Many of us think about will power when it comes to sticking to a diet or exercise program. Some think of the reserve needed to get through a difficult life season. A person may call on all their will power to finish strenuous requirements of an educational or professional pursuit. A dear person may be reaching into their storehouse of will power to manage depression or on-going health problems. Regardless, will power is required to succeed.
Will power is an energy that flows from your power center. This center is created by God and given to each person. Did you know you have one of these? Your power center's essence and eternal replenishment flows from God's very own, personal power center. How incredible! This means if a person is plugged into God and relying on Him, they have endless power available to live daily life. This is important since life can get pretty difficult. Problems get complicated. Relationships can get dicey. Stress mounts. Energy drain is a reality.
If your life is like mine, you know you need power to live each day well. You have so many balls to keep in the air, decisions to make, results to ensure, and responsibilities. Many things that happen are out of your control, which makes living even more challenging. In fact, honestly? The only thing you have full control over is your will.
Being truly honest, you know you have control over what you might say or do next. Your power center – your will – gives you options. Because God is the source of your power, it is God who gives you options. And options are what you need. This realization keeps you out of a "victim mentality," especially when your life falls apart or gets trespassed by extreme problems or suffering.
It is by using your will that you build your life. Every moment, you decide what kind of person you will be the next moment. All those moments chain together to form your life. That can be good – or not so good. It's your decision.
Is it best to ignore?
How to give it a shot – and hope.
Do you set a boundary?
How to use your time and energy.
Engaging a creative, fresh approach.
Choosing the character to practice.
With your will, you set priorities and pathways, forming new habits and patterns. With your will, you break old ones! You are the only person on the planet who can use your will to accomplish anything. A will powered by God is a fabulous resource.
Solving problems, getting results, and maintaining a balanced, healthy life isn't always about changing your situation. That's good news, because many times you can't. Running doesn't help. Pretending is unhealthy. So why not begin to live with the power God designed you to have. Partner with God.
The single best way to use your free will and get control of your life is to take responsibility for your power center and partner with God. He will help you be all He created you to be. Engage your will power to enable a well-lived life.
Line your perspective up with God's.
Think according to the Scriptures.
Apply scriptural principles no matter how you feel.
Be a living sacrifice. This is really hard until you actually do it. Then somehow, as you simply surrender, being a living sacrifice becomes much easier.
Embrace the privilege of waiting in each "pause" of life. Most living is done in a pause. Why not empower your pauses with God's refreshing resources?
Eagerly expect to receive from God. He is faithful.
Rest in the Lord and praise Him before you see progress or intervention.
Gain an indomitable will power by prayerfully practicing surrender to God each day.
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text:
\section{Introduction}\label{SEC:Intro}

Consider the following problem

{\footnotesize
\begin{equation} \label{EQN:RDFHN}
\begin{cases}
\partial_t v(t,\xi) = \Delta \gamma(v(t,\xi)) - I^{ion} (v(t,\xi)) - f(\xi) v(t,\xi) + F(t,\xi) +  v(t,\xi) \partial_t W(t) \;, \xi \in \mathcal{O}\\
v(0,\xi) =v_0(\xi) \;, \\
\gamma(v(t,\xi)) = 0 \;, \text{on} \; (0,T) \times \partial \mathcal{O} \;.
\end{cases}
\end{equation}
}

$\gamma: \mathbb{R} \rightarrow \mathbb{R}$ being a monotone, increasing continuous function,  $v=v(t, \xi)$  represents the {\it transmembrane electrical potential}, $\mathcal{O} \subset \RR^d$, $d=2,3,$ is a bounded and open set with smooth boundary $\partial \mathcal{O}$. We indicate with $\Delta_\xi$  the Laplacian operator with respect to the spatial variable $\xi$, while $\varepsilon$ and $\delta$ are positive constants representing phenomenological coefficients, $f(\xi)$ is a given external forcing term, while  $I^{ion}$ is the {\it Ionic current} and, according with the FitzHugh-Nagumo model, it equals $I^{ion}(v)= v(v-a)(v-1)$, $v_0$, $w_0 \, \in L^2(\mathcal{O})$, namely it represents a cubic non-linearity. Also $F$ is a bounded term needed to treat the general controlled equation in next section.

Equation \eqref{EQN:RDFHN} with linear diffusion, i.e. $\gamma(x)=x$, is the well-known FitzHugh--Nagumo (FHN) equation. FHN equation is a reaction--diffusion equation, first introduced by Hodgkin and Huxley in \cite{HH} and then simplified in \cite{FH,Nag}. The model has been proposed to provide a rigorous, yet simplified, analysis of electrical impulses dyanmics along a nerve axon, see, e.g., \cite{Rin}, where the propagation of the transmembrane potential on the nerve axon is represented by a cubic nonlinear reaction term, possibily perturbed by a noisy one, see, e.g., \cite{BCDP,CDPFHN,Rin,Tuc}.

The random perturbation represents the effect of noisy input currents within neurons, their source being the random opening/closing actions of  ion channels, see, e.g., \cite{Tuc}. Moreover, in two-dimensional and three-dimensional settings,  equation \eqref{EQN:RDFHN} plays also a relevant role in statistical mechanics,  under the name of \textit{Ginzburg-Landau} equation, as well as concerning phase transition models of \textit{Ginzburg-Landau} type, see, e.g., \cite{Die}.

The general case where $\gamma$ is a monotone function corresponds to an anomalous--diffusive FitzHugh--Nagumo (FHN) equation, see \cite{Iqb}, also describing phase transitions in porous media, see, e.g., \cite{Pan,Sid}.

\begin{Remark}
In what follows we shall focus on the mathematical setting behind the Stochastic FitzHugh-Nagumo (FHN) model, without
entering into details about the neuro-biological justification of parameters characterizing it. Appropriate details, as well as
in depth analysis of the existing literature on the subject, will be provided later.
\end{Remark}

We assume that $W$ is a $H^{-1}:=H^{-1}(\mathcal{O} )-$cylindrical Wiener processes, such that
\[
\begin{split}
W(t,\xi) &= \sum_{n= 1}^\infty \mu_n e_n \beta_n(t) \,,\quad t \geq 0\,,\xi \in \mathcal{O}\,,\\
\end{split}
\]
where $\{\beta_n\}_{n \geq 1}$  is a sequence of mutually independent standard Brownian motion defined on a filtered probability space $\left (\Omega,\mathcal{F},\left \{\mathcal{F}_t\right \}_{t \geq 0},\mathbb{P}\right )$, while  $\{e_n\}_{\geq 1}$ is an orthonormal basis in $H^{-1}$ and $\mu_n \in \RR$.

Since the Laplacian operator $\Delta_\xi$ is a linear operator in $L^2\left (\mathcal{O}\right )$, and $-\Delta_\xi$ is self--adjoint, then  there exists a complete orthonormal system $\{\bar{e}_k\}_{k\geq 1}$ in $L^2\left (\mathcal{O}\right )$ of eigenfunctions of $-\Delta_\xi$, and we shall indicate the corresponding sequence of eigenvalues denoted by $\{\bar{\lambda}_k\}_{k\geq 1}$. Therefore, we have
\[
\Delta_\xi \bar{e}_k = -\bar{\lambda}_k \bar{e}_k\,,\quad k \in \NN\;.
\]

Also, we set
\begin{equation}\label{EQN:OpG}
Gv = I^{ion}(v) = v(v-a)(v-1) \;,
\end{equation}
and note that $G$ is monotonically nondecreasing.

The present paper addresses the problem of existence and uniqueness of a strong solution, in a sense to be better specified in a while, to equation \eqref{EQN:RDFHN}. We stress that this is not a trivial problem as the nonlinear operator $\Delta \gamma$ is naturally defined on the space $H^{-1}$ whereas the nonlinear polynomial perturbation $I^{ion}$ is not $m-$accreative on the same space. In order to solve above problem we will transform the original equation, via a \textit{rescaling transformation}, to a random PDE. It turns out that the existence and uniqueness of transformed random PDE can be treated by the theory of nonlinear semigroup in $L^1$.

We will further consider the problem of existence of an optimal control for the nonlinear FHN equation. Again, in order to solve the problem we will apply a \textit{rescaling transformation} to obtain a corresponding random PDE. As already emerged in \cite{BCDP,CDPFHN}, the nonlinear polynomial term implies that standard minimization argument does not apply. Therefore, existence of an optimal control is achieved using \textit{Ekeland's variational principle}. First order conditions of optimality are given in terms of dual stochastic backward equation, see, e.g, \cite{BCDP,Bre}, whereas, due to the applied \textit{rescaling transformation} are expressed in terms of a random backward dual equation which allows to simplify the setting also giving more insights on the derived optimal controller.

The present paper is structured as follows: Section \ref{SEC:MS} introduces main notation used thorough the paper. Section \ref{SEC:E!} addresses the problem of proving existence and uniqueness for the state equation whereas in Section \ref{SEC:OC} the problem of the existence for an optimal control is considered.

\subsection{Main notations}\label{SEC:MS}

In what follows we will denote by $| \cdot|$, resp. $\langle \cdot,\cdot\rangle$, the norm, resp. scalar product, $\RR^d$. Also, $L^p\left (\mathcal{O}\right ) =: L^p$, for $1 \leq p \leq \infty$, is the standard space of $p-$Lebesgue measurable function over the domain $\mathcal{O} \subset \RR^d$, with corresponding norm defined as $|\cdot|_p$. For the case $p=2$, we will further denote by $\langle \cdot,\cdot\rangle_2$ the scalar product in $L^2$. The space $H^1(\mathcal{O})=: H^1$ is the Sobolev space $\left \{u \in L^2\,:\,\partial_\xi u(\xi) \in L^2\right \}$, endowed with standard norm $\|u\|_{H^1}^2:= \int_{\mathcal{O}} \left (|u|^2 + |\nabla u|^2 \right )d\xi$. The dual of the space $H^1$ will be denoted as $H^{-1}$ equipped with corresponding norm $|\cdot|_{-1}$.

Similarly, we will denote by $W^{n,p}(\mathcal{O})=:W^{n,p}$, $n \in \NN$, $1\leq p \leq \infty$, the standard Sobolev space of $p-$integrable functions with $p-$integrable $n-$order derivatives. Coherently, $W^{1,p}([0,T];H^{-1})$ will be the space of absolutely continuous function $u:[0,T]\to H^{-1}$ such that both $u$ and $\frac{d}{dt}u \, \in L^p([0,T];H^{-1})$. Further, given a Banach space $X$, $L^p([0,T];X)$ is the space of $X-$valued Bochner $p-$integrable functions on the interval $[0,T]$. Also, $C([0,T];X)$, resp. $C^1([0,T];X)$, denotes the space of continuous, resp. continuously differentiable, functions $u:[0,T]\to X$.

