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