safety-critical-contextual-control-riemannian

name	safety-critical-contextual-control-riemannian
version	1.0.0
description	Safety-critical contextual control via online Riemannian optimization with world models — Penalized Predictive Control (PPC) framework for provably safe control using black-box simulators and score-based density estimation.
category	systems-engineering
tags	["safety-critical control","Riemannian optimization","world models","predictive control","online optimization","barrier functions","cyber-physical systems"]
source	arXiv:2604.19639v1
authors	["Tongxin Li"]
date	2026-04-21T00:00:00.000Z

Safety-Critical Contextual Control via Online Riemannian Optimization with World Models

Overview

Penalized Predictive Control (PPC) — a framework that unifies black-box simulator-based world models with online density estimation to achieve provable safety in contextual control problems. The key innovation is replacing Lipschitz-continuity assumptions with barrier curvature λ, enabling tighter safety bounds.

Core Architecture: Simulator–Planner Decomposition

┌──────────────┐         score ŝ_t = ∇_u ln p̂_t        ┌──────────────┐
│   Simulator  │ ───────────────────────────────────────▶ │   Planner    │
│ (World Model)│                                          │ (Controller) │
│              │ ◀─────────────────────────────────────── │              │
│  Compresses  │         action u_t                       │ Minimizes    │
│  feasibility │                                          │ free energy  │
│  manifold to │                                          │ functional   │
│  density p̂_t │                                          │              │
└──────────────┘                                          └──────────────┘

Simulator (World Model):

• Compresses the feasibility manifold into a score-based density p̂_t(u)
• Transmits the score ŝ_t = ∇_u ln p̂_t to the Planner
• Does NOT need to be differentiable — black-box is sufficient

Planner (Controller):

• Minimizes a free energy functional via Gibbs-Boltzmann distribution
• Combines contextual density from simulator with task objective
• Produces safe actions through probabilistic sampling

Key Mathematical Framework

1. Penalized Predictive Control (PPC) Objective

The planner solves:

min_u  J(u) − λ ln p̂_t(u)

Where:

• J(u) = task cost function
• p̂_t(u) = estimated feasibility density from simulator
• λ = barrier curvature parameter (controls safety margin)

2. Gibbs-Boltzmann Policy

The optimal policy is:

π*(u|s) = exp(−J(u)/λ) · p̂_t(u) / Z

Where Z is the partition function (normalization constant).

3. Safety Bound Theorem

Key Result: The distance from the true feasibility manifold is bounded by: