Internal Stabilization by Noise of the Navier–Stokes Equation

We show that the Navier-Stokes equation in O C Rd, d = 2, 3, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller V (t, ξ) = Σ Ni=1 Vi(t)ψi(ξ) ̇Βi(t), ξ ε O, where {Βi} Ni=1 are independent Brownian motions and {ψi}Ni=1 is a system of functions on O with support in an arbitrary open subset O0 C O. The stochastic control input {Vi}Ni=1 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].