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 $u \in C([0,T], \mathbb{R})$ $$\varphi'(t) = A \varphi(t) + N u(t) \varphi(t)$$ 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 $\psi \in V$ does this imply that $\varphi([0,T]) \subset V$?