It's a question I haven't thought of. It's evidence the solution of damped harmonic oscillation can be represented by:
$$x(t) = e^{\textstyle -\delta\,t}\,\left[c_1\,\cos(\omega_e\,t)+c_2\,\sin(\omega_e\,t)\right]$$
where the initial ODE was $$0 = \ddot{x} + \beta\,\dot{x}+m\,{\omega_0}^2$$ as well as $\delta = \frac{\beta}{2\,m}$ plus $\omega_e = \sqrt{{\omega_0}^2-\delta^2}$
with starting conditions $x(0) = x_0$ and $\dot{x}(0) = 0$ this further specifies to:
$$x(t) = e^{\textstyle -\delta\,t}\,\left[x_0\,\cos(\omega_t\,t)+\dfrac{\delta\,x_0}{\omega_e}\,\sin(\omega_t\,t)\right]$$
This is all open access, but I've found no one addressing ending conditions.
So I ask myself can a value $x_{\textstyle t = \infty}$ be determined? I'm stunting since $\displaystyle \lim_{t\to\infty}x(t)$ seems not probable.