# Boundary condition in perpetual american option problem

I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.

$$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t$$

where $$D_0$$ is the dividend. Basically the problem tries to find optimal stopping time to exercise the option i.e. we try to optimize the value function $$V(S)$$ w.r.t stopping time $$\tau$$.

$$V(S) = \sup_{\tau \in T}\ \mathbb{E}[e^{-r\tau}\max(S_\tau -K,\ 0)]$$

Once we have the Value function $$V(S)=AS^\beta$$, we use the boundary condition at optimal time to get relation between $$A$$ and $$S^*$$.

$$V(S^{*})=S^*-K$$

To get another relation (slide 12) for solving for both $$A$$ and $$S^*$$ we try to maximize $$A$$ so as to increase the value function as much as possible. I do not see any justification for this.

Contrary to the derivation in the above reference, another reference (equation 6) uses smooth pasting condition i.e.

$$V'(S^*) = f'(S*)$$ ; $$\ \ f(S) = S-K$$

The difference in the techniques to obtain the second relation between $$S^*$$ and $$A$$ leads to different solutions. I'm not convinced with the argument of maximizing $$A$$. So if anybody can justify it, it'd be great. Also which solution is actually optimal?

Thanks :)