722 research outputs found
About least-squares type approach to address direct and controllability problems
- We discuss the approximation of distributed null controls for partial
differential equations. The main purpose is to determine an approximation of
controls that drives the solution from a prescribed initial state at the
initial time to the zero target at a prescribed final time. As a non trivial
example, we mainly focus on the Stokes system for which the existence of
square-integrable controls have been obtained in [Fursikov \& Imanuvilov,
Controllability of Evolution Equations, 1996]) via Carleman type estimates. We
introduce a least-squares formulation of the controllability problem, and we
show that it allows the construction of strong convergent sequences of
functions toward null controls for the Stokes system. The approach consists
first in introducing a class of functions satisfying a priori the boundary
conditions in space and time-in particular the null controllability condition
at time T-, and then finding among this class one element satisfying the
system. This second step is done by minimizing a quadratic functional, among
the admissible corrector functions of the Stokes system. We also discuss
briefly the direct problem for the steady Navier-Stokes system. The method does
not make use of any duality arguments and therefore avoid the ill-posedness of
dual methods, when parabolic type equation are considered
Fast global null controllability for a viscous Burgers' equation despite the presence of a boundary layer
In this work, we are interested in the small time global null controllability
for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment
[0,1]. The second-hand side is a scalar control playing a role similar to that
of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two
controls (namely the interior one u(t) and the boundary one y(t,0)). In this
setting, we show that small time global null controllability still holds by
taking advantage of both hyperbolic and parabolic behaviors of our system. We
use the Cole-Hopf transform and Fourier series to derive precise estimates for
the creation and the dissipation of a boundary layer
Numerical controllability of the wave equation through primal methods and Carleman estimates
This paper deals with the numerical computation of boundary null controls for
the 1D wave equation with a potential. The goal is to compute an approximation
of controls that drive the solution from a prescribed initial state to zero at
a large enough controllability time. We do not use in this work duality
arguments but explore instead a direct approach in the framework of global
Carleman estimates. More precisely, we consider the control that minimizes over
the class of admissible null controls a functional involving weighted integrals
of the state and of the control. The optimality conditions show that both the
optimal control and the associated state are expressed in terms of a new
variable, the solution of a fourth-order elliptic problem defined in the
space-time domain. We first prove that, for some specific weights determined by
the global Carleman inequalities for the wave equation, this problem is
well-posed. Then, in the framework of the finite element method, we introduce a
family of finite-dimensional approximate control problems and we prove a strong
convergence result. Numerical experiments confirm the analysis. We complete our
study with several comments
On the null-controllability of the heat equation in unbounded domains
We make two remarks about the null-controllability of the heat equation with
Dirichlet condition in unbounded domains. Firstly, we give a geometric
necessary condition (for interior null-controllability in the Euclidean
setting)which implies that one can not go infinitely far away from the control
region without tending to the boundary (if any), but also applies when the
distance to the control region is bounded. The proof builds on heat kernel
estimates. Secondly, we describe a class of null-controllable heat equations on
unbounded product domains. Elementary examples include an infinite strip in the
plane controlled from one boundary and an infinite rod controlled from an
internal infinite rod. The proof combines earlier results on compact manifolds
with a new lemma saying that the null-controllability of an abstract control
system and its null-controllability cost are not changed by taking its tensor
product with a system generated by a non-positive self-adjoint operator.Comment: References [CdMZ01, dTZ00] added, abstract modifie
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