Unsafe Probabilities and Risk Contours for Stochastic Processes using Convex Optimization
This work addresses safety analysis for stochastic systems, providing interpretable risk contours, but it is incremental as it builds on existing barrier certificate methods.
The paper tackles the problem of estimating the maximal probability of unsafety for trajectories of stochastic processes relative to hazard sets by formulating it as a convex optimization problem, achieving nonconservative estimates under certain conditions and demonstrating the approach with examples.
This paper proposes an algorithm to calculate the maximal probability of unsafety with respect to trajectories of a stochastic process and a hazard set. The unsafe probability estimation problem is cast as a primal-dual pair of infinite-dimensional linear programs in occupation measures and continuous functions. This convex relaxation is nonconservative (to the true probability of unsafety) under compactness and regularity conditions in dynamics. The continuous-function linear program is linked to existing probability-certifying barrier certificates of safety. Risk contours for initial conditions of the stochastic process may be generated by suitably modifying the objective of the continuous-function program, forming an interpretable and visual representation of stochastic safety for test initial conditions. All infinite-dimensional linear programs are truncated to finite dimension by the Moment-Sum-of-Squares hierarchy of semidefinite programs. Unsafe-probability estimation and risk contours are generated for example stochastic processes.