Safety Certificate for Stochastic Fractional-Order Dynamical Systems

Certifying the safety of stochastic systems with long-range memory is challenging because future risk can depend on the entire state history. Existing safety certificates commonly assume Markovian dynamics, and their safety and persistent-feasibility guarantees do not carry over when infinite histories are unavailable to an online safety certificate. In this paper, we study how to quantify risk and ensure safety when only limited history information is available to the online certificate. First, we propose a Bellman-like backward recursion that constructs lower bounds on the value and action-value functions, and hence on long-term risk, using only a finite-memory representation. Second, we derive constraints on control actions that guarantee long-term safety while ensuring persistent feasibility. By combining offline-computed lower bounds with these action constraints, we obtain an efficient online safety certificate whose certified action set remains nonempty at every decision step. Then, we apply the framework to stochastic fractional-order dynamical systems, whose long-range memory makes existing Markov-based risk evaluation and safety certification techniques inapplicable. Finally, we use numerical simulations to demonstrate the effectiveness of the proposed framework and compare with existing techniques.