Stabilization of nonlinear systems by neural Lyapunov approximators and Sontag's formula

This note describes a method to train a Neural Network so that it approximates a control Lyapunov function for a nonlinear system in affine form. The network is trained in a physics-informed fashion, as the training data are generated by enforcing the negativity of the orbital derivative of the clf along the system trajectories in a large set of collocation points. Positive-definiteness of the clf is guaranteed by the choice of the network structure. The network is then used to derive a stabilizing control law based on the well-known Sontag's formula. The validity of the proposed approach is illustrated through numerical examples.