Constrained non-linear estimation and links with stochastic filtering
Louis-Pierre Chaintron, Laurent Mertz, Philippe Moireau, Hasnaa Zidani
TL;DR
The paper addresses state estimation for finite‑dimensional, constrained, non‑smooth (sweeping) dynamics by linking deterministic Mortensen observers to stochastic filtering through large‑deviation limits. It develops a penalisation approach to approximate the constrained dynamics by smooth ones, proves convergence of the penalised value function $\mathscr{V}^\kappa$ to the constrained value $\mathscr{V}$, and characterises $\mathscr{V}$ as a viscosity solution of a Hamilton–Jacobi–Bellman equation with distinct Neumann boundary conditions due to non‑reversibility. It further connects to stochastic filtering by showing small‑noise limits reproduce Kalman‑Bucy behavior in the linear case and yield large‑deviation principles that concentrate the filter on minimisers of $\mathscr{V}$, both for penalised and for reflected dynamics (with boundary effects). The analysis exposes how boundary conditions and inward pointing drift affect uniqueness and how dual/limit formulations can bypass non‑uniqueness, providing quantitative convergence rates and numerical illustrations. These results bridge deterministic observer theory and stochastic filtering under state constraints, with implications for robust real‑time estimation in constrained environments.
Abstract
This article studies the problem of estimating the state variable of non-smooth subdifferential dynamics constrained in a bounded convex domain given some real-time observation. On the one hand, we show that the value function of the estimation problem is a viscosity solution of a Hamilton Jacobi Bellman equation whose sub and super solutions have different Neumann type boundary conditions. This intricacy arises from the non-reversibility in time of the non-smooth dynamics, and hinders the derivation of a comparison principle and the uniqueness of the solution in general. Nonetheless, we identify conditions on the drift (including zero drift) coefficient in the non-smooth dynamics that make such a derivation possible. On the other hand, we show in a general situation that the value function appears in the small noise limit of the corresponding stochastic filtering problem by establishing a large deviation result. We also give quantitative approximation results when replacing the non-smooth dynamics with a smooth penalised one.