We shall also introduce $C_W([0,T];H)$ the space of all $H^{-1}$--valued $\left (\mathcal{F}_t\right )$--adapted processes such that $X \in C\left ([0,T];L^2\left (\Omega;H^{-1}\right )\right )$, that is $X$ satisfies
\[
\sup_{t \in [0,T]} \mathbb{E}|X(t)|^2_{-1} <\infty\,.
\]

In an analogous manner $L^2_W([0,T];H^{-1})$ is the space of all $H^{-1}$--valued $\left (\mathcal{F}_t\right )$--adapted processes such that $X \in L^2\left ([0,T];L^2\left (\Omega;H^{-1}\right )\right )$, that is $X$ satisfies
\[
\int_{0}^T \left ( \mathbb{E}|X(t)|^2_{-1}\right )^2 dt<\infty\,.
\]

At last $L^2_W(\Omega;C\left ([0,T];H^{-1}\right ))$ denotes the space of all $H^{-1}$--valued $\left (\mathcal{F}_t\right )$--adapted  and continuous processes such that
\[
\mathbb{E} \sup_{t \in [0,T]} \left |X(t)\right |^2_{-1}<\infty\,.
\]

Above definition are still in place if instead of $H^{-1}$ we consider a general Hilbert space $H$. It is also known that there is a natural embedding of $L^2_W(\Omega;C\left ([0,T];H^{-1}\right ))$ into the space $C_W([0,T];H^{-1})$, see, e.g. \cite[Chapter 1]{BDPR}.

We therefore can rewrite equation \eqref{EQN:RDFHN} as
\begin{equation}\label{EQN:MainFHN}
\begin{cases}
dX(t)-[\Delta(\gamma( X(t)))-G(X(t)) + fX(t) + F]dt =XdW(t),\\
X(0) = x_0  \in H^{-1}\, ,\quad  t \in [0,T]\, ,
\end{cases}
\;.
\end{equation}

We will assume the following to hold.

\begin{Hypothesis}\label{HYP:1}
\begin{description}
\item[(i)] $\gamma:\RR \to \RR$ with $\gamma(0)=0$ is continuous, monotonically non--decreasing and there exists $C>0$ such that
\[
(\gamma(x_1) - \gamma(x_2))(x_1-x_2) \geq C(x_1 - x_2)^2\,,\quad \forall \, x_1,\,x_2 \in \RR\,,
\]
\item[(ii)] $F \in L^\infty((0,T)\times \mathcal{O})$, $\mathbb{P}-$a.s. and it is progressively measurable w.r.t. $(0,T) \times \Omega \times \mathcal{B}(\mathcal{O})$; $f \in L^\infty(\mathcal{O})$, and $f \geq 0$ $a.e.$ in $\mathcal{O}$;
\item[(iii)] $W$ is a $H^{-1}:=H^{-1}(\mathcal{O})-$cylindrical Wiener processes, that is,
\[
W = \sum_{j=1}^\infty \mu_j e_j \beta_j\,,
\]
with
\[
\sum_{j=1}^\infty \mu_j^2 |e_j|^2_{L^\infty(\mathcal{O})} < \infty\,,
\]
see, \cite[pag. 22]{BDPR}.
\end{description}
\end{Hypothesis}

Then, we can state the notion of solution to equation \eqref{EQN:MainFHN} that we will consider in subsequent analysis.

\begin{Definition}\label{DEF:SS}
Let $x \in H^{-1}$, we say that the process
\[
X \in L^2_W \left (\Omega;C\left ([0,T];H^{-1}\right )\right ) \cap L^2_W([0,T];L^2)\,,
\]
is a \textit{strong solution} to \eqref{EQN:MainFHN} if $X(t): [0,T] \to H^{-1}$ is $\mathbb{P}-$a.s. continuous and $\forall$ $t \in [0,T]$
\[
X(t) = x + \int_0^t \left (\Delta(\gamma(X(s))) - G(X(s)) + fX(s) + F(s)\right )ds + \int_0^t X(s) dW(s), .
\]
\end{Definition}

\section{Existence for the state equation}\label{SEC:E!}

The main problem in proving existence and uniqueness for a solution to equation \eqref{EQN:MainFHN} is that the operator $G$ in not m-accretive on the space $H^{-1}$ and so basic existence results in \cite{BDPR,BDPR3} are not applicable in the present case. It turns out that the proper space one has to consider to successfully treat equation \eqref{EQN:MainFHN} is the space $L^1$, which, in turn, is not the proper one if one has to deal with SPDEs such as \eqref{EQN:MainFHN}.

To overcome such a stalemate,  we follow \cite{BR,BR2}. In particular, we apply the transformation $X=e^Wy$, which allows to reduce the stochastic equation \eqref{EQN:MainFHN} to a random PDE that can be treated with analytical techniques. In fact, the random equation can be successfully solved  by exploiting the theory of nonlinear semigroup in $L^1$. As noted in \cite{BR}, we have still to face the problem that, because of the non regularity of the term $W$, the general theory cannot be applied straightforward to the resulting random PDE. Therefore, for $\epsilon>0$, we shall consider a suitable sequence of regular approximations $W_\epsilon$ of $W$, to first establish a priori estimates for solutions $y_\epsilon$ of the associated $W_\epsilon-$approximating problem, and then to show that, in  the limit $\epsilon \to 0$, we obtain both existence and uniqueness of the solution for the original equation.

The following theorem constitutes the main result of this section.

\begin{Theorem}\label{THM:E!}
There is a unique strong solution to equation \eqref{EQN:MainFHN} $X=e^Wy$ which satisfies
\[
Xe^{-W} \in W^{1,2}([0,T];H^{-1})\,,\quad \mathbb{P}-a.s.
\]
\end{Theorem}

In order to prove  Theorem \ref{THM:E!} we need some auxiliary lemmas. In particular, let us then introduce the transformation
\begin{equation}\label{EQN:Resca}
X(t)=e^{W(t)} y(t) \,,\quad t \geq 0\,,
\end{equation}
so that by an application of the It\^{o} formula we obtain the random equation
\begin{equation}\label{EQN:RPDE}
\begin{split}
&\frac{\partial}{\partial t} y + e^{-W} G(e^W y) - e^{-W} \Delta\gamma(e^W y) + fy + \mu y = e^{-W}F \,\\
&y(0,\xi) = x(\xi)\,,\quad \xi \in \mathcal{O}\,,\\
&y(t) \in H^1_0(\mathcal{O})\,,\, t \in (0,T)
\end{split}
\end{equation}
with
\[
\mu = \frac{1}{2} \sum_{n=1}^\infty \mu_n^2 e_n^2\,,
\]
see, e.g. \cite{BR,BR2,BRrs}.

Following \cite{BR}, we prove the existence of a unique strong solution to equation \eqref{EQN:RPDE} by
first considering an approximating problem. In particular, let us denote by $\beta^\epsilon (t) := (\beta * \rho_\epsilon)(t)$, where $\rho_\epsilon(t) = \frac{1}{\epsilon}\rho\left (\frac{t}{\epsilon}\right )$ is a standard mollifier and $\rho \in C^\infty_0$, then we have that $\beta^\epsilon \in C^1([0,T];\RR)$. Setting
\[
W_\epsilon(t,\xi) = \sum_{n= 1}^\infty \mu_n e_n \beta^\epsilon(t) \,,\quad t \geq 0\,,\xi \in \mathcal{O}\,.\\
\]
we thus have that $W_\epsilon \in C^1([0,T] \times \mathcal{O})$. Moreover
\[
W_\epsilon(t,\xi) \to W(t,\xi) \,\quad \mbox{uniformly in} \quad (t,\xi) \in [0,T]\times \xi\,,
\]
as $\epsilon \to 0$.

For each $\epsilon >0$, let us thus consider the approximating equation associated to \eqref{EQN:RPDE}
\begin{equation}\label{EQN:RPDEApp}
\begin{split}
&\frac{\partial}{\partial t} y_\epsilon + e^{-W_\epsilon} G_\epsilon(e^{W_\epsilon} y_\epsilon) - e^{-W_\epsilon} \Delta\left (\gamma(e^{W_\epsilon} y_\epsilon + \epsilon e^{W_\epsilon}y_\epsilon\right ) + fy + \mu y_\epsilon =  e^{-W_\epsilon} F\,\\
&y_\epsilon(0,\xi) = x(\xi)\,,\quad \xi \in \mathcal{O}\,,\\
&y_\epsilon(t) \in H^1_0(\mathcal{O})\,,\, t \in (0,T)
\end{split}
\end{equation}
where $G_\epsilon$ is the Yosida approximation of $G$, that is
\begin{equation}\label{EQN:YosApp}
G_\epsilon := \frac{1}{\epsilon} \left (I - (I + \epsilon G)^{-1}\right )\,, \quad \epsilon >0\,.
\end{equation}

Note that, $G_\epsilon$ is monotoniccaly nondecreasing,  Lipschitzian and
\[
\lim_{\epsilon \to 0} G_\epsilon(z) = G(z)\,,\, \forall z \in \mathbb{R}
\]
uniformly on compacts.

Defining $z_\epsilon := e^{W_\epsilon}y_\epsilon$,  equation \eqref{EQN:RPDEApp} becomes
\begin{equation}\label{EQN:RPDEAppZ}
\begin{split}
&\frac{\partial}{\partial t} z_\epsilon + G_\epsilon(z_\epsilon) - \Delta\left (\gamma(z_\epsilon)+\epsilon z_\epsilon\right ) + fz + \left ( \mu - \frac{\partial}{\partial t} W_\epsilon\right ) z_\epsilon = F_\epsilon\,, \quad \mbox{in} \, (0,T) \times \mathcal{O}\,,\\
&z_\epsilon(0,\xi) = x(\xi)\,,\quad \xi \in \mathcal{O}\,,\\
&\gamma(z_\epsilon(t)) + \epsilon z_\epsilon(t)\in H^1_0(\mathcal{O})\,,\, t \in (0,T)
\end{split}
\end{equation}
where
$F_\epsilon := e^{-W_\epsilon} F$.

\begin{Lemma}
Let $x \in H^{-1} \cap L^1$ with $\gamma(x) \in H^1_0$, then for each $\epsilon >0$ equation \eqref{EQN:RPDEApp} has a unique solution such that
\[
y_\epsilon \in W^{1,\infty} \left ([0,T];H^{-1}\right ) \cap L^\infty \left (0,T;H^1_0\right ).
\]

\end{Lemma}
\begin{proof}
Let us first prove existence and uniqueness of a solution to equation \eqref{EQN:RPDEAppZ} in the space $H^{-1}$. For a fixed $\epsilon >0$, let us define the operator $A:D(A)\subset H^{-1} \to H^{-1}$ as
\begin{equation}\label{EQN:OpA}
\begin{split}
Az &= - \Delta\left (\gamma(z) + \epsilon z\right ) + fz + G_\epsilon(z) + \mu z\,,\\
D(A) &= \left \{z \in L^2 \,:\, \gamma(z) \in H^1_0 \right \}\,.,
\end{split}
\end{equation}

We equip the space $H^{-1}$ with the scalar product
\[
\langle y,z\rangle_{-1} := {}_{H^1}\langle \left (-\Delta\right )^{-1} y,z\rangle_{H^{-1}} \,,
\]
where $\left (-\Delta\right )^{-1} y = x$ indicates  the solution to the Dirichlet problem $-\Delta x = y$ in $\mathcal{O}$, $x \in H^1_0$.

Taking into account that $G_\epsilon$ is Lipschitz continuous in $L^2$ and since
\[
z \mapsto -\Delta(\gamma(z)+\epsilon z)\,,
\]
is $m-$accreative in the space $H^{-1}$, see, e.g., \cite[p. 68]{Bar1}, we have that, for a suitable $\alpha = \alpha_\epsilon$, it holds
\[
\langle (A+\alpha I)z - (A+\alpha I)\bar{z},z-\bar{z}\rangle_{-1} \geq0\,
\]
which implies $(A+\alpha I)$ to be accretive in $H^{-1}$.

