$\int_0^t e^{sA}\cos(\omega s)ds$ with $A$ matrix

Question

Let $A$ be a singular square matrix and $\omega,t\in\mathbb{R}^{*+}$. How to compute the following integral?

$$I = \int_0^t e^{sA}\cos(\omega s)\,\mathrm{d}s$$

Since I am looking for a numerical solutions, I am open to approximations. For example, $$ I = \sum_{k=0}^\infty \int_0^t \dfrac{(sA)^k}{k!} \cos(\omega s)\,\mathrm{d}s = \sum_{k=0}^\infty \dfrac{A^k}{k!}\int_0^t s^k \cos(\omega s)\,\mathrm{d}s$$

so I can truncate the above sum but I'm wondering if there is a more ingenious approach.

@metamorphy I thought of it but I am stuck with $e^{sA}e^{i\omega s}$ since it does not equal $e^{sA+i\omega s}$. — anderstood, Jan 30 '19 at 10:55
Well, $e^X e^Y=e^{X+Y}$ for commuting matrices $X, Y$. So if your $\omega$ is just a scalar, then $e^{sA} e^{i\omega s}=e^{s(A+i\omega I)}$. — metamorphy, Jan 30 '19 at 11:02

score 1 · Accepted Answer · answered Jan 30 '19 at 13:37

1

As suggested by @metamorphy in the comments, it suffices to write

$$ I = \operatorname{Re}\left(\int_0^t e^{s(A+i\omega I_n)}\,\mathrm{d}s\right)$$ and the computation boils down to the integral of a matrix exponential as addressed in this post.

answered Jan 30 '19 at 13:37

anderstood

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$\int_0^t e^{sA}\cos(\omega s)ds$ with $A$ matrix

1 Answers1