text:
\section{Introduction} Let $T>0$ some fixed time, $\mathcal{O}$ and $H$ two real separable Hilbert spaces and $V$ a reflexive Banach space with $H'$ and $V'$ the dual spaces of $H$ and $V$ respectively. Consider the following initial value problem involving a controlled SPDE:
\begin{equation}\label{coeq1}
du(t)=(A(t)u(t)+B(u(t),\Phi(t)))dt+\Xi(u(t))dW(t),\,  u(0)=u_0.
\end{equation}
where $B:V\times\mathcal{O}\times\Omega\rightarrow V'$ and $\Xi: H\times\Omega\rightarrow L_2(U;H)$ are measurable maps and the map $\Phi:[0,T]\times\Omega\rightarrow\mathcal{O}$  is progressively measurable which will play the role of the control, $B$ satisfies a locally monotone condition (see condition {\bf A2} below) $A:[0,T]\times V\times\Omega\rightarrow V'$ is  progressively measurable such that $A(t,\cdot,\omega)$  with $(t,\omega)\in[0,T]\times\Omega$ is a linear operator, $\left\{W(t)\right\}_{t\geq0}$ be a cylindrical Wiener process. For more details about the assumptions on the coefficients see the Section 2.\\
\indent The typical optimal control problem is the minimization of  $\mathcal{J}(\Phi)$ where $\mathcal{J}$ is a cost functional (in our case it is given by (\ref{cost})) with $\Phi$ belonging to  $\mathcal{U},$ the set $\mathcal{U}$ will denote the  admissible controls associated with the controlled initial value problem (\ref{coeq1}). The present work deals with deriving the maximum principle (or  Pontryagin's maximum principle) for that control problem, more specifically, we will provide necessary conditions for optimality for that optimal control problem. To attain our objective we use the adjoint backward stochastic partial differential equation (\ref{BSE}) associated with (\ref{coeq1}). This argument is well known (see  \cite{AH3} for example), combining this with some ideias from \cite{AH2} and \cite{HL} we get necessary conditions for optimality.\\
\indent It is important to remark that the use of  backward stochastic differential equations (BSDEs) for deriving the maximum principle for forward controlled stochastic equations was first discussed by Bismut \cite{B}. Several authors showed the relation between BSDEs and the maximum principle for stochastic differential equations, see \cite{AH2}, \cite{AH3}, \cite{O}, \cite{P}, and \cite{YZ} for example.\\
\indent It is important to mention that the works \cite{AH2} , \cite{AH3} and  \cite{FT} use BSDEs to derive the maximum principle in infinite dimension but the results of these papers cannot be applied in the study of the equation in (\ref{coeq1}) because they assume that the derivative of the map $B(\cdot,\cdot)$ in  (\ref{coeq1}) with respect to the variable $u$ and with respect to the variable $\Phi$ is uniformly bounded (which implies that the map $B(\cdot,\cdot)$ is Lipschitz continuous), whereas in the present work we assume that the map $B(\cdot,\cdot)$ is locally monotone and its derivative satisfies {\bf A8} and {\bf A9}.\\
\indent Another important question in the control theory is to study the existence of optimal control, we do not include it in this paper. This question was studied in \cite{CMF} where the existence of optimal control was demonstrated under the assumption that the map $B(\cdot,\cdot)$ satisfies a locally monotone condition and additional conditions.\\
\indent The present article is organized in the following way: in Section 2, we present the basic spaces, the norms, properties and notations which we are going to work with in the subsequent sections. In Section 3, we formulate the optimal control problem and provide estimates to linear equation associated with the equation (\ref{coeq1}). The Section 4 is devoted to study the Adjoint Equation, we demonstrate the existence and uniqueness of a solution and some useful estimates which we will use in the next section.The Section 5 is dedicated to state the goal of the present work: Necessary  Conditions for Optimal Control.
\section{Notation}
Let $H$ be a real separable Hilbert space.  Let $V$ be a
reflexive Banach space. Identify $H$ with its dual $H'$ and denote the dual of $V$ by $V'$. Let
$$
V\subset H\cong H'\subset V'
$$
where the inclusions are assumed to be dense and compact. The triad $(H,V,V')$ is
known as a \textit{Gelfand triple}. We will denote by $\|\cdot\|_V,$ $\|\cdot\|,$ $\|\cdot\|_{V'}$ the norms in $V,$ $H,$ and $V'$ respectively. The inner product in $H$ and the duality scalar product between $V$ and $V'$ will be denoted by $(\cdot,\cdot)$ and $\langle\cdot,\cdot\rangle$ respectively. Furthermore we will denote by $\|\cdot\|_{\mathcal{O}}$ the norm in $\mathcal{O}$ and by $L_2(U;H)$  the space of all Hilbert-Schmidt operators from $U$ to $H$ with inner product  $\langle\cdot,\cdot\rangle_2$ and norm $\|\cdot\|_2$, for $U$ a real separable Hilbert space.\\
\indent Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. Let $\left\{W_t\right\}_{t\geq0}$ be a $R-$cylindrical Wiener process on $U$ with its completed natural filtration $\mathcal{F}_t,$ with $t\geq0.$\\
\indent To simplify notation we use the letter $\mathbb{T}$ for the interval $[0,T].$ $\mathbb{E}(X)$ denotes the mathematical expectation of the random variable $X.$ We abbreviate ``\textit{almost surely} $\omega\in\Omega.$" to  a.s.\\
\indent Let $B$ be a Banach space with norm $\|\cdot\|_B$ and let $\mathcal{B}(B)$ denote the Borel $\sigma-$algebra of $B$. The space $L^2(\Omega\times\mathbb{T};B)$ is the set of all $\mathcal{F}\otimes\mathcal{B}(\mathbb{T})-$measurable processes $u:\Omega\times\mathbb{T}\rightarrow B$ which are $\mathcal{F}_t-$ adapted and $\mathbb{E}(\int_{\mathbb{T}}\|u\|^2_Bdt)<\infty.$ The constant $c_{HV}$ is such that $\|v\|^2\leq c_{HV}\|v\|^2_V$ for all $v\in V$.\\
\indent In order to get solutions to (\ref{coeq1}), we state the following conditions on the coefficients:
Suppose there exist positive constants  $\theta,$ $K$ and a positive adapted process $f\in L^1(\Omega\times\mathbb{T};\mathbb{R})$ such that the following conditions hold for all $v,$ $v_1,$ $v_2\in V,$ $\Phi\in\mathcal{O}$ and a.e. $(t,\omega)\in\mathbb{T}\times\Omega$.
\begin{enumerate}
\item[{\bf A1})]  (Hemicontinuity) The map $s\rightarrow\langle B(v_1+sv_2,\Phi),v\rangle+\langle A(v_1+sv_2,v\rangle$ is continuous on $\mathbb{R}.$
\item[{\bf A2})] (Local monotonicity)
$$\begin{array}{rl}
2\langle B(v_1,\Phi)-B(v_2,\Phi),v_1-v_2\rangle&+\|\Xi(v_1)-\Xi(v_2)\|_2^2\leq\\
&(K+\rho(v_2))\|v_1-v_2\|^2,
\end{array}
$$
where $\rho: V\rightarrow[0,+\infty)$ is a mensurable function and locally bounded in $V.$
\item[{\bf A3})] (Coercivity)
$$
2\langle A(t)v,v\rangle\leq-\theta\|v\|_V^2
$$
and
$$
2\langle B(v,\Psi),v\rangle+\|\Xi(v)\|_2^2\leq K\|v\|^2+f(t).
$$
\item[{\bf A4})] (Growth)
$$
\|A(t)v\|^2_{V'}+\|B(v,\Psi)\|^2_{V'}+\leq (f(t)+K\|v||^2_V)(1+\|v\|^2).
$$
\end{enumerate}
In this work, we understand that the stochastic process $u_{\Phi}$ is a solution to
the problem in (\ref{coeq1}) in the following sense.
\begin{definition} Let $u_0$ be a  random variable which does not depend on $W(t).$ The
stochastic process $(u_{\Phi}(t))_{t\in\mathbb{T}}\in L^2(\Omega\times\mathbb{T};V),$ $\mathcal{F}_t-$ adapted, with a.s. sample paths continuous in $H$, is a solution to (\ref{coeq1}) if it satisfies the equation:
\begin{equation}\label{iska}\begin{array}{rl}
(u_{\Phi}(t),v)\!=&(u_0,v)+\!\displaystyle \int_0^t\left\langle B(u_{\Phi}(s),\Phi(s)),v\right\rangle ds+\displaystyle \int_0^t(A(s)u_{\Phi}(s)),v)ds+\\
&+\displaystyle \int_0^t(v,\Xi(u_{\Phi}(s))dW(s))
\end{array}
\end{equation}
a.s. for all $v\in V$ and $t\in\mathbb{T}.$
\end{definition}
\indent {\bf Uniqueness} means indistinguishability.\\
We need the following existence of solutions theorem, the result is a particular case of \cite[Th. ~1.1]{LR} .
\begin{theorem}\label{exsth} Let $u_0\in L^4(\Omega, V)$. Suppose that {\bf A1}-{\bf A4} is satisfied and there is a constant $C$ such that
\begin{equation}\begin{array}{rl}
\|\Xi(v)\|^2_2\leq& C(f(t)+\|v\|^2), \ \ \ v\in V;\\
\rho(v)\leq &C(1+\|v\|_V^2)(1+\|v\|^2)\ \ \ v\in V.
\end{array}
\end{equation}
The problem (\ref{iska}) has a unique solution $u_{\Phi}$ which  has a.s. sample paths continuous in $H$ and satisfies
\begin{equation}\label{estiska}
\mathbb{E}\left[\sup_{t\in\mathbb{T}}\|u_{\Phi}(t)\|^2+\int_0^T\|u_{\Phi}(t)\|^2_Vdt\|\right]\leq C,
\end{equation}
where $C$ is a constant dependent of $T$ and $u_0.$
\end{theorem}
\begin{proof} The proof follows from theorem 1.1 of \cite{LR} because our assumptions on coefficients of the equation in (\ref{coeq1}) satisfies the conditions of that theorem.
\end{proof} \hfill $\Box$
\section{Formulation of the Control Problem and Estimates}
Let $\mathcal{O}_1$ be a fixed convex subset of $\mathcal{O}.$ The map $\Phi\in L^2(\Omega\times\mathbb{T};\mathcal{O})$ such that $\Phi(t)\in \mathcal{O}_1$ a.e., a.s. is called \textit{admissible control}. The set of admissible controls will be denoted by $\mathcal{U}.$
Let us now define the {\it cost functional}:
\begin{equation}\label{cost}
\mathcal{J}(\Phi):=\mathbb{E}\left[\displaystyle \int_0^T\Bigl(\mathcal{L}(u_{\Phi}(s),\Phi(s))\Bigr)ds+(\mathcal{K}(u_{\Phi}(T)))\right],\ \Phi\in\mathcal{U}
\end{equation}
whenever the integral in (\ref{cost}) exists and is finite. We will assume that the mappings $\mathcal{L}:H\times\mathcal{O}\rightarrow\mathbb{R}_+$ and $\mathcal{K}:H\rightarrow\mathbb{R}_+$ are measurable and satisfy the following conditions:
\begin{enumerate}
\item[{\bf H1})] we assume that $\mathcal{L},$ $\mathcal{K}$ satisfies
$$\begin{array}{rl}|\mathcal{L}(u_1,\Phi_1)-\mathcal{L}(u_2,\Phi_2)|&\leq C_{\mathcal{L}}(\|u_1-u_2\|^2+\|\Phi_1-\Phi_2\|^2_{\mathcal{O}}),\\
|\mathcal{K}(u_1)-&\mathcal{K}(u_2)|\leq C_{\mathcal{K}}\|u_1-u_2\|^2,\end{array}$$
for all  $u_1,\ \  u_2\in H,$ $\Phi_1,\ \ \Phi_2\in\mathcal{O}$ and $C_{\mathcal{L}},$ $C_{\mathcal{K}}$ are positive constants.
\item[{\bf H}2)]  the mappings $\mathcal{L}(\cdot,\cdot)$ and $\mathcal{K}(\cdot)$ are Fr\'echet differentiable and their derivatives $\mathcal{L}_u(\cdot,\cdot),$ $\mathcal{L}_\Phi(\cdot,\cdot)$ and $\mathcal{K}'(\cdot)$ satisfies
$$\begin{array}{rl}
\|\mathcal{L}_u(u,\Phi)\|&\leq k_{\mathcal{L}}\|u\|,\ \mathcal{L}_u(0,\Phi)=0,\ \|\mathcal{L}_\Phi(u,\Phi)\|\leq k_{\mathcal{L}}\|\Phi\|_{\mathcal{O}} \textmd{ and}\\
\|\mathcal{K}'(u)\|&\leq k_{\mathcal{K}}\|u\|
\end{array}
$$
for all $\Phi\in\mathcal{O},$ $u\in H$ and $k_{\mathcal{L}},$ $k_{\mathcal{K}}$ positive constants.

\end{enumerate}
\indent We will assume the following additional conditions for the equation (\ref{coeq1}) which are useful to demonstrate the existence and uniqueness of the solution to linear equation (\ref{linequ}). Let $u,\ v,\ v_1,\ v_2$ be in $V$ and $\Phi,\,\Phi_1\in\mathcal{O}$, suppose that there are positive constants $\theta_1$ $K_2,$ $K_3$, $K_4$, $K_5$ and $K_6$ such that:
\begin{enumerate}
\item[{\bf A5})]
The mapping $\Xi(\cdot)$ is Fr\'echet differentiable and its derivative $\Xi'(u)\in L^2(H,L_2(U;H)).$
\item[{\bf A6})] $B:V\times\mathcal{O}\rightarrow V',$ is a mapping such that the map $B(\cdot,\Phi)$ is Fr\'echet differentiable and we denote by $B_u(u,\Phi)$ its Fr\'echet derivative at the point $u.$
\item[{\bf A7})] For  $u\in V$ fixed, we have
$$\begin{array}{rl}
\langle B_u(u,\Phi)(v),v_1-v_2\rangle\leq K_2\|v_1-v_2\|_V\|v\|\rho_1(u),
\end{array}
$$
where $\rho_1: H\rightarrow[0,+\infty)$ is a measureable function and locally bounded in $H.$
\item[{\bf A8})]
$
\langle (B_u(u,\Phi)-B_u(v,\Phi_1))(w),z\rangle \leq K_3\|z\|_V\rho_1(u-v)\|w\|
$
\item[{\bf A9})]

2\langle B_u(v_1,\Phi)(v),v\rangle+2\|\Xi_u(v_1)(v)\|_2^2\leq\theta_1(\rho_1(v_1))^2\|v\|^2.

