2,017 research outputs found
On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations
The aim of this work is to present some strategies to solve numerically controllability problems for the two-dimensional heat equation, the Stokes equations and the Navier-Stokes equations with Dirichlet boundary conditions. The main idea is to adapt the Fursikov-Imanuvilov formulation,
see [A.V. Fursikov, O.Yu. Imanuvilov: Controllability of Evolutions Equations, Lectures Notes Series, Vol. 34, Seoul National University, 1996]; this approach has been followed recently for the onedimensional heat equation by the first two authors. More precisely, we minimize over the class of
admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up near the final time. The associated optimality conditions can be viewed as a differential system in the three variables x1, x2 and t that is second–order in time and fourth–order
in space, completed with appropriate boundary conditions. We present several mixed formulations of the problems and, then, associated mixed finite element Lagrangian approximations that are easy to handle. Finally, we exhibit some numerical experiments.Ministerio de Economía y CompetitividadCentre national de la recherche scientifiqu
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
The control of PDEs: some basic concepts, recent results and open problems
These Notes deal with the control of systems governed by some PDEs. I will mainly consider time-dependent problems. The aim is to present some fundamental results, some applications and some open problems related
to the optimal control and the controllability properties of these systems.
In Chapter 1, I will review part of the existing theory for the optimal control of partial differential systems. This is a very broad subject and there have been so many contributions in this field over the last years that we will have to limit considerably the scope. In fact, I will only analyze a few questions concerning some very particular PDEs. We shall focus on the Laplace, the
stationary Navier-Stokes and the heat equations. Of course, the existing theory allows to handle much more complex situations. Chapter 2 is devoted to the controllability of some systems governed by linear time-dependent PDEs. I will consider the heat and the wave equations. I will try to explain which is the meaning of controllability and which kind of controllability properties can be expected to be satisfied by each of these PDEs. The main related results, together with the main ideas in their proofs, will be recalled.
Finally, Chapter 3 is devoted to present some controllability results for other time-dependent, mainly nonlinear, parabolic systems of PDEs. First, we will revisit the heat equation and some extensions. Then, some controllability
results will be presented for systems governed by stochastic PDEs. Finally, I
will consider several nonlinear systems from fluid mechanics: Burgers, NavierStokes, Boussinesq, micropolar, etc. Along these Notes, a set of questions (some of them easy, some of them more intrincate or even difficult) will be stated. Also, several open problems will be mentioned. I hope that all this will help to understand the underlying basic concepts and results and to motivate research on the subject
Local null controllability of a class of non-Newtonian incompressible viscous fluids
We investigate the null controllability property of systems that
mathematically describe the dynamics of some non-Newtonian incompressible
viscous flows. The principal model we study was proposed by O. A.
Ladyzhenskaya, although the techniques we develop here apply to other fluids
having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum
Principle, we utilize a bootstrapping argument to prove that sufficiently
smooth controls to the forced linearized Stokes problem exist, as long as the
initial data in turn has enough regularity. From there, we extend the result to
the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to
compute the states and a control, which we prove to converge in an appropriate
sense. We finish the work with some numerical experiments
Theoretical and numerical local null controllability of a Ladyzhenskaya-Smagorinsky model of turbulence
This paper deals with the control of a differential turbulence model of the Ladyzhenskaya–Smagorinsky kind. In the equations we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the system is locally null-controllable, with distributed controls locally supported in space. The proof relies on rather well known arguments. However, some specific difficulties are found here because of the occurrence of nonlocal nonlinear terms. We also present an iterative algorithm of the quasi-Newton kind that provides a sequence of states and controls that converge towards a solution to the control problem. Finally, we give the details of a numerical approximation and we illustrate the behavior of the algorithm with a numerical experiment.Dirección General de Enseñanza Superio
- …