The discrete ARE is

Or,

We can get three equations from the discrete ARE. These are

$\displaystyle 0.25p_3-p_1+1-\frac{0.25p_3^2}{0.1+p_3}=0$
$\displaystyle 0.4p_3-0.5p_2+0.5-\dfrac{0.5p_2p_3+0.4p_3^2}{0.1+p_3}=0$
$\displaystyle p_1+1.6 p_2 -0.36p_3+1-\frac{p_2^2+1.6p_2p_3+0.64p_3^2}{0.1+p_3}=0$

Since the above three equations comprises three unknown parameters, these parameters can be solved uniquely, as

$\displaystyle p_1= 1.0238, \quad p_2=0.5513, \quad p_3 = 1.9811$

The optimal control law can be found out as

The optimal cost can be found as