\item[{\bf A10})
$
\|A(t)(v)\|_{V'}^2+\|B_u(u,\Phi)(v)\|^2_{V'}+\|\Xi_u(v)\|^2_{V'}\leq (1+K_4\|v\|^2_V)(1+\|u\|^2).
$
\item[{\bf A11})] $B:V\times\mathcal{O} \times\Omega\rightarrow V',$ is a mapping such for each $u\in V$ the mapping $\Phi\in \mathcal{O}\rightarrow B(u,\Phi)\in V'$ is Fr\'echet differentiable and we denote by $B_{\Phi}(u,\Phi)$ its Fr\'echet derivative at the point $\Phi$ which  satisfies:
$$
\|B_\Phi(u,\Phi)(\Upsilon)-B_\Phi(v,\Phi)(\Upsilon_1)\|_{V'}\leq K_5\|\Upsilon-\Upsilon_1\|_{\mathcal{O}}
$$
for all  $\Upsilon,\ \ \Upsilon_1$ $\in\mathcal{O}.$
\item[{\bf A12})]  $\|\Xi'(v)(w)-\Xi'(u)(w)\|_2 \leq  K_6\|w\|,$\ \  $\Xi'(0)=0.$
\end{enumerate}
\indent Our control problem is to minimize (\ref{cost}) over $\mathcal{U}.$ When $\Phi^{\star}\in\mathcal{U}$ satisfies
\begin{equation}\label{minizacost}
\mathcal{J}(\Phi^{\star})=\inf_{\Phi}\mathcal{J}(\Phi)
\end{equation}
is called an \textit{optimal control}. The corresponding solution $u_{\Phi^{\star}}$ of (\ref{coeq1}) and the pair $(u^{\star},\Phi^{\star})$ are called respectively an \textit{optimal solution} and \textit{optimal pair} of the optimal problem control (\ref{coeq1}) and (\ref{minizacost}).\\
\indent Let $\Phi^{\star}$ be  an optimal control,  the corresponding solution  $u_{\Phi^{\star}}$ of (\ref{coeq1}) will be denoted by $u^{\star}:=u_{\Phi^{\star}}$. Now we will consider the initial value problem involving a linear SPDE:
\begin{equation}\begin{array}{ll}\label{linequ}
dP(t)=&A(t)P(t)dt+(B_u(u^{\star}(t),\Phi^{\star})(P(t)))dt+\\
&+(B_{\Phi}(u^{\star}(t),\Phi^{\star})(\Phi(t)))dt+\Xi'(u^{\star}(t))(P(t))dW(t),\\
P(0)&=0.
\end{array}
\end{equation}
\indent Our first objective is to demonstrate the existence and uniqueness of the solution to (\ref{linequ}) (see Theorem \ref{TEUC}). We need to prove this fact because we have not found in the literature a result that can be applied to obtain existence and uniqueness of the solution of the equation (\ref{linequ}) whose coefficients satisfy the conditions {\bf A5} -{\bf A12}. To demonstrate that fact we use well known arguments and some ideas from \cite{HL}. With this result in hand we can demonstrate important estimates for that solution which will be used to obtain the maximum principle in Sect. 5.\\
\indent  Previous definitions are necessary, let  $\{u_{\Phi}(t)\}_ {t\in\mathbb{T}}$ be a $V$-valued stochastic process such that $\sup_{t\in\mathbb{T}}\|u_{\Phi}(t)\|^2<\infty$ and $\int_0^T\|u_{\Phi}(t)\|^2_Vdt<\infty$ for a.e.\\
\indent For each $M,$ a nonnegative integer, we define the following stopping times:
$$
\mathcal{T}_M^{u_{\Phi}}:=\left\{\begin{array}{lr}
T,\textmd{ if }\sup_{t\in\mathbb{T}}\|u_{\Phi}(t))\|^2<M&\\
\inf\left\{t\in\mathbb{T}:\sup_{t\in\mathbb{T}}\|u_{\Phi}(t)\|^2\geq M\right\},\textmd{otherwise}
\end{array}
\right.
$$
and
$$
\mathcal{S}_M^{u_{\Phi}}:=\left\{\begin{array}{lr}
T,\textmd{ if }\int_0^T\|u_{\Phi}(t))\|^2_Vdt<M&\\
\inf\left\{t\in\mathbb{T}:\int^t_0\|u_{\Phi}(s)\|^2_Vds\geq M\right\},\textmd{otherwise.}
\end{array}
\right.
$$
We define $\mathcal{T}_M:=\min\{\mathcal{T}_M^{u_{\Phi}},\mathcal{S}_M^{u_{\Phi}}\}$ and  $u^M(t):=u(t\wedge\mathcal{T}_M).$\\
\indent From the properties of the stopping time it holds that $\lim_{M\rightarrow\infty}\mathcal{T}_M=T$  (see the appendix of \cite{HL}).\\
\indent For  $M,$  integer nonnegative we consider the following initial value problem
\begin{equation}\label{linstop}\begin{array}{rl}
d P^{M}(t)=&A(t)P^M(t)dt+B_u((u^\star)^M(t),\Phi^\star(t))(P^{M}(t))dt+\\
+&B_{\Phi}((u^{\star})^M(t),\Phi^\star(t))\Phi(t)dt+\Xi'((u^{\star})^M(t))(P^{M}(t))dW(t),\\
P^M(0)&=0.
\end{array}
\end{equation}
\indent In order to demonstrate the existence of the solution of (\ref{linequ}), first we will demonstrate the existence of the solution of (\ref{linstop}). With this end we will use the Faedo-Galerkin's method. Let $\{w_k\}_{k=1}^{\infty}$ be a complete orthonormal basis for $H$ and orthogonal in $V.$ For each $n\in \mathbb{Z}^+,$ we consider $H_n:=span\left\{w_1,w_2,\ldots,w_n\right\}$ equipped with the norm induced from $H.$ We define by $\Pi_n: H\rightarrow H_n$ the orthogonal projection such that, $\Pi_nh:=\sum_{i=1}^n(h,w_i)w_i$ for $h\in H,$ we can extend $\Pi$ to a projection operator $\Pi'_n:V'\rightarrow V_n'$ by $\Pi '_nv:=\sum_{i=1}^n\left\langle v,w_i\right\rangle w_i$ with $V_n=H_n=V_n'.$ Analogously if $\left\{v_i\right\}_{i=1}^{\infty}$ be the set of eigenfunctions of $R, $ $K_n:=span\left\{v_1,v_2,\dots,v_n\right\}$ we may to define the projection operator $P_n$ from $K$ into $K_n$ such that $P_nh:=\sum_{i=1}^n(h,v_i).$ We define the following truncations:
\begin{equation}\begin{array}{rl}
(u_{\Phi})_n:=\Pi_n u_{\Phi},&( B_u)_n(u,\Phi):=\Pi_nB_u(u,\Phi),\\
 (B_{\Phi})_n(u,\Phi):=&\Pi_n B_{\Phi}(u,\Phi),\, (\Xi')_n(u):=\Pi_n\Xi'(u),\\
 A_n(t)(u):=&\Pi'_nA(t)(u)\, W^n:=P_nW
\end{array}
\end{equation}
for $t\in\mathbb{T}$, $u\in V$.\\
\indent Analogously  we can assume that it is possible to define the truncation $\Phi_n$ for $\Phi\in\mathcal{O}.$\\
\indent Now, we will consider the following finite dimensional stochastic differential equation:
\begin{equation}\label{linaprostop}\begin{array}{rl}
dP^M_n(t)=&\left(A_n(t)P^M_n(t)+(B_{u})_n(((u^\star)^M)_n(t),\Phi^\star(t))(P^M_n(t))\right)dt+\\
&(B_{\Phi})_n(((u^\star)^M)_n(t),\Phi^\star(t))(\Phi_n(t))dt) +\\
&(\Xi')_n(((u^\star)^M)_n(t))(P^M_n(t))dW^n(t),
\end{array}
\end{equation}
a.s., $t\in\mathbb{T}$ with initial condition $P^M_n(0)=0.$
\begin{theorem}\label{TEUC} Suppose that $\rho_1(u)\leq\|u\|$ and conditions {\bf A1}-{\bf A12} are satisfied, then, there exists a unique
solution $(Z(t))_{t\in\mathbb{T}}$ to the equation (\ref{linequ}) which is a $V-$valued,
$\mathcal{F}\times\mathcal{B}_{\mathbb{T}}-$measurable process and adapted to the filtration
$(\mathcal{F}_t)_{t\in\mathbb{T}}.$ Furthermore, there exists a constant $c$ such that
\begin{equation}\label{estimatlinequ}\begin{array}{rl}
\mathbb{E}(\exp\left(-\int_0^T\rho_1(u^\star(t))dt\right)&\|P(T)\|^2)+\\
\mathbb{E}\int_0^T&(\exp\left(-\rho_1(u^\star(s))ds\right)\|P(t)\|^2_Vdt\leq\\
c\mathbb{E}(\int_0^T&\|u^\star(t)\|^2dt+\mathbb{E}\int_0^T\|\Xi'(u^\star(t)\|^2_2dt),\\
\mathbb{E}(\left(\exp-\int_0^T\rho_1(u^\star(t))dt\right)^2&\|P(T)\|^2)+\\
\mathbb{E}(\int_0^T&(\exp\left(-\rho_1(u^\star(s))ds\right)\|P(t)\|^2_Vdt)^2\leq\\
c(\mathbb{E}(\int_0^T&\|u^\star(t)\|^2dt)^2 +\mathbb{E}(\int_0^T\|\Xi'(u^\star(t)\|^2_2dt))^2
\end{array}
\end{equation}
\end{theorem}
\begin{proof}We can apply the theory of finite dimensional It\^o equation (see \cite{KR1} Sect. 3.3 ) to the equation (\ref{linaprostop}) to demonstrate that there exists a unique solution $P^M_n,$ which has almost surely continuous trajectories in $H.$\\ \indent Now we need estimates to the equation (\ref{linaprostop}). Using the It\^o formula we get
$$\begin{array}{rl}
\|P^M_n(t)\|^2=&2\int_0^t\left\langle A_n(t)P^M_n(s), P^M_n(s)\right\rangle ds+\\
&2\int_0^t\left\langle (B_u)_n(((u^\star)^M)_n(s),\Phi^\star(s))(P^M_n(s)),P^M_n(s)\right\rangle ds+\\
&2\int_0^t\left\langle (B_{\Phi})_n(((u^\star)^M)_n(s),\Phi^\star(t))(\Phi_n(s),P^M_n(s)\right\rangle ds+\\
&\int_0^t\|(\Xi_u)_n(((u^\star)^M)_n(s))(P^M_n(s))\|^2_2 ds+\\
&2\int_0^t(P^M_n(s),(\Xi')_n(((u^\star)^M)_n(s))(P^M_n(t))dW^n(s)).
\end{array}
$$
Using the equation above, the It\^o formula with $e(t)=\exp(-\int_0^t(\rho_1(((u^{\star})^M))_n)^2(s)ds)$ and from our assumptions \textbf{A10} , \textbf{A11} and  \textbf{A12} we have
$$\begin{array}{rl}
\mathbb{E}(e(t)\|P^M_n(t)\|^2)&+\frac{\theta}{2}\mathbb{E}(\int_0^t e(s)\|P^M_n(t)\|^2_Vdt)\leq c [\mathbb{E}(\int_0^te'(s)\|P^M_n(s)\|^2ds)+\\
&\mathbb{E}\int_0^te(s)(\rho_1((u^\star)^M)_n(s))^2\|P^M_n(s)\|^2ds+\\
&\mathbb{E}\int_0^te(s)\|\Phi(s)\|^2_{\mathcal{O}}ds+K_6\mathbb{E}(\int_0^te(s)\|P^M_n(s)\|^2ds)]
\end{array}
$$
Where $c$ is  a constant independent of $n$ and $M$. The last inequality and the Gronwall's inequality imply that
\begin{equation}
\mathbb{E}(e(t)\|P^M_n(t)\|^2)+\frac{\theta}{2}\mathbb{E}(\int_0^T e(t)\|P^M_n(t)\|^2_Vdt)\leq c\mathbb{E}(\int_0^T\|\Phi(s)\|^2_{\mathcal{O}}ds),
\end{equation}
where $c$ is  a constant independent of $n.$\\
From the inequality above and using the assumption about $\rho_1$ we obtain
\begin{equation}\begin{array}{rl}\label{estimatlinear}
\mathbb{E}(\int_0^t\|P^M_n(t)\|^2_Vdt)&\leq\\
\mathbb{E}(&(e(T))^{-1}\int_0^te(s)\|P^M_n(s)\|^2_Vds)\leq\\
c\mathbb{E}(&\exp( \int_0^T\|u^\star(t\wedge\mathcal{T}_M)(t)\|^2)dt)\mathbb{E}(\int_0^T\|\Phi(s)\|^2_{\mathcal{O}}ds)\leq\\
c\mathbb{E}(&\int_0^T\|\Phi(s)\|^2_{\mathcal{O}}ds),
\end{array}
\end{equation}
where $c$ is a constant that depends on $M.$\\
Analogously we can prove that
\begin{equation}
\mathbb{E}(\|P^M_n(t)\|^2)^2\leq c
\end{equation}
and
\begin{equation}
\mathbb{E}(\|P^M_n(t)\|^4)+\mathbb{E(}\int_0^T \|P^M_n(t)\|^2_Vdt)^2\leq c
\end{equation}
where $c$ is a constant independent of $n$ but dependent of $M,$ $T$ and $\Phi$.\\
Thus, the inequality (\ref{estimatlinear}) implies that the sequence $(P_n^M)$ is bounded in the space $L^2(\Omega\times\mathbb{T};V).$ Then, there exists a subsequence of $(P_n^M)$ relabeled the same and $P^M \in L^2(\Omega\times\mathbb{T};V)$ such that letting $n\rightarrow\infty$ we have
\begin{equation}\label{weakconv2}
(P_n^M)\rightharpoonup P^M.
\end{equation}
\indent Now, we will prove that
\begin{equation}\label{convT}
(B_u)_n (((u^\star)^M)_n,\Phi^\star)(P^M_n)\rightharpoonup B_u((u^\star)^M,\Phi^\star)(P^M)
\end{equation}
in $L^2(\Omega\times\mathbb{T};V').$ Let $v\in V$ and $v_n=\Pi_n v.$ To demonstrate (\ref{convT}), first we observe that
\begin{equation}\label{convTI}\begin{array}{rl}
