See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92.
Given an instantaneous cost function, $g(x, u)$, we can form a quadratic approximation of this cost function with a Taylor expansion about $x_0(t)$, $u_0(t)$: $g(x,u) \approx g(x_0, u_0) + \frac{\partial g}{\partial x}\bar{x} + \frac{\partial g}{\partial u}\bar{u} + \frac{1}{2}\bar{x}^T \frac{\partial^2 g}{\partial x^2}\bar{x} + \bar{x} \frac{\partial^2 g}{\partial x \partial u} \bar{u} + \frac{1}{2} \bar{u}^T \frac{\partial^2 g}{\partial u^2}\bar{x}$
with $\bar{x} = x - x_0$ and $\bar{u} = u - u_0$. But the quadratic approximation has to be in this format: $g(x,u) \approx \bar{x}^T Q \bar{x} + \bar{u}^T R \bar{u}$
How to derive the $Q$ and $R$ matrices from the second-order Taylor expansion?