Large Deviations in Safety-Critical Systems with Probabilistic Initial Conditions

TL;DR

This work addresses the challenge of quantifying rare safety events in dynamical systems when initial conditions are uncertain. It extends Large Deviations theory to incorporate stochastic initial states by deriving a conditional initial-state distribution given the instanton reaching an unsafe region and formulates a MAP pathway that jointly optimizes the initial state, hitting time, and path deviations, via a variational framework. Key contributions include a principled expression for the initial-state density conditioned on unsafe hitting, and a computational pathway to identify the most probable collision configurations through a MAP/ML comparison, demonstrated on a high-dimensional spacecraft-collision problem. The approach enables more accurate rare-event estimates and supports design of importance-sampling strategies and avoidance controls for safety-critical applications.

Abstract

We often rely on probabilistic measures--e.g. event probability or expected time--to characterize systems safety. However, determining these quantities for extremely low-probability events is generally challenging, as standard safety methods usually struggle due to conservativeness, high-dimension scalability, tractability or numerical limitations. We address these issues by leveraging rigorous approximations grounded in the principles of Large Deviations theory. By assuming deterministic initial conditions, Large Deviations identifies a single dominant path as the most significant contributor to the rare-event probability: the instanton. We extend this result to incorporate stochastic uncertainty in the initial states, which is a common assumption in many applications. To that end, we determine an expression for the probability density of the initial states, conditioned on the instanton--the most dominant path hitting the unsafe region--being observed. This expression gives access to the most probable initial conditions, as well as the most probable hitting time and path deviations, leading to an unsafe rare event. We demonstrate its effectiveness by solving a high-dimensional and non-linear problem: a space collision.

Figures (4)

• Figure 1: Geometric description of the problem.
• Figure 2: Conjunction geometry and MAP path solution.
• Figure 3: Numerical solution of $\lambda$ superposed with the solution of (29b), and boundary conditions (30a) and (30b).
• Figure 4: Comparison between the action $S$ and magnitude of the deviation $||w||$ exerted by the ML and MAP paths.