Constrained Nonlinear Estimation and Links with Stochastic Filtering

Abstract. This article studies the problem of estimating the state variable of nonsmooth 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 supersolutions have different Neumann type boundary conditions. This intricacy arises from the nonreversibility in time of the nonsmooth 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 nonsmooth 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 nonsmooth dynamics with a smooth penalized one. We eventually provide a numerical illustration of the estimation problem in a one-dimensional configuration with zero drift coefficient.