\langle(B_u)_n (((u^\star)^M)_n,\Phi^\star)(P^M_n),v\rangle&=\langle B_u(((u^\star)^M)_n,\Phi^\star)(P^M_n),v_n\rangle=\\
\langle(B_u(((u^\star)^M)_n,\Phi^\star)(P^M_n),v_n\rangle&-\langle(B_u(((u^\star)^M)_n,\Phi^\star)(P^M_n),v\rangle+\\
\langle B_u(((u^\star)^M)_n,\Phi^\star)(P^M_n),v\rangle&-\langle B_u((u^\star)^M,\Phi^\star)(P^M_n),v\rangle+\\
\langle B_u((u^\star)^M,\Phi^\star)(P^M_n),v\rangle&-\langle B_u((u^\star)^M,\Phi^\star)(P^M),v\rangle+\\
&\langle B_u((u^\star)^M,\Phi^\star)(P^M),v\rangle.
\end{array}
\end{equation}
 We need to estimate each term in the last inequality. From \textbf{A7} and the assumption about $\rho_1$, we have
 \begin{equation}\label{convT1}\begin{array}{rl}
 \mathbb{E}(\int_0^T|\langle(B_u(((u^\star)^M)_n(t)&,\Phi^\star(t))(P^M_n(t)),v_n\rangle-\\
 \langle B_u(((u^\star)^M)_n(t),&\Phi^\star(t))(P^M_n(t)),v\rangle|dt)\leq\\
  c\|v_n-v\|_V(\mathbb{E}&(\int_0^T\|(P^M)_n)\|^2dt)^{1/2}(\mathbb{E}(\int_0^T\|(u^\star)^M_n\|^2dt))^{1/2},
 \end{array}
 \end{equation}
where $c$ is a generic constant.\\
Using \textbf{A8}  and the assumption about $\rho_1$, we have
 \begin{equation}\label{convT2}\begin{array}{rl}
 \mathbb{E}(\int_0^T\langle B_u(((u^\star)^M)_n(t),&\Phi^\star(t))(P^M_n(t)),v\rangle-\\
 &\langle B_u((u^\star)^M(t),\Phi^\star(t))(P^M_n(t)),v\rangle dt)\leq\\
 &c\|v\|_V(\mathbb{E}(\int_0^T\|P^M)_n(t)dt\|^2)^{1/2}\times\\
 &(\mathbb{E}(\int_0^T\|((u^\star)^M)_n(t)-(u^\star)^M(t)\|^2dt))^{1/2},
\end{array}
 \end{equation}
 where $c$ is a generic constant.\\
 And from (\ref{weakconv2}) we have
 \begin{equation}\label{convT3}
  B_u((u^\star)^M,\Phi^\star)(P^M_n)\rightharpoonup B_u((u^\star)^M,\Phi^\star)(P^M).
 \end{equation}
Using (\ref{convT1}), (\ref{convT2}), (\ref{convT3})  in (\ref{convTI}) we obtain
$$\begin{array}{rl}
\lim_{n\rightarrow\infty}\int_0^T&\langle(B_u)_n (((u^\star)^M)_n(t),\Phi^\star(t))(P^M_n)(t),\lambda(t)\rangle dt=\\
&\int_0^T\langle B_u ((u^\star)^M(t),\Phi^\star(t))P^M(t),\lambda(t)\rangle dt
\end{array}
$$
for all $\lambda=v\phi,$ with $v\in V,$ $\phi$ a $V-$valued process, $\mathcal{F}\times\mathcal{B}(\mathbb{T})-$measurable, $\mathcal{F}_t-$adapted and for a.e. $(\omega,t)$ bounded. As the maps $\lambda$ with these properties are dense in $L^2(\Omega\times\mathbb{T};V)$ we get (\ref{convT}).\\
Using \textbf{A11} it is possible to demonstrate that
\begin{equation}\label{weakconv3}
B_\Phi((u^\star)^M_n,\Phi^\star)(\Phi_n)\rightharpoonup B_\Phi((u^\star)^M,\Phi^\star)(\Phi)
\end{equation}
\indent Using (\ref{convT}), (\ref{weakconv3}) and taking the limit $n\rightarrow\infty$ in  (\ref{linaprostop}) we obtain
 \begin {equation}\begin{array}{rl}
 (P^M(t),v)=&\displaystyle \int_0^t(A(s),P^M(s))ds+\int_0^t(B_u((u^{\star})^M(s),\Phi^{\star}(s))(P^M(s)),v)ds+\\
 &\displaystyle \int^t_0(B_{\Phi}((u^{\star})^M(s),\Phi^{\star}(s))(\Phi(s)),v)ds+\\
 &\displaystyle \int_0^t(v,\Xi'(u^{\star}(s)(P^M(s)))dW(s)).
 \end{array}
 \end{equation}
 From this and arguing as \cite{HL} it is possible to conclude that there is a process $\{P^M_{\Phi}(t)\}_{t\in\mathbb{T}}$ in $L^2(\Omega\times\mathbb{T};V)$ which has a.s. continuous trajectories in $H$ and satisfies  (\ref{linstop}) a.s for all $v\in V,$ $t\in\mathbb{T}.$ Using well known methods we can demonstrate that that process is the unique solution.\\
 \indent We proceed to demonstrate the existence of the solution of  (\ref{linequ}), we will use the argument of \cite{HL}. Let $\Omega_M=\{\omega\in\Omega: P^M_{\Phi}(\omega,\cdot)\textmd{ satisfies }(\ref{linstop})\textmd{ for all }t\in\mathbb{T},\, v\in V\textmd{ and has continuous trajectories in }H\},$  $\Omega'=\bigcap_{M=1}^{\infty}\Omega_M,$  and $S=\bigcup_{K=1}^{\infty}\bigcup_{1\leq M\leq K}\{\omega\in\Omega':\mathcal{T}_M=T\textmd{ and there is }t\in \mathbb{T}, P^M_{\Phi}(\omega,t)\neq P^K_{\Phi}(\omega,t)\},$ it is possible to demonstrate that $\mathbb{P}(S)=0.$ \\
 \indent We define $\Omega''=\bigcup_{M=1}^{\infty}\{\mathcal{T}_M=T\},$ then $\mathbb{P}(\Omega'\bigcap\Omega'')\setminus S)=1.$ For $\omega\in\Omega'\bigcap\Omega'')\setminus S$ there is a natural number $M_0$ such that $\mathcal{T}_M=T$ for all $M\geq M_0.$ Thus, $(u^{\star})^M(t)= (u^{\star})^{M_0}(t)$ for all $t\in\mathbb{T}$ and for all $M\geq M_0.$ From (\ref{linstop}) we have
  \begin {equation}\label{lineqend}\begin{array}{rl}
 (P^M(t),v)=&\displaystyle \int_0^t(A(s),P^M(s))ds+\int_0^t(B_u(u^{\star}(s),\Phi^{\star}(s))(P^M(s)),v)ds+\\
 &\displaystyle \int^t_0(B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))(\Phi(s)),v)ds+\\
 &\displaystyle \int_0^t(v,\Xi'(u^{\star}(s)(P^M(s)))dW(s)),
 \end{array}
 \end{equation}
 for all $M\geq M_0,$ and all $t\in\mathbb{T},$ $v\in V.$ Then,
 $$
 \lim_{M\rightarrow\infty}\int_0^T\|P^M(t)-P^M_0(t)\|_V^2dt=0
 $$
 and
 $$
 \lim_{M\rightarrow\infty}\|P^M(t)-P^M_0(t)\|^2dt=0
 $$
 for all  $t\in\mathbb{T}.$\\
 \indent For each $t\in\mathbb{T}$ we define
 $$
 P(\omega,t):=P^M_0(\omega,t)\lim_{M\rightarrow\infty}P^M(\omega,t).
 $$
 Thanks to (\ref{lineqend}) we obtain
  \begin {equation}\label{lineqend1}\begin{array}{rl}
 (P(t),v)=&\displaystyle \int_0^t(A(s),P(s))ds+\int_0^t(B_u(u^{\star}(s),\Phi^{\star}(s))(P(s)),v)ds+\\
 &\displaystyle \int^t_0(B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))(\Phi(s)),v)ds+\\
 &\displaystyle \int_0^t(v,\Xi'(u^{\star}(s)(P(s)))dW(s)),
 \end{array}
 \end{equation}
 for all $\omega\in\Omega'\bigcap\Omega'')\setminus S),$ $t\in\mathbb{T},$ $v\in V.$ This demonstrates the existence of the solution of (\ref{linequ}). The inequalities in (\ref{estimatlinequ}) and unicity are demonstrated as  in \cite{HL}.
\end{proof} \hfill $\Box$
\section{The Adjoint Equation}
It is known that there is a connection between BSDEs and the maximum principle for stochastic differential equations. Thus, it seems natural to consider that there is the same connection with the equation (\ref{coeq1}). With this objective, let us first consider the \textit{Hamiltonian}:
\begin{equation}\label{hamilt}
\begin{array}{rl}
\mathcal{H}:&V\times\mathcal{O}\times V\times L_2(U;H)\rightarrow\mathbb{R},\\
\mathcal{H}(u,\Phi,v,Z)&:=\mathcal{L}(u,\Phi)+\langle B(u,\Phi),v\rangle+\langle\Xi(u),Z\rangle_2
\end{array}
\end{equation}
Now, we consider the adjoint equation associated with (\ref{coeq1}) which is given by the following terminal value problem involving a BSPDE:
\begin{equation}\label{BSE}\begin{array}{rl}
-dv^{\Phi}(t)=&(A^{\star}v^{\Phi}(t)+\nabla_u\mathcal{H}(u^{\Phi}(t),\Phi(t),v^{\Phi}(t),Z^{\Phi}(t)))dt-Z^{\Phi}(t)dW(t),\\
v^{\Phi}(T)=&\nabla\mathcal{K}(u^{\Phi}(T)).
\end{array}
\end{equation}
Where $\nabla$ denotes the gradient.\\
According to (\ref{hamilt}) we can rewrite (\ref{BSE}) in the following way:
\begin{equation}\label{BSER}
\begin{array}{rl}
dv^{\Phi}(t)=&(-A^{\star}v^{\Phi}(t)-\nabla_u\mathcal{L}(u^{\Phi}(t),\Phi(t))-\nabla_u\langle B(u^{\Phi}(t),\Phi(t),v^{\Phi}(t))\rangle+\\
-&\nabla_u\langle\Xi(u^{\Phi}(t)),Z^{\Phi}(t)\rangle_2)dt+Z^{\Phi}(t)dW(t),\ \ 0\leq t\leq T,\\
v^{\Phi}(T)=&\nabla\mathcal{K}(u^{\Phi}(T)).
\end{array}
\end{equation}
\begin{definition}In this work we understand that the $\mathcal{F}_t-$adapted pair $(v,Z)$ in $L^2(\Omega\times\mathbb{T};H)\times L^2(\Omega\times\mathbb{T};L_2(U;H))$ is a solution of (\ref{BSER}) if the following is satisfied
\begin{equation}\label{BSEI}\begin{array}{rl}
(v^{\Phi}(t),y)=&(v^{\Phi}(T),y)+\int^T_t(\langle A^{\star}v^{\Phi}(s),y\rangle+(\nabla_u\mathcal{L}(u^{\Phi}(s),\Phi(s)),y)_Vds+\\
&\int^T_t\langle B_u(u^{\Phi}(s),\Phi(t))(y),v^{\Phi}(s)\rangle+(\Xi_u(u^{\Phi}(s))(y),Z^{\Phi}(s))_2)ds-\\
&\int^T_t(y,Z^{\Phi}(s)dW(s))
\end{array}
\end{equation}
a.s. for all $y\in V$ and $t\in\mathbb{T}.$
\end{definition}
Let $M>0$ be a  natural number, given $\Phi\in\mathcal{U},$ let $u^{\Phi}$ be the corresponding solution to SPDE in (\ref{coeq1}) and $u^{\Phi,M}:=(u^{\Phi})^M.$\\
\indent In order to prove that there is a solution (\ref{BSE}) we will consider the following terminal value problem:
\begin{equation}\label{BSERstp}\begin{array}{rl}
dv^{\Phi, M}(t)&=(-A^{\star}v^{\Phi,M}(t)-\nabla_u\mathcal{L}({u^{\Phi,M}(t)},\Phi(t))-\\
&\nabla_u\langle B(u^{\Phi,M}(t),\Phi(t)),v^{\Phi,M}(t)\rangle-\nabla_u\langle\Xi(u^{\Phi,M}(t)),Z^{\Phi,M}(t)\rangle_2)dt+\\
&Z^{\Phi,M}(t)dW(t),\\
v^{\Phi,M}(T)=&\nabla\mathcal{K}(u^{\Phi}(T)).
\end{array}
\end{equation}
\indent To demonstrate that there is a solution to the equation (\ref{BSERstp}) we will use the Faedo-Galerkin's method.  Thus, for each $n\in\mathbb{Z}^+,$ we will consider the projected equation corresponding with (\ref{BSERstp}):
\begin{equation}\label{BSERn}\begin{array}{rl}
d(v^{\Phi,M})_n(t)=&(-A^{\star}_n(v^{\Phi,M})_n(t)-(\nabla_u\mathcal{L})_n(((u^{\Phi,M}))_n(t),\Phi_n(t))-\\
&(\nabla_u)_n\langle B_n(((u^{\Phi,M}))_n(t),\Phi_n(t),(v^{\Phi})_n(t))\rangle+\\
&(\nabla_u)_n\langle\Xi_n(((u^{\Phi,M}))_n(t)),(Z^{\Phi})_n(t)\rangle_2)dt+\\
&(Z^{\Phi})_n(t)dW^n(t),\\
(v^{\Phi})_n(T)=&(\nabla\mathcal{K})_n((u^{\Phi})(T)).
\end{array}
\end{equation}
\begin{proposition}
Suppose that {\bf H1} - {\bf H2}  and {\bf A8} - {\bf A12} is satisfied. There is a solution $(v^{\Phi,M},Z^{\Phi,M})$ to (\ref{BSERstp}) for each $M=1,2,\ldots$ and
\begin{equation}\label{estimequM}
\mathbb{E}(\|v^{\Phi,M)}\|^2(t)+\mathbb{E}(\int^T_t\|v^{\Phi,M}\|_V^2ds)+\mathbb{E}(\int^T_t\|Z^{\Phi,M}\|^2_2ds)\leq c\ \
\end{equation}
with $t\in\mathbb{T}$, to a constant $c$ dependent of $M.$
\end{proposition}
{\bf Proof:}  It is possible to demonstrate that the coefficients of the equation (\ref{BSERn}) satisfy the conditions of the Theorem 3.4 of   \cite{BD}. Thus, let $((v^{\Phi,M})_n,(Z^{\Phi,M})_n)$ be a solution  to the equation (\ref{BSERn}).\\

