I'm trying to program something like this article. On page 3, there's the following equation (eq. 6) that should be expressible in closed-form:
$$\int_0^te^{(A(t-t'))} M e^{(A^T(t-t'))}dt'$$
where $A$ is a nilpotent matrix. Both $A$ and $M$ are constant matrices and, for this specific case, of size $5 \times 5$. I've tried integrating by parts, but after substituting $x = t - t'$ and simplifying, I was basically right back at the start:
$$A \left( \int_0^te^{(Ax)} M e^{(A^Tx)}dx \right) + \left( \int_0^te^{(Ax)} M e^{(A^Tx)}dx \right) A^T = e^{(Ax)} M e^{(A^Tx)} - M$$
which is basically just applying
$$\int u'v + \int uv' = [uv]$$
How should I tackle this? Thanks in advance!