I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point.
For the given control problem, HJB eq. is defined as follows:
$\displaystyle (\lambda - \mu_1)\frac{\partial v}{\partial x_1} + (\mu_1 - \mu_2)\frac{\partial v}{\partial x_2} + c_1 x_1 + c_2 x_2 =0$
However, the author then defines ( in the context of perturbation analysis) by Taylor expansion:
$\displaystyle G_1 V + \frac{\epsilon^2}{2}G_2 V + \frac{\epsilon^3}{6}G_3 V .. + c_1 x_1 + c_2 x_2=0$
Then comes the part that I am confused about:
$\displaystyle G_1 = (\lambda - \mu_1)\frac{\partial v}{\partial x_1} + (\mu_1 - \mu_2)\frac{\partial v}{\partial x_2}$
and $\displaystyle G_2=(\lambda + \mu_1)\frac{\partial v^2}{\partial x_1^2} + (\mu_1 + \mu_2)\frac{\partial v^2}{\partial x_2^2} - 2 \mu_1 \frac{\partial v^2}{\partial x_1 x_2} $
I do get that $G_1 $is first order approximation but I have no idea how $G_2$ is derived the way it is. Any help is appreciated, thanks in advance!