Moreover, for $\lambda>0$ sufficiently large, we also have $\mathcal{R}\left ((\lambda + \alpha)I + A\right )=H^{-1}$, so that $A$ is quasi-m-accretive. In other words, for $f \in H^{-1}$ the equation
\begin{equation}\label{EQN:Za}
(\lambda + \alpha)\left (- \Delta\right )^{-1}z + \gamma(z) + \epsilon z + \left (- \Delta\right )^{-1}\left (G_\epsilon(z) + fz -\mu z\right )= \left (- \Delta\right )^{-1}\tilde{f}\,,
\end{equation}
has a unique solution in $z \in L^2$. Indeed, introducing the operators
\[
\begin{split}
B:L^2 \to L^2\,,\quad Bz := \gamma(z)\,,
\end{split}
\]
and
\[
\begin{split}
\Gamma&: L^2 \to L^2\,,\\
\Gamma z &= (\lambda + \alpha)\left (- \Delta\right )^{-1}z + \left (- \Delta\right )^{-1}\left (G_\epsilon(z) + fz -\mu z\right )\,,
\end{split}
\]
we see that equation \eqref{EQN:Za} can be rewritten as
\begin{equation}\label{EQN:zOper}
\epsilon z + Bz + \Gamma z = \left (- \Delta\right )^{-1}\tilde{f}\,.
\end{equation}
Since $B$ is m-accretive and $\Gamma$ is m-accretive and continuous in $L^2$, it follows, see, e.g., \cite[p.104]{Bar1}, that $\mathcal{R}(\epsilon I + B + \Gamma)=L^2$, so that equation \eqref{EQN:zOper} admits a unique solution $z$ in $L^2$. Moreover, since  $\gamma(z) + \epsilon z \in H^1$ and the inverse map of $z \mapsto \gamma(z) + \epsilon z$ is Lipschitz, then  $z \in D(A)$.
It follows that, applying \cite[Lemma A.1, Corollary A.2]{BR}, see also \cite[Sec. 4]{Bar1},  $z_\epsilon$ is a strong solution to equation \eqref{EQN:RPDEAppZ} in $W^{1;\infty}([0,T];H^{-1})$. In addition, by \cite[Corollary A.2]{BR}, we also have
\[
\gamma(z_\epsilon) + \epsilon z_\epsilon - \left (- \Delta\right )^{-1} G_\epsilon(z) \in L^\infty(0,T;H^1_0)\,,
\]
and
\[
\left |\left (- \Delta\right )^{-1}G_\epsilon(z)\right |_2 \leq C_\epsilon |z|_{-1}\,,
\]
so that, since $z_\epsilon \in W^{1;\infty}([0,T];H^{-1})$, we obtain
\[
\gamma(z_\epsilon) + \epsilon z_\epsilon \in L^\infty(0,T;L^2)\,,
\]
and consequently $z_\epsilon \in L^\infty(0,T;L^2)$. Moreover we have that
\[
\left ( - \Delta\right )^{-1}G_\epsilon(z_\epsilon) \in L^\infty(0,T;H^1_0)\,,
\]
which implies that $\gamma(z_\epsilon) + \epsilon z_\epsilon \in L^\infty(0,T;H^1_0)$ and consequently $z_\epsilon \in L^\infty(0,T;H^1_0)$.

\end{proof}

\begin{Lemma}\label{LEM:2}

Taking $x \in D(A)$, then $y_\epsilon \in L^\infty((0,T)\times \mathcal{O})$, and it holds
\begin{equation}\label{EQN:IneqY}
\begin{split}
&\sup_{\epsilon} \left \{ |y_\epsilon|_{L^\infty((0,T)\times \mathcal{O})} \right \} \leq C(1+|x|_\infty)\,.\\
\end{split}
\end{equation}
\end{Lemma}
\begin{proof}
Let $\alpha \in C^1([0,T])$, such that $\alpha(0) = 0$ and $\alpha'\geq 0$. Then, defining  $M:= (1+|x|_\infty)$, we have
\[
\begin{split}
&\frac{\partial}{\partial t} \left (y_\epsilon- M \alpha(t)\right ) + e^{-W_\epsilon} \left (G_\epsilon(e^{W_\epsilon} (y_\epsilon))- G_\epsilon(e^{W_\epsilon}(M +  \alpha(t)))\right ) - f y_\epsilon +\\
&- e^{-W_\epsilon} \Delta\left (\gamma(e^{W_\epsilon} y_\epsilon)+\epsilon e^{W_\epsilon}y_\epsilon\right ) + \\
&+ e^{-W_\epsilon}\Delta \left ( \gamma( e^{W_\epsilon}(M+\alpha(t)))+ \epsilon e^{-W_\epsilon}(M+\alpha(t))\right )+\\
&+ \mu (y_\epsilon-M-\alpha(t)) = \tilde{F}_\epsilon -\alpha'\,,
\end{split}
\]
with
\[
\begin{split}
\tilde{F}_\epsilon &:= - e^{-W_\epsilon}G_\epsilon(e^{W_\epsilon}(M +  \alpha(t)))-(M+\alpha(t))+ f y_\epsilon +\\
&+ e^{-W_\epsilon} \Delta \gamma(e^{W_\epsilon}(M+\alpha(t))) + \epsilon(M+\alpha(t))e^{-W_\epsilon}\Delta(e^{W_\epsilon})+ e^{-W_\epsilon}F\,,
\end{split}
\]
with $\alpha$ such that $F_\epsilon - \alpha' \leq 0$.

Following \cite[Lemma 3.3]{BR}, we  first assume that
\begin{equation}\label{EQN:CondL1}
\frac{\partial}{\partial t} y_\epsilon \,,\quad \Delta\left (\gamma(e^{W_\epsilon} y_\epsilon)+\epsilon e^{W_\epsilon}y_\epsilon\right ) \in L^1((0,T)\times \mathcal{O})\,,
\end{equation}
then, denoting by
\[
\begin{split}
J(t)&:= - \int_{\mathcal{O}} e^{-W_\epsilon}\left [ \left (\Delta(\gamma(e^{W_\epsilon}y_\epsilon)+\epsilon e^{W_\epsilon}y_\epsilon\right ) \right .+\\
&- \left .\Delta\left (\gamma(e^{W_\epsilon}(M+\alpha(t))) + \epsilon e^{W_\epsilon}(M+\alpha(t))\right )\right ]sign(y_\epsilon -M -\alpha(t))^+ d\xi\,,
\end{split}
\]
we have
\[
\int_0^t J(s) ds \geq - (\gamma'(e^{|W|_\infty}M)+1)e^{|W|_\infty}(|\Delta W|_\infty + |\nabla W|_\infty^2)\int_0^t |(y_\epsilon - (M+\alpha(s))))^+|_1 ds\,.
\]
Moreover, by hypothesis \ref{HYP:1}, it follows that $G_\epsilon$ is monotone, so that

{\footnotesize
\[
\begin{split}
&\int_0^t \int_{\mathcal{O}} e^{-W_\epsilon} \left (G_\epsilon(e^{W_\epsilon} (y_\epsilon))- G_\epsilon(e^{W_\epsilon}(M +  \alpha(t)))\right ) sign(y_\epsilon - M - \alpha(s))^+ ds d\xi = \\
&\int_0^t \int_{\mathcal{O}} e^{-W_\epsilon} \left (G_\epsilon(e^{W_\epsilon} (y_\epsilon))- G_\epsilon(e^{W_\epsilon}(M +  \alpha(t)))\right ) sign(G_\epsilon(y_\epsilon) - G_\epsilon(M + \alpha(s)))^+ ds d\xi \geq 0\,,
\end{split}
\]
}

and since
\[
\begin{split}
&\int_{\mathcal{O}} \frac{\partial}{\partial t}\left (y_\epsilon - M -\alpha(s)\right ) sign(y_\epsilon - M -\alpha(s))^+ d\xi =\\
&\frac{d}{dt}|(y_\epsilon(t) - M -\alpha(t))^+|_1 \,, \mbox{a.e.} t \in (0,T)\,,
\end{split}
\]
by \cite[Lemma 3.3]{BR}, we conclude that
\[
|(y_\epsilon(t) - M -\alpha(t))^+|_1 =0\,,
\]
if $\tilde{F}_\epsilon \leq \alpha'$ a.e. in $(0,T)\times \mathcal{O}$. Moreover, for a suitable $\alpha$, it also holds
\[
y_\epsilon \leq M+\alpha(t)\,,\quad  \mbox{a.e.}\, \mbox{in} (0,T)\times \mathcal{O}\,,
\]
and
\[
y_\epsilon \geq -M-\alpha(t)\,,\quad  \mbox{a.e.}\, \mbox{in} (0,T)\times \mathcal{O}\,,
\]
and  inequalities \eqref{EQN:IneqY} follows.

Using the approximating scheme described in \cite[Lemma 3.3]{BR}, we have \eqref{EQN:IneqY} without requiring the condition \eqref{EQN:CondL1}, and the claim follows.
\end{proof}

\begin{Lemma}\label{LEM:1}
Let $x \in D(A)$, then there exists an increasing function $C:[0,\infty) \to (0,\infty)$ such that
\[
\sup_{t \in [0,T]} |y_\epsilon(t)|^2_2 + \int_0^T \int_{\mathcal{O}} |\nabla \gamma(e^{W_\epsilon}y_\epsilon)|^2 d\xi ds \leq C(1+|x|_\infty)\,,\quad \forall \epsilon \in (0,1]\,.
\]
\end{Lemma}
\begin{proof}
In what follows we will use the following
\[
\begin{split}
&\int_{\mathcal{O}} j(y_\epsilon(t)) d\xi = \int_0^t {}_{H^{-1}} \left \langle \frac{d y_\epsilon}{ds}(s),\gamma(y_\epsilon(s))\right \rangle_{H^1_0} ds + \int_{\mathcal{O}} j(x) d\xi\,,\\
&{}_{H^{-1}} \left \langle \Delta \gamma(e^{W_\epsilon}y_\epsilon), e^{-W_\epsilon}\gamma(y_\epsilon) \right \rangle= - \int_{\mathcal{O}} \nabla \gamma(e^{W_\epsilon}y_\epsilon) \cdot \nabla(e^{-W_\epsilon}\gamma(y_\epsilon)) d\xi\,,
\end{split}\,,
\]
with $j(r) = \int_0^r \gamma(s)ds$, $r \in \RR^+$.

