Let $a, b \in \mathbb R^n$ and $f, g \in L^1 [0,1]$. Assume for all $h \in AC[0,1 ]$ (space of absolutely continuous functions) following integral equality holds
$$ \int_{0}^{1} \langle f(t) , h(t) \rangle \; dt + \int_{0}^{1} \langle g(t) , h' (t) \rangle \; dt + \langle h(0) , a \rangle + \langle h(1) , b \rangle = 0 $$
My question : Can we simplify the above expression more? In the sense above can be equivalently written in the form of an ODE in terms of $f,g,a,b$?
Note that all functions are from $[0,1]$ to $\mathbb R^n$. However we can assume they are just real valued function for simplicity. I was thinking about trying $h = \text{constant or exponential functions.}$