Consider an ODE $$ \dot x(t) = A(t) x(t) + B(t) b $$ $$ x(0) = 0 $$ for $A(t) \in \mathbb{R}^{n\times n}$, $B(t) \in \mathbb{R}^{n\times m}$, $b \in \mathbb{R}^m$
Let's denote by $S(t) : \mathbb{R}^m \rightarrow \mathbb{R}^n$ taking $b$ to $x(t)$. This map is linear thus we can talk about its transpose $S^T(t)$.
Question: Is there a nice formula for $S^T(t)y$ where $y\in \mathbb{R}^n$?
Attempted solution that does not give satisfactory answer:
Denote $U(t) \in \mathbb{R}^{n \times n}$ the solution matrix for the homogenous problem i.e. $\dot U(t) = A(t) U(t)$, $U(0) = I$.
Then $x(t)$ can be expressed as: $$ x(t) = U(t) \int_0^t U^{-1}(s) B(s) b ds $$ Thus $S(t)$ is indeed linear map, give by: $$ S(t) = U(t) \int_0^t U^{-1}(s) B(s) ds $$ and its transpose is: $$ S^T(t) = \left( \int_0^t B^T(s) U^{-T}(s) ds \right) U^T(t) $$
Now for $y_0 \in \mathbb{R}^n$, I want to compute $y(t) = S^T(t)y_0$. Ideally, get an ODE for $y(t)$ and solve it.
By differentiating the expression for $S^T(t)$ we can show that it satisfies this equation: $$ \dot S^T(t) = B^T(t) + S^T(t) A^T(t) $$ Unfortunately, we can't conclude that $y(t) = S^T(t)y_0$ satisfies $$ \dot y(t) = B^T(t)y(t) + A^T(t) y(t) $$ which is only meaningful if $n=m$, but it is a type of an answer I would like to get i.e. nice ODE for $y(t)$ I can solve.
Suppose we have the following linear time-varying (LTV) system
$$ {\dot {\bf x}} (t) = {\bf A} (t) \, {\bf x} (t) + {\bf B} (t) \, {\bf u} (t)$$
with initial state ${\bf x} (t_0) =: {\bf x}_0 = {\bf 0}$. Integrating, we obtain
$$ {\bf x} (t) = \underbrace{{\bf \Phi} (t, t_0) \, {\bf x}_0}_{= {\bf 0}} + \int_{t_0}^t {\bf \Phi} (t, \tau) \, {\bf B} (\tau) \, {\bf u} (\tau) \, {\rm d} \tau $$
where ${\bf \Phi} (t, t_0)$ denotes the state-transition matrix from $t_0$ to $t$. If the input signal is constant, i.e., ${\bf u} (t) = {\bar {\bf u}}$, then
$$ {\bf x} (t) = \color{blue}{\left( \int_{t_0}^t {\bf \Phi} (t, \tau) \, {\bf B} (\tau) \, {\rm d} \tau \right)} {\bar {\bf u}}$$
Note that if $\bf A$ and $\bf B$ were time-invariant, the integrals above would be convolution integrals with matrix exponentials. For details, take a look at section 2.6 of Antsaklis & Michel.