13,782 research outputs found
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
Destabilization of free convection by weak rotation
This study offers an explanation of a recently observed effect of
destabilization of free convective flows by weak rotation. After studying
several models where flows are driven by a simultaneous action of convection
and rotation, it is concluded that the destabilization is observed in the cases
where centrifugal force acts against main convective circulation. At relatively
low Prandtl numbers this counter action can split the main vortex into two
counter rotating vortices, where the interaction leads to instability. At
larger Prandtl numbers, the counter action of the centrifugal force steepens an
unstable thermal stratification, which triggers Rayleigh-B\'enard instability
mechanism. Both cases can be enhanced by advection of azimuthal velocity
disturbances towards the axis, where they grow and excite perturbations of the
radial velocity. The effect was studied considering a combined
convective/rotating flow in a cylinder with a rotating lid and a parabolic
temperature profile at the sidewall. Next, explanations of the destabilization
effect for rotating magnetic field driven flow and melt flow in a Czochralski
crystal growth model were derived
Relaminarisation of Re_{\tau} = 100 channel flow with globally stabilising linear feedback control
The problems of nonlinearity and high dimension have so far prevented a
complete solution of the control of turbulent flow. Addressing the problem of
nonlinearity, we propose a flow control strategy which ensures that the energy
of any perturbation to the target profile decays monotonically. The
controller's estimate of the flow state is similarly guaranteed to converge to
the true value. We present a one-time off-line synthesis procedure, which
generalises to accommodate more restrictive actuation and sensing arrangements,
with conditions for existence for the controller given in this case. The
control is tested in turbulent channel flow () using full-domain
sensing and actuation on the wall-normal velocity. Concentrated at the point of
maximum inflection in the mean profile, the control directly counters the
supply of turbulence energy arising from the interaction of the wall-normal
perturbations with the flow shear. It is found that the control is only
required for the larger-scale motions, specifically those above the scale of
the mean streak spacing. Minimal control effort is required once laminar flow
is achieved. The response of the near-wall flow is examined in detail, with
particular emphasis on the pressure and wall-normal velocity fields, in the
context of Landahl's theory of sheared turbulence
Traveling Wave Fronts and Localized Traveling Wave Convection in Binary Fluid Mixtures
Nonlinear fronts between spatially extended traveling wave convection (TW)
and quiescent fluid and spatially localized traveling waves (LTWs) are
investigated in quantitative detail in the bistable regime of binary fluid
mixtures heated from below. A finite-difference method is used to solve the
full hydrodynamic field equations in a vertical cross section of the layer
perpendicular to the convection roll axes. Results are presented for
ethanol-water parameters with several strongly negative separation ratios where
TW solutions bifurcate subcritically. Fronts and LTWs are compared with each
other and similarities and differences are elucidated. Phase propagation out of
the quiescent fluid into the convective structure entails a unique selection of
the latter while fronts and interfaces where the phase moves into the quiescent
state behave differently. Interpretations of various experimental observations
are suggested.Comment: 46 pages, 11 figures. Accepted for publication in Phys. Rev.
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