\indent Now we need to obtain estimates to the solution $((v^{\Phi,M})_n,(Z^{\Phi,M})_n).$ To do this we employ the It\^o formula to that solution and to the function $e^{r(t)}$, with $r(t)=(\int_0^t1+2K_6+\frac{K_2^2}{\theta}((\rho_1((u^{\Phi,M})^n(s))^2))ds$ and taking expectation we get
$$\begin{array}{rl}
\mathbb{E}(e^{r(t)}&\|(v^{\Phi,M})_n(t)\|^2)+\mathbb{E}(\int_t^Te^{r(s)}\|(Z^{\Phi,M})_n(s)\|^2_2ds)=\mathbb{E}(e^{r(T)}\|(v^{\Phi,M})_n(T)\|)+\\
&2\mathbb{E}(\int_t^Te^{r(s)}\langle A^{\star}  (v^{\Phi,M})_n(s), (v^{\Phi,M})_n(s)\rangle ds)+\\
&2\mathbb{E}(\int_t^Te^{r(s)}((\nabla_u)_n\mathcal{L}((u^{\Phi})_n(s),\Phi_n(s)),(v^{\Phi,M})_n(s))ds)+\\
&2\mathbb{E}(\int_t^Te^{r(s)}((\nabla_u)_n\langle B((u^{\Phi,M})_n(s),\Phi_n(s),(v^{\Phi,M})_n(s))\rangle,(v^{\Phi,M})_n(s))ds)+\\
&2\mathbb{E}(\int_t^Te^{r(s)}((\nabla_u)_n(\Xi((u^{\Phi})_n(s)),(Z^{\Phi,M})_n(s))_2 ,(v^{\Phi,M})_n(s))ds)-\\
&\mathbb{E}(\int_t^Tr'(s)e^{r(s)}\|(v^{\Phi,M})_n(s)\|^2ds).
\end{array}
$$
From this and using  {\bf A8}-{\bf A12} we have
$$
\begin{array}{rl}
\mathbb{E}(e^{r(t)}&\|(v^{\Phi,M})_n(t)\|^2)+\frac{1}{2}\mathbb{E}(\int_t^Te^{r(s)}\|(Z^{\Phi,M})_n(s)\|^2_2ds)+\\
&\frac{\theta}{2}\mathbb{E}(\int_t^Te^{r(s)}\|(v^{\Phi,M})_n(s)\|_V^2ds)\leq \mathbb{E}(e^{r(T)}\|v^{\Phi,M}(T)\|)+\\
&\mathbb{E}(\int_t^Te^{r(s)}K_{\mathcal{L}}\|(u^{\Phi,M})_n(s)\|^2ds)+\mathbb{E}(\int_t^Te^{r(s)}\|(v^{\Phi,M})_n(s)\|^2ds)+\\
&\frac{K_2^2}{\theta}\mathbb{E}(\int_t^Te^{r(s)}(\rho_1((u^{\Phi})_n(s)))^2\|(v^{\Phi,M})_n(s)\|^2ds)+\\
&2\mathbb{E}(\int_t^Te^{r(s)}K_6\|(v^{\Phi,M})_n(s)\|^2ds)-\\
&\mathbb{E}(\int_t^Tr'(s)e^{r(s)}\|(v^{\Phi,M})_n(s)\|^2ds).
\end{array}
$$
Since that $r'(t)=1+2K_6+\frac{K^2_2}{\theta}(\rho_1((u^{\Phi,M})^n(t)))^2$ from the last inequality we have
$$
\begin{array}{rl}
\mathbb{E}(e^{r(t)}&\|v^{\Phi,M}(t)\|^2)+\frac{1}{2}\mathbb{E}(\int_t^Te^{r(s)}\|Z^{\Phi,M}(s)\|^2_2ds)+\\
&\frac{\theta}{2}\mathbb{E}(\int_t^Te^{r(s)}\|v^{\Phi,M}(s)\|_V^2ds\leq\mathbb{E}e^{r(T)}\|v^{\Phi,M}(T)\|.\\
\end{array}
$$
Then,
 $$
 \mathbb{E}(e^{r(t)}\|(v^{\Phi,M})_n(t)\|^2)+\mathbb{E}(\int_t^Te^{r(s)}\|(v^{\Phi,M})_n(s)\|_V^2ds+\mathbb{E}(\int_t^Te^{r(s)}\|(Z^{\Phi,M})_n(s)\|^2_2ds) \leq c
 $$
 where $c$ is a constant independent of $n$.\\
 Using the same argument as the demonstration of Theorem \ref{TEUC}, we can demonstrate that
 $$
 \mathbb{E}\|(v^{\Phi,M})_n(t)\|^2+\mathbb{E}(\int_t^T\|(v^{\Phi,M})_n(s)\|_V^2ds+\mathbb{E}(\int_t^T\|(Z^{\Phi,M})_n(s)\|^2_2ds) \leq c
 $$
 for a constant $c$ which depends on $M$ but independent of $n.$\\
\indent Since the sequences $((v^{\Phi,M})_n)$ and  $((Z^{\Phi,M})_n)$ are bounded in the space $L^2(\Omega\times\mathbb{T};V)$ and $L^2(\Omega\times\mathbb{T}:L_2(U;H))$ respectively, there exists subsequences of $((v^{\Phi,M})_n)$ and $((Z^{\Phi,M})_n)$ relabeled the same and $v^M \in L^2(\Omega\times\mathbb{T};V)$ and $Z^M\in L^2(\Omega\times\mathbb{T}:L_2(U;H))$ such that letting $n\rightarrow\infty$ we have
\begin{equation}\label{weakconv4}
\begin{array}{rl}
((v^{\Phi,M})_n)\rightharpoonup v^M,\\
((Z^{\Phi,M})_n)\rightharpoonup Z^M.
\end{array}
\end{equation}
Following the same argument as the proof of the Theorem \ref{TEUC}, we can demonstrate the following weak convergence $(\nabla_u\mathcal{L})_n(((u^{\Phi,M}))_n,\Phi_n)\rightharpoonup (\nabla_u\mathcal{L})((u^{\Phi,M}),\Phi),$ $(\nabla_u)_n\langle B(((u^{\Phi,M}))_n,\Phi_n),(v^{\Phi,M})_n\rangle \rightharpoonup (\nabla_u)\langle B(u^{\Phi,M},\Phi),(v^{\Phi,M})\rangle,$ $(\nabla_u)_n\langle\Xi(((u^{\Phi,M}))_n),(Z^{\Phi,M})_n\rangle_2\rightharpoonup(\nabla_u)\langle\Xi((u^{\Phi,M}),(Z^{\Phi,M})\rangle_2$ as $n\rightarrow\infty.$\\
Letting $n\rightarrow\infty$ in (\ref{BSERn}) we obtain a solution $(v^{\Phi,M},Z^{\Phi,M})$ to equation (\ref{BSERstp}). \hfill $\Box$
\begin{theorem}\label{BSExUn} Given $u\in V,$ and $\Phi\in\mathcal{O}.$ Under the assumptions of the last proposition, there is a unique solution $(v^{\Phi},Z^{\Phi})$ to (\ref{BSE}).
\end{theorem}
{\bf Proof:}
With the goal to demonstrate that there is a solution to equation (\ref{BSE},) we will use the same argument as in the proof of Theorem \ref{TEUC}, let $\Omega_M=\{\omega\in\Omega:(v^{\Phi,M}(\omega,\cdot),Z^{\Phi,M}(\omega,\cdot))\textmd{ for all }t\in\mathbb{T},v\in V \}.$ Set $\Omega'=\bigcap_{M=1}^{\infty}\Omega_M$ and $S=\bigcup_{K=1}^{\infty}\bigcup_{1\leq M\leq K}\{\omega\in\Omega':\mathcal{T}_M=T\textmd{ and there is }t\in\mathbb{T},\  (v^{\Phi,M}(\omega,t),Z^{\Phi,M}(\omega,t)) \neq(v^{\Phi,K}(\omega,t),Z^{\Phi,K}(\omega,t))\},$ it is possible to demonstrate that $\mathbb{P}(S)=0.$\\
\indent Let $\Omega"=\bigcup_{M}\{\mathcal{T}_M=T\},$ it is possible to demonstrate that $\mathbb{P}(((\Omega'\cap\Omega"))\setminus S)=1.$ Let  $\omega\in(\Omega'\cap\Omega"))\setminus S,$ for this $\omega$ there is a natural number $M_0$ such that
$\mathcal{T}_M=T$ for all $M\geq M_0$ implying that $u^{\Phi,M}(s)=u^{\Phi}(s)$ for all $s\in\mathbb{T}$ and for all $M\geq M_0.$ The equation (\ref{BSERstp}) implies
\begin{equation}\label{BSEIntSt}\begin{array}{rl}
(v^{\Phi,M}(t),y)\!\!&=(v^{\Phi,M}(T),y)+\int^T_t(\langle A^{\star}v^{\Phi,M}(s),y\rangle+(\nabla_u\mathcal{L}(u^{\Phi}(s),\Phi(s)),y)_Vds\\
\!\!\!\!&+\int^T_t\langle B_x(u^{\Phi}(s),\Phi(t))(y),v^{\Phi,M}(s)\rangle+(\Xi_x(u^{\Phi}(s))(y),Z^{\Phi,M}(s))_2)ds\\
&-\int^T_t(y,Z^{\Phi}(s)dW(s))
\end{array}
\end{equation}
a.s. for all $y\in V$ and $t\in\mathbb{T}.$
Hence
$$\begin{array}{rl}
\|(v^{\Phi,M}-v^{\Phi,M_0})(t)\|^2&=\int_0^T\|(v^{\Phi,M}-v^{\Phi,M_0})(s)\|^2_Vds=\\
&\int_0^T\|(Z^{\Phi,M}-Z^{\Phi,M_0})(s)\|^2_2ds=0
\end{array}
$$
for all $t\in\mathbb{T}.$ Thus, taking $v^{\Phi}(\omega,t)=v^{\Phi,M_0}(\omega,t)$ and $Z^{\Phi}(\omega,t)=Z^{\Phi,M_0}(\omega,t)$ for $t\in\mathbb{T}$ and $\omega\notin S,$ we obtain the solution $(v^{\Phi},Z^{\Phi})$ to equation  (\ref{BSE}).
 \hfill $\Box$
\section{A stochastic maximum principle}
Let $\Phi^{\star}$ be an optimal control and let $u^{\Phi^{\star}}$ be the corresponding solution of (\ref{coeq1}). This solution will be denoted briefly by $u^{\star}$. Let $\Phi$ be such that $\Phi^{\star}+\Phi\in\mathcal{U}.$ For a given $0\leq\epsilon\leq1$ consider the control:\\
$$
\Phi_{\epsilon}(t)=\Phi^{\star}(t)+\epsilon\Phi(t),\ \ \ t\in\mathbb{T},
$$
to which will be associated the  solution of (\ref{coeq1}), $u^{\Phi_{\epsilon}},$ which will be denoted by $u_{\epsilon}$ for short.\\
Now, we will obtain some important estimates that we will use to obtain the maximum principle.
\begin{lemma}\label{estpm} With the notation above we have
\begin{equation}\label{estimat3}
\sup_{t\in\mathbb{T}}\mathbb{E}(e(t)(\|u_{\epsilon}-u^{\star})(t)\|^2)=O(\epsilon^2),
\end{equation}
where $e(t)=e^{-\int_0^tK+\rho(u^{\star}(s))ds}$ and $K$ is the constant from {\bf A2}.
\end{lemma}
 \begin{proof} Using the It\^o's formula and taking the expectation,  we get
 $$\begin{array}{rl}
 \mathbb{E}(e(t)\|\!\!\!\!&(u_{\epsilon}-u^{\star})(t)\|^2)\leq\\
 &2\mathbb{E}(\int_0^te(s)\langle B(u_{\epsilon}(s),\Phi_{\epsilon}(s))-B(u^{\star}(s),\Phi^{\star}(s)),u_{\epsilon}-u^{\star}(t)\rangle ds))+\\
&2\mathbb{E}(\int_0^te(s)\langle A(s)(u_{\epsilon}-u^{\star})(s),(u_{\epsilon}-u^{\star})(s)\rangle ds)+\\
&\mathbb{E}(\int_0^te(s)\|\Xi(u_{\epsilon}(s),\Phi_{\epsilon}(s))-\Xi(u^{\star}(s),\Phi^{\star}(s))\|^2_2ds)+\\
&2\mathbb{E}(\int_0^te'(s))\|(u_{\epsilon}-u^{\star})(t)\|^2ds)\leq\\
&2\mathbb{E}(\int_0^te(s)\langle B(u_{\epsilon}(s)),\Phi_{\epsilon}(s))-B(u_{\epsilon}(s),\Phi^{\star}(s)),(u_{\epsilon}-u^{\star})(s)\rangle ds))+\\
&2\mathbb{E}(\int_0^te(s)\langle B(u_{\epsilon}(s),\Phi^{\star}(s))-B(u^{\star}(s)),(\Phi^{\star}(s),(u_{\epsilon}-u^{\star})(s)\rangle ds))+\\
&\mathbb{E}(\int_0^te'(s)\|(u_{\epsilon}-u^{\star})(s)\|^2ds)+\mathbb{E}(\int_0^te(s)\|\Xi(u_{\epsilon}(s))-\Xi(u^{\star}(s))\|^2_2ds)+\\
&2\mathbb{E}(\int_0^t\langle A(s)(u_{\epsilon}-u^{\star})(s),(u_{\epsilon}-u^{\star})(s)\rangle ds)
\end{array}
 $$
Since {\bf A11}, we have
$$\begin{array}{rl}
\mathbb{E}(\int_0^t\|B(u_{\epsilon}(s))\!\!\!\!&,\Phi_{\epsilon}(s))-B(u_{\epsilon}(s),\Phi^{\star}(s))\|^2_{V'}ds)=\\
\mathbb{E}(\int_0^t\|\int_0^1\!\!\!\!&B_{\Phi}(u_{\epsilon}(s),\Phi^{\star}(s)+r(\Phi_{\epsilon}(s)-\Phi^{\star}(s)))(\Phi_{\epsilon}(s)-\Phi^{\star}(s))dr\|^2_{V'}ds)\leq\\
&K_5\epsilon^2\mathbb{E}(\int_o^t\|\Phi(t)\|^2_{\mathcal{O}}dt).
\end{array}
$$
Combining these two inequalities above,  {\bf A2} and {\bf A3},  we have
$$
 \mathbb{E}(e(t)\|(u_{\epsilon}-u^{\star})(t)\|^2)+\frac{\theta}{2}\mathbb{E}(\int_0^te(s)\|(u_{\epsilon}-u^{\star})(s)\|^2_Vds)\leq\epsilon^2C,\\
 $$
 where $C$ represents a generic constant. Thus, (\ref{estimat3}) follows from the last inequality.
\end{proof}
  \hfill $\Box$ \\
 \indent Let $0\leq r\leq1$ and $0<\epsilon<1.$ Set $e_1(t):=e^{-\int_0^t(\rho_1(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s))))^2ds}$ we need to prove the property
\begin{equation} \label{specP}
 \mathbb{E}(e_1^{-4}(T)<\infty.
\end{equation}
It is possible to demonstrate that under special assumptions  this property is satisfied. To prove it we use a similar argument as in \cite{HL}, subsection  2.5.
 \begin{lemma} With the notation above, let $K$ and $\theta$ the constants appearing in {\bf A3}. Suppose that $\gamma:=\sup_{u\in H}\|\Xi(u)\|^2_2<\infty$ and $(K-2\theta)^2-48\gamma c_{HV}^2>0.$ Then, the property (\ref{specP}) is satisified.
 \end{lemma}
 \begin{proof} The proof is similar as in the proof of the Theorem 2.5.1 in \cite{HL}.
 \end{proof} \hfill $\Box$ \\
 Now we will return to estimates to obtain the maximum principle.
 \begin{lemma}\label{estimatLemma} Assume the conditions of the last lemma and {\bf A5}-{\bf A12}.  Let $\Delta_{\epsilon}(t)=\frac{u_{\epsilon}(t)-u^{\star}(t)-\epsilon P(t)}{\epsilon}$ where $P$ is the solution to the equation (\ref{linequ}). Then,
\begin{equation}\label{estimat4}
\lim_{\epsilon\rightarrow0}\sup_{t\in\mathbb{T}}\mathbb{E}(\|\Delta_{\epsilon}(t)\|^2)=0
\end{equation}
 \end{lemma}
 \begin{proof} According to the notation we have
 $$\begin{array}{rl}
 \Delta_{\epsilon}(t)&=\int_0^tA(s)\Delta_{\epsilon}(s)ds+\\
 &\int_0^t\frac{1}{\epsilon}(B(u_{\epsilon}(s),\Phi_{\epsilon}(s))-B(u^{\star}(s),\Phi^{\star}(s)))-B_u(u^{\star}(s),\Phi^{\star}(s))P(s)ds+\\
 &\int_0^t\frac{1}{\epsilon}(\Xi(u_{\epsilon}(s))-\Xi(u^{\star}(s)))-\Xi_u(u^{\star}(s))P(s))dW(s)-\\
 &\int_0^t B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))\Phi(s)ds=\int_0^tA(s)\Delta_{\epsilon}(s)ds+\\
 &\int_0^t\frac{1}{\epsilon}(B(u_{\epsilon}(s),\Phi_{\epsilon}(s))-B(u^{\star}(s),\Phi_{\epsilon}(s)))-B_u(u^{\star}(s),\Phi^{\star}(s))P(s)ds+\\
 &\int_0^t\frac{1}{\epsilon}(B(u^{\star}(s),\Phi_{\epsilon}(s))-B(u^{\star}(s),\Phi^{\star}(s)))-B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))\Phi(s)ds+\\
 &\int_0^t\frac{1}{\epsilon}(\Xi(u_{\epsilon}(s))-\Xi(u^{\star}(s)))-\Xi_u(u^{\star}(s))P(s))dW(s),
 \end{array}
 $$
 for $t\in\mathbb{T}.$ Using the It\^o's formula with $e_1(t)=e^{-\int_0^t(\rho_1(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s))))^2ds}$ for $r\in[0,1]$and taking the expectation, we have
 $$\begin{array}{rl}
 \mathbb{E}(e_1(t)\|\Delta_{\epsilon}(t)\|^2)+\theta\mathbb{E}(\int_0^te_1(s)\|\Delta_{\epsilon}(s)\|^2_Vds)&\leq\mathbb{I}+\mathbb{II}+\mathbb{III}+\\
 &\mathbb{E}(\int_0^te_1'(s)\|\Delta_{\epsilon}(s)\|^2ds),
 \end{array}
 $$
 where
 $$\begin{array}{rl}
 \mathbb{I}=&2\mathbb{E}(\int_0^te_1(s)\langle \frac{1}{\epsilon}(B(u_{\epsilon}(s),\Phi_{\epsilon}(s))-\\
 &B(u^{\star}(s),\Phi_{\epsilon}(s)))-B_u(u^{\star}(s),\Phi^{\star}(s))P(s), \Delta_{\epsilon}(s)\rangle)ds),\\
 \mathbb{II}=&2\mathbb{E}(\int_0^te_1(s)\langle \frac{1}{\epsilon}(B(u^{\star}(s),\Phi_{\epsilon}(s))-\\
 &B(u^{\star}(s),\Phi^{\star}(s)))-B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))\Phi(s),\Delta_{\epsilon}(s)\rangle ds)\\
 \mathbb{III}=&\mathbb{E}(\int_0^te_1(s)\|\frac{1}{\epsilon}(\Xi(u_{\epsilon}(s))-\Xi(u^{\star}(s)))-\Xi_u(u^{\star}(s))P(s)\|^2_2ds)
 \end{array}
 $$
We need to estimate each term above. Using our assumption about $\rho_1$ we get
$$\begin{array}{rl}
\mathbb{I}&=2\mathbb{E}(\int_0^te_1(s)\langle \frac{1}{\epsilon}\int_0^1B_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)),\Phi_{\epsilon}(s))(u_{\epsilon}(s)-u^{\star}(s))dr, \\
