Neural controller for incremental stability of unknown continuous-time systems

This work primarily focuses on synthesising a controller that guarantees an unknown continuous-time system to be incrementally input-to-state stable (δ-ISS). In this context, the notion of δ-ISS control Lyapunov function (δ-ISS-CLF) for the continuous-time system is introduced. Combined with the controller, the δ-ISS-CLF guarantees that the system is incrementally stable. As the paper deals with unknown dynamical systems, the controller as well as the δ-ISS-CLF are parametrised using neural networks. The data set used to train the neural networks is generated from the state space of the system by proper sampling. Now, to give a formal guarantee that the controller makes the system incrementally stable, we develop a validity condition by having some Lipschitz continuity assumptions and incorporate the condition into the training framework to ensure a provable correctness guarantee at the end of the training process. Finally, we demonstrate the effectiveness of the proposed approach through several case studies.