Detectability and global observer design for 2D Navier-Stokes equations with uncertain inputs

TL;DR

This paper develops detectability conditions and designs a global observer for 2D Navier-Stokes equations with uncertain inputs, demonstrating convergence despite small perturbations and measurement uncertainties.

Contribution

It introduces simulation-friendly detectability conditions and a generic class of observation operators for 2D NSE, along with a global observer design that handles uncertain inputs.

Findings

Observer converges despite small initial perturbations.

Detectability conditions include pointwise and spatial average measurements.

Numerical experiments confirm robustness to uncertain inputs.

Abstract

We present simulation friendly detectability conditions for 2D Navier-Stokes Equation (NSE) with periodic boundary conditions, and describe a generic class of ``detectable'' observation operators: it includes pointwise evaluation of NSE's solution at interpolation nodes, and spatial average measurements. For ``detectable'' observation operators we design a global infinite-dimensional observer for NSE with uncertain possibly destabilizing inputs: in our numerical experiments we illustrate $H^1$-sensitivity of NSE to small perturbations of initial conditions, yet the observer converges for known and uncertain inputs.