\right)^{K}\nonumber
\end{eqnarray}
with components
\begin{eqnarray}
\widetilde{T}_{i}:\mathcal{ML}^{m}\left(\Omega\right)\rightarrow
\mathcal{ML}^{0}\left(\Omega\right)\nonumber
\end{eqnarray}
defined through
\begin{eqnarray}
\widetilde{T}_{i}\left(\textbf{u}\right):x\rightarrow \left(I\circ
S\right)\left(F\left(\cdot,u_{1},...,u_{K},...,\mathcal{D}^
{\alpha}u_{i},...\right)\right)\left(x\right)\mbox{,
}x\in\Omega\nonumber
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{D}^{\alpha}f=\left(I\circ
S\right)\left(D^{\alpha}f\right)\mbox{,
}f\in\mathcal{C}^{m}_{nd}\left(\Omega\right).\nonumber
\end{eqnarray}
Since each $\mathcal{D}^{\alpha}u_{i}$ is continuous on an open
and dense set of $\Omega$, it follows by the continuity of the
$F_{i}$ that $\widetilde{\textbf{T}}\textbf{u}\in
\mathcal{ML}^{0}\left(\Omega\right)^{K}$ for every $\textbf{u}\in
\mathcal{ML}^{m}\left(\Omega\right)^{K}$.  Moreover, since
(\ref{LSCChar}) and (\ref{USCChar}) imply that a function
continuous at a point $x\in\Omega$ is invariant under the
operators $I$ and $S$, at $x$, it follows that
\begin{eqnarray}
&&\forall\mbox{
}\textbf{u}\in\mathcal{ML}^{m}\left(\Omega\right)\mbox{
:}\nonumber\\
&&\exists\mbox{ }\Gamma\subset\Omega\mbox{ :}\label{TExtTRel}\\
&&\Gamma\mbox{ is closed nowhere dense :}\nonumber\\
&&x\in\Omega\setminus\Gamma\Rightarrow
\widetilde{\textbf{T}}\left(\textbf{u}\right)\left(x\right)=
\textbf{T}\left(x,D\right)\textbf{u}\left(x\right)\nonumber
\end{eqnarray}
On the space $\mathcal{ML}^{m}\left(\Omega\right)^{K}$ consider
the equivalence relation induced by $\widetilde{\textbf{T}}$
through
\begin{equation}\label{TEquiv}
\textbf{u}\sim_{\widetilde{\textbf{T}}}\textbf{v}\Leftrightarrow
\widetilde{\textbf{T}}\textbf{u}=\widetilde{\textbf{T}}\textbf{v}
\end{equation}
We denote the quotient space
$\mathcal{ML}^{m}\left(\Omega\right)^{K}/\sim_{\widetilde{\textbf{T}}}$
by $\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$.
With the mapping $\widetilde{\textbf{T}}$ we now associate a
mapping
\begin{equation}\label{PulbackOp}
\widehat{\textbf{T}}:\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)
\rightarrow \mathcal{ML}^{0}\left(\Omega\right)^{K}
\end{equation}
which assigns to each equivalence class in
$\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$ its
image under the operator $\widetilde{\textbf{T}}$, that is,
\begin{eqnarray}
&&\forall\mbox{ }\textbf{U}\in\mathcal{ML}^{m}
_{\widetilde{\textbf{T}}}\left(\Omega\right)\mbox{ :}\nonumber\\
&&\forall\mbox{ }\textbf{u}\in\textbf{U}\mbox{ :}\label{TExtTRel2}\\
&&\widehat{\textbf{T}}\left(\textbf{U}\right)=\widetilde{\textbf{T}}\left(\textbf{u}\right)\nonumber
\end{eqnarray}
The generalized version of equation (\ref{PDEDef}) is
\begin{equation}\label{GenPDE}
\widehat{\textbf{T}}\textbf{U}=\textbf{f},
\end{equation}
where $\textbf{U}$ is an equivalence class under the equivalence
$\sim_{\widetilde{\textbf{T}}}$.  The mapping
$\widehat{\textbf{T}}$ is injective, but it is in general not
surjective.

