Internal Stabilization by Noise of the Navier--Stokes Equation

One shows that the Navier-Stokes equation in O⊂Rd, d = 2, 3, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller V (t, ξ) = N i=1 Vi(t)ψi(ξ) ˙βi(t), ξ ∈ O, where {βi}N i=1 are independent Brownian motions and {ψi}N i=1 is a system of functions on O with support in an arbitrary open subset O0 ⊂ O. The stochastic control input {Vi}N i=1 is found in feedback form. The corresponding result for the linearized Navier-Stokes equation was established in [2]. 2000 Mathematics Subject Classification AMS: 35Q30, 60H15, 35B40 Key words: Navier-Stokes equation, feedback controller, stochastic process, Brownian motion.

• Type: Journal
• Year: 2011
• Where: SIAMCO
• Authors: Viorel Barbu, Giuseppe Da Prato
• Keywords: Internal Noise Controller, Navier-Stokes Equation, SIAMCO 2011, Software Engineering, Unstable Equilibrium Solution