Thus, multiplying equation \eqref{EQN:RPDEApp} by $\gamma(y_\epsilon)$ and integrating over $(0,t) \times \mathcal{O}$ we obtain
\begin{equation}\label{EQN:BouGg}
\begin{split}
&\int_{\mathcal{O}} j(y_\epsilon(t)) d\xi + \int_0^t \int_{\mathcal{O}} \left [ \nabla \gamma(e^{W_\epsilon}y_\epsilon) + \epsilon \nabla(e^{W_\epsilon}y_\epsilon)) \cdot \nabla(\gamma(y_\epsilon)e^{-W_\epsilon})\right ] d\xi ds \leq \\
&\leq \int_{\mathcal{O}} j(x) d\xi - \int_0^t \int_{\mathcal{O}} \left (e^{-W_\epsilon} G_\epsilon(e^{W_\epsilon} y_\epsilon) + f y_\epsilon\right )\gamma(y_\epsilon) d\xi ds\,.
\end{split}
\end{equation}

Concerning the last integral in the right hand side of equation \eqref{EQN:BouGg}, we have recalling that $G_\epsilon$ is the Yosida approximant of $G$ and using the monotonicity of $\gamma$ and $G_\epsilon$ that Lemma \ref{LEM:2},

\begin{equation}\label{EQN:BouGg2}
\begin{split}
&\int_0^t \int_{\mathcal{O}} \left (e^{-W_\epsilon} G_\epsilon(e^{W_\epsilon} y_\epsilon) + f y_\epsilon \right )\gamma(y_\epsilon) d\xi ds \geq  \\
&\geq C\int_0^t \int_{\mathcal{O}} e^{-W_\epsilon} G_\epsilon(e^{W_\epsilon} y_\epsilon)y_\epsilon d\xi ds \geq  \\
&\geq C\int_0^t \int_{\mathcal{O}} e^{-W_\epsilon} y_\epsilon d\xi ds \,,
\end{split}
\end{equation}
while the other terms in equation \eqref{EQN:BouGg} can be studied as done in \cite[Lemma 3.3]{BR}, so that the claim follows by Lemma \ref{LEM:2}.
\end{proof}

\begin{Lemma}\label{LEM:E!RPDE}
There is a unique solution to equation \eqref{EQN:RPDE} with

\begin{equation}\label{EQN:BoundSolY}
\begin{split}
&y \in W^{1,2}([0,T];H^{-1}) \cap L^\infty((0,T) \times \mathcal{O})\,,\\
&y \in L^2(0,T;H^{-1})\,,\quad \gamma(e^W y) \in L^2(0,T;H^1_0)\,.
\end{split}
\end{equation}

Moreover, the process $y$ is $\left (\mathcal{F}_t\right )_{t \geq 0}-$adapted.

\end{Lemma}
\begin{proof}
Let us first prove uniqueness. Let $y_1$ and $y_2$ be two solutions to equation \eqref{EQN:RPDE}, and let $\bar{y} := y_1 - y_2$. Then it holds
\begin{equation}\label{EQN:RPDE!}
\begin{split}
&\frac{\partial}{\partial t} \bar{y} + e^{-W} \left (G(e^W y_1)-G(e^W y_2)\right ) + f \bar{y}+ \\
&\qquad - e^{-W} \Delta\left (\gamma(e^W y_1) - \gamma(e^W y_2)\right ) + \mu \bar{y} = 0\, \quad \mbox{in} \quad (0,T) \times \mathcal{O}\\
&\bar{y}(0,\xi) = 0\,,\quad \xi \in \mathcal{O}\,.\\
\end{split}
\end{equation}

We can rewrite equation \eqref{EQN:RPDE!} as
\begin{equation}\label{EQN:RPDE!2}
\begin{split}
&\frac{\partial}{\partial t} \bar{y} + (-\Delta)(\bar{y}\eta)= - e^{-W} \left (G(e^W y_1)-G(e^W y_2)\right ) - e^{-W} \Delta(e^{-W})\bar{y}\eta+\\
&- f \bar{y} - 2 \nabla(e^{-W}) \cdot \nabla(e^W \bar{y}\eta)- \mu \bar{y} = 0\,,\\
\end{split}
\end{equation}
where we have denoted for short
\[
\eta :=
\begin{cases}
\frac{\left (\gamma(e^W y_1) - \gamma(e^W y_2)\right )}{e^W \bar{y}} & \{(t,\xi) \,:\, \bar{y}(t,\xi) \not = 0\}\,,\\
0 & \{(t,\xi) \,:\, \bar{y}(t,\xi) = 0\}\,.\\
\end{cases}
\]

Multiplying equation \eqref{EQN:RPDE!2} by $(-\Delta)^{-1}\bar{y}$, we obtain

\begin{equation}\label{EQN:StimaF}
\begin{split}
&\frac{1}{2}|\bar{y}|_{-1}^2 + \int_0^t \int_{\mathcal{O}} \eta \bar{y}^2 ds d\xi =\\
&= \int_0^t \int_{\mathcal{O}} e^{-W} \left (G(e^W y_1)-G(e^W y_2)\right )(-\Delta)^{-1}\bar{y} ds d\xi +\\
&- \int_0^t \int_{\mathcal{O}} e^W \Delta(e^{-W})\bar{y}\eta (-\Delta)^{-1}\bar{y} ds d\xi+\\
&-2 \int_0^t \int_{\mathcal{O}} \nabla(e^{-W}) \cdot \nabla(e^W \bar{y}\eta)(-\Delta)^{-1}\bar{y} ds d\xi+\\
&-\int_0^t \int_{\mathcal{O}} f  \bar{y}(-\Delta)^{-1}\bar{y} ds d\xi - \int_0^t \int_{\mathcal{O}} \mu  \bar{y}(-\Delta)^{-1}\bar{y} ds d\xi +\,.
\end{split}
\end{equation}

Concerning the first integral in the right hand side of equation \eqref{EQN:StimaF}, we have using the fact that, for $\alpha \in [0,1]$ it holds
\[
G(e^W y_1)-G(e^W y_2) = G'(\alpha e^W y_1 + (1-\alpha) e^W y_2) e^W \bar{y}\,,
\]
we infer that, denoting for short $\vartheta = \alpha e^W y_1 + (1-\alpha) e^W y_2$,
\[
\begin{split}
&\left | \int_{\mathcal{O}} e^{-W} \left (G(e^W y_1)-G(e^W y_2)\right )(-\Delta)^{-1}\bar{y} d\xi\right |=\\
&=\left | \int_{\mathcal{O}} G'(\vartheta)\bar{y}(-\Delta)^{-1}\bar{y} d\xi\right | \leq C |\bar{y}|_2 |\bar{y}|_{-1}\,,
\end{split}
\]
whereas other terms can be treated as in \cite[Theorem 2.2]{BR}. So that, we have
\[
\frac{d}{dt}|\bar{y}|_{-1}^2 \leq C |\bar{y}|_{-1}^2 \,, \, \mbox{a.e.}\, t >0\,,
\]
from which it follows that $\bar{y} = 0$, and, by Lemma \ref{LEM:1}, it holds
\[
|y(t)|_\infty + \int_0^t \int_{\mathcal{O}} |\nabla \gamma(y(s))|^2 d\xi ds \leq C(1+ |x|_\infty)\,,
\]
so that, see \cite[Theorem 2.2]{BR}, we further have
\[
\begin{split}
&y \in W^{1,2}([0,T];H^{-1}) \cap L^\infty((0,T) \times \mathcal{O})\,.\\
\end{split}
\]

As regard existence, by Lemma \ref{LEM:1}--\ref{LEM:2}, we have that $(\gamma(e^{W_\epsilon}y_\epsilon))$ is bounded in $L^2(0,T;H^1_0)$, $(y_\epsilon)$ is bounded in $L^\infty(0,T;L^2) \cap L^\infty ((0,T) \times \mathcal{O}) \cap L^2(0,T;H^1_0)$, and  $\left (\frac{d y_\epsilon}{dt}\right )$ is bounded in $L^2(0,T;H^{-1})$. Thus, by Aubin compactness theorem, $(y_\epsilon)$ is compact in each $L^2(0,T;L^2(\mathcal{O}))$.
It follows that, for fixed $\omega \in \Omega$, along a subsequence, which we still denote by $\{\epsilon\} \to 0$ for the sake of clarity, we have
\begin{equation}\label{EQN:ConSubs}
\begin{split}
y_\epsilon \to y & \quad \mbox{strongly}\, \mbox{in} \, L^2((0,T);L^2)\,,\\
& \quad\mbox{weak-star}\, \mbox{in} \, L^2((0,T);L^2)\,,\\
& \quad\mbox{strongly}\, \mbox{in} \, L^\infty((0,T)\times \mathcal{O})\,,\\
& \quad\mbox{weakly}\, \mbox{in} \, L^2((0,T);H^1_0)\,,\\
\gamma(e^{W_\epsilon}y_\epsilon) \to \eta & \quad \mbox{weakly}\, \mbox{in} \, L^2((0,T);H^1_0)\,,\\
\frac{d y_\epsilon}{dt} \to \frac{d y}{dt} & \quad \mbox{weakly}\, \mbox{in} \, L^2((0,T);H^{-1})\,,\\
W_\epsilon \to W & \quad \mbox{in} \, C((0,T)\times \mathcal{O})\,.\\
\end{split}
\end{equation}

Since the map $z \mapsto \gamma(z)$ is maximal monotone, by \eqref{EQN:ConSubs} we have that $\eta = \gamma(e^Wy)$.

Then, since it holds
\[
|(1+\epsilon G)^{-1}(e^{W_\epsilon} y_\epsilon) - e^{W_\epsilon} y_\epsilon|\leq \epsilon |G_\epsilon(e^{W_\epsilon} y_\epsilon)| \leq C \epsilon\,,
\]
and
\[
(1+\epsilon G)^{-1}(e^{W_\epsilon} y_\epsilon) \to y  \quad \mbox{strongly}\, \mbox{in} \, L^2((0,T);L^2)\,,\\
\]
then, for $\epsilon \rightarrow 0$, we get
\begin{equation}\label{EQN:ConvGe}
\begin{split}
G_\epsilon \left (e^{W_\epsilon} y_\epsilon\right ) \to \zeta & \quad \mbox{weakly}\, \mbox{in} \, L^2((0,T);H^1_0)\,.\\
\end{split}
\end{equation}

Thus, again from the fact that $G:\RR \to \RR$ is maximal monotone it follows that it is also closed and therefore we have that $\zeta = G(e^W y)$.

Therefore, by letting $\epsilon \to 0$, from equation \eqref{EQN:RPDEApp} we obtain
\[
\begin{split}
&\frac{dy}{dt} + e^{-W} G(e^Wy) + fy - e^{-W} \Delta \gamma(e^Wy) + \mu y = e^{-W} F\,,\quad \mbox{in} \, (0,T) \times \mathcal{O}\,,\\
&y(0)=x\,.
\end{split}
\]

Then, by the  uniqueness result already proved,
we also have that the sequence $(y_\epsilon)$ is independent of $\omega \in \Omega$, implying that
$y$ is $\left (\mathcal{F}_t\right )-$adapted, ending the proof.
\end{proof}

We can finally prove that it exists a unique strong solution $X$ to equation \eqref{EQN:MainFHN} which satisfies
\[
Xe^{-W} \in W^{1,2}([0,T];H^{-1})\,\quad \mathbb{P}-a.s.
\]

\begin{proof}[Proof of Theorem \ref{THM:E!}]
Using \cite[Lemma 8.1]{BR2} we have the equivalence between the stochastic PDE \ref{EQN:MainFHN} and the random PDE \ref{EQN:RPDE} via the rescaling transformation \ref{EQN:Resca}, so that existence and uniqueness of a solution $X$ in the sense of Definition \ref{DEF:SS} follows by Lemma \ref{LEM:E!RPDE}.
\end{proof}

\section{The optimal control problem}\label{SEC:OC}

In this section we will focus the attention to a controlled version of equation \eqref{EQN:RDFHN}. We  denote by $\mathcal{X}=L^2_{ad}\left ((0,T) \times  \mathcal{O}\right )$ the space of all $\mathscr{F}_t - $adapted processes $u : [0,T ] \to \RR^d$, and
we consider the following optimal control problem
\begin{equation}\label{EQN:P}
  \tag{P}
\mbox{Minimize} \,  \mathbb{E} \left [ \int_0^T \int_{\mathcal{O}} \left |v(t,\xi) -v_1(\xi)\right |^2  + \frac{\alpha}{2} |u(t,\xi)|^2 d\xi dt + \int_{\mathcal{O}} \left |v(T,\xi) -v_2(\xi)\right |^2 d\xi \right ]\,,
\end{equation}
subject to $u \in \mathcal{U}$ and

{\footnotesize
\begin{equation} \label{EQN:CFHN}
\begin{cases}
\partial_t v(t,\xi) - \Delta \gamma(v(t,\xi)) + I^{ion} (v(t,\xi)) + f(\xi) v(t,\xi) = u(t,\xi) +  v(t,\xi) \partial_t W(t) \;, \mbox{in}\, (0,T)\times \mathcal{O}\\
v(0,\xi) =v_0(\xi)\,,\quad \xi \in \mathcal{O} \;, \\
v(t,\xi) = 0 \;, \text{on} \; (0,T) \times \partial \mathcal{O} \;.
\end{cases}
\end{equation}
}

Here
\[
\begin{split}
\mathcal{U} &:= \left \{ u \in L^2_{ad}((0,T)\times \mathcal{O} \times \Omega) \right .\,:\\
&\left . \qquad \quad   \,|u(t,\xi,\omega)|\leq M\,\quad \mbox{a.e.} (t,\xi,\omega) \in (0,T)\times \mathcal{O}\times \Omega\right \}\,,
\end{split}
\]
$M>0$ being a suitable constant, while $v_1$, $v_2 \in L^2_{\mathcal{F}_0}(\Omega)$ and $\alpha>0$ are given.

