Theory and algorithms to learn, propagate, and exploit uncertainty for stochastic optimal control of dynamical systems

Non-Gaussian uncertainty frequently arises in learning and control problems involving stochastic dynamical systems, particularly in autonomous vehicles, UAVs, satellites, and robotics. In this dissertation, we propose a new framework that leverages characteristic functions that provides a frequency-domain representation of random variables. The dissertation is structured into three key areas. First, we address model-based stochastic optimal control for linear systems with non-Gaussian noise, demonstrating that characteristic functions can be used to enforce chance constraints and control systems toward desired distributions. Second, we explore data-driven stochastic control, utilizing empirical characteristic functions to handle systems with unknown disturbances. In addition, we derive several metrics of the cost distribution through characteristic functions, which facilitates further exploration in reinforcement learning. Finally, we utilize characteristic functions in neural network verification by propagating by propagating distributions through ReLU activation functions. While this analytical propagation shows promise, we reveal its limitations in higher dimensions and propose a sampling-based approach to verification that maintains guarantees. The core novelty of this dissertation is the creation of a set of mathematical tools and methods that can be used to address difficult problems in stochastic optimal control, neural net verification, and reinforcement learning. These methods and tools are designed to facilitate learning, propagation, and exploitation of uncertainty in autonomous dynamical system, well beyond state-of-the-art approaches.