We consider the space $\mathcal{ML}^{0}\left(\Omega\right)^{K}$
equipped with the product uniform convergence structure when each
copy of $\mathcal{ML}^{0}\left(\Omega\right)$ carries the Uniform
Order Convergence Structure.
\begin{definition}\label{ProdUCSML0}
A filter $\mathcal{U}$ on
$\mathcal{ML}^{0}\left(\Omega\right)^{K}\times
\mathcal{ML}^{0}\left(\Omega\right)^{K}$ belongs to
$\mathcal{J}^{K}_{o}$ whenever $\left(\pi_{i}\times\pi_{i}\right)
\left(\mathcal{U}\right)\in\mathcal{J}_{o}$ for every $i=1,...,K$.
Here $\pi_{i}:\mathcal{ML}^{0}\left(\Omega\right)^{K} \rightarrow
\mathcal{ML}^{0}\left(\Omega\right)$ denotes the projection along
the $i$th coordinate.
\end{definition}
$\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$
carries the initial uniform convergence structure
$\mathcal{J}_{\widetilde{\textbf{T}}}$ with respect to the
injective mapping
\begin{eqnarray}\label{InjMap}
\widehat{\textbf{T}}:\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)
\rightarrow \mathcal{ML}^{0}\left(\Omega\right)^{K}
\end{eqnarray}
\begin{definition}\label{InitCSMLmT}
A filter $\mathcal{U}$ on $\mathcal{ML}^{m}_
{\widetilde{\textbf{T}}}\left(\Omega\right)\times
\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$
belongs to $\mathcal{J}_{\widetilde{\textbf{T}}}$ whenever
$\left(\widetilde{\textbf{T}}\times
\widetilde{\textbf{T}}\right)\left(\mathcal{U}\right)$ belongs to
$\mathcal{J}^{K}_{o}$.
\end{definition}
From the injectivity of $\widehat{\textbf{T}}$  we obtain
\begin{eqnarray}
&&\widehat{\textbf{T}}:
\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)
\hookrightarrow \mathcal{ML}^{0}\left(\Omega\right)^{K}\mbox{ is a
uniformly continuous embedding}\label{TEmbedding}
\end{eqnarray}

The completion of $\mathcal{ML}^{0}\left(\Omega\right)$ can be
constructed as a set of Hausdorff continuous functions.  In fact,
it is the set $\mathbb{H}_{nf}\left(\Omega\right)$ consisting of
all nearly finite Hausdorff continuous functions, equipped with a
suitably chosen uniform convergence structure, see \cite{vdWalt3}
for details.  Hence it follows by Theorem \ref{ProdCom} that the
completion of $\mathcal{ML}^{0}\left(\Omega\right)^{K}$ is
$\mathbb{H}_{nf}\left(\Omega\right)^{K}$, equipped with the
product uniform convergence structure.  Moreover, it now follows
by (\ref{TEmbedding}) and Theorem \ref{SubspUCSComp} that the
completion of $\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}
\left(\Omega\right)$ is homeomorphic to a subspace
$\mathbb{H}_{\widetilde{\textbf{T}}}\left(\Omega\right)$ of
$\mathbb{H}_{nf}\left(\Omega\right)$.  We will not give details
concerning Hausdorff continuous functions.  For details concerning
Hausdorff continuous functions, we refer the reader to
\cite{Anguelov}.

\section{Approximation Results}

We now again consider a system of $K$ nonlinear PDEs of the form
(\ref{PDEDef}) through (\ref{PDEOperator}). Recall that the Order
Completion Method for single nonlinear PDEs of the form
(\ref{PDE1}) through (\ref{PDE2}) is based on the simple
approximation result (\ref{Approx1}).  This result is follows from
the fact that every continuous function can be approximated
locally by a sequence of polynomials.  In this section we extend
this result to the general $K$-dimensional case, for $K\geq 1$
arbitrary but given.

