Dynamic Programming

yamdizzle

How do we solve optimization problems with infinite horizon. I tried to look online for some guidance but nothing but just problems and no solution methods. For example how can I solve:

maximize a_t [itex]\in[/itex][0,1]
[itex]\sum\frac{-2a_t}{3}[/itex]+log(S_T)
where sum goes from 0 to T-1
subject to: s_t+1 = s_t *(1+a_t)

Some sources say we can use backwards induction but doesn't really tell me how I can do so.
Can someone explain the methodology or direct me to somewhere that explains it.

Thanks