Internal Stabilization by Noise of the Navier–Stokes Equation

Barbu, Viorel; Prato, Giuseppe Da

doi:10.1137/09077607x

Cited by 16 publications

(12 citation statements)

References 15 publications

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“…By the law of the iterated logarithm and arguing similarly as in [4,Lemma 3.4], it follows that there exists a constant C Γ > 0 such that Γ(t) = e θW (t)−θ1t e −(m S + 1 100 )t ≤ C Γ e −(m S + 1 100 )t , ∀t > 0, P − a.s.,…”

mentioning

confidence: 89%

Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise

Munteanu¹

2020

Evolution Equations &Amp; Control Theory

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Here we study the problem of boundary feedback stabilization to unbounded trajectories for semi-linear stochastic heat equation with cubic nonlinearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. The simpleform feedback allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument its stability is proved. The approach requires the initial data to be a random variable implying the fact that the solution of the random equation is not adapted. Thus, one cannot recover the solution of the initial stochastic equation from the random one. Hence, the designed feedback controller stabilizes the associated random equation and not the original stochastic equation. Anyway, it stabilizes its random version.

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confidence: 89%

Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise

Munteanu¹

2020

Evolution Equations &Amp; Control Theory

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“…, u N ) is a control function expressed with a feedback formulation. In [25] and [1], where d = 2, and d = 3, respectively, the feedback control is obtained from the solution of a finite-dimensional Riccati equation while a stochastic-based stabilization technique is employed in [5], in the case of an internal control, which avoids the difficult computation problems related to infinite-dimensional Riccati equations. The procedure employed in [3] for a boundary control resembles the form of stabilizing noise controllers designed in [5].…”

Section: Evrad M D Ngom Abdou Sène and Daniel Y Le Rouxmentioning

confidence: 99%

Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller

Ngom¹,

Sène²,

Roux³

2015

EECT

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This paper presents a global stabilization for the two and threedimensional Navier-Stokes equations in a bounded domain Ω around a given unstable equilibrium state, by means of a boundary normal feedback control. The control is expressed in terms of the velocity field by using a non-linear feedback law. In order to determine the feedback control law, we consider an extended system coupling the equations governing the perturbation with an equation satisfied by the control on the domain boundary. By using the Faedo-Galerkin method and a priori estimation techniques, a stabilizing boundary control is built. This control law ensures a decrease of the energy of the controlled discrete system. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions.

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“…A more delicate problem arises in the case of Dirichlet boundary control system 20) where u ∈ L 2 loc (0, ∞; L 2 (∂O)). In order to represent (2.20) into form (2.11), we consider first the so-called Dirichlet map y = Du which is defined as the solution to the Dirichlet problem…”

Section: Nonlinear Parabolic-like Systemsmentioning

confidence: 99%