A natural assumption on the function
$\textbf{F}:\Omega\times\mathbb{R}^{M}\rightarrow\mathbb{R}^{K}$,
and hence the PDE-operator $\textbf{T}\left(x,D\right)$, and the
righthand term $\textbf{f}$ is that, for every $x\in\Omega$
\begin{equation}\label{Assumption}
\textbf{f}\left(x\right)\in\textrm{int}R_{x}\mbox{ where
}R_{x}=\{\textbf{F}\left(x,\xi_{1\alpha},...,\xi_{i\alpha},...\right)\mbox{
: }\xi_{i\alpha}\in\mathbb{R}\mbox{, }i=1,...,K\mbox{,
}|\alpha|\leq m\}
\end{equation}
Note that (\ref{Assumption}) is is of a technical nature, and
hardly a restriction on the class of PDEs considered.  In fact,
every linear PDE, and also most nonlinear PDEs, satisfy
(\ref{Assumption}).  It is necessary condition for the existence
of a classical solution to (\ref{PDEDef}) in a neighborhood of
$x$.
\begin{theorem}\label{Approx}
Consider a system of PDEs of the form (\ref{PDEDef}) through
(\ref{PDEOperator}) that also satisfies (\ref{Assumption}).  For
every $\epsilon>0$ there exists a closed nowhere dense set
$\Gamma_{\epsilon}\subset\Omega$ with zero Lebesgue measure, and a
function
$\textbf{U}_{\epsilon}\in\mathcal{C}^{m}\left(\Omega\setminus\Gamma_{\epsilon}\right)^{K}$
with components $U_{\epsilon,1},...,U_{\epsilon,K}$ such that
\begin{equation}\label{ApEq}
f_{i}\left(x\right)-\epsilon\leq
T_{i}\left(x,D\right)\textbf{U}_{\epsilon}\left(x\right)\leq
f_{i}\left(x\right)\mbox{, }x\in\Omega\setminus\Gamma_{\epsilon}
\end{equation}
\end{theorem}
\begin{proof}\\
Let
\begin{equation}\label{AE3}
\Omega=\bigcup_{\nu\in\mathbb{N}}C_{\nu}
\end{equation}
where, for $\nu\in\mathbb{N}$, the compact sets $C_{\nu}$ are
$n$-dimensional intervals
\begin{equation}
C_{\nu}=[a_{\nu},b_{\nu}]
\end{equation}
with $a_{\nu}=\left(a_{\nu,1},...,a_{\nu,n}\right)$,
$b_{\nu}=\left(b_{\nu,1},...,b_{\nu,n}\right)\in\mathbb{R}^{n}$
and $a_{\nu,i}\leq b_{\nu,i}$ for every $i=1,...,n$.We also assume
that $C_{\nu}$, with $\nu\in\mathbb{N}$ are locally finite, that
is,
\begin{equation}\label{AE2}
\forall x\in\Omega\exists V_{x}\subseteq\Omega\mbox{ a
neighborhood of }x\mbox{ : }\{\nu\in\mathbb{N}\mbox{ :
}C_{\nu}\cap V_{x}\neq\emptyset\}\mbox{ is finite}
\end{equation}
We also assume that the interiors of $C_{\nu}$, with
$\nu\in\mathbb{N}$, are pairwise disjoint.  We note that note that
such $C_{\nu}$ exist, see \cite{Forster}.\\
Let us now take $\epsilon>0$ given arbitrary but fixed.  Let us
take $\nu\in\mathbb{N}$ and apply Lemma \ref{ApproxLemma} to each
$x_{0}\in C_{\nu}$.  Then we obtain $\delta_{x_{0}}>0$ and
$P_{x_{0},1},...,P_{x_{0},K}$ polynomial in $x\in\mathbb{R}^{n}$
such that
\begin{equation}
f_{i}\left(x\right)-\epsilon\leq
T_{i}\left(x,D\right)\textbf{P}_{x_{0}}\left(x\right)\leq
f\left(x\right)\mbox{,
}x\in\Omega\cap\overline{B}\left(x_{0},\delta_{x_{0}}\right)\mbox{
and }i=1,...,K
\end{equation}
where $\textbf{P}_{x_{0}}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{K}$
is the $K$-dimensional vector valued function with components
$P_{x_{0},1},...,P_{x_{0},K}$.  Since $C_{\nu}$ is compact, it
follows that
\begin{eqnarray}
&&\exists\delta>0\mbox{ : }\nonumber\\
&&\forall x_{0}\in C_{\nu}\mbox{ : }\nonumber\\
&&\exists P_{x_{0},1},...,P_{x_{0},K}\mbox{ polynomial in }x\in\mathbb{R}^{n}\mbox{ : }\label{AE}\\
&&\|x-x_{0}\|\leq\delta\Rightarrow
f_{i}\left(x\right)-\epsilon\leq
T_{i}\left(x,D\right)\textbf{P}_{x_{0}}\left(x\right)\leq
