Why is positive semi definiteness a sufficient condition for Lyapunov stability of the Hopfield networks?
The fact that it's a Lyapunov function means that
$$
\frac{dL}{dt} = \frac{\partial L}{\partial x} \cdot \frac{dx}{dt} \leq 0
$$
If $I-W$ is positive semidefinite, $L$ is bounded from below by $-\frac{1}{2} b^T (I-W)^{-1} b $.
This means that the network will end up in a minimum of $L$ and cannot keep moving for ever.