9 research outputs found
Some Controllability Results for Linearized Compressible Navier-Stokes System
In this article, we study the null controllability of linearized compressible
Navier-Stokes system in one and two dimension. We first study the
one-dimensional compressible Navier-Stokes system for non-barotropic fluid
linearized around a constant steady state. We prove that the linearized system
around , with is
not null controllable by localized interior control or by boundary control. But
the system is null controllable by interior controls acting everywhere in the
velocity and temperature equation for regular initial condition. We also prove
that the the one-dimensional compressible Navier-Stokes system for
non-barotropic fluid linearized around a constant steady state , with is not
null controllable by localized interior control or by boundary control for
small time Next we consider two-dimensional compressible Navier-Stokes
system for barotropic fluid linearized around a constant steady state We prove that this system is also not null controllable by
localized interior control
Null Controllability for Wave Equations with Memory
We study the memory-type null controllability property for wave equations
involving memory terms. The goal is not only to drive the displacement and the
velocity (of the considered wave) to rest at some time-instant but also to
require the memory term to vanish at the same time, ensuring that the whole
process reaches the equilibrium. This memory-type null controllability problem
can be reduced to the classical null controllability property for a coupled
PDE-ODE system. The later is viewed as a degenerate system of wave equations,
the velocity of propagation for the ODE component vanishing. This fact requires
the support of the control to move to ensure the memory-type null
controllability to hold, under the so-called Moving Geometric Control
Condition. The control result is proved by duality by means of an observability
inequality which employs measurements that are done on a moving observation
open subset of the domain where the waves propagate
Local controllability to trajectories for non-homogeneous incompressible Navier-Stokes equations *
Abstract The goal of this article is to show a local exact controllability to smooth (C 2 ) trajectories for the density dependent incompressible Navier-Stokes equations. Our controllability result requires some geometric condition on the flow of the target trajectory, which is remanent from the transport equation satisfied by the density. The proof of this result uses a fixed point argument in suitable spaces adapted to a Carleman weight function that follows the flow of the target trajectory. Our result requires the proof of new Carleman estimates for heat and Stokes equations