f\left(x\right)\mbox{,
}x\in\overline{B}\left(x_{0},\delta\right)\cap \Omega\nonumber
\end{eqnarray}
where $i=1,...,K$.  Subdivide $C_{\nu}$ into $n$-dimensional
intervals $I_{\nu,1},...,I_{\nu,\mu}$ with diameter not exceeding
$\delta$ such that their interiors are pairwise disjoint.  If
$a_{j}$ with $j=1,...,\mu$ is the center of the interval
$I_{\nu,j}$ then by (\ref{AE}) there exists
$P_{a_{j},1},...,P_{a_{j}},K$ polynomial in $x\in\mathbb{R}^{n}$
such that
\begin{equation}\label{AE1}
f_{i}\left(x\right)-\epsilon\leq
T_{i}\left(x,D\right)\textbf{P}_{a_{j}}\left(x\right)\leq
f_{i}\left(x\right)\mbox{, }x\in I_{\nu,j}
\end{equation}
where $i=1,...,K$.  Now set
\begin{equation}
\Gamma_{\nu,\epsilon}=C_{\nu}\setminus\left(\left(\bigcup_{j=1}
^{\mu}\textrm{int}I_{\nu,j}\right)\cup\textrm{int}C_{\nu}\right)
\end{equation}
that is, $\Gamma_{\nu,\epsilon}$ is a rectangular grid generated
as a finite union of hyperplanes.  Furthermore, using (\ref{AE1}),
we find
\begin{equation}\label{AE4}
\textbf{U}_{\nu,\epsilon}\in\mathcal{C}^{m}\left(C_{\nu}\setminus\Gamma_{\nu,\epsilon}\right)
\end{equation}
such that
\begin{equation}\label{AE5}
f_{i}\left(x\right)-\epsilon\leq
T_{i}\left(x,D\right)\textbf{U}_{\nu,\epsilon}\left(x\right)\leq
f_{i}\left(x\right)\mbox{, }x\in
C_{\nu}\setminus\Gamma_{\nu,\epsilon}
\end{equation}
In view of (\ref{AE2}) it follows that
\begin{equation}
\Gamma_{\epsilon}=\bigcup_{\nu\mathbb{N}}\Gamma_{\nu,\epsilon}\mbox{
is closed nowhere dense and
}\textrm{mes}\left(\Gamma_{\epsilon}\right)=0
\end{equation}
From (\ref{AE3}), (\ref{AE4}) and (\ref{AE5}) we obtain
(\ref{ApEq}).
\end{proof}\\
The above proof relies on the following lemma which is in fact the
basic approximation result.
\begin{lemma}\label{ApproxLemma}
Consider a system of PDEs of the form (\ref{PDEDef}) through
(\ref{PDEOperator}) that also satisfies (\ref{Assumption}).  Then
\begin{eqnarray}
&&\forall\mbox{ }x_{0}\in\Omega\mbox{, }\epsilon>0\mbox{
:}\nonumber\\
&&\exists\mbox{ }\delta>0\mbox{, }P_{1},...,P_{K}\mbox{ polynomial
in $x\in\mathbb{R}^{n}$ :}\label{ALemEq}\\
&&x\in B\left(x_{0},\delta\right)\cap\Omega\mbox{, }1\leq i\leq k
\Rightarrow f_{i}\left(x\right)-\epsilon\leq
T_{i}\left(x,D\right)\textbf{P}\left(x\right)\leq
f_{i}\left(x\right)\nonumber
\end{eqnarray}
Here $\textbf{P}$ is the $K$-dimensional vector valued function
with components $P_{1},...,P_{K}$.
\end{lemma}
\begin{proof}\\
For any $x_{0}\in\Omega$ and $\epsilon>0$ small enough it follows
by (\ref{Assumption}) that there exist
\begin{equation}
\xi_{i\alpha}\in\mathbb{R}\mbox{ with }i=1,...,K\mbox{ and
}|\alpha|\leq m
\end{equation}
such that
\begin{equation}
F_{i}\left(x_{0},...,\xi_{i\alpha},...\right)=f_{i}\left(x_{0}\right)-\frac{\epsilon}{2}
\end{equation}
Now take $P_{1},...,P_{K}$ polynomials in $x\in\mathbb{R}^{n}$
that satisfy
\begin{equation}
D^{\alpha}P_{i}\left(x_{0}\right)=\xi_{i\alpha}\mbox{ for
}i=1,...,K\mbox{ and }|\alpha|\leq m
\end{equation}
Then it is clear that
\begin{equation}
T_{i}\left(x,D\right)\textbf{P}\left(x_{0}\right)-f_{i}\left(x_{0}\right)=-\frac{\epsilon}{2}
\end{equation}
where $\textbf{P}$ is the $K$-dimensional vector valued function
on $\mathbb{R}^{n}$ with components $P_{1},...,P_{K}$.  Hence
(\ref{ALemEq}) follows by the continuity of the $f_{i}$ and the
$F_{i}$.
\end{proof}\\

