Variation of parameters for ODES with distributions as coefficients

Question

In my work, I encounter the following type of equation. Consider a non-homogeneous system \begin{equation} X'=A(t)X+f(t),\;\;\;\;\;\;\;\;(1) \end{equation} where $X$ is a $n$-dimensional vector valued function of the real variable $t$ and $A(t)$ is an $n\times n$ matrix valued integrable (on $\mathbb{R}$) function of $t$. If $f$ is smooth and if a fundamental matrix solution of the homogeneous equation $\Phi$ is known, then variation of parameter gives us the following solution to (1) $$ X=\Phi(t)\int^t \Phi^{-1}(t') f(t')dt'. $$ What about if $f$ is a distribution or, less generally, belongs to a Sobolev space $H^\gamma$ with negative $\gamma$? I have fairly certain that the same formula works in the distribution sense but I would like to have a reference with some theorems (perhaps giving some restrictions on $f$) to know for sure it is the case.

Answer 1

If $f$ is a distribution (1) may not be meaningful pointwise, so presumably what you mean is a weak solution, i.e. $X$ should be a distribution such that for any test function $\phi$,

$$ \langle \phi, A X + f - X' \rangle = \langle A^T\phi + \phi', X \rangle + \langle \phi, f \rangle = 0 $$

I think you'll want to assume $A$ is smooth as a function of $t$, so that $A^T \phi$ is a test function.

Write $X(t) = \Phi(t) Y(t)$ where $\Phi(t)$ is a fundamental matrix, so $\Phi' = A \Phi$. Then this becomes

$$ \langle (\Phi^T \phi)', Y \rangle + \langle \phi, f \rangle = 0 $$ But $$ \langle (\Phi^T \phi)', Y \rangle = - \langle \Phi^T \phi, Y' \rangle = - \langle \phi, \Phi Y' \rangle$$ so we get

$$ Y' = \Phi^{-1} f $$

and thus

$$X(t) = \Phi(t) Y(t) = \Phi(t) \int^t \Phi^{-1}(s) f(s)\; ds $$