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