Time-to-reach Bounds for Verification of Dynamical Systems Using the Koopman Spectrum

In this work, we present a novel Koopman spectrum-based reachability verification method for nonlinear systems. Contrary to conventional methods that focus on characterizing all potential states of a dynamical system over a presupposed time span, our approach seeks to verify the reachability by assessing the non-emptiness of estimated time-to-reach intervals without engaging in the explicit computation of reachable set. Based on the spectral analysis of the Koopman operator, we reformulate the problem of verifying existence of reachable trajectories into the problem of determining feasible time-to-reach bounds required for system reachability. By solving linear programming (LP) problems, our algorithm can effectively estimate all potential time intervals during which a dynamical system can enter (and exit) target sets from given initial sets over an unbounded time horizon. Finally, we demonstrate our method in challenging settings, such as verifying the reachability between non-convex or even disconnected sets, as well as backward reachability and multiple entries into target sets. Additionally, we validate its applicability in addressing real-world challenges and scalability to high-dimensional systems through case studies in verifying the reachability of the cart-pole and multi-agent consensus systems.