Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C\_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t) + Z u(t) \varphi(t)$$
with $\varphi(0)=\varphi\_0 \in D(A)$ has a unique solution.


I would like to know if the following is true: 


Let $\varphi\_0 \in V \cap D(A)$ where $V$ is a closed subspace of $X$. If we have that 
$\langle \varphi'(t) , \psi \rangle =0$ for all $\psi \in V^{\perp}$ does this imply that $\varphi([0,T]) \subset V$?



Evolution equation invariance of sets
If $\varphi'(t)$ is orthogonal to $V^\perp$ then it belongs to $V^{\perp\perp}=V$, hence $$\varphi(s)=\varphi(0)+\int\_0^s\varphi'(t)dt \in V.$$


