$($a$)$ Write down the fundamental equations for the dynamic programming problem:

$ma{x}_{y_t\text{i}\text{n}\text{R}}{\displaystyle \underset{t=0}{\overset{T}{}}}(x_t-y_t^2),x_{t+1}=2(x_i+y_t),t=0,1,$dots,$T-1,x_0=0$

$($b$)$ Using the recursive formulation technique, prove that the value function for the problem is

given by

$V_{T-n}(x)=(2^{n+1}-1)x+{\displaystyle \underset{t=0}{\overset{n}{}}}(2^j-1{)}^2,n=0,1,$dots,$T$

Determine the optimal controls $y_t=y_t^{**}$ and the maximum value $V=V_0(0).$