Internal Stabilization by Noise of the Navier–Stokes Equation

Abstract

We show that the Navier–Stokes equation in $\mathcal{O}\subset R^d$, $d=2,3$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V(t,\xi)=\sum_{i=1}^{N}V_i(t)\psi_i(\xi)\dot{\beta}_i(t)$, $\xi\in\mathcal{O}$, where $\{\beta_i\}_{i=1}^N$ are independent Brownian motions and $\{\psi_i\}_{i=1}^N$ is a system of functions on $\mathcal{O}$ with support in an arbitrary open subset $\mathcal{O}_0\subset\mathcal{O}$. The stochastic control input $\{V_i\}_{i=1}^N$ is found in feedback form. The corresponding result for the linearized Navier–Stokes equation was established in [E. Barbu, The internal stabilization by noise of the linearized Navier–Stokes equation, ESAIM Control Optim. Calc. Var., to appear].