I am facing an issue with the adjoint system.
I have this PDE system, which is considered as strctured by $a$
\begin{align} \begin{cases} \lambda P + \frac{\partial (\Gamma_0 P)}{\partial a} + \frac{\partial (\Gamma_1(a,x) P) }{\partial x} = -(L(a,x) + F(a,x) + d_1) P + \tilde{G}Q,\\ (\lambda + \tilde{G} + d_2)Q = L(a,x) P,\\ P(0,x) = \frac{2}{\Gamma_0} \int_0^\infty \int_0^\infty f(a,x,y) \, P(a,y) \, da \, dy. \end{cases} \label{Lösungsmodell nach t} \end{align}
and I have to get the adjoint system, which should look like that:
\begin{align} \begin{cases} \lambda \phi - \Gamma_0 \frac{\partial \phi}{\partial a} - \Gamma_1(a,x) \frac{\partial \phi}{\partial x} - 2 \int_0^\infty \phi(0,y) f(a,y,x) \, da \, dy. \\ = -(L(a,x) + F(a,x) + d_1) \phi + L(a,x) \psi,\\ (\lambda + \tilde{G} + d_2)\psi = \tilde{G}_\phi \end{cases} \end{align}
where they say it was normalised by this condition here:
\begin{align} \int_0^{\infty} \int_0^{\infty}(\phi(a,x)P(a,x)+\psi(a,x)Q(a,x))\,da\,dx=1 \end{align}
I know that this is a really specific question but I am hoping for any answers. I don't understand how to get the adjoint system and also how to use this normalisation condition.
Looking forward to any suggestions. Thanks :)