A Two-Part Controller Synthesis Approach for Nonlinear Stochastic Systems Perturbed by Lévy Noise Using Renewal Theory and HJB-Based Impulse Control

We are motivated by the lack of discussion surrounding methodological control design procedures for nonlinear shot and L\'evy noise stochastic systems to propose a hierarchical controller synthesis method with two parts. The first part is a primitive pattern-learning component which recognizes specific state sequences and stores in memory the corresponding control action that needs to be taken when the sequence has occurred. The second part is a modulation control component which computes the optimal control action for a pattern when it has occurred for the first time. Throughout our presentation of both components, we provide a self-contained discussion of theoretical concepts from Poisson processes theory, renewal theory, and impulse control, all of which are necessary as background. We demonstrate application of this controller to the simplified, concrete case studies of fault-tolerance and vehicle congestion control.