On an optimal control problem for a parabolic equation with an integral condition and controls in coefficients

In this paper, an optimal control problem for a parabolic equation with an integral boundary condition and controls in coefficients is considered. Let it be required to minimize the functional
$$ J(\nu)=\int_0^{\mathfrak{l}}|u(x;T;\nu)-y(x)|^2dx $$
on the solutions $u=u(x,t)=u(x,t;\nu)$ of the boundary value problem
\begin{gather*} u_t-(k(x,t)u_x)_x+q(x,t)u=f(x,t),\quad (x,t)\in\mathcal{Q}_T=\{(x,t): 0<x<\mathfrak{l},\ 0<t\leqslant T\}\\ u(x,0)=\varphi(x),\ 0\leqslant x\leqslant \mathfrak{l},\\ u_x(0,t)=0, \quad k(l,t)u_x(\mathfrak{l},t)=\int_0^{\mathfrak{l}}H(x)u_x(x,t)dx+g(t),\quad 0<t\leqslant T, \end{gather*}
corresponding to all allowable controls $\nu=\nu(x,t)=(k(x,t),q(x,t))$ from the set
\begin{gather*} V=\{\nu(x,t)=(k(x,t),q(x,t))\in H=W_2^1(\mathcal{Q}_T)\times L_2(\mathcal{Q}_T): 0<v<k(x,t)\leqslant\mu,\\ |k_x(x,t)|\leqslant\mu_1,\ |k_t(x,t)|\leqslant\mu_2\quad |q(x,t)|\leqslant\mu_3 \text{ a.e. on }\mathcal{Q}_T\}. \end{gather*}
Here, $l, T, v, \mu, \mu_1, \mu_2, \mu_3>0$ are given numbers and $y(x), \varphi(x)\in W_2^1(0,\mathfrak{l})$, $H(x)\in \mathring{W}_2^1(0,\mathfrak{l})$, $f(x,t)\in L_2(\mathcal{Q}_T)$, and $g(t)\in W_2^1(0,T)$ are known functions.
The work deals with problems of correctness in formulating the considered optimal control problem in the weak topology of the space $H=W_2^1(\mathcal{Q}_T)\times L_2(\mathcal{Q}_T)$. Examples showing that this problem is incorrect in the general case in the strong topology of the space $H$ are presented. The objective functional is proved to be continuously Frechet differentiable and a formula for its gradient is found. A necessary condition of optimality is established in the form of a variational inequality.