currently I'm working on an optimal control problem and I have what I think is a really simple question about it,
The variable that I'm looking to optimize is $\lambda$ and it lies in an $X^d$ space which can be the real $\mathbb{R}^d$ space or any Functional space. Its minimum $\bar{\lambda}$ satisfies the next inequality:
$2 \beta (\bar {\lambda}, \lambda - \bar {\lambda})_{X^d} + \int_{\Omega} \phi'(\bar{u},f) p (\lambda - \bar {\lambda}) \ dx \geq 0 \ \ , \ \forall \lambda \geq 0$
Where $\phi$ and $p$ are functions, such as $u(\bar{\lambda})= \bar{u}$.
$\beta >0$ is a constant, and $\phi (u,f)$ is a functional dependant on the functions $u$ and $f$. $\bar{\lambda}$ is the minimizer such that the function $u(\lambda)$ reaches its minimum.
Now, I want to proof that, in the $X^d = \mathbb{R}^d$ case, this is the same as the system:
$$\mu _i = 2 \beta \bar{\lambda_i} + \int_{\Omega} p \phi '_i (\bar{u},f) \ dx $$ $$\mu _i \geq 0, \ \ \ \lambda_i \geq 0, \ \ \ \mu _i \lambda_i = 0$$
for $i=1,..d$
Note that $u,f,p$ are functions defined in $\mathbb{R}^2$ that depend on the coordinates $x,y$.
Can you please help me?