My calculus skills are a bit rusty. Can somebody help me to solve the following integral?
$$\int_{t_0}^t f(s)f(s)^\top ds$$
with $$f(s)=e^{A(t-s)}Q$$
where $A$ is an invertible matrix.
2026-04-09 18:14:25.1775758465
Solve Multivariate Integral with Matrix Exponential
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Note that $\int_{t_0}^t f(s)f(s)^Tds=e^{tA}.\int_{t_0}^te^{-sA}QQ^Te^{-sA^T}ds .e^{tA^T}$.
If $e^{-sA}Q$ and $A$ are normal and if $A+A^T$ is invertible, then $\int_{t_0}^te^{-sA}QQ^Te^{-sA^T}ds=Q^T\int_{t_0}^te^{-s(A+A^T)}dsQ=Q^T(A+A^T)^{-1}(e^{-t_0(A+A^T)}-e^{-t(A+A^T)})Q$