Stochastic Barrier Certificates in the Presence of Dynamic Obstacles

2026-04-22

AI Summary

This work proposes a safety verification method for stochastic dynamical systems operating in the presence of dynamic obstacles, aiming to guarantee—within a finite time horizon and with high probability—that system trajectories remain within a prescribed safe set. The approach introduces time-varying stochastic barrier certificates that explicitly characterize time-dependent unsafe regions and leverages the Bellman optimality principle to model temporal structure, thereby yielding a certifiable lower bound on the probability of safety. By restricting the barrier certificates to polynomial form, the synthesis problem is cast as a convex sum-of-squares (SOS) optimization, enabling efficient computation. Experimental results demonstrate that, compared to existing methods, the proposed framework provides tighter, less conservative safety guarantees for nonlinear systems, achieving both higher accuracy and improved scalability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems

Abstract

Safety of stochastic dynamic systems in environments with dynamic obstacles is studied in this paper through the lens of stochastic barrier functions. We introduce both time-invariant and time-varying barrier certificates for discrete-time, continuous-space systems subject to uncertainty, which provide certified lower bounds on the probability of remaining within a safe set over a finite horizon. These certificates explicitly account for time-varying unsafe regions induced by obstacle dynamics. By leveraging Bellman's optimality perspective, the time-varying formulation directly captures temporal structure and yields less conservative bounds than state-of-the-art approaches. By restricting certificates to polynomial functions, we show that time-varying barrier synthesis can be formulated as a convex sum-of-squares program, enabling tractable optimization. Empirical evaluations on nonlinear systems with dynamic obstacles show that time-varying certificates consistently achieve tight guarantees, demonstrating improved accuracy and scalability over state-of-the-art methods.

Problem

• stochastic barrier certificates
• dynamic obstacles
• safety verification
• stochastic dynamic systems
• time-varying unsafe regions

Innovation

• stochastic barrier certificates
• dynamic obstacles
• time-varying safety guarantees
• sum-of-squares programming
• Bellman optimality