Suppose, we have two functions, $x_1(t)$ and $x_2(t)$, and they have the following relationship:
\begin{align} x_1(t) &= x_2(t), &t \leq t_1\\ x_2(t) &= x_1(t) + e^{A(t-t_1)}b, &t > t_1 \end{align}
where $b$ is a constant vector. Now, it is known that:
\begin{equation} x_1(t_{f1}) = x_f \end{equation}
i.e. at $t=t_{f1}>t_1$, the function $x_1$ has a value equal to $x_f$. My question is: using the above equations, is there a quick way to find the time $t_{f2}$ such that $x_2(t_{f2}) = x_f$? I am looking for something like a closed form solution, if its exists, that will express $t_{f2}$ in terms of $t_{f1}$ and a correction factor.
In case its helpful, $x_1(t) = e^{At} x_0$ where $x_0$ is constant vector.