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

2025-07-30

AI Summary

This paper addresses the probabilistic invariance problem for controlled diffusion systems over a given connected, bounded Lipschitz domain. Methodologically, it introduces a novel necessary and sufficient condition based on the score vector field—the gradient of the log-density—thereby pioneering the integration of this statistical learning concept into stochastic control invariance analysis and unifying finite- and infinite-horizon settings. The condition is explicitly verifiable and enables either the constructive synthesis of all Markovian controllers ensuring invariance or a rigorous proof of their nonexistence; for finite horizons, it further incorporates terminal target-set constraints. The approach combines log-likelihood gradient analysis with the solution of Dirichlet boundary-value problems to achieve a complete characterization of invariance. Semi-analytical and numerical experiments validate the criterion’s effectiveness, computational tractability, and constructive nature.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Intelligent Robots: Behavior Learning & Control

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

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.

Problem

• Determine conditions for controlled set invariance in diffusions
• Develop score-based test for Markovian controller existence
• Solve Dirichlet problems to verify set invariance feasibility

Innovation

• Score-based test for set invariance
• Dirichlet boundary value problems
• Markovian controllers characterization