364 research outputs found
Internal exponential stabilization to a nonstationary solution for 1D Burgers equations with piecewise constant controls
International audienceThe feedback stabilization of the Burgers system to a nonstationary solution using a finite number of internal piecewise constant controls is considered. Estimates for the number of needed controls are derived. In the particular case of no constraint on the support of the control a better estimate is derived, so the possibility of getting an analogous estimate for the general case is discussed.That possibility is suggested by the results of some numerical simulations
Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations
International audienceThe feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the controla better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examplesare presented illustrating the stabilizing effect of the feedback control, and suggesting that the existence of an estimatein the general case analogous to that in the particular one is plausible
On the Fattorini Criterion for Approximate Controllability and Stabilizability of Parabolic Systems
In this paper, we consider the well-known Fattorini's criterion for
approximate controllability of infinite dimensional linear systems of type
. We precise the result proved by H. O. Fattorini in
\cite{Fattorini1966} for bounded input , in the case where can be
unbounded or in the case of finite-dimensional controls. More precisely, we
prove that if Fattorini's criterion is satisfied and if the set of geometric
multiplicities of is bounded then approximate controllability can be
achieved with finite dimensional controls. An important consequence of this
result consists in using the Fattorini's criterion to obtain the feedback
stabilizability of linear and nonlinear parabolic systems with feedback
controls in a finite dimensional space. In particular, for systems described by
partial differential equations, such a criterion reduces to a unique
continuation theorem for a stationary system. We illustrate such a method by
tackling some coupled Navier-Stokes type equations (MHD system and micropolar
fluid system) and we sketch a systematic procedure relying on Fattorini's
criterion for checking stabilizability of such nonlinear systems. In that case,
the unique continuation theorems rely on local Carleman inequalities for
stationary Stokes type systems
Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?
The contents of this paper is twofold. First, important recent results concerning finite element
methods for convection-dominated problems and incompressible flow problems are described that
illustrate the activities in these topics. Second, a number of, in our opinion, important problems in
these fields are discussed
On Right Continuity in at All the Points of Energy-regularized Solutions for the 3D Navier-Stokes Equations
In this note I provide the notion of energy-regularized solutions
(ER-solutions) of the 3D Navier-Stokes equations. These solutions can be
obtained via the standard Galerkin arguments. I prove that each ER-solution for
the 3D Navier-Stokes system satisfies Leray-Hopf property. Moreover, each
ER-solution is rightly continuous in the standard phase space endowed with
the strong convergence topology
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