\section{Existence of Generalized Solutions}

We now proceed to establish both the existence and uniqueness of
generalized solutions to the generalized equation (\ref{GenPDE}).
This result appears as an application of the theory developed in
Sections 2 and 3, and the results concerning completions of
uniform convergence spaces contained in Appendix A.

The uniqueness of the solution should not be interpreted as
implying that any meaningful solutions are disregarded.  Indeed,
recall that the elements of the space
$\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$ are
equivalence classes under the equivalence (\ref{TEquiv}).  Hence,
if there is, for instance, a solution $\textbf{U}\in
\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$,
then $\textbf{U}$ is the equivalence class containing all possible
solutions $\textbf{u}\in\mathcal{ML}^{m}\left(\Omega\right)^{K}$
to (\ref{PDEDef}).

Consider now the following commutative diagram:\\
\\
\\
\begin{math}
\setlength{\unitlength}{1cm} \thicklines
\begin{picture}(13,6)

\put(3.4,5.4){$\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$}
\put(5.1,5.5){\vector(1,0){6.4}}
\put(11.6,5.4){$\mathcal{ML}^{0}\left(\Omega\right)^{K}$}
\put(7.8,5.9){$\widehat{\textbf{T}}$}
\put(3.8,5.2){\vector(0,-1){3.5}} \put(4.7,1.2){\vector(1,0){6.7}}
\put(3.4,1.1){$\mathbb{H}_{\widehat{\textbf{T}}}\left(\Omega\right)$}
\put(11.6,1.1){$\mathbb{H}_{nf}\left(\Omega\right)^{K}$}
\put(3.5,3.4){$\phi$} \put(12.2,3.4){$\varphi$}
\put(12.0,5.2){\vector(0,-1){3.5}}
\put(7.8,1.4){$\widehat{\textbf{T}}^{\sharp}$}

\end{picture}
\end{math}\\
\\
Here $\phi$ and $\varphi$ are the uniformly continuous embeddings
associated with the completions
$\mathbb{H}_{\widehat{\textbf{T}}}\left(\Omega\right)$ and
$\mathbb{H}_{nf}\left(\Omega\right)$, and
$\widehat{\textbf{T}}^{\sharp}$ is the extension of
$\widehat{\textbf{T}}$ achieved through uniform continuity.  The
existence of generalized solutions will follow from Theorem
\ref{Approx1} and (\ref{TEmbedding}).

