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I need help finding,

$$\lim_{t\to\infty}\int_0^t \exp((t-s)A)g(s)\,\mathrm{d}s$$ when $$\lim_{t\to\infty} |g(t)|=g_0$$

Here A is a nxn matrix, whose eigenvalues satisfy $$\Re(\alpha_j)<0$$ and g(t) is a vector.

Please help me I feel stuck. When A is just a complex number $A=\alpha$ $$\lim_{t\to\infty}\int_0^t \exp((t-s)\alpha) g(s)\mathrm{d}s=\lim_{t\to\infty} exp(t\alpha)\int_0^t\exp((-s)\alpha) g(s)\mathrm{d}s$$

What is the $$\lim \int_0^t\exp(-s\alpha) g(s)\mathrm{d}s ?$$

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    What is A, t, g(s)...?? The limit when what goes where? Please do explain yourself.2012-11-27
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    Before you tackle the general case, what happens in the one dimensional case ($A$ is a constant, $g$ is a complex valued function)? Next, what happens in the higher dimensional case when $g$ is always an eigenvector of $A$? What if $A$ is diagonalizable?2012-11-27
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    It is the Laplace transform of $g(s)$.2012-11-27
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    @pragabhava Where does is go when Lim g(0)-g_0?2012-11-27

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