Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation $$y'(t)=A^*y(t).$$
I would like to understand whether something like this is true in a more general context:
Given the linear ODE in a Hilbert space $H$
$$y'(t)=Ay(t)+By(t)\kappa(t)$$ where $A$ is the generator of a $C_0$ semigroup and $B$ a bounded operator $B:H \rightarrow H$ is bounded and $\kappa \in C_c^{\infty}$ a scalar function.
The solution for some initial state $y_0$ can be expressed in terms of a flow
$y(t)=\Phi(t)y_0.$ Thus, $\Phi(t):H \rightarrow H$ and one can consider the adjoint operator $\Phi(t)^*.$
It is very tempting to assume that the adjoint flow operator solves
$$y'(t)=A^*y(t)+B^*y(t) \kappa(t).$$
Is this true? The problem is that the solution can often only be expressed as a mild or weak solution.
The weak solution for example reads for $x \in D(A^*)$
$$\langle \Phi(t)y_0,x\rangle = \langle y_0,x \rangle + \int_0^t \langle \Phi(s)y_0, A^*x+\kappa(s) B^*x \rangle ds$$
and I do not see how to infer anything from this formualtion about $\Phi(t)^*.$