\begin{theorem}\label{SolEx}
For every $\textbf{f}\in\mathcal{C}^{0}\left(\Omega\right)^{K}$
that satisfies (\ref{Assumption}), there exists a unique
$\textbf{U}\in \mathbb{H}_{\widetilde{\textbf{T}}}
\left(\Omega\right)^{K}$ such that
\begin{equation}\label{ExtPDEII}
\widehat{\textbf{T}}^{\sharp}\textbf{U}=\textbf{f}
\end{equation}
\end{theorem}
\begin{proof}
First let us show existence.  For every $n\in\mathbb{N}$, Theorem
\ref{Approx} yields a closed nowhere dense set
$\Gamma_{n}\subset\Omega$ and a function
$\textbf{u}_{n}\in\mathcal{C}^{m}\left(\Omega\right)$ that
satisfies
\begin{equation}
x\in\Omega\setminus\Gamma_{n}\Rightarrow
f_{i}\left(x\right)-\frac{1}{n}\leq
T_{i}\left(x,D\right)\textbf{u}_{n}\left(x\right)\leq
f_{i}\left(x\right)\mbox{, }i=1,...,K\label{Approx2}
\end{equation}
Since $\Gamma_{n}$ is closed nowhere dense we associate
$\textbf{u}_{n}$ with a function
$\textbf{v}_{n}\in\mathcal{ML}^{m}\left(\Omega\right)$ in a unique
way.  Indeed, consider for instance the function
\begin{eqnarray}\nonumber
\textbf{w}_{n}:x\mapsto\left\{\begin{array}{ll}
  \textbf{u}_{n}\left(x\right) & \mbox{ if }x\in\Omega\setminus\Gamma \\
  0 & \mbox{ if }x\in\Gamma \\
\end{array}\right.
\end{eqnarray}
Now let $\textbf{v}_{n}$ be the $K$-dimensional vector valued
function with components $v^{i}_{n}=\left(I\circ
S\right)\left(w_{n}^{i}\right)$.\\
Denote by $\textbf{V}_{n}$ the
equivalence class generated by $\textbf{v}_{n}$ under the
equivalence relation $\sim_{\widetilde{\textbf{T}}}$.  Applying
(\ref{TExtTRel}), (\ref{TExtTRel2}) and (\ref{Approx2}) one finds
that the sequence
$\left(\widehat{\textbf{T}}\left(\textbf{V}_{n}\right)_{i}\right)$
is increasing in $\mathcal{ML}^{0}\left(\Omega\right)$ and has
upper bound $f_{i}$, for each $i=1,...,K$.  It is clear that for
each $i=1,...,K$, $f_{i}$ is in fact the least upper of the
sequence
$\left(\widehat{\textbf{T}}\left(\textbf{V}_{n}\right)_{i}\right)$.
Hence each $\left(\widehat{\textbf{T}}
\left(\textbf{V}_{n}\right)_{i}\right)$ converges to $f_{i}$ in
$\mathcal{ML}^{0}\left(\Omega\right)$ so that the sequence
$\left(\widehat{\textbf{T}}\left(\textbf{V}_{n}\right)\right)$
converges to $\textbf{f}$ in
$\mathcal{ML}^{0}\left(\Omega\right)^{K}$.  It now follows that
$\left(\textbf{V}_{n}\right)$ is a Cauchy sequence in
$\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)$ so
that there exists
$\textbf{U}\in\mathbb{H}_{\widehat{\textbf{T}}}\left(\Omega\right)$
that satisfies (\ref{ExtPDEII}).\\
Since the mapping
$\widehat{\textbf{T}}:\mathcal{ML}^{m}_{\widetilde{\textbf{T}}}\left(\Omega\right)
\rightarrow\widehat{\textbf{T}}$ is a uniformly continuous
embedding by (\ref{TEmbedding}), the uniqueness of the solution
$\textbf{U}$ found above now follows by Theorem \ref{EmbExt}.
\end{proof}

\section{Conclusion}

It has been shown that the Order Completion Method for systems of
nonlinear PDEs can be naturally considered within the framework of
uniform convergence spaces.  With respect to the previous
existence and regularity results obtained by the Order Completion
Method, we have not made any improvement.  The class of systems of
nonlinear PDEs treated is the same as that considered in
\cite{Obergugenberger and Rosinger}, that is, we considered all
continuous PDEs.

However, the generalized solutions obtained are elements of the
completion of a space of equivalence classes of piecewise smooth
functions, which are assimilated with Hausdorff continuous
functions.  In order to obtain meaningful solutions, for instance
classical solutions, a great deal of hard analysis is required.
Within the framework of uniform convergence spaces as applied
here, we have a sound topological foundation at our disposal.