reachability analysis

This work addresses the challenge of efficiently and reliably computing closed-loop reachable sets for unknown nonlinear dynamical systems. The authors propose a data-driven safety verification framework that integrates Koopman operator theory with conformal prediction. By leveraging neural networks to learn a lifting map, the nonlinear system is approximated as a linear one in an augmented space, where a linear controller is designed to track reference trajectories. Reachable sets computed in this lifted space are then mapped back to the original state space using neural network verification tools. To account for model mismatch, conformal prediction is employed to quantify approximation errors and provide statistically valid probabilistic coverage guarantees. This approach achieves, for the first time, scalable, generalizable, and probabilistically sound reachability analysis without repeated computation in high-dimensional systems—including an 11-dimensional Hopper, a 28-dimensional Swimmer, and a 12-dimensional quadrotor—demonstrating significantly improved coverage, computational efficiency, and reduced conservativeness.

How can piecewise-linear trajectory tracking be achieved for unknown nonlinear systems with time-varying dynamics—without prior system knowledge—while ensuring robustness against abrupt dynamic changes? Method: We propose a data-driven online control framework that (i) identifies local system dynamics in real time via small-perturbation excitation and local system identification; (ii) analytically derives the state reachable set without model assumptions, leveraging an upper bound on the local growth rate; and (iii) synthesizes a receding-horizon closed-loop controller based on reachable-set prediction. Contribution/Results: The approach avoids global modeling and parametric structural assumptions, enabling rapid adaptation to sudden dynamic shifts. Experimental validation across multiple unknown nonlinear systems demonstrates stable waypoint-sequence tracking, confirming strong robustness and generalization capability under unmodeled dynamics and disturbances.

This work addresses the problem of certifying safety reachability prior to executing a fixed control sequence under model mismatch and sparse single-step transition data. By constructing a set-membership envelope of model errors and propagating reachability tubes via zonotopes, certification is granted only when safety constraints are satisfied. The key contribution lies in uncovering a trilemma among trajectory inclusion, finite projection width, and model error, leading to a condition-dependent certification mechanism that avoids overconfidence in regions lacking data. The approach integrates component-wise Lipschitz bounds, set-membership modeling, and lower-bound analysis of projection width. Evaluated on two benchmark systems, it outperforms calibrated baselines by rejecting certification for sequences unsupported by data while recovering valid certification when relevant data and sufficient collision-avoidance margins are present.

This work addresses the challenge of providing provably safe control in settings with unknown or stochastic system dynamics and continuous state-action spaces, where existing methods often fall short. The paper introduces the first framework that integrates conformal prediction with reachability analysis to establish probabilistic safety guarantees. Specifically, conformal prediction is employed to construct valid uncertainty sets for the unknown dynamics, within which reachability analysis is performed to verify safety and guide the training of safe reinforcement learning policies. This approach overcomes the limitations of traditional methods that rely on known or deterministic models, offering theoretically grounded probabilistic safety bounds for nonlinear systems. Evaluated across seven tasks spanning four domains—inverted pendulum, lane keeping, drone control, and safe navigation—the method achieves state-of-the-art provable safety guarantees while maintaining high average returns.

Computing discrete-time reachable sets for nonlinear systems under neural network (NN) controllers remains computationally challenging due to the trade-off between accuracy and efficiency. Method: This paper proposes a temporal refinement strategy that dynamically identifies critical time steps for high-precision symbolic queries, while employing efficient concrete simulations for remaining intervals—thereby jointly optimizing computational efficiency and bound tightness. It integrates symbolic analysis, numerical simulation, adaptive time-step scheduling, and an error-balancing heuristic. Contribution/Results: To our knowledge, this is the first approach to introduce explicit temporal granularity into reachability analysis, moving beyond conventional spatial discretization paradigms. Evaluated on multiple NN-controlled benchmarks, it reduces computation time by 20–70% over baseline methods while preserving equivalent approximation accuracy.

This work addresses the exponential growth in data requirements caused by smoothness assumptions in high-dimensional nonlinear Hamiltonian systems. It proposes a data-efficient, target-oriented reachability control method that uniquely integrates symplectic geometric inductive bias with trajectory recurrence on energy manifolds. By composing demonstrably verified local trajectory segments through a chaining strategy, the approach achieves control without relying on global smoothness. The established sufficient conditions for reachability tie sample complexity to the system’s intrinsic geometric and dynamical properties rather than its state dimensionality, thereby substantially overcoming the data bottleneck that plagues conventional methods in high-dimensional settings.

This work proposes a data-driven reachability analysis method for nonlinear dynamical systems without explicit models, leveraging denoising diffusion probabilistic models. The approach learns the system’s state distribution from trajectory data and constructs reachable sets as sublevel sets of a non-conformity score derived from reconstruction errors. By integrating the Learn Then Test framework, it calibrates the decision threshold to guarantee a Probably Approximately Correct (PAC) error bound. This is the first method to combine diffusion models with PAC guarantees, requiring no prior knowledge of the system dynamics and scaling effectively to high-dimensional settings. Experiments on the Duffing oscillator, a planar quadrotor, and a high-dimensional reaction-diffusion system demonstrate that the empirical miss rate consistently remains below the theoretical PAC bound, significantly outperforming conventional grid-based or polynomial methods.

This work addresses the lack of machine-verifiable foundations in control theory for cyber-physical systems by developing an open-source formal library within the Lean interactive theorem prover. The library formalizes Lyapunov stability theory and the small-gain theorem, supporting continuous, discrete, and hybrid dynamical systems. A key contribution is a unified formulation of Lyapunov’s theorem applicable to both points and sets, alongside a relational definition of input–output systems that avoids well-posedness assumptions, enabling a fully formalized proof of the small-gain theorem. Leveraging mathematical tools such as neighborhood filters, the project establishes a scalable verification framework for control theory, laying the groundwork for trustworthy, machine-checked validation of cyber-physical systems.

This work addresses the challenge of proving non-reachability in vector addition systems (VAS), where existing methods rely on symmetric forward–backward reasoning that does not generalize well to asymmetric models such as branching VAS. The paper proposes a purely forward construction that generates semilinear inductive invariants directly from the initial configuration, eliminating the need for backward reasoning and ensuring natural alignment between the invariants and the system’s structure. This approach constitutes the first fully forward method for constructing semilinear inductive invariants, capable of producing invariants with periodic structure in cyclic VAS. By doing so, it significantly enhances the applicability of such invariants to asymmetric infinite-state systems and establishes a theoretical foundation for extending these techniques to branching VAS.

This work addresses the challenges of transferability and computational feasibility in discrete abstraction for symbolic model checking of cyber-physical systems by proposing a conservatism-first, four-step modular workflow to construct finite-state abstractions of closed-loop dynamical systems. The approach integrates state partitioning, conservative transition construction, spurious behavior elimination, and specification semantics lifting, enabling composable and replaceable subroutine design. Transition relations are built using axis-aligned bounding boxes, polyhedra, and sampling with PAC coverage certificates, combined with certified erasure and counterexample-guided refinement. Reliable lifting of LTL specifications is achieved through may–must semantics. Evaluation across three case studies demonstrates that the workflow effectively balances abstraction accuracy and verification efficiency while clearly revealing the impact of different design choices on the outcomes.