&\Delta_{\epsilon}(s)\rangle ds)+2\mathbb{E}(\int_0^te_1(s)\langle-B_u(u^{\star}(s),\Phi^{\star}(s))P(s), \Delta_{\epsilon}(s)\rangle ds)=\\
&2\mathbb{E}(\int_0^te_1(s)\langle\int_0^1B_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)),\Phi_{\epsilon}(s))(\Delta_{\epsilon}(s)), \Delta_{\epsilon}(s)\rangle ds)+\\
&2\mathbb{E}(\int_0^te_1(s)\langle\int_0^1 B_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)),\Phi_{\epsilon}(s))P(s)- B_u(u^{\star}(s),\Phi^{\star}(s))P(s),\\
&\Delta_{\epsilon}(s)\rangle drds)=\\
&2\mathbb{E}(\int_0^te_1(s)\langle\int_0^1B_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)),\Phi_{\epsilon}(s))(\Delta_{\epsilon}(s)),\Delta_{\epsilon}(s)\rangle drds)+\\
&2\mathbb{E}(\int_0^te_1(s)\langle \int_0^1B_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)),\Phi_{\epsilon}(s))P(s)-B_u(u^{\star}(s),\Phi^{\star}(s))P(s),\\
& \Delta_{\epsilon}(s)\rangle drds)\leq\\
&\frac{\theta}{4}\mathbb{E}(\int_0^te_1(s)\|\Delta_{\epsilon}(s)\|^2_Vds)+\frac{8K_2^2}{\theta}\mathbb{E}(\int_0^te_1(s)(\rho_1(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s))))^2\times\\
&\|\Delta_{\epsilon}(s)\|^2ds)+\\
&\frac{8}{\theta}(\mathbb{E}(\sup_{t\in\mathbb{T}}\|P(t)\|^4))^{1/2(}\mathbb{E}(\int_0^te_1(s)\|(u_{\epsilon}-u^{\star})(s)\|^2ds)^2)^{1/2}.
\end{array}
$$
\indent About the second term
$$\begin{array}{rl}
\mathbb{II}&=2\mathbb{E}(\int_0^te_1(s)\langle\int_0^1(B_{\Phi}(u^{\star}(s),\Phi^{\star}+r(\Phi_{\epsilon}(s)-\Phi^{\star}(s)))-\\
&B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))(\Phi(s)), \Delta_{\epsilon}(s)\rangle drds)\leq\\
&\frac{4}{\theta}\mathbb{E}(\int_0^te_1(s)\int_0^1\|B_{\Phi}(u^{\star}(s),\Phi^{\star}+r(\Phi_{\epsilon}(s)-\Phi^{\star}(s)))-\\
&B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))(\Phi(s))\|^2_{V'}drds)+\\
&\frac{\theta}{4}\mathbb{E}(\int_0^te_1(s)\|\Delta_{\epsilon}(s)\|^2_Vds).
\end{array}
$$
Now we will estimate the third term
$$\begin{array}{rl}
\mathbb{III}&\leq\mathbb{E}(\int_0^te_1(s)\|\int_0^1\Xi_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)))(\Delta_{\epsilon}(s))dr\|^2_2ds)+\\
&\mathbb{E}(\int_0^te_1(s)\|\int_0^1\Xi_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)))(P(s))-\\
&\Xi_u(u^{\star}(s))(P(s))dr\|^2_2ds)\leq\\
&\mathbb{E}(\int_0^te_1(s)\|\int_0^1\Xi_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)))(P(s))-\\
&\Xi_u(u^{\star}(s))(P(s))dr\|^2_2ds)+\\
&K_6\mathbb{E}(\int_0^te_1(s)\|\Delta_{\epsilon}(s)\|^2ds).
\end{array}
$$
Thus,
\begin{equation}\label{estimafinal1}\begin{array}{rl}
 \mathbb{E}(e_1(t)&\|\Delta_{\epsilon}(t)\|^2)+\frac{\theta}{2}\mathbb{E}(\int_0^te_1(s)\|\Delta_{\epsilon}(s)\|^2_Vds)\leq\\
 &\frac{4}{\theta}\mathbb{E}(\int_0^te_1(s)\int_0^1\|B_{\Phi}(u^{\star}(s),\Phi^{\star}+r(\Phi_{\epsilon}(s)-\\
 &\Phi^{\star}(s)))-B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))(\Phi(s))\|^2_{V'}drds)+\\
 &\frac{8}{\theta}(\mathbb{E}(\sup_{t\in\mathbb{T}}\|P(t)\|^4))^{1/2}(\mathbb{E}(\int_0^te_1(s)\|u_{\epsilon}(s)-u^{\star}(s)\|^2ds)^2)^{1/2}+\\
&\mathbb{E}(\int_0^te_1(s)\|\int_0^1\Xi_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)))(P(s))-\\
&\Xi_u(u^{\star}(s))(P(s))dr\|^2_2ds)+K_6\mathbb{E}(\int_0^te_1(s)\|\Delta_{\epsilon}(s)\|^2ds).
 \end{array}
 \end{equation}
\indent Using the Lemma \ref{estpm}, it is possible to demonstrate that $\mathbb{E}(\int_0^te_1(s)\|u_{\epsilon}(s)-u^{\star}(s)\|^2ds)^2=O(\epsilon^2).$ Also, using the assumptions {\bf A11}-{\bf A12} and dominated convergence it is possible to demonstrate that $\mathbb{E}(\int_0^t\|\int_0^1\Xi_u(u^{\star}(s)+r(u_{\epsilon}(s)-u^{\star}(s)))(P(s))-\Xi_u(u^{\star}(s))(P(s))dr\|^2_2ds\rightarrow0$ and $\mathbb{E}(\int_0^te_1(s)\int_0^1\|B_{\Phi}(u^{\star}(s),\Phi^{\star}+r(\Phi_{\epsilon}(s)-\Phi^{\star}(s)))-B_{\Phi}(u^{\star}(s),\Phi^{\star}(s))(\Phi(s))\|^2_{V'}drds)\rightarrow0$ as $\epsilon\rightarrow0^+.$ Combining these facts in the inequality (\ref{estimafinal1}) and Gronwall's inequality we obtain
\begin{equation}\label{estimatfinal2}
\sup_{t\in\mathbb{T}}\mathbb{E}(e_1(t)\|\Delta_{\epsilon}(t)\|^2)\rightarrow0,\ \textmd{as } \epsilon\rightarrow0^+.
\end{equation}
Analogously we can demonstrate
\begin{equation}\label{estimatfinall3}
\mathbb{E}((e_1(t))^2\|\Delta_{\epsilon}(t)\|^4)\rightarrow0,\ \textmd{as } \epsilon\rightarrow0^+.
\end{equation}
In order to demonstrate (\ref{estimat4}) we observe that
$$\mathbb{E}(\|\Delta_{\epsilon}(t)\|^2)\leq(\mathbb{E}(e_1(t)\|\Delta_{\epsilon}(t)\|^2 ))^{1/2}(\mathbb{E}(e_1^{-4}(t)))^{1/4}(\mathbb{E}(e_1^2(t)\|\Delta_{\epsilon}(t)\|^4 ))^{1/4}$$
Hence, using (\ref{specP}), (\ref{estimatfinal2}) and (\ref{estimatfinall3}) we obtain (\ref{estimat4}).
\end{proof} \hfill $\Box$
\begin{theorem} We assume {\bf A5}-{\bf A12}. Then, for each $\epsilon>0,$
\begin{equation}\label{estPrinMax}\begin{array}{rl}
\mathcal{J}(\Phi_{\epsilon})-\mathcal{J}(\Phi^{\star})&=
\epsilon\mathbb{E}(\mathcal{K}'(u^{\star}(T),P(T)))+\\
&\epsilon\mathbb{E}(\int_0^T(\mathcal{L}_u(u^{\star}(t),\Phi^{\star}(t)), P(t))dt)+\\
&\mathbb{E}(\int_0^T(\mathcal{L}(u^{\star}(t),\Phi_{\epsilon}(t))-\mathcal{L}(u^{\star}(t),\Phi^{\star}(t)))dt)+o(\epsilon).
\end{array}
\end{equation}
\end{theorem}
{\bf Proof:}  Since
\begin{equation}\label{estimfunct}\begin{array}{rl}
\mathcal{J}(\Phi_{\epsilon})-\mathcal{J}(\Phi^{\star})=&\mathbb{E}(\mathcal{K}(u_{\epsilon})(T)-\mathcal{K}(u^{\star})(T))+\\
\mathbb{E}(\int_0^T\mathcal{L}(u_{\epsilon}(s),\Phi_{\epsilon}(s))-\mathcal{L}(u^{\star}(s),\Phi^{\star}(s)ds).
\end{array}
\end{equation}
We need to calculate each term above. Since
$$
\mathcal{K}(u_{\epsilon}(T))-\mathcal{K}(u^{\star}(T))=\int_0^1\epsilon(\mathcal{K}'(u^{\star}(T)+r\epsilon(\Delta _{\epsilon}(T)+P(T)), \Delta _{\epsilon}(T)+P(T)))dr
$$
which implies
$$
\frac{\mathcal{K}(u_{\epsilon}(T))-\mathcal{K}(u^{\star}(T))}{\epsilon}=\int_0^1(\mathcal{K}'(u^{\star}(T)+r\epsilon(\Delta _{\epsilon}(T)+P(T)), \Delta _{\epsilon}(T)+P(T)))dr.
$$
Using the Lemma ( \ref{estimatLemma}) and the dominated convergence theorem we have
$$
\mathbb{E}(\frac{\mathcal{K}((u_{\epsilon}(T))-\mathcal{K}(u^{\star}(T)))}{\epsilon})\rightarrow  \mathbb{E}(\mathcal{K}'(u^{\star}(T),P(T))), \textmd { as } \epsilon\rightarrow0^+.
$$
In particular, we have
\begin{equation}\label{estimat3}
\mathbb{E}(\mathcal{K}(u_{\epsilon})(T)-\mathcal{K}(u^{\star})(T))=\epsilon\mathbb{E}(\mathcal{K}'(u^{\star}(T),P(T)))+o(\epsilon).
\end{equation}
Now we need to estimate the second factor on the right hand side of (\ref{estimfunct}), first we will observe that
$$\begin{array}{rl}
\mathcal{L}(u_{\epsilon}\!\!\!\!&(t),\Phi_{\epsilon}(t))-\mathcal{L}(u^{\star}(t),\Phi^{\star}(t))=\\
\!\!\!&\mathcal{L}(u_{\epsilon}(t),\Phi_{\epsilon}(t))-\mathcal{L}(u^{\star}(t),\Phi_{\epsilon}(t))+\mathcal{L}(u^{\star}(t),\Phi_{\epsilon}(t))-\mathcal{L}(u^{\star}(t),\Phi^{\star}(t))=\\
&\int_0^1\epsilon((\mathcal{L}_u(u^{\star}(t)+r\epsilon(\Delta _{\epsilon}(t)+P(t)),\Phi^{\star}(t)+\epsilon\Phi(t)), \Delta _{\epsilon}(t)+P(t))dr+\\
&\mathcal{L}(u^{\star}(t),\Phi_{\epsilon}(t))-\mathcal{L}(u^{\star}(t),\Phi^{\star}(t)),
\end{array}
$$
for all $t\in\mathbb{T}.$\\
By applying Lemma  \ref{estimatLemma},  the continuity and boundedness of $\mathcal{L}_u$ in  {\bf H2} and the  dominated convergence theorem we have
$$\begin{array}{rl}
\mathbb{E}\!\!\!\!&(\int_0^T\int_0^1((\mathcal{L}_u(u^{\star}(t)+r\epsilon(\Delta _{\epsilon}(t)+P(t)),\Phi^{\star}(t)+r\epsilon\Phi(t)), \Delta _{\epsilon}(t)+P(t))drdt)\\
&\rightarrow\mathbb{E}(\int_0^T((\mathcal{L}_u(u^{\star}(t),\Phi^{\star}(t)),P(t))dt) \textmd{ as }\epsilon\rightarrow0^+.
\end{array}
$$
Then,
\begin{equation}\label{esmt2}\begin{array}{rl}
\mathbb{E}&(\int_0^T\mathcal{L}(u_{\epsilon}(s),\Phi_{\epsilon}(s))-\mathcal{L}(u^{\star}(s),\Phi^{\star}(s))ds)=\\
&\epsilon\mathbb{E}(\int_0^T((\mathcal{L}_u(u^{\star}(t),\Phi^{\star}(t)),P(t))dt)+\\
&\mathbb{E}(\int_0^T(\mathcal{L}(u^{\star}(t),\Phi_{\epsilon}(t))-\mathcal{L}(u^{\star}(t),\Phi^{\star}(t)))dt)+o(\epsilon).
\end{array}
\end{equation}
The theorem follows from ( \ref{estimat3}) and \ref{esmt2}).
 \hfill $\Box$ \\
The last theorem will play a crucial importance to get the maximum principle. In order to obtain this we need the following Lemmas, the proof follows the ideas of (\cite{AH2}).
\begin{lemma} Suppose that {\bf A1}-{\bf A12}, {\bf H1} and {\bf H2} are satisfied. Then, the solution $(v^{\star},Z^{\star})$ to the equation (\ref{BSE}) satisfies
\begin{equation}\label{inequPrM1}\begin{array}{rl}
\epsilon\mathbb{E}&\left(v^{\star}(T),P(T)\right)+\epsilon\mathbb{E}\left[\int_0^T \mathcal{L}_u(u^{\star}(t),\Phi^{\star}(t))P(t)dt
\right]+\\
&\mathbb{E}\left[\int_0^T\mathcal{H}(u^{\star}(t),\Phi_{\epsilon}(t),v^{\star}(t),Z^{\star}(t))-\mathcal{H}(u^{\star}(t),\Phi^{\star}(t),v^{\star}(t),Z^{\star}(t)) dt\right]-\\
&\mathbb{E}\left[\int_0^T (v^{\star}(t),B(u^{\star}(t),\Phi_{\epsilon}(t))-B(u^{\star}(t),\Phi^{\star}(t)))dt\right]\geq o(\epsilon).
\end{array}
\end{equation}
\end{lemma}
{\bf Proof:}   Let us  first observe that $\mathcal{J}(\Phi_{\epsilon})-\mathcal{J}(\Phi^{\star})\geq0$ because $\Phi^{\star}$ is an optimal control. The proof follows from this observation, (\ref{estPrinMax}) and (\ref{hamilt}).
 \hfill $\Box$
\begin{lemma} Suppose that {\bf A1}-{\bf A12} are satisfied. We have
\begin{equation}\label{inequPrM2}\begin{array}{rl}
\mathbb{E}\left(v^{\star}(T),P(T)\right)&=-\mathbb{E}\left[\int_0^T \mathcal{L}_u(u^{\star}(t),\Phi^{\star}(t))P(t)dt
\right]+\\
&\mathbb{E}\left[\int_0^T (v^{\star}(t),B_\Phi(u^{\star}(t),\Phi^{\star}(t))\Phi(t)dt
\right].
\end{array}
\end{equation}
\end{lemma}
{\bf Proof:}  See \cite{AH2} Lemma 6.2.
 \hfill $\Box$
\begin{theorem}Suppose that {\bf A1}-{\bf A12}, {\bf H1} and {\bf H2} are satisfied. Given an optimal pair $(u^{\star},\Phi^{\star}).$ Then, there is a unique pair $(v^{\star},Z^{\star})$ solving the corresponding adjoint BSE (\ref{BSE}) such that the following variational inequality is satisfied
\begin{equation}\label{varioequa}
\left( \nabla_{\Phi}\mathcal{H}(u^{\star}(t),\Phi^{\star}(t),v^{\star}(t),Z^{\star}(t)),\Phi^{\star}-\Phi \right)_{\mathcal{O}}\leq 0,
\end{equation}
for all $\Phi\in \mathcal{O}_1,$ a.e. $t\in\mathbb{T},$ a.s.
\end{theorem}
{\bf Proof:}  Given $(u^{\star},\Phi^{\star})$ an optimal pair, thanks to the Theorem \ref{BSExUn}, there is a unique solution $(v^{\star},Z^{\star})$ to (\ref{BSE}).\\
Now, we will demonstrate (\ref{varioequa}). From (\ref{inequPrM1}) and  (\ref{inequPrM2}) we have
\begin{equation}\begin{array}{rl}
\epsilon\mathbb{E}&\left[\int_0^T (v^{\star}(t),B_\Phi(u^{\star}(t),\Phi^{\star}(t))\Phi(t))dt
\right]+\\
&\mathbb{E}\left[\int_0^T\mathcal{H}(u^{\star}(t),\Phi_{\epsilon}(t),v^{\star}(t),Z^{\star}(t))-\mathcal{H}(u^{\star}(t),\Phi^{\star}(t),v^{\star}(t),Z^{\star}(t)) dt\right]-\\
&\mathbb{E}\left[\int_0^T (v^{\star}(t),B(u^{\star}(t),\Phi_{\epsilon}(t))-B(u^{\star}(t),\Phi^{\star}(t)))dt\right]\geq o(\epsilon)
\end{array}
\end{equation}
Furthermore, by using the continuity and boundedness of $B_{\Phi}$ in {\bf A11} and the dominated convergence Theorem, we have
$$\begin{array}{rl}
\frac{1}{\epsilon}\!\!\!\!&\mathbb{E}\left[\int_0^T(v^{\star}(t),-(B(u^{\star}(t),\Phi_{\epsilon}(t))-B(u^{\star}(t),\Phi^{\star}(t)))+\right.\\
&\left.\epsilon B_\Phi(u^{\star}(t),\Phi^{\star}(t)),\Phi(t))dt\right]=\\
&\mathbb{E}\left[\int_0^T(v^{\star}(t),-\int_0^1(B_{\Phi}(u^{\star}(t),\Phi^{\star}(t)+r(\Phi_{\epsilon}(t)-\Phi^{\star}(t)),\Phi(t)))\right.dr+\\
&\left. B_\Phi(u^{\star}(t),\Phi^{\star}(t))(\Phi(t))dt\right]\rightarrow0, \textmd{ as }\epsilon\rightarrow0.
\end{array}
$$
Therefore,
$$
\mathbb{E}\left[\int_0^T\mathcal{H}(u^{\star}(t),\Phi_{\epsilon}(t),v^{\star}(t),Z^{\star}(t))-\mathcal{H}(u^{\star}(t),\Phi^{\star}(t),v^{\star}(t),Z^{\star}(t)) dt\right]\geq o(\epsilon).
$$
Hence, dividing by $\epsilon$ and letting  $\epsilon\rightarrow0$ we obtain
$$
\mathbb{E}\left[\int_0^T(\nabla_{\Phi}\left(u^{\star}(t),\Phi^{\star}(t),v^{\star}(t),Z^{\star}(t)),\Phi(t) \right)_{\mathcal{O}}dt\right]\geq0.
$$
Finally, (\ref{varioequa}) follows by arguing as in (\cite{AH2}).
 \hfill $\Box$

