mlsanityData to 14 Oct 2026

Set Invariance with Probability One for Controlled Diffusion: Score-based Approach

arXiv 2507.22385 ↗First posted 30 Jul 2025math.OC cs.LG cs.SYcited by 0

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Abstract (arXiv)

Given a controlled diffusion and a connected, bounded, Lipschitz set, when is it possible to guarantee controlled set invariance with probability one? In this work, we answer this question by deriving the necessary and sufficient conditions for the same in terms of gradients of certain log-likelihoods -- a.k.a. score vector fields -- for two cases: given finite time horizon and infinite time horizon. The deduced conditions comprise a score-based test that provably certifies or falsifies the existence of Markovian controllers for given controlled set invariance problem data. Our results are constructive in the sense when the problem data passes the proposed test, we characterize all controllers guaranteeing the desired set invariance. When the problem data fails the proposed test, there does not exist a controller that can accomplish the desired set invariance with probability one. The computation in the proposed tests involve solving certain Dirichlet boundary value problems, and in the finite horizon case, can also account for additional constraint of hitting a target subset at the terminal time. We illustrate the results using several semi-analytical and numerical examples.

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References · 16

  1. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  2. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  3. 2021Score-based Generative Modeling in Latent Spacecited by 636
  4. 2016Control Barrier Function Based Quadratic Programs for Safety Critical Systemscited by 287
  5. 2022Score-based Generative Modeling of Graphs via the System of Stochastic Differential Equationscited by 283
  6. 2022Convergence of score-based generative modeling for general data distributionscited by 181
  7. 2022Safety Certification for Stochastic Systems via Neural Barrier Functionscited by 22
  8. 2019A Barrier Function Approach to Finite-Time Stochastic System Verification and Controlcited by 14
  9. 2020Control Barrier Functions for Stochastic Systemscited by 14
  10. 2018Stochastic reachability of a target tube: Theory and computationcited by 7
  11. 2022Control Barrier Functions for Stochastic Systems and Safety-critical Control Designscited by 4
  12. 2022Reach-Avoid Analysis for Polynomial Stochastic Differential Equationscited by 3
4 more
  1. 2021Safety-Critical Control of Stochastic Systems using Stochastic Control Barrier Functionscited by 2
  2. 2020On a notion of stochastic zeroing barrier functioncited by 1
  3. 2022Mean viability theorems and second-order Hamilton-Jacobi equationscited by 1
  4. 2023Safety Under Uncertainty: Tight Bounds with Risk-Aware Control Barrier Functionscited by 1

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