Internal stabilization of the Oseen–Stokes equations by Stratonovich noise

Abstract

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.

Stabilization of Navier--Stokes Equations by Oblique Boundary Feedback Controllers

“…Instead, we propose an explicit feedback controller of the form (1.4) easy to implement into system. It should be said that this construction resembles the form of stabilizable noise controllers recently designed in the author's works [4], [5], [6], [8], [9], which seem to be, however, more robust to stochastic perturbations.…”

Improved stability for linear SPDEs using mixed boundary/internal controls

“…C is got by setting C := Cφ i , φ j 1≤i,j≤n ; 3. We choose to write our controls as column vector in the order β 1 , α, β 2…”

“…The family D φ is given by 2i π sin (jy) i.e. 2 π sin (y) , 2 π sin (2y) , 4 π sin (y) , 4 π sin (2y) . We explicitely compute the matrices R 4×2…”

“…For the case m = 3, we consider O = (0, π)3 , O 0 = (0, π) × 0, π and Γ 1 = {0} × (0, π)2 . Again, the eigen-functions are generically given by a triple (i, j, k) ∈ (N * ) 3 by setting φ(x, y, z) = 2 sin(ix) sin(jy) sin(kz) (ordered first according to the distance to origin, then according to the first, then the second component).…”