UC Berkeley

Reachability analyzes a dynamic system’s abilities to reach goals or maintain safety. This analysis plays an essential role in various safety-critical applications. Previous reachability theory characterizes the success or failure of reachability tasks. However, this does not tell us the degree to which the goal will be achieved, or the safety will be maintained. Our new reachability formulation aims to provide measures for each goal-reaching and safety.

This dissertation introduces three bodies of work. The first presents state-constrained reachability problems that provide the goal-reaching or safety metrics, and the corresponding Hamilton-Jacobi (HJ) frameworks. The HJ frameworks guarantee performance and safety metrics for general nonlinear systems with non-convex constraints. Despite this advantage, its computational complexity is exponential in the state dimension. Thus, it is not scalable for high-dimensional systems. In order to alleviate this computational complexity, the second presents efficient Hopf-Lax theory, which provides analytic solutions to HJ partial differential equations (PDEs) for state-constrained reachability problems, and the third presents reinforcement-learning approaches.