Given a global nonlinear state feedback which globally stabilises an equilibrium, the aim of this article is to modify the local behaviour of the trajectories in order to get local optimality with respect to a given quadratic cost. A sufficient condition is given in terms of Linear Matrix Inequalities (LMIs) to design a locally optimal and globally stabilising control law. This approach is illustrated on an academic inverted pendulum model in order to stabilise its upper equilibrium point. An extension of the main result is then given to address the problematic cases. Moreover, the cases in which the previous LMI condition failed to be satisfied is addressed and a new sufficient condition is then given (which is not anymore linear).