In what follows we are going to treat the problem (P) by a {\it rescaling procedure} which allows us to reduce it to a random optimal control problem.

\begin{Theorem}\label{THM:E!4}
Let hypothesis \ref{HYP:1} holds, then, for $T$ sufficiently small, there exists at least one optimal pair $ \left (u^*,v^*\right ) $ solution to problem \eqref{EQN:P}.
\end{Theorem}
\begin{proof}
As in Section \ref{SEC:E!}
we will apply the rescaling transformation $y := e^{-W}v$ so that the optimal control problem \eqref{EQN:P} reads
\begin{equation}\label{EQN:P2}  \tag{P2}
\begin{split}
\mbox{Minimize} \, &\mathbb{E} \left [\int_0^T \int_{\mathcal{O}} \left |e^W y(t,\xi) -v_1(\xi)\right |^2  + \frac{\alpha}{2} |u(t,\xi)|^2 d\xi dt \right ] +\\
&+ \mathbb{E} \left [\int_{\mathcal{O}} \left |e^W y((T,\xi) -v_2(\xi)\right |^2 d\xi \right ] \,,
\end{split}
\end{equation}
subject to
\begin{equation} \label{EQN:CFHNPDE}
\begin{split}
&\partial_t y - e^{-W} \Delta \gamma(e^W y) + e^{-W} G(e^W y)+fy+\mu y = e^{-W}u \;, \mbox{in}\, (0,T)\times \mathcal{O}\,,\\
& y = 0 \,,\quad \mbox{in} \quad (0,T) \times \partial \mathcal{O}\,.
\end{split}
\end{equation}

Existence and uniqueness for a solution to equation \eqref{EQN:CFHNPDE} follows from Lemma \ref{LEM:E!RPDE}.

Applying Ekeland's variational principle, see, e.g. \cite{Eke} or also \cite{CDPFHN,BCDP}, there exists a sequence $\{u_\epsilon\}\subset \mathcal{U}$ such that
\begin{equation}\label{novemberrain1}
\begin{split}
\Psi(u_\epsilon) &\leq \inf \{ \Psi(u) \, ; u \in \mathcal{U}\}+ \epsilon\, ,\\
\Psi(u_\epsilon) &\leq \Psi(u) + \sqrt{\epsilon}\left |u_\epsilon - u\right |\, ,\quad \forall \, u \in \mathcal{U}\, ,
\end{split}
\end{equation}
or equivalently it holds
\begin{equation}\label{novemberrain2}
u_\epsilon = \arg \min_{u \in \mathcal{U}} \{  \Psi(u) + \sqrt{\epsilon}\left |u_\epsilon - u\right |_{\mathcal{U}} \}\, .
\end{equation}

By the standard maximum principle for problem \eqref{novemberrain1}, we have
\begin{equation}\label{EQN:MaxPEk}
\begin{cases}
&\partial_t y_\epsilon - e^{-W} \Delta \gamma (e^W y_\epsilon) + e^{-W} G(e^W y_\epsilon) + fy_\epsilon+\mu y_\epsilon = \frac{1}{\alpha} e^{-W}(p_\epsilon + \theta_\epsilon)\,\\
&\partial_t p_\epsilon + e^{W} \gamma'(e^W y_\epsilon) \Delta(e^{-W}p_\epsilon) + e^W G'(e^W y_\epsilon)p_\epsilon - f p_\epsilon - \mu p_\epsilon = 2(e^W y_\epsilon -v_1)\,\\
&y_\epsilon (0) = y_0 \,,\quad p_\epsilon(T) = 2 (e^W y_\epsilon(T) - v_2)\,,\\
\end{cases}
\end{equation}

where $|\theta_\epsilon|_{L^2(\Omega \times \mathcal{O} \times (0,T))}\leq \sqrt{\epsilon}$.
Indeed, by \eqref{novemberrain1}, it follows $\Psi'(u_\epsilon) = \theta_\epsilon$, yelding \eqref{EQN:MaxPEk}.

By equation \eqref{EQN:MaxPEk} we have $\mathbb{P}-a.s.$,
\begin{equation}\label{EQN:MaxPEkDiff}
\begin{split}
&\partial_t \left (y_\epsilon - y_\lambda\right ) - e^{-W} \Delta \left ( \gamma (e^W y_\epsilon) - \gamma (e^W y_\lambda)\right ) + \\
&- e^{-W} \left ( G(e^W y_\epsilon)- G(e^W y_\lambda)\right )+f (y_\epsilon - y_\lambda)+\mu (y_\epsilon - y_\lambda) =\\
&= \frac{1}{\alpha} e^{-W}\left ((p_\epsilon -p_\lambda) + (\theta_\epsilon-\theta_\lambda)\right )\,,\\
\end{split}
\end{equation}
hence, multiplying equation \eqref{EQN:MaxPEkDiff} by $(-\Delta)^{-1}(y_\epsilon - y_\lambda)$ and integrating over $\mathcal{O}$, it holds
\begin{equation}\label{EQN:Est1}
\begin{split}
&\frac{1}{2}|y_\epsilon - y_\lambda|_{-1}^2 + \int_0^t \int_{\mathcal{O}} \eta (y_\epsilon - y_\lambda)^2 d\xi ds =\\
&= - \int_0^t \int_{\mathcal{O}} e^{-W} \left (G(e^W y_\epsilon)-G(e^W y_\lambda)\right )(-\Delta)^{-1}(y_\epsilon - y_\lambda)d\xi ds +\\
&- \int_0^t \int_{\mathcal{O}} e^W \Delta(e^{-W})(y_\epsilon - y_\lambda)\eta (-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi ds+\\
&-2 \int_0^t \int_{\mathcal{O}} \nabla(e^{-W}) \cdot \nabla(e^W (y_\epsilon - y_\lambda)\eta)(-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi ds+\\
&-\int_0^t \int_{\mathcal{O}} f (y_\epsilon - y_\lambda)(-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi ds +\\
&-\int_0^t \int_{\mathcal{O}} \mu  (y_\epsilon - y_\lambda)(-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi ds +\\
&+\int_0^t \int_{\mathcal{O}} \frac{e^{-W}}{\alpha} (p_\epsilon -p_\lambda)(-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi ds +\\
&+\int_0^t \int_{\mathcal{O}} \frac{e^{-W}}{\alpha} (\theta_\epsilon -\theta_\lambda)(-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi ds \,,
\end{split}
\end{equation}
where $\eta$ is defined as in \eqref{EQN:StimaF}.

While, the first four integrals in the right hand side of equation \eqref{EQN:Est1} can be bounded similarly as done proving Lemma \ref{LEM:E!RPDE}, see \eqref{EQN:StimaF}, the last two terms can be treated exploiting the Young inequality
\begin{equation}\label{EQN:EstPT}
\begin{split}
\left |\int_{\mathcal{O}} \frac{e^{-W}}{\alpha} (p_\epsilon -p_\lambda)(-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi \right | &\leq C |p_\epsilon -p_\lambda|_2^2 |y_\epsilon - y_\lambda|_{-1} \leq \\
&\leq C_1 |p_\epsilon -p_\lambda|_2^2 + C_2 |y_\epsilon - y_\lambda|_{-1}\,,\\
\left |\int_{\mathcal{O}} \frac{e^{-W}}{\alpha} (\theta_\epsilon -\theta_\lambda)(-\Delta)^{-1}(y_\epsilon - y_\lambda) d\xi \right | &\leq C |\theta_\epsilon -\theta_\lambda|_2^2 |y_\epsilon - y_\lambda|_{-1} \leq \\
&\leq C |y_\epsilon - y_\lambda|_{-1} + \epsilon + \lambda\,,
\end{split}
\end{equation}
as to  obtain
\begin{equation}\label{EQN:Est2}
\begin{split}
&\frac{1}{2}\frac{d}{dt} |y_\epsilon(t) - y_\lambda(t)|_{-1}^2 \leq C_1 |y_\epsilon(t) -y_\lambda(t)|_{-1}^2 + C_2 |p_\epsilon(t) -p_\lambda(t)|_{2}^2 + \epsilon + \lambda \,.
\end{split}
\end{equation}

