Nonlinear and locally optimal controllers design for input affine locally controllable systems

Given a global nonlinear state feedback which stabilizes globally an equilibrium, the aim of this paper is to modify the local behavior 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 (LMI) to design a locally optimal and globally stabilizing control law. This approach is illustrated on an academic inverted pendulum model in order to stabilize 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).