Set Invariance with Probability One for Controlled Diffusion: Score-based Approach
This work addresses a foundational control theory problem for stochastic systems, providing a rigorous test for set invariance, which is incremental in extending score-based methods to controlled diffusions.
The paper tackles the problem of guaranteeing controlled set invariance with probability one for a diffusion process, deriving necessary and sufficient conditions based on score vector fields to certify or falsify the existence of Markovian controllers for both finite and infinite time horizons.
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.