Applying the Gronwall lemma and taking the mean value, we have
\begin{equation}\label{EQN:Est2bis}
\begin{split}
&\mathbb{E} |y_\epsilon(t) - y_\lambda(t)|_{-1}^2 \leq C \mathbb{E} \int_0^t |p_\epsilon(s) -p_\lambda(s)|_{2}^2 ds + \epsilon + \lambda \,,.
\end{split}
\end{equation}
Regarding the second equation in \eqref{EQN:MaxPEk},  we obtain
\begin{equation}\label{EQN:MaxPEkDiff2}
\begin{split}
&\partial_t (p_\epsilon - p_\lambda) + e^{W} \left (\gamma'(e^W y_\epsilon) \Delta(e^{-W} p_\epsilon) - \gamma'(e^W y_\epsilon) \Delta(e^{-W}p_\lambda)\right )+ \\
&+e^W \left (G'(e^W y_\epsilon)p_\epsilon - DG(e^W y_\lambda) p_\lambda \right )-f (p_\epsilon - p_\lambda) -\mu (p_\epsilon - p_\lambda) = 2(e^W (y_\epsilon -y_\lambda))\,.\\
\end{split}
\end{equation}
Then, multiplying  equation \eqref{EQN:MaxPEkDiff2} by $(p_\epsilon - p_\lambda)$ and integrating over $\mathcal{O}$, we obtain
\[
\begin{split}
&\frac{1}{2}|p_\epsilon(t) - p_\lambda(t)|_2^2 = |y_\epsilon(T) - y_\lambda(T)|_2^2 +\\
&\int_t^T \int_{\mathcal{O}} e^{W} \left (\gamma'(e^W y_\epsilon) \Delta(e^{-W}p_\epsilon) - \gamma'(e^W y_\lambda) \Delta(e^{-W}p_\lambda)\right )(p_\epsilon(s) - p_\lambda(s)) d\xi ds + \\
&+\int_t^T\int_{\mathcal{O}} e^W \left (G'(e^W y_\epsilon)p_\epsilon(s) - G'(e^W y_\lambda)p_\lambda(s)\right )(p_\epsilon(s) - p_\lambda(s)) d\xi ds+\\
 &-\int_t^T \int_{\mathcal{O}}\mu (p_\epsilon(s) - p_\lambda(s))^2 d\xi ds+\\
 &-\int_t^T \int_{\mathcal{O}} 2(e^W (y_\epsilon -y_\lambda))(p_\epsilon(s) - p_\lambda(s)) d\xi ds \,.
\end{split}
\]
Rearranging terms above, we further have
\begin{equation}\label{EQN:Est3}
\begin{split}
&\frac{1}{2}|p_\epsilon(t) - p_\lambda(t)|_2^2 = |y_\epsilon(T) - y_\lambda(T)|_2^2 +\\
&\int_t^T \int_{\mathcal{O}} e^{W} \gamma'(e^W y_\epsilon) \left (\Delta(e^{-W}p_\epsilon) - \Delta(e^{-W}p_\lambda)\right )(p_\epsilon(s) - p_\lambda(s)) d\xi ds + \\
  &+\int_t^T \int_{\mathcal{O}} e^{W} \Delta(e^{-W}p_\lambda) \left (\gamma'(e^W y_\epsilon) - \gamma'(e^W y_\lambda))\right )(p_\epsilon(s) - p_\lambda(s)) d\xi ds + \\
&+\int_t^T\int_{\mathcal{O}} e^W G'(e^W y_\epsilon) \left (p_\epsilon(s) - p_\lambda(s)\right )^2 d\xi ds+\\
&+\int_t^T\int_{\mathcal{O}} e^W p_\lambda(s) \left (G'(e^W y_\epsilon) - G'(e^W y_\lambda)\right )(p_\epsilon(s) - p_\lambda(s)) d\xi ds+\\
 &-\int_t^T \int_{\mathcal{O}}\mu (p_\epsilon(s) - p_\lambda(s))^2 d\xi ds+\\
 &-\int_t^T \int_{\mathcal{O}} 2(e^W (y_\epsilon -y_\lambda))(p_\epsilon(s) - p_\lambda(s)) d\xi ds \,. \\
\end{split}
\end{equation}
Therefore, by using the Young inequality, we have
\begin{equation}\label{EQN:Est3bis}
\begin{split}
&|p_\epsilon(t) - p_\lambda(t)|_2^2 \leq |y_\epsilon(T) - y_\lambda(T)|_2^2 +\\
&C \int_t^T |p_\epsilon(s) - p_\lambda(s)|_2^2 ds + C \int_t^T |p_\lambda (s)|_2 |p_\epsilon(s) - p_\lambda(s)|_2 ds +\\
&+C \int_t^T |y_\epsilon(s) - y_\lambda(s)|_2 |p_\lambda (s)|_2 |p_\epsilon(s) - p_\lambda(s)|_2 ds \leq \\
&\leq C\left ( \int_t^T |p_\epsilon(s) - p_\lambda(s)|_2^2 ds + \int_t^T |y_\epsilon(s) - y_\lambda(s)|_2^2 ds\right )\,,
\end{split}
\end{equation}
where we have used $C$ to denote possibly different constants as to simplify notation.

Taking the expectation in equation \eqref{EQN:Est3bis} and combining it with equation \eqref{EQN:Est2bis}, we thus have
\begin{equation}\label{EQN:EstComb}
\begin{split}
&\mathbb{E} |y_\epsilon(t) - y_\lambda(t)|_{-1}^2 + |p_\epsilon(t) - p_\lambda(t)|_2^2 \leq \\
&\leq C \left ( \mathbb{E} \int_0^t |p_\epsilon(s) -p_\lambda(s)|_{2}^2 ds + \epsilon + \lambda \right ) +\\
&+C\left ( \int_t^T |p_\epsilon(s) - p_\lambda(s)|_2^2 ds + \int_t^T |y_\epsilon(s) - y_\lambda(s)|_2^2 ds\right )\,,
\end{split}
\end{equation}
so that, if $T$ is small enough, we can infer that
\begin{equation}\label{EQN:EstFinal}
\begin{split}
&\mathbb{E} |y_\epsilon(t) - y_\lambda(t)|_{-1}^2 + |p_\epsilon(t) - p_\lambda(t)|_2^2 \leq C \left (\epsilon + \lambda \right ) \,,
\end{split}
\end{equation}
implying that  $(y_\epsilon,p_\epsilon)$ is a Cauchy sequence, therefore
,along a subsequence still denoted by $\{\epsilon\} \to 0$ for the sake of clarity, we have
\begin{equation}\label{EQN:ConSubs2}
\begin{split}
y_\epsilon \to y & \quad\mbox{weakly}\, \mbox{in} \, L^2((0,T);H^1_0)\,,\\
p_\epsilon \to p & \quad\mbox{weakly}\, \mbox{in} \, L^2((0,T)\times \mathcal{O})\,,\\
u_\epsilon := \frac{1}{\alpha} e^{-W}(p_\epsilon + \theta_\epsilon) \to u^* & \quad\mbox{weakly}\, \mbox{in} \, L^2((0,T)\times \mathcal{O} \times \Omega)\,.
\end{split}
\end{equation}

Letting then $\epsilon \to 0$ in the first equation in \eqref{EQN:MaxPEk}, we have
\[
\partial_t y^* - e^{-W} \Delta \gamma (e^W y^*) - e^{-W} G(e^W y^*)+\mu y^* = e^{-W} u^*\,,\\
\]
hence, since $\Psi$ is lower--semicontinuous, previous computations give us:
\[
\Psi(u^*) = \inf_{u \in \mathcal{U}} \Psi(u)\,,
\]
and the claimed existence result follows.

\end{proof}

\begin{Theorem}[Necessary condition of optimality]\label{THM:NC}
Let be $(v^*,u^*)$ an optimal pair for problem \eqref{EQN:P}, then if $\alpha > 0$ it holds
\[
u^*(t,\xi) = \frac{1}{\alpha} P_U(-e^{-W}p(t,\xi))\,,\quad \mbox{a.e. on} \quad (0,T) \times \mathcal{O} \times \Omega\,,
\]
where $p$ is the solution to the dual backward equation \eqref{EQN:DualBack} and
\begin{equation}\label{EQN:Proj}
P_U(v) =
\begin{cases}
M & v \geq M\,,\\
v & |v| \leq M\,\\
-M & v \leq M\,.\\
\end{cases}
\end{equation}
\end{Theorem}
\begin{Remark}
 We would like to underline that in literature about stochastic control problem, the first order conditions of optimality (the {\it Pontryagin maximum principle})
are expressed in terms of dual stochastic backward equation, see, e.g, \cite{BCDP,Bre}. Here, instead, optimality conditions are given in terms of a random backward dual equation which allows to simplify the setting also giving more insights on the derived optimal controller.
\end{Remark}
\begin{proof}
We provide the result exploiting the rescaling transformation $y := e^{-W}X$, hence proving necessary condition for the problem \eqref{EQN:P2}.

Let $(y^*,u^*)$ be an optimal pair for problem \eqref{EQN:P2}, therefore we have that for any $u \in \mathcal{U}$, defining $u^\lambda := u^* + \lambda \bar{u} = u^* + \lambda (u - u^*)$, $\lambda \geq 0$, by the optimality of $u^*$ it must hold,
\[
\frac{1}{\lambda} \left (\Psi(u^\lambda) - \Psi(u^*)\right ) \geq 0\,.
\]

By the G\^{a}teaux differentiability of $\Psi$ it follows, taking the limit as $\lambda \to 0$,
\begin{equation}\label{EQN:VarCont}
\begin{split}
&\mathbb{E}\left [ \int_0^T \int_{\mathcal{O}} \left(e^W y^*(t,\xi) -v_1(\xi)\right) z(t) + \alpha u^* \bar{u} d\xi dt \right ]+\\
&+ \mathbb{E}\left [\int_{\mathcal{O}} \left (e^W y^*(T,\xi) -v_2(\xi)\right) z(T) d\xi \right ] \geq 0\,,
\end{split}
\end{equation}
being $z$ the solution to the system in variation defined as
\begin{equation}\label{EQN:SysVar}
\begin{cases}
&\dot{z}(t) - \Delta \left (\gamma'(e^W y^{*}) z(t)\right ) + G'(e^W y^{*})z(t) + \mu z(t) = e^{-W} u^*  \,,\\
&\gamma'(e^W y^{*}) z(t) \in H^1_0(\mathcal{O})\,,\, t \in (0,T)\\
& z(0)=0\,.
\end{cases}
\end{equation}
Therefore, introducing the backward dual system
\begin{equation}\label{EQN:DualBack}
\begin{split}
\dot{p}(t) &= - \Delta \gamma' (e^W y^{*})p - G'(e^W y^{*})p_\lambda (t) + \mu p(t) + 2(e^W y^{*} - v_1)\,,\\
p(T) &= 2(y^*(T)-v_2)\,,
\end{split}
\end{equation}
and exploiting equations \eqref{EQN:SysVar}--\eqref{EQN:DualBack} together with equation \eqref{EQN:VarCont},
we have
\begin{equation}\label{EQN:FeedB}
\begin{split}
\mathbb{E}\left [ \int_0^T \int_{\mathcal{O}} \left \langle e^{-W}p(t) + \alpha u^*, \bar{u} \right \rangle dt \right ]\geq 0\,,
\end{split}
\end{equation}
which gives
\[
u^*(t,\xi) = \frac{1}{\alpha} P_U(-e^{-W}p(t,\xi))\,,\quad \mbox{a.e. on} \quad (0,T) \times \mathcal{O} \times \Omega\,,
\]
where $P_U$
is the projection operator
defined  in \eqref{EQN:Proj}.
\end{proof}

\begin{Theorem}[The bang--bang principle]
Let be $(v^*,u^*)$ an optimal pair for problem \eqref{EQN:P} and let $\alpha = 0$, then it holds
\begin{equation}\label{EQN:BangBang}
u^*=
\begin{cases}
-M & \text{ if } p > 0\\
\in [-m,M] & \text{ if } p = 0\\
M & \text{ if } p <0 \,.
\end{cases}
\end{equation}
where $p$ is the solution to the dual backward equation \eqref{EQN:DualBack}.
\end{Theorem}
\begin{proof}
Proceeding as in Theorem \ref{THM:NC} with obtain the equivalent of equation \eqref{EQN:FeedB}to be
\begin{equation}\label{EQN:FeedBBB}
\begin{split}
\mathbb{E}\left [ \int_0^T \int_{\mathcal{O}} \left \langle e^{-W}p (t) , \bar{u} \right \rangle dt\right ] \geq 0\,,
\end{split}
\end{equation}
which yields equation \eqref{EQN:BangBang}, and the claim follows.
\end{proof}
\begin{Remark}
 By \eqref{EQN:BangBang} it follows that if $\mid v^* -v_1 \mid >0 $ a.e. on $(0,T) \times \mathcal{O} \times \Omega$ then the optimal controller $u^*$ is a {\it bang-bang} controller, namely $\mid u^* \mid =M$ a.e. on $(0,T) \times \mathcal{O} \times \Omega$.
\end{Remark}

