Exponentially convergent data assimilation algorithm for Navier-Stokes equations

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier-Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in ℝ
²
. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate the results we apply the proposed algorithm to the reconstruction a chaotic fluid flow from incomplete and noisy data.