redpajama_set_name: RedPajamaArXiv


text:
Q: Как сделать реляционный INSERT в  "родительскую и дочернею " базы одновременно Всем привет. Есть таблица tbl_test (вопросы) и есть таблица tbl_ans(ответы).
tbl_test:

*

*id

*title

tbl_ans:

*

*id

*title

*right(yes, no)

*test_id (foreign key)

Хочу сделать для удобства клиента форму создания теста и форму создания ответов одной формой. Вопрос как сделать так чтобы при создании теста, создавались одновременно и ответы? Можно ли в принципе сотворить такое? Или только сначала тесты потом ответы? Надеюсь вы поняли о чем я.
Проект делается на YII.
Спасибо.
A: public function actionCreate()  {
    $Test=new Test;
    $Ans=new    Ans;
    if(isset($_POST['Test']))       {
        $Test->attributes=$_POST['Test'];
        if($model->save())
            $this->redirect(array('view','id'=>$Test->id));
    }
    //И тут заполняем новый экземпляр класса Ans  как и с
    //Test   или в блоке   if
    //$Ans->attributes=
    // С данными полученными с формы мы можем делать что хотим
    ));     }

A: Смотрите в сторону SQL деклараций on update cascade и on insert cascade - это по сути триггеры которые делают вставку/апдейт в связанные таблицы. Некоторые православные прогеры считают что каскадные вставки это зло - так что смотрите. Если все правильно сделать получается иногда удобно.

redpajama_set_name: RedPajamaStackExchange


text:
Cisco, Bosch, ABB joint venture targets smart home opportunity
by Sean Buckley |
Cisco has teamed with Bosch and ABB to form a joint venture to develop and operate what they say is an open software platform targeting smart home devices and applications.
The aim of the platform is to facilitate the exchange of data between different manufacturers' devices. What's more, the planned platform will make it possible to provide a number of services related to household devices, such as energy management, security technology and entertainment.
Besides developing and operating a software platform, the three companies said they plan to invite an array of appliance electronics manufacturers, home automation vendors and service providers to join a business ecosystem. The business ecosystem will aim to facilitate collaboration and incorporate a wide range of user requirements when developing the software platform.
Set to be housed in Germany, the three companies expect the joint venture will begin operations at the beginning of next year.
These three companies are hardly alone in their desires and will be challenged by other players like Google (NASDAQ: GOOG). Following its acquisition of Nest, Google launched its "Works with Nest" developer program in June to encourage partner companies such as Whirlpool, Mercedes Benz and garage door manufacturer Chamberlain Group to come up with new apps that work with the Nest products already in the home.
While still a relatively nascent segment, smart home networking is on the rise. According to ABI Research, growing competition, new applications and feature innovation will help drive the worldwide smart home controller market to $6 billion by 2019.
However big the smart home and overall Internet of Things (IoT) concept could be, a separate ABI report says the other concern will be cybersecurity. A number of variables, including safety considerations, consumer privacy and data protection, will have to be dealt with as more smart devices emerge in the home.
- see the release
AT&T, Cisco serve up new cloud-based conferencing tool for business customers
Cisco's service provider routing market share rebounds with APAC, EMEA growth, says Synergy
Cisco's Chambers slams Obama's net neutrality stance
Cisco commits $1B to build Intercloud network, adds 30 new partners
Sean Buckley