\newpage

redpajama_set_name: RedPajamaArXiv


Reverse osmosis (R/O) is a water treatment process in which water is forced through a semi-permeable membrane that has very small holes or "pores". Clean water passes through and impurities that are too big to pass through the membrane are left behind and flushed away.
Do I Need a Reverse Osmosis Unit?
It is presumed in this document that the water you are using meets all health regulations and is known to be safe. If your municipality or utility supplies your drinking water, it is likely that you do not need a reverse osmosis unit. Municipally supplied drinking water is microbiologically safe. It is treated to meet health and aesthetic requirements, and is subject to routine testing for microbiological contamination.
If you obtain drinking water from a private supply such as well, it may not be safe from microbiological, chemical, or other types of contamination. Drinking water from private sources should be tested periodically to determine if treatment is required; and, if so, for what specific contaminants or minerals.
Water chemistry is complex and no single water treatment device can be used to remove all types of substances from water. Different drinking water treatment devices have their own advantages and disadvantages. Each household must individually determine if there is a need for additional water treatment. If this is the case, determine the unit or combination of units best suited for your water needs.
Reverse osmosis systems purify water by forcing pressurized water through a very fine, plastic membrane. If the raw water being treated comes from a well or another private source, disinfection and pre-filters (to remove chlorine and/or particulates/sediment) may be needed in advance of the R/O unit to remove contaminants that can foul or damage the membrane.
During the initial filtration stage, tap water or well water (pressurized by a booster pump) is passed through a particle filter (a pre-filter) that removes silt, sediment, sand, and clay particles that might clog the R/O membrane.
The water is then forced through an activated carbon filter that traps minerals and contaminants such as chromium, mercury, copper, chloramine and pesticides. It also removes chlorine, which is important, as chlorine will shorten the life of the membrane.
Water is transferred under pressure into the R/O module, allowing only clean water to pass through the small pores in the membrane. Impurities unable to pass through the membrane are left behind and flushed down the drain.
Treated water is then sent to a storage tank.
Treated water is passed through an activated carbon filter before use to further improve the water's taste and smell.
Water that contains manganese, hydrogen sulphide or iron should be pre-treated to extend the life of the membrane. A dealer can recommend the pre-treatment needed.
Note: Reverse Osmosis units produce no noise other than the sound of water discharging into the drain (usually a sink or a floor drain).
How Do I Know What Size Unit to Buy?
R/O units are rated according to the amount of treated water produced per day. For example, one type of unit produces 50 litres of treated water per day under its design conditions. Such a unit is generally rated with 60 psi water line pressure, a water temperature of 25° C (77° F), normal dissolved solids and 2 atmospheres of pressure. In reality, conditions frequently vary. Line pressure is often lower, water will frequently be colder than 25° C and backpressure in the storage tank will likely reduce the performance of the unit. Consequently, you should examine water conditions and buy a larger rated unit than needed if any of the above problems are noted.
Where Do I Buy Reverse Osmosis Units?
Reverse osmosis units are sold at some department and hardware stores, and by water equipment dealers. Companies can be found listed under "Water"or "Water Companies" in the Yellow Pages. You may wish to talk to a variety of vendors to compare features.
How Much Do Reverse Osmosis Units Cost?
Reverse osmosis unit prices vary, from $400 for a portable or undersink unit to $2,500 for a larger, stationary (basement) unit where a booster pump and a pressure system are installed. Replacement pre-filters range in price from approximately $100 – $200 each.
Who Installs the RO Unit?
Many R/O units are designed to operate on the kitchen counter. Some of the larger units are connected under the counter or in the basement. The unit will need to be hooked up to the water line and a discharge-to-the-drain line. Larger units may require professional installation, where a special water supply line is run from the main household water line. If you are unsure about installation, contact a plumber or mechanical contractor.
Reverse osmosis can remove dissolved solids, salts, minerals that cause hardness, organic chemicals and other impurities. It can improve the taste of water for people who do not like the taste of dissolved mineral solids.
Treated water will not produce scale in kettles and coffee makers. Because sodium and potassium are removed, people on a medically prescribed sodium- or potassium-restricted diet may benefit. R/O units may also remove contaminants such as chromium, mercury and nitrates. Before purchasing a R/O unit, check the certification and literature for the particular model to verify exactly what it can and cannot remove.
Is Reverse Osmosis - Treated Water Safe to Drink?
Reverse osmosis treatment systems remove minerals like calcium and magnesium from drinking water. In Canada, water is a minor source of such minerals when compared to foods. If you consume a reasonably balanced diet, you do not need to take a mineral supplement when drinking water treated with a reverse osmosis system. Low levels of minerals in drinking water may be a concern for people living in countries with very hot climates.
Is the Water Source Safe?
A R/O unit should be used only with drinking water that is known to be safe to drink. Although reverse osmosis can remove microbiological contaminants, R/O does not disinfect the water to drinking standards. A flaw or tear to the membrane could allow untreated water to flow through the unit without removing disease-causing organisms. Remember if you are unsure of the quality of your water, get it tested. If you have any doubts about the safety of your water, then it should be disinfected before using a R/O unit.
How Much Water Does the Unit Use? Is It Water-Efficient?
Some R/O units can produce 4 litres per minute and others will produce 30 to 94 litres per day. In operating a R/O unit, a large amount of incoming water is used to produce the final treated water. This unusable water (called brine) contains contaminants that the R/O unit has removed. The amount of brine created will depend on the quality of the incoming water.
Operating a reverse osmosis unit is not water-efficient — and the amount of water used depends on the quality of the incoming water. In some cases, where water is free of dissolved solids, two litres of water may be needed to produce one litre of finished water. In other cases, 4 or 5 litres of water may be used to produce one litre of treated water.
A family of four might need 40 to 80 litres of water to produce 8 to 16 litres of treated water for drinking and cooking per day. This would cost about ten to twenty cents per day for water. If more treated water is desired, then more water would be used. This will also increase water and wastewater bills, or create a higher demand on your well and septic system.
R/O units will not operate efficiently at water pressures below 40 – 45 psi. If the pressure is too low, as in the case in many rural private systems; in an apartment on a higher floor of a building; or at the end of a long water line serving several units, a booster pump should be installed to increase pressure.
Do I Need to Maintain the Unit?
Reverse osmosis units must be maintained as per the manufacturer's recommendations. Typically the sediment pre-filter and the activated carbon pre-filters should be changed at least annually. However, these prefilters may need to be changed as often as once every six months if the water entering the unit contains sand, large amounts of chlorine or other substances that impair filter efficiency. Although membranes in a well-maintained unit can last for several years, the membrane may need to be replaced more frequently than the manufacturers suggested timeline.


There are a lot of choices when it comes to picking a flight controller for your quadcopter. If you are looking for a low cost way to auto-stabilize your quadcopter, this is the guide for you!
Posted in How-to, Users Projects by Russell 137 Comments.


GE Power has announced that NTPC and Tata Chemicals have selected its technology to upgrade two coal-fired boilers in India due to its low nitrogen oxide (NOx) firing system.
Image: GE Power to deliver low NOx Boiler Technology for thermal plants in India. Photo courtesy of General Electric.
The technology provided by GE Power will be implemented for the first time in India at NTPC's 2×490 thermal plant in Dadri, Uttar Pradesh, and the 136TPH Boiler Tata project in Mithapur, Gujarat.
GE Power's technology helps reduce NOx generation by up to 40% from current levels in these units.
More than 170GW of India's coal-fired fleet operates on sub-critical levels and the implementation of the low NOx boiler technology can help the country reduce its NOx by up to 50%.
GE Power also has the technology to cut NOx levels even further by using secondary measures such as selective catalytic reductions (SCRs), which together with the low NOx boiler technology, can reach reductions of more than 95%.
GE Power India managing director Andrew DeLeone said: "This is the first order by any utility or industrial company in India for the firing system modification to reduce the NOx generation at the primary source of combustion boilers.
"The success of these projects will benchmark the technology and technical specifications for future low NOx firing system modification in India.
Under new coal plant emission regulations in India, firing systems at all utility boilers as well as industrial and captive plant boilers should modified to improve NOx emissions.
By using this technology at its primary source of emissions reduction, customers experience shorter outage time and lower operating costs.
The contract strengthens GE Power's position as a major AQCS technology provider in India.
Earlier in 2018, GE Power's steam power business has successfully completed NTPC's first wet flue gas desulphuriation (WFGD) unit at Vindhyachal Power Plant and a contract to install a new WFGD at NTPC's coal plant in Telangana.
Banks have invested $1.9tn in fossil fuels since 2015 – who tops the list?


Mr. Ernst joined Price Edwards in June of 2006 after receiving his Bachelor's degree in Finance from the University of Oklahoma. Everest became a part of the retail division in which he is responsible for the marketing and leasing of various retail properties within the Price Edwards portfolio. He also works very closely with the retail staff in tenant representation. Tenant representation includes T-Mobile, La Madeleine, True Value, Save A Lot, Homeland, Bank of Oklahoma, Yogurtland, Menchie's, Casey's General Stores, Sprint, Tiff's Treats, MOD Pizza and Cedarwood Development. Everest is a 12-year retail real estate veteran with leasing and sales experience throughout the state of Oklahoma.
Everest earned the distinction of leasing associate by receiving his real estate license in the summer of 2006. Everest is a member of the International Council of Shopping Centers, the Commercial Real Estate Council, and the Chainlinks Retail Advisors.


