Blending Neural Control Density Functions for Stabilization and Safety

Recent work on Neural Network-based methods for nonlinear control use Lyapunov Functions to obtain controllers with guarantees of stability. However, Lyapunov-based methods are fundamentally limited: they cannot be used for smooth blending with formal Region of Attraction (RoA) expansion guarantees, and also fail to certify stability when unstable equilibria or saddle points are present. Density functions provide an alternate stability certificate, and address these limitations by certifying almost everywhere stability, and enable smooth blending of controllers. Learning valid density certificates is challenging due to integrability constraints, and the effect of density-based blending controllers on RoAs is not well understood. In this work, we provide the first guarantee that controllers blended with density functions yield RoAs containing the union of the RoAs achieved by the constituent controllers. Then, we propose a novel exponential characterization of density functions that provably satisfies the integrability condition, and introduce Neural Control Density Functions (NCDFs), that leverage this new parameterization. We also extend NCDFs for synthesizing safe-stable controllers by combining NCDFs with control barrier functions (NCDF-CBFs). Our experiments show that blended controllers obtain superior RoAs to state-of-the-art methods like Neural Lyapunov Control and Sum-of-Squares based techniques.