redpajama_set_name: RedPajamaCommonCrawl


text:
In-Brief: May 24th, 2018
Prostate and Lung Cancers Expected to Be Most Common Cancers Among HIV+ Persons by 2030
Based on current trends in cancer diagnoses, the National Cancer Institute (NCI) projects that prostate and lung cancers will be most commonly diagnosed cancers among persons living with HIV (PLWH) in the U.S. by 2030. In a recent report published in the Annals of Internal Medicine, researchers from NCI and the Centers for Disease Control and Prevention (CDC) note that total annual new cancer diagnoses among U.S. PLWH are expected to decline 18% from approximately 8150 cases in 2010 to 6690 by 2030 – largely the result of and 80% decrease in the number of AIDS-defining cancer (ADCs) – such as Kaposi's sarcoma and non-Hodgkin lymphoma – arising from the earlier diagnosis and major advances in the treatment of HIV infection since the mid- to late 1990s. During this period, the proportion of cancer deaths from ADCs versus non-AIDS defining cancers (NADCs) has fallen dramatically. The researchers note, however, that with the aging of the HIV-infected population in the U.S., the incidence of some NADCs associated with aging and lifestyle will likely increase, with the total number of NADC diagnoses rising from an estimated 5420 in 2010 to 5980 in 2030. By that time, prostate cancer and lung cancer are expected to be the most frequently diagnosed cancers among PLWH, with 1590 and 1030 cases, respectively.
In another study published in the Journal of the American Academy of Dermatology, Danish researchers reported that HIV infection may significantly increase the risk of skin cancer. They found that Danish PLWH had about a 1.8 times higher rate of basal cell carcinoma (BCC) and a 5.4 times high risk of squamous cell carcinoma (SCC) than the general population. The study also indicated that PLWH's lowest ever CD4 T-cell count – but not their current CD4 count – was associated with the increased risk of SCC.
HIV Transmission
Meta-Analysis: Spitting Presents No Risk & Biting Negligible Risk of HIV Transmission
Spitting presents no risk and biting only a negligible risk of HIV transmission, according to a meta-analysis published in the journal HIV Medicine. According to the article, "The perceived threat of HIV transmission through spitting and biting is evidenced by the increasing use of 'spit hoods' by Police Forces in the U.K." In addition, the U.K. parliament has considered adopting a bill that would increase penalties for assaults on emergency workers, citing the risk of communicable disease transmission as one justification.
For their analysis, an international research team reviewed 742 abstracts, yielding 32 articles for full?text review and 13 case reports/series. The researchers found no reported cases of HIV transmission related to spitting and nine cases identified following a bite, in which the majority (6 of 9) occurred between family, in fights involving serious wounds (3 of 9), or to untrained first?aiders placing fingers in the mouth of someone having a seizure (2 of 9). Only 4 of the cases were classified as highly plausible or confirmed transmission, and none of the cases related to emergency workers.
"There is no risk of transmitting HIV through spitting, and the risk through biting is negligible," the research team concluded. "Post?exposure prophylaxis is not indicated after a bite in all but exceptional circumstances. Policies to protect emergency workers should be developed with this evidence in mind."
Clinical Trial to Track Outcomes of Kidney Transplants Between HIV+ Donors and Recipients
The first large-scale clinical trial to study kidney transplantations between HIV+ donors and HIV+ recipients has begun at 19 clinical centers across the U.S. The purpose of the HOPE in Action Multicenter Kidney Study is to determine the safety of HIV+ to HIV+ kidney transplants by evaluating recipients for potential transplant-related and HIV-related complications following surgery. The study is sponsored by Johns Hopkins University in collaboration with the National Institute of Allergy and Infectious Diseases.
While organ transplants between donors and recipients with HIV have been successfully completed in South Africa since 2008, such transplants were illegal in the U.S. until the passage of the HIV Organ Policy Equity (HOPE) Act in 2013. The HOPE Act allows U.S. transplant teams with an approved research protocol to transplant organs from donors with HIV into qualified recipients with end-stage organ failure who are also living with HIV, a practice that may shorten the time people living with HIV wait to receive a transplant. The transplantation of organs from donors who have HIV to individuals uninfected with HIV remains illegal in the U.S.
The Hope in Action clinical trial will track the clinical outcomes of 160 kidney transplants. All transplant recipients in the study will be living with HIV; 80 of them will receive kidneys from deceased donors who had HIV, and 80 will receive kidneys from HIV-uninfected deceased donors serving as the control group. Health care teams and study participants will be made aware of the HIV status of the organ donor. The study began in mid-April 2018 and is expected to be completed by August 2022.
"A kidney transplant can mean a second chance at a normal life for a person with end-stage renal disease who may otherwise need hours-long dialysis sessions, multiple times per week," notes Johns Hopkins researcher Christine Durand, the principal investigator for the clinical trial. "If proven safe and effective in our study, kidney transplants between people with HIV may result in people living with HIV receiving donated organs sooner and the overall organ transplant waiting list shrinking — to the benefit of everyone who needs a kidney transplant, regardless of HIV status."
Social Determinants and Health Equity
Kaiser Brief Examines Role of Social Determinants in Promoting Health and Health Equity
A new Kaiser Family Foundation (KFF) issue brief provides an overview of social determinants of health – the "conditions in which people are born, grow, live, work and age that shape health" – and their role in health equity. Social determinants include factors like socioeconomic status, education, neighborhood and physical environment, employment, and social support networks, as well as access to health care. "Addressing social determinants of health is important for improving health and reducing longstanding disparities in health and health care," according to the brief.
"There are a growing number of initiatives to address social determinants of health within and outside of the health care system," the brief notes. "Outside of the health care system, initiatives seek to shape policies and practices in non-health sectors in ways that promote health and health equity. Within the health care system, there are multi-payer federal and state initiatives as well as Medicaid-specific initiatives focused on addressing social needs. These include models under the Center for Medicare and Medicaid and Innovation, Medicaid delivery system and payment reform initiatives, and options under Medicaid. Managed care plans and providers also are engaged in activities to identify and address social needs."
Despite these advances, "Many challenges remain to address social determinants of health, and new directions pursued by the Trump Administration could limit resources and initiatives focused on these efforts," according to the brief. "The Trump Administration is pursuing a range of new policies and policy changes, including enforcing and expanding work requirements associated with public programs and reducing funding for prevention and public health. These changes may limit individuals' access to assistance programs to address health and other needs and reduce resources available to address social determinants of health."
Access to PrEP
Latino Commission on AIDS Seeks Information on PrEP Providers Serving Hispanic/Latinx Communities
Although approximately one-quarter of all new U.S. HIV diagnoses in 2015 were among Latinos, this group accounted for only 12% of the prescriptions for HIV pre-exposure prophylaxis (PrEP) issued between 2012 and 2015, according to the Latino Commission on AIDS (LCA). To help address this disparity in PrEP access, LCA is requesting assistance to develop list of PrEP providers throughout the U.S., including Puerto Rico and the U.S. Virgin Islands, who serve Hispanic/Latinx communities. If you'd like to help, you can complete a brief form with the name of medical providers, clinics, and community centers that are providing services to Hispanic/Latinx communities – and the different populations they work with.
Fact Sheets and Other Resources
Information on Investigational HIV Drugs from AIDSinfo
If you want to learn about new HIV medicines in development, then it's worth checking out the AIDSinfo Drug Database, which has fact sheets on more than 60 investigational HIV vaccines and drugs used to treat HIV infection itself and a wide range of opportunistic infections and other illnesses that may affect people living with HIV. These investigational drugs and vaccines are currently being studied in clinical trials and have not yet been approved by the Food and Drug Administration (FDA) for sale in the U.S. to treat or prevent HIV and related conditions. Database users can browse by drug names or do filtered searches by drugs' FDA approval status – which includes a category for investigational drugs – as well as drug classes and the conditions for which particular drugs are used. AIDSinfo provides information for both patients and health professionals. Each patient fact sheet includes an overview of the drug and information on clinical trials studying the drug. A corresponding health professional fact sheet includes more detailed, scientific information about the drug, including details on the pharmacology of the drug and on adverse events identified in clinical trials. For example, the latest investigational drug fact sheets added to the database include VRC-HIVADV014-00-VP and VRC-HIVDNA016-00-VP, both of which are therapeutic HIV vaccines. AIDSinfo's lefitolimod, romidepsin, and VAC-3S fact sheets have recently been updated based on research that was presented at the 2018 Conference on Retroviruses and Opportunistic Infections (CROI).
New and Updated CDC Reports, Fact Sheets, and Flyers
During the past few weeks, CDC has published several new reports, fact sheets, and flyers related to HIV, affected populations, and harm reduction. These include:
Diagnoses of HIV Infection among Adults and Adolescents in Metropolitan Statistical Areas – United States and Puerto Rico, 2016 – This 87-page surveillance report focuses on HIV diagnoses in Metropolitan Statistical Areas (MSAs) with breakdowns by gender, age, race-ethnicity, and transmission category. Information on HIV prevalence rates is also provided.
Access to Syringe Services Programs – Kentucky, North Carolina, and West Virginia, 2013-2017 – This research report, published in MMWR, describes the expansion of syringe services programs (SSPs) in Kentucky, North Carolina, and West Virginia between 2013 and 2017, and the care that has been provided by these programs.
How to Clean Your Syringes – This illustrated fact sheet with detailed step-by-step instructions for rinsing and disinfecting syringes.
Used Syringes Are Dangerous – This illustrated flyer that users can customize to include contact information for their local health department.
HIV Among Native Hawaiians and Other Pacific Islanders in the United States and 6 Dependent Areas – This updated fact sheet has information on HIV diagnoses, prevalence, prevention challenges, and what CDC is doing to address in this population group.


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Home — Essay Samples — Science — Biology — Potassium ion (K+), an Inorganic Macro-Nutrient for Plant Growth
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Potassium ion (K+), an Inorganic Macro-Nutrient for Plant Growth
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Potassium ion (K+) is one of the essential inorganic macro-nutrients for plant growth and the fourth abundant mineral on lithosphere. Although, plants constitute 2-10% of total dry weight and abundantly present cation in the cytosol with a narrow range (i.e. 60-150 mM) (Ashley, Grant, & Grabov, 2006).Plants utilize K+ ions for different fundamental processes like cell elongation, osmoregulation, control of membrane polarization and electrical neutralization of anionic groups. Potassium is also involved in maintainace the Ph of the cytosol, which is important for proper functioning of most of the enzymes. In addition to cytosol, potassium is also present in certain organelles i.e. vacuoles, nucleus and chloroplast(Wang, Zheng, Shen, & Guo, 2013).
Storage of potassium in vacuole is helpful for maintaining the concentration of K+ in cytosol through these two compartments. Despite its importance and abundance in the cell, optimum concentration of K+ is maintained for the proper functioning of the cell. For example, Plant tolerance to salinity (Shi, Ishitani, Kim, & Zhu, 2000) and drought stress (Gupta, Berkowitz, & Pier, 1989) correlated with maintainace of K+ ion concertation in the cytoplasm. Moreover, it is also important for the modulation of membrane potential and neutralization of anions. homeostasis of K+ in the cytosol is maintained by complex transport system consisting of transporters and channels (Leigh, 2001).This system mediates the absorption and translocation of K+ from soil into various part of plant. Almost 35 genes have been reported to code for K+ transport proteins (15 channels and 20 transporters) in Arabidopsis thaliana. (Mäser et al., 2001; Véry & Sentenac, 2003).
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Potassium channels are multimeric proteins containing trans-membrane segments, that are quantified on the basis of pore (P) domains. To make a part of channel conduction pathway four P domains are associated with functional multimeric proteins. Highly conserved motif with an amino acid sequence "GYGD/E". is present in the P domain of potassium channel. Based on topology,15 selective K+ channels are classified into three families in Arabidopsis thaliana, which includes one potassium inward rectifier,9 voltage gated ion channels and, 5 Tandem-Pore K+ channels (TPK) (Gambale & Uozumi, 2006). Similarly, Potassium transporters are also classified into three families including KEA family of K+ efflux antiporters (six members), KUP/HAK/KT family of K+ uptake permeases (thirteen members) and Trk/HKT family of high-affinity K+ transporters (one member) (Rodríguez-Navarro & Rubio, 2006). However, various potassium channels and transporters are well characterized in O. sativa and A. thaliana, but there is no information is available on potassium transporters and channels in Vigna radiata.
Mung bean is an important legume crop, diploid (2n=2x=22) pulse crop, grown under tropical and subtropical regions. The seeds of mung bean are an excellent source of carbohydrates, fats, proteins and micronutrients (Nair et al., 2013). The production of mung bean is increase annually, mostly in Asia its center of diversity. Adaptation in extreme environmental conditions and long-term evolution make a mung bean a rich pool stress tolerance gene (Jain et al., 2013; Varshney et al., 2013). The availability of genomic and transcriptomics resources makes a valuable opportunity for the analysis of comparative and evolutionary analysis in mung bean. K+ is important micronutrient which greatly contribute to various physiological and biochemical process correlate with biotic and abiotic stress tolerances in plants(Kant & Kafkafi, 2002). However, in mung bean our standing about the role of potassium in whole plant stress response mechanism is limited. This study was designed to explore the phylogenetic relation of various K+ transporters and channels in Vigna radiata, Oryza sativa Cicer arietinum and Arabidopsis thaliana. and to study the effect and possible role of potassium in abiotic stress responses.
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Regen ist schön est une chanson allemande interprétée par la chanteuse française Mireille Mathieu sorti en 1973 sous le label Ariola. Cette chanson a une version française Pleure mon cœur sorti en 1972 sur l'album Mireille Mathieu.

