The internal stabilization by noise of the linearized Navier-Stokes equation

Abstract

Abstract. One shows that the linearized Navier-Stokes equation in O⊂Rd , d ≥ 2, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller Mathematics Subject Classification. 35Q30, 60H15, 35B40.

Internal stabilization of the Oseen–Stokes equations by Stratonovich noise

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One designs a Stratonovich noise feedback controller with support in an arbitrary open subset O 0 of O which exponentially stabilizes in probability, that is with probability one, the Oseen-Stokes systems in a domain O ⊂ R d , d = 2, 3. This completes the stabilization results from the author's work [6] which is concerned with design of an Ito noise stabilizing controller.

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“…Taking into account that, by (7), e −A s t is an exponentially stable semigroup on X s , without loss of generality we may assume that Re A s x, x ≥ γ|x| 2 H . (Otherwise, proceeding as in [6], we replace the scalar product x, y by Qx, y where Q is the solution to the Lyapunov equation A s Q + QA * s = γI.) Then, applying Ito's formula in (25), we see that…”

Internal Stabilization by Noise of the Navier–Stokes Equation

“…This work is a continuation of [2] where such a result is proved for the linearized Navier-Stokes equation associated with (1.3). The previous treatment of internal stabilization of Navier-Stokes equations ( [1], [3]) is based on the stabilization by a linear feedback provided by the solution of an algebraic infinite dimensional Riccati equation associated with the Stokes-Oseen operator A .…”

Stabilization of Navier–Stokes Flows

“…The results of Sect. 2.4 and, in particular, Theorem 2.7 were first established for the linearized Navier-Stokes equations in [14], but the treatment extended mutatis mutandis to the present general case. The results of Sect.…”

“…[18], while Theorem 4.5 on the stochastic boundary tangential stabilization was established in Barbu [14]. The results of Sect.…”