2021/09/24 by Tyler Lekang, Lekang, Tyler, Andrew Lamperski +1 · 1 citation
Environmental Science · Mathematics · Physics and Astronomy · #FOS: Mathematics #Groundwater flow and contamination studies #Markov Chains and Monte Carlo Methods #Model Reduction and Neural Networks #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2109.12198
openalex publication_date 2021/09/24 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
This paper is motivated by the problem of quantitatively bounding the convergence of adaptive control methods for stochastic systems to a stationary distribution. Such bounds are useful for analyzing statistics of trajectories and determining appropriate step sizes for simulations. To this end, we extend a methodology from (unconstrained) stochastic differential equations (SDEs) which provides contractions in a specially chosen Wasserstein distance. This theory focuses on unconstrained SDEs with fairly restrictive assumptions on the drift terms. Typical adaptive control schemes place constraints on the learned parameters and their update rules violate the drift conditions. To this end, we extend the contraction theory to the case of constrained systems represented by reflected stochastic differential equations and generalize the allowable drifts. We show how the general theory can be used to derive quantitative contraction bounds on a nonlinear stochastic adaptive regulation problem.