Stochastic stability of master--slave synchronization for a class of PDEs with Burgers-type nonlinearity

Abstract

Master--slave synchronization reconstructs the evolving state of a reference (or master) system by continuously correcting a second system, the slave, using observations of the master. In applications, the observations are often contaminated by noise, so the corrections injected into the slave are themselves uncertain. In this paper we aim at quantifying the effect of noisy observations on master--slave synchronization for a family of evolution equations that combine a quadratic transport term with a polynomial linear differential operator. The family includes the Burgers, Kuramoto--Sivashinsky, Kawahara, Benney--Lin and Nikolaevskiy equations, among others. Under periodic boundary conditions, a Fourier--Galerkin approximation of order $K$ yields a finite-dimensional master--slave system in which white observation noise of intensity $\sigma$ is modeled through an Itô stochastic differential equation. In this framework, we first establish local exponential stability of noiseless synchronization for sufficiently strong scalar coupling. Then we compare the stochastic synchronization error with its noiseless counterpart resulting from the same initial slave state, isolating in this way the effect of noise. Assuming bounded reference paths, the mean-square deviation between the noiseless and noisy synchronization errors is $\mathcal{O}({\sigma }^2)$ on finite time intervals. Under additional regularity assumptions, we also bound the error between the finite-resolution stochastic slave model and the infinite-dimensional deterministic master field, explicitly separating spectral approximation error, deterministic synchronization error and observation noise. Numerical experiments for the Kawahara and Nikolaevskiy equations confirm the predicted noise scaling.