Notes et références

Chanson de 1973
Chanson interprétée par Mireille Mathieu
Chanson de schlager

redpajama_set_name: RedPajamaWikipedia


Recently I was looking for a way to add to my koken site the possibility to change between different languages. I found a plugin but in order to use it it is necessary to change quite a few files in the server php code - not bad but difficult to maintain.
A different approach was to translate the content directly on the client side - the browser. One framework I found was i18next. With i18nextify they offer a way to add easily translations to any page. So I wrote a small koken plugin that integrates i18nextify. You can find it here.
Furthermore the plugin offers the possibility to save the translation files in the browser of your visitor and therefore avoid retrieving them from the server on every page reload.
The Readme on github describes how to get everything set up.
If you want to bring things a bit further you can think about using a translation management system. This is (in most cases an online) platform on which you upload your translation files and translate your content in a nice dashboard rather than editing json files in a text editor locally on your computer. With them you can also easily add new languages.
Hint: Please note that this plugin is about translating what your site visitor sees - not about translating the admin dashboard.
If you like to try Koken with a CDN but are looking for a free service (*) you might run across 2 companies: CLOUDFLARE and incapsula (although incapsula doesn't support SSL in the free version).
Next you need to set up a page rule. Add as URL eg. yoursite.com/* and as property "Cache Level: Cache Everything". After that all requests that users make on your page will be routed through the namespace server of cloudflare. Webspace (per default only images, no html code) traffic will be cached, e-mail and other traffic will be sent directly to your "own" webspace.
It might be necessary to tell cloudflare to not cache the admin-area of koken so that it works properly (otherwise eg. you might not see update notes for new releases). To do so add a page rule in the dashboard: Add eg. http://www.yoursite.com/admin as url and set the cache level to Bypass.
After some days go to your dashboard and check that on the Analytics tabs the "Cached Requests" are not 0 as otherwise your Cache is not set up properly. Certain pages that offer website speed measurements also mention if code is served through a CDN, so that would be another option to verify your setup.
Please be aware that if you are running a statistics tool within your domain that then inside the statistics tool the locations/countries of the cloudflare servers instead of those of your page visitors will show up. This is due to the fact that CLOUDFLARE is a reverse proxy. There are several ways to fix this: Please check eg. this CLOUDFLARE support site, a second support site or the Piwik help.
If you use a shared certificate on your koken-installation it will be passed automatically to CLOUDFLARE.
(*) There are also plenty of other companies that offer CDN services but with a cost like SecureCDN (formerly MaxCDN), Coral, HostEurope, Cloudtront.
The camera and the lenses are part of the Micro Four Thirds standard. Micro Four Thirds Photography is relatively new - the standard was originally released by Panasonic together with Olympus in 2008.
There are quite a few tutorials out there about how to integrate the open source analytics tool Piwik (which is in my opinion a pretty good alternative to Google analytics) into Koken and they are working fine.
Unfortunately they have the disadvantage that you can't see how much time a user spends on your site unless he moves between different pages. Instead of the specific time in your Piwik statistics you can only see a visiting time of 0 seconds in these casepleases.
This line sends a heartbeat to Piwik so that Piwik knows that the visitor is still viewing the site. Per default the heartbeat is sent every 15 seconds.
4. In the bottom area paste the code that you can find on your Piwik administration page under "Tracking code"
5. Add the above mentioned heartbeat-code under the line that includes the word "trackPageView" Please click here for a complete code sample or check the following screenshot.
Please see this link if you wish more information about the Piwik heartbeat funtion.
Although there's still much that can be improved you can already give primitive.nextgen a try by checking the github repository.
I'm using the image libraries primitive.js / primitive.lol for showing images in a different way.
Credits for the libraries go to Ondřej Žára and Michael Fogleman. Please see the licenses on the respective pages.
In order to see the images please go to the masterpieces album.
If you want to improve user experience on smartphones then you might think about using the Photoswipe library. This enables swiping through the images (once you click on one to open the gallery), furthermore it preloads images so that when the user swipes to the next image it is already there. This makes him feel as if the photos were stored on his local device.
For Koken there already exists a plugin on github. Just download, unzip and load into the /storage/plugins folder of your Koken installation. Then enable it over the Koken admin page and that's it.
For more infos ab about the plugin please check here.
A Content Delivery Network (CDN) is a network of servers distributed over the whole world. They mirror the page content of your Koken page. This has one big advantage: Let's take for example that your hoster sits in Europe and a possible visitor from Australia connects to your site. Without a CDN all the files (and especially the heavy images) would need to go all the long way from Europe to Australia which would result in a long waiting time for the user until the data arrives on his device.
This scenario can be improved by using a CDN. This stores the images on the CDN-servers. So if the CDN fe. has a server in Australia (or at least closer than Europe) then the Australian user gets the images from the Australian server which is obviously faster.
It only took minutes for me to get everything set up and the difference in image load speed is really notable comparing the page with and without CDN. (although I didn't make any measurements yet).
- In your Koken admin panel go to Settings -> Plugins and enable the plugin (CDN for images). Perhaps you need to wait a bit or reload the page until the plugin appears.
- Leave all the check boxes selected unless the first one and wait.
- Go to your homepage and make sure that everything is working correctly. You should already see that the images are loading faster now. In your KeyCDN admin-panel under Reporting -> Analytics you can see under "Cache Hits" what percentage of files is served by the CDN.
- In your KeyCDN add a Zonealias. As Alias enter fe. cdn.myexample.com and as zone select the previously created one.
- Login into your webserver admin panel and go to the DNS settings page.
- Wait at least 5 minutes until the changes apply globally.
KeyCDN even lets you add a certificate for your CDN (fe. a Let's Encrypt certificate). So if your page is accessed via https then the user gets a valid certificate. Please be aware that after configuring the certificate you need to recreate the zonealias.
You can improve your Koken site performance by making sure that KeepAlive is enabled. KeepAlive is a webserver feature that allows the transmissions of several files at the same time and not one after another. For this you could edit either your webserver config (fe. apache or nginx) or just add the following lines to your .htaccess in the root-folder of your Koken-installation.


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Understanding Red Hat security ratings
Red Hat Product Security rates the impact of security issues found in Red Hat products using a four-point scale (Low, Moderate, Important, and Critical), as well as Common Vulnerability Scoring System (CVSS) base scores. These provide a prioritized risk assessment to help you understand and schedule upgrades to your systems, enabling informed decisions on the risk each issue places on your unique environment.
The four-point scale tells you how serious Red Hat considers an issue to be, helping you judge the severity and determine what the most important updates are. The scale takes into account the potential risk based on a technical analysis of the exact flaw and its type, but not the current threat level; a given rating will not change if an exploit or worm is later released for a flaw, or if one is available before the release of a fix.
Severity Rating
This rating is given to flaws that could be easily exploited by a remote unauthenticated attacker and lead to system compromise (arbitrary code execution) without requiring user interaction. These are the types of vulnerabilities that can be exploited by worms. Flaws that require an authenticated remote user, a local user, or an unlikely configuration are not classed as Critical impact.
Important impact
This rating is given to flaws that can easily compromise the confidentiality, integrity, or availability of resources. These are the types of vulnerabilities that allow local users to gain privileges, allow unauthenticated remote users to view resources that should otherwise be protected by authentication, allow authenticated remote users to execute arbitrary code, or allow remote users to cause a denial of service.
Moderate impact
This rating is given to flaws that may be more difficult to exploit but could still lead to some compromise of the confidentiality, integrity, or availability of resources, under certain circumstances. These are the types of vulnerabilities that could have had a Critical impact or Important impact but are less easily exploited based on a technical evaluation of the flaw, or affect unlikely configurations.
This rating is given to all other issues that have a security impact. These are the types of vulnerabilities that are believed to require unlikely circumstances to be able to be exploited, or where a successful exploit would give minimal consequences.
A Red Hat security advisory can contain fixes for more than one vulnerability and for packages for more than one product (such as both Red Hat Enterprise Linux 5 and 6). Each issue in an advisory has an impact rating for each product. The overall severity of an advisory is the highest severity out of all the individual issues, across all the products the advisory targets. For simplicity, advisories only show the overall severity (except for kernel advisories, which list the severity of each issue). The advisories contain links to the relevant entries in Red Hat's bug-tracking system, where you can examine individual impacts and additional commentary.
When a technology—enabled and most likely used by default—completely blocks the exploitation of a particular vulnerability across all architectures, we will adjust the severity level. When a technology reduces the risk of a vulnerability, we may adjust the severity level and give an explanation of the decision in the bug-tracking entry.
Common Vulnerability Scoring System (CVSS)
Common Vulnerability Scoring System (CVSS) base scores provide additional guidance about a vulnerability, giving a detailed severity rating by scoring the constant aspects of a vulnerability: Access Vector, Access Complexity, Authentication, Confidentiality, Integrity, and Availability.
CVSS version 2 base metrics are available for all vulnerabilities from 2009 onwards, as well as selected earlier vulnerabilities. In 2016 Red Hat adopted the CVSS v3 standard, and all vulnerabilities going forward will use version 3 for scoring. These scores are found on the CVE pages (linked to from the References section of each Red Hat Security Advisory) and also from our Security Measurements page.
This information is very useful, but Red Hat does not solely use the CVSS rating to determine the priority with which flaws are fixed. It is used as a guideline to identify key metrics of a flaw, but the priority for which flaws are fixed is determined by the overall impact of the flaw using the aforementioned four-point scale.
CVSS v3 Base Metrics
The CVSS v3 base metrics group covers the constant aspects of a vulnerability:
Attack Vector (AV) - Expresses the "remoteness" of the attack and how the vulnerability is exploited.
Attack Complexity (AC) - Speaks to how hard the attack is to execute and what factors are needed for it to be successful. (The older Access Complexity metric is now split into Attack Complexity and User Interaction.).
User Interaction (UI) - Determines whether the attack require an active human to participate or if the attack can be automated.
Privileges Required (PR) - Documents the level of user authentication required for attack to be successful (replaces older Authentication metric).
Scope (S) - Determines whether an attacker can affect a component that has a different level of authority.
Confidentiality (C) - Determines whether data can be disclosed to non-authorized parties and, if so, to what level.
Integrity (I) - This measures how trustworthy the data is and how far it can be trusted to not be modified by unauthorized users.
Availability (A) - This metric is concerned with data or services being accessible to authorized users when they need to access it.
A formula translates these measurements into a single, numerical base score, ranging from 0.0 (no risk) to 10.0 (highest risk). Refer to Common Vulnerability Scoring System v3.0: User Guide for detailed descriptions of the base metrics.
How Red Hat Uses CVSS v3 Base Metrics
We aim to comply with the CVSS v3 standard when allocating base scores. There are circumstances where it may not be obvious why a particular score was chosen. The following are some common vulnerability types with our interpretation and typical scoring:
It is not always possible to know how third-party applications use libraries, making it difficult to give accurate CVSS v3 base metrics for flaws in system libraries. When rating these flaws, we take into account the way the library is used by applications that we ship, but also look at other popular software that may use the library in a way that would cause an issue to be of higher severity, and adjust the score appropriately.
Web Browsers (and associated plug-ins)
There is a class of flaws in web browsers and their plug-ins that can allow a malicious web page to execute arbitrary code when the victim accesses the page. We rate these flaws as having Critical severity, even though they need some user interaction. Scoring these issues using CVSS v3 gives a base score of 6.3 (for example CVE-2016-1953) with the base metrics of CVSS:3.0/AV:N/AC:L/PR:N/UI:R/S:U/C:L/I:L/A:L. Because the desktop is running as a non-privileged user, the CVSS v3 Confidentiality, Integrity, and Availability impacts are all rated as Low not High.
Other Common Scores
Flaws where a local, unprivileged user can cause a kernel to crash (denial of service) typically score 6.2 (for example, CVE-2014-8173).
Flaws that allow a local, unprivileged user to escalate their privileges to root typically score 8.4 (for example, CVE-2014-9322).
Cross-site scripting flaws are generally scored as having low impact to Integrity, with no impact to Confidentiality or Availability, in line with scoring from other vendors and NVD. A typical score for these flaws is 6.1 (for example, CVE-2009-0153).
Base Score Variations Across Products
It is common for a given CVE-named vulnerability to have several different CVSS base metrics, depending on the product, version, and architecture. Examples of this include:
A vulnerability that only affected some architectures. For example, CVE-2007-6694 only affected PowerPC.
A vulnerability that is mitigated by source code protection mechanisms on some platforms. For example, CVE-2009-1252 could have led to arbitrary code execution on Red Hat Enterprise Linux 4, but it is only a denial of service on Red Hat Enterprise Linux 5.
A vulnerability that affected more than one application. For example, CVE-2009-0352 affected both Firefox (web browser) and Thunderbird (mail reader), but had a lower CVSS score and severity for Thunderbird.
If CVSS v3 base scores are significantly different across products, we give them separately wherever possible. If we do not split the score, we report the metric that gives the highest CVSS v3 base score (the worst-case outcome).
Differences Between NVD and Red Hat Scores
For open source software shipped by multiple vendors, the CVSS base scores may vary for each vendor's version, depending on the version they ship, how they ship it, the platform, and even how the software is compiled. This makes scoring of vulnerabilities difficult for third-party vulnerability databases, such as NVD, who can give only a single CVSS base score to each vulnerability.
These differences can cause the scores to vary widely. For example, NVD rates Firefox flaws as having High impact metrics because the Firefox application is also available for Microsoft Windows, where it is common that the user is running Firefox with administrator privileges. For Red Hat Enterprise Linux, we use Low impact metrics, as Firefox is most likely to be run as an unprivileged user.
For these reasons, we recommend that, whenever possible, you use a CVSS base score provided by Red Hat in preference to a score from a third party. Please let us know if you believe we have given an incorrect CVSS v3 base score for a particular vulnerability. We are happy to discuss the severity and update it if needed.


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What is the birthstone for August? It seems that people born in this month are controversial, as there are so many stones appropriate for it. Currently accepted opinion is that August Birthstone is Peridot. However, contemporary and ancient charts alike contain other stones that fit people born in August. The US diagram dated 1912 also includes Sardonyx. This list matches with the British dated 2013. The current US catalog of Birthstones also includes Spinel. At the same time, the ancient sources tell us that Carnelian, Moonstone, and Topaz also correlate to the month of August. We'd like to describe Peridot because exactly that gem is in the US and British listings alike.
In fact, Peridot is chemically Chrysolite. This name derived from the ancient Greek ?????? or more accurately chrysós. This word means "gold." The second part of the name is ?????, that means "stone." Summing up both parts, we can get "Golden Stone." The American jewelers used to call this stone Peridot, and its variety, Olivine. As you can guess, the latter takes its name from the color of an olive.
History & Origin
As many other gems, the first specimens of this stone were obtained in upper Egypt. However, Ceylon (now Sri Lanka) and Pegu (presently part of Myanmar) can be considered a birthplace of Peridot. But the deposits of this gemstone also locate in Brazil, Scotland, and Mexico. The specimens called "Olivine" of volcanic origin. But some varieties of this stone came to Earth from space as the particles of meteorites. In fact, nobody knows the indeed place of origin of this gem. There are no reliable sources that prove that Chrysolite mentioned in the ancient books was Peridot or Olivine. For the centuries, jewelers have been taking Chrysoberyl for Chrysolite. But true Chrysolite softer than then Chrysoberyl. But it wins in luster and color compared to the rival.
There are too many opinions concerning the actual color of August Birthstone. It depends on what exactly stone we mean. If we're talking about Olivine, it's a yellowish emerald-green variety of Chrysolite. At the same time, more yellowish Chrysolite, resembling Tourmaline it's indeed Peridot. The color of Olivine also defines as olive-green, whence its name. When you purchase jewelry, you must take into account, that yellowish-green gemstones, called "Olivine" are in fact the Russian demantoids, more known as "Green Garnets."
Magical & Healing Properties
We can meet various mentions of Peridot (under the name Chrysolite) in several ancient sources, Medieval treatises, and even the Bible. Diodorus Siculus said that Chrysolite is the gemstone which is capable of changing its color during the day and becoming exceptionally bright at night. Thereby, he emphasizes the ability of this stone to supervise its owner. This gem was set into the Ring of Hermes during the Ceremony of the Beetle (Scarabaeus). As described, he presented a vessel (most likely a model) made of Chrysolite. The ancients realized the divine origin of this gemstone. Therefore, they regarded Peridot as a deity. Modern sources tell us that Peridot fits Leo and correlates with Gladiolus, a birth flower of August. Read more about Peridot.


23All the crowds were amazed and said, 'Can this be the Son of David?' 24But when the Pharisees heard it, they said, 'It is only by Beelzebul, the ruler of the demons, that this fellow casts out the demons.' 25He knew what they were thinking and said to them, 'Every kingdom divided against itself is laid waste, and no city or house divided against itself will stand. 26If Satan casts out Satan, he is divided against himself; how then will his kingdom stand? 27If I cast out demons by Beelzebul, by whom do your own exorcists cast them out? Therefore they will be your judges. 28But if it is by the Spirit of God that I cast out demons, then the kingdom of God has come to you. 29Or how can one enter a strong man's house and plunder his property, without first tying up the strong man? Then indeed the house can be plundered. 30Whoever is not with me is against me, and whoever does not gather with me scatters. 31Therefore I tell you, people will be forgiven for every sin and blasphemy, but blasphemy against the Spirit will not be forgiven.