Convection-Enabled Boundary Control of a 2D Channel Flow
Résumé
We consider the incompressible Navier-Stokes
equations in a two-dimensional channel. The tangential
and normal velocities are assumed to be periodic in the
streamwise (horizontal) direction. Moreover, we consider
no-slip boundary conditions on the tangential velocity at the
top and bottom walls of the channel, and normal velocity
actuation at the top and bottom walls. For an arbitrarily
large Reynolds number, we design the boundary control
inputs to achieve global exponential stabilization, in the L2
sense, of a chosen parabolic Poiseuille profile. Moreover,
we design the control inputs such that they have zero
mean, but non-zero cubic mean. The zero-mean property
is to ensure that the conservation of mass constraint is
verified. The non-zero cubic mean property is the key to
exploiting the stabilizing effect of nonlinear convection and
achieving global stabilization independently of the size of
the Reynolds number. This paper is not only the first work
where a closed-form feedback law is proposed for global
stabilization of parabolic Poiseuille profiles for arbitrary
Reynolds number but is also the first generalization of the
Cardano-Lyapunov formula, designed initially to stabilize
scalar-valued convective PDEs, to a vector-valued convec-
tive PDE with a divergence-free constraint on the state.
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