Risk-Gated Contractive Dynamics Under Stochastic Receipts: Radius Bounds, Compositional Guarantees, and Runtime Implications

Risk-Gated Contractive Dynamics Under Stochastic Receipts: Radius Bounds, Compositional Guarantees, and Runtime Implications

This work introduces a risk-gated control framework for contractive dynamical systems operating under execution noise, model–reality mismatch, and logic-level safety envelopes. Instead of requiring uniformly negative expected Lyapunov progress, the method uses Conditional Value-at-Risk (CVaR) to eliminate candidate control steps with heavy-tailed regression events, enabling provably bounded error radii even when mean-based conditions fail.

The paper provides:

Main Contributions

• A stochastic perturbed-contraction radius theorem guaranteeing bounded steady-state error under receipts noise.

• A finite-sample CVaR gate, with explicit sample complexity and concentration bounds ensuring safe acceptance decisions.

• A monotonicity result showing that tighter risk margins reduce the achievable radius.

• A compositional logic product theorem ensuring stability when combining numerical control with LTL safety shields / runtime assurance.

• A heavy-tailed counterexample demonstrating how mean-based tests can catastrophically underestimate risk while CVaR avoids divergence.

Why this matters

Modern autonomous systems increasingly mix:

• learning-based planners,

• runtime safety shields,

• degraded-mode fallback logic, and

• uncertain execution environments.

Traditional Lyapunov or contraction tools often assume sub-Gaussian noise or bounded mean regressions. This work provides risk-aware performance guarantees that hold under distributions with rare but severe failures—a scenario relevant for robotics, embedded autonomy, formal runtime assurance, and safety-critical optimization.

Experiments

A lightweight scalar example validates:

• radius preservation under light-tailed noise,

• radius shrinkage under risk gating for heavy tails,

• compositional behavior with logic-level enforcement.