How can I improve my grammar? What grammar books do you recommend? What books are the best for learning English grammar?
These are the questions I receive from my students and online all the time. So, I have decided to compile a list of the best books for English grammar. You will find here the reference books as well as grammar courses for different goals and levels.
It has been my Go-To English grammar book throughout years. Not only it is a comprehensive guide to English grammar rules but also a practical guide to English speaking. It's THE BEST book for you if you want to understand how English works.
Michael Swan has also written practice books with exercises. See below for Grammar Scan and Oxford Grammar Course.
A big part of the book is about the vocabulary which is often confusing, with numerous examples.
There are not some many attempts in the ELT world to connect spoken and written language together. This is one of the best guides to go to if you want to learn how English is used on a daily basis.
They explore 'grammar and vocabulary together'. This is the approach we are creating our courses in as I believe it's the best way to learn English these days.
The book is based on 3000 most used words in English.
What I like about this guide is it's so practical. If you feel like grammar references make you fall asleep, try this one. Many English teachers use it to learn to explain grammar in simple terms so it is really a great help.
It is organised around words, not the abstract grammar system.
I am foreseeing questions and list some books teaching you American grammar. While we still don't have an officially established grammar standard other than the British one, there are some grammar differences between these two accents of English along with the more obvious vocabulary ones.
I can't give an objective recommendation here but I chose 2 books based on the opinions of the people I know and positive reviews on Amazon. If you know something amazing written about American grammar, please comment below!
'English Usage Rules Explained in Plain English.
A collection of practical exercises from one of the best British grammarians, Michael Swan. The book will teach you all the important structures using real-life examples. There are three levels: Basic, Intermediate, and Advanced.
It is an upgraded version of the long-loved 'How English Works' – I have used this book a lot. I like the humour in the texts used for the exercises and logical structure.
Very in-depth guides, good for systematic studies.
Betty Azar uses a more intuitive, communicative, approach to grammar learning. The book goes step by step from one reflexive exercise to another involving you in a journey. I know some people who could only learn English grammar with the help of this book. If you learn better through doing, try this book – maybe it's your learning style.
All skills involved, including speaking and listening.
It is one of my favourites to use with students. I like the approach of exploring the most frequent structures and word combinations instead of learning abstract rules.
It is truly a book written to teach you how to USE English, not just learn ABOUT English. Each chapter describes how to use TOP WORDS in English in everyday context – with examples, set expressions, idioms, phrasal verbs, and numerous examples.
You can use it without a teacher – the structure is very clear.
Despite the popularity of Raymond Murphy's books, I must admit I am not a huge fan. Yes, they are systematic and easy to follow, but I have seen numerous cases when it was used as a core book for many English courses. I am not sure it's the right approach.
If you enjoy learning with this book, of course, go on. It gives you a well-structured understanding of English grammar and a lot of practice. But please remember that knowing grammar doesn't mean you can speak the language – practise speaking and listening with authentic materials even more.
There are versions translated in many languages.
If you have been learning English for a while, you will need to go ahead with complex structures like verb patterns, conditionals, and so on. This book is a real help with it. It's not too long, and you get a pretty good idea about everything in a short time.
I also recommend it to those who are preparing for English exams (like IELTS or CAE). Along with the books written specially about exams, this one will help you with completing the gaps and creating a clear grammar system in your head.
There are handy grammar tables for the things you need to remember – no need for note-taking.
This book will come handy if you want to check your knowledge of English grammar. Like, really. In depth.
It is a collection of diagnostic tests in 3 levels testing all grammar aspects including confusing vocabulary, spelling, and spoken grammar. If you make a mistake, you can go to the corresponding chapter in 'Practical English Usage' to learn more.
There is an online version available (which is much cheaper, by the way).
Great help for those who are preparing for IELTS or Cambridge Exams. They are brief and highly practical. If you have a good level of English, it's a great way to go through the tricky grammar points before your exam.
Not too academic – written in simple language, concentrate only on the necessary points.
Even if you are not exactly preparing for any English exam at the moment, this book is the most compact and practical way to learn Advanced English grammar. It's all in one place: with explanations, exercises, idioms, and phrasal verbs. It helps a lot if you were missing something and need to complete the gaps.
Lists of the minimum you need to know: idioms, phrasal verbs etc.
Again, it is a good idea to use an exam preparation book if you need to revise English grammar fast. If you ARE preparing for the Cambridge exam though, you can't do without these guides. I love their approach of teaching you grammar, vocabulary, style, and writing skills all in one. They never disappoint.
If you are learning English for your job, it makes sense to practise your grammar using business vocabulary and concentrate only on what will be useful for your day-to-day operations, not everything.
Business Grammar Builder will teach you not only grammar but functions too – it's a good reference guide for business communication. They use real articles for exercises and you will practise grammar in dialogues (!), which is great and quite rare.
Many exercises are recorded on a CD to help you with pronunciation too.
I could go on and on but I don't want to confuse you. This list is surely not complete – my goal was to share the books which have worked well for my students. However, we all learn differently and you are free to choose what fits you the best and what will lead you to your goal.
Do you have favourite books about English grammar? Did I miss something truly useful, in your opinion? Share your ideas in the comments!


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Kamran Asdar Ali | UT Austin
Dr. Kamran Asdar Ali (University of Texas at Austin) was elected the Vice President of the Association of Asian Studies (AAS)! Dr. Ali is currently an AIPS Institutional Trustee and was the AIPS President from 2011-2017. His term as AAS Vice President starts in March of 2021, and he will become the President of AAS in March 2022. To read more about AAS and their recent election, please check out their website.
Maria-Magdalena Pruss | Leibniz-Zentrum Moderner Orient
Dr. Maria-Magdalena Pruss won the DAVO (German Middle East Studies Association) Award 2020 for best dissertation in the fields of contemporary studies on the Middle East, North Africa,and other parts of the Islamic World for her 2019 dissertation "Islamic Modernism in Colonial Punjab: The Anjuman-i Himayat-i Islam, 1834-1923." Congratulations, Dr. Pruss!
Hassan Abbas | National Defense University
Dr. Abbas recently published a policy brief with the Center for Global Policy (Washington, DC) on "U.S. Policy Toward South Asia: Ideas and Choices for the Next Administration." In the report, Dr. Abbas outlines key issues facing the incoming US administration and makes policy suggestions for navigating COVID recovery, India-Pakistan, and the rising regional influence of China.
Waqas Butt | UTSC
Dr. Waqas Butt received a short-term lecturing and research grant from AIPS in 2015 and has recently published several articles from that research. Click on these links to read more about Dr.
Mushtaq Bilal | Binghamton University
Mushtaq Bilal published "Reading Mohsin Hamid's Exit West as a World Novel" in the July issue of the Journal of World Literature. From the abstract, Bilal writes "This paper investigates how Mohsin Hamid's fiction challenges the so-called "natural law" of world literature according to which a work of literature has to be first born into a national literature before circulating into world literature."
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October 11, 2018 (8:25am-3:15pm)
Parlour Room 638, Madison Concurse Hotel
with an all symposium reception to follow in Madison Ballroom (6:00-7:30pm)
Funded by a grant from CAORC
AIPS hosted its third Junior Scholars Conference on October 11, 2018. This conference showcased the new research being done by junior scholars (both PhDs and graduate students with ABD status) in the field of Pakistan Studies in United States. The conference concluded with a reception hosted by AIPS.
Eight conference participants were selected through a competitive process:
(Paper titles and abstracts subject to change)
Ahsan Kamal, Ph.D. Candidate, Department of Sociology, University of North Carolina at Chapel Hill
Paper Title: Saving Sindhu: River Defense Movements Along the Indus River in Pakistan (Abstract)
Amna Qayyum, Ph.D. Candidate, Department of History, Princeton University
Paper Title: Co-operatives and Contraceptives: Family Planning and Theories of Rural Development in Comilla, East Pakistan (Abstract)
Sohaib Baig, Ph.D. Candidate, Department of History, University of California-Los Angeles
Paper Title: What Makes a Hanafi? The Juristic Boundaries of the Hanafi School of Law across Thatta and Medina in the 18th Century (Abstract)
Michelle Grisé, Associate Policy Researcher, RAND Corporation
Paper Title: Inside the Nuclear Labyrinth: Understanding the Consequences of Nuclear Weapons Development in Pakistan, 1972-1998 (Abstract)
Mashail Malik, Ph.D. Candidate, Department of Political Science, Stanford University
Paper Title: Political Violence and Ethnocentric Trust in Multiethnic Megacities: The Case of Karachi (Abstract)
Neelum Sohail, Ph.D. Candidate, Department of History, Tufts University
Paper Title: Lord of the District and Next to God: Policing and Governance in the Punjab (Abstract)
Shehram Mokhtar, Ph.D. Candidate, School of Journalism and Communication, University of Oregon
Paper Title: Gender & Sexuality of the Other: Under the Gaze of Transnational Documentary (Abstract)
Ghazal Asif, Ph.D. Candidate, Department of Anthropology, Johns Hopkins University
Paper Title: Women's Work: Ethnography of a Pakistani Vocational Center (Abstract)
October 20, 2016 (10:00am -5:30pm)
Conference Room 5, Madison Concurse Hotel
with a reception to follow in Senate A/B (5:30 - 6:30pm)
AIPS is hosted its second Junior Scholars Conference on October 20, 2016. This conference showcased the new research being done by junior scholars (both PhDs and graduate students with ABD status) in the field of Pakistan Studies in United States. The conference concluded with a reception hosted by AIPS.
Seven conference participants were selected through a competitive process
Paper titles and abstracts subject to change
Mashal Saif, Assistant Professor, Department of Philosophy, Clemson University
Paper Title: Sovereignty between God and the State: Insulting Muhammad in Contemporary Pakistan (Abstract)
Sahar Khan, PhD Candidate, Department of Political Science, University of California - Irvine
Paper Title: Ontological Security: Explaining Continued State-sponsorship of Militancy in Pakistan (Abstract)
Shayan Rajani, PhD Candidate, Department of History, Tufts University
Paper Title: Obstructing Geography: Resisting British Interventionism in Early Nineteenth Century Sindh (Abstract)
Faiza Moatasim, Visiting Assistant Professor, Department of Asian Studies, Hamilton College
Paper Title: Negotiating Nonconformity: The Politics of Encroachment in the Planned Modern City of Islamabad (Abstract)
Maira Hayat, PhD Candidate, Department of Anthropology, University of Chicago
Paper Title: Big Companies, Small Bureaucrats: Water and Governance in Pakistan (Abstract)
Abida Bano, PhD Candidate, Department of Political Science, Western Michigan University
Paper Title: Women's Representation in Local Democracy: Formal and Informal Institutions in Khyber Pakhtunkhwa (Abstract)
Saad Gulzar, PhD Candidate, Department of Political Science, New York University
Paper Title: Why Do Citizens Become Politicians? Experimental Evidence on Candidacy (Abstract)
October 22, 2015 (8:30 am - 5:30pm)
Pyle Center, University of Wisconsin-Madison
with a reception to follow in Pyle Center Lee Lounge (5:30 - 6:30 pm)
AIPS is hosted its first Junior Scholars Conference on October 22, 2015. This conference showcased the new research being done by junior scholars (both recent PhDs and graduate students with ABD status) in the field of Pakistan Studies in the United States. The conference was open to the public and concluded with a joint reception co-hosted by the American Institute of Bangladesh Studies for its Conference on Bengali Mangalakāvya and Related Literature, which was held on the same day.
Ten conference participants were selected through a competitive process.
Paper titles and abstracts subject to change.
Majed Akhter, Assistant Professor of Geography, Indiana University - Bloomington
Paper Title: The long hydraulic partition: Producing national space in the Indus Basin (Abstract)
Andrew Amstutz, PhD candidate in History, Cornell University
Paper Title: Urdu in Karachi before Pakistan: Language Politics & Indian Ocean Commercial Connections, 1914-1946 (Abstract)
Yelena Biberman, Assistant Professor, Government Department, Skidmore College
Paper Title: How We Know What We Know about Pakistan: New York Times News Production, 1954-1971 (Abstract)
Elizabeth Bolton, PhD Candidate in Radio-Television-Film, University of Texas at Austin
Paper Title: Khabarnak: Satirizing Current Affairs Television in Pakistan (Abstract)
Waqas H. Butt, PhD candidate in Anthropology, University of California, San Diego
Paper Title: Waste and Its Infrastructures: Networks of Work in Contemporary Lahore (Abstract)
Abdul Haque Chang, PhD candidate in Anthropology, University of Texas at Austin
Paper Title: Different Histories and Geographies of the Indus Delta (Abstract)
Mariam Durrani, PhD candidate in Educational Linguistics and Anthropology, University of Pennsylvania
Paper Title: Desi/Brown as Mediatized Muslim Youth Personae: The Online and Offline Lives of Transnational Pakistanis (Abstract)
Isabel Huacuja Alonso, Postdoctoral Fellow in History, University of Texas at Austin
Paper Title: Awaz de Kahan: Nur Jehan's Radio Voice and the Making and Unmaking of Borders (Abstract)
Faris Ahmed Khan, Lecturer, Department of Anthropology, Brandeis University
Paper Title: Contextualizing Khwaja Sira Rights: Toward a History of the Pakistani State's Gender and Sexuality Politics (Abstract)
Farhan Navid Yousaf, PhD candidate in Sociology, University of Connecticut
Paper Title: Women's Trafficking in Pakistan: Intersecting Vulnerabilities and Repeated Exploitation (Abstract)
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