I am studying control problems with Neumann control following the notes in Exact and Approximate Controllability for Distributed Parameter Systems: A Numerical Approach (Glowinksi, Lions and He). The central problem here consists of a parabolic equation in a domain $\Omega\times (0,T)$
$$ \frac{\partial y}{\partial t} + \mathcal{A}y = 0, $$
with trivial initial condition $y(0,\cdot)=0$ and control $v$ applied on a subset $\Sigma_0$ of the boundary $\Sigma = \partial\Omega\times (0,T)$,
$$ \frac{\partial y}{\partial n_{\mathcal{A}}} = v \text{ on } \Sigma_0,\quad \frac{\partial y}{\partial n_{\mathcal{A}}} = 0 \text{ on } \Sigma-\Sigma_0. $$
The Neumann control problem considered is defined by
$$ \inf_{v\in L^2(\Sigma_0)}\left(\frac{1}{2}\int_{\Sigma_0} v^2 d\Sigma + \frac{1}{2}k\|y(T;v)-y_T\|^2_{L^2(\Omega)} \right), $$
where $k$ is a positive number arbitrarily large and $y_T$ is a given state (we try to reach $y_T$ in time $T$ but we can only get to some state $y(T;v)$ as close as possible to $y_T$).
In order to solve the Neumann control problem we define a functional $J_k:L^2(\Sigma_0)\rightarrow \mathbb{R}$ as follows,
$$J_k(v) = \frac{1}{2}\int_{\Sigma_0} v^2 d\Sigma + \frac{1}{2}k\|y(T;v)-y_T\|^2_{L^2(\Omega)}. $$
At this point the authors compute its derivative making no comment about the procedure:
$$(J_k'(v),w)_{L^2(\Sigma_0)} = \int_{\Sigma_0} (v+p)wd\Sigma \quad \forall v,w\in L^2(\Sigma_0)$$
where the adjoint state function $p$ is obtained via the solution of the adjoint state equation
$$ -\frac{\partial p}{\partial t} +\mathcal{A}^*p = 0 \quad \text{in }\Omega\times (0,T) $$
with boundary and initial conditions
$$ \frac{\partial p}{\partial n_{\mathcal{A}^*}} = 0 \text{ in } \Sigma = \partial\Omega\times (0,T),\quad p(T) = k(y(T)-y_T). $$
I know how to compute the functional derivative when the functional itself is of the form
$$ J(v) = \int_{\Omega} L(x,v(x),\nabla v(x))dx $$
using the well-known Euler-Lagrange equations, but considering the functional $J_k(v)$ there is this extra term related to the $L^2(\Omega)$-norm that does not let me operate as I am used to. I tried to develop the classical proof of the E-L equations for this functional $J_k(v)$ yet I could not find a solution. I also tried to compute the weak formulation of the adjoint problem using $y$ as test but, if I made no mistakes, the only relation I could get was $\int_\Omega ky(T)(y(T)-y_T) dx = 0$ and I do not know how to link it to my original problem.
Thank you very much.