172 research outputs found
On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient
We present some results concerning the controllability of a quasi-linear parabolic equation (with linear principal part) in a bounded domain of with Dirichlet boundary conditions. We analyze the controllability problem with distributed controls (supported on a small open subset) and boundary controls (supported on a small part of the boundary). We prove that the system is null and approximately controllable at any time if the nonlinear term grows slower than at infinity (generally, in this case, in the absence of control, blow-up occurs). The proofs use global Carleman estimates, parabolic regularity, and the fixed point method
Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem
We study a system of nonlinear partial differential equations resulting from
the traditional modelling of oil engineering within the framework of the
mechanics of a continuous medium. Recent results on the problem provide
existence, uniqueness and regularity of the optimal solution. Here we obtain
the first necessary optimality conditions.Comment: 9 page
An inverse source problem for the heat equation and the enclosure method
An inverse source problem for the heat equation is considered. Extraction
formulae for information about the time and location when and where the unknown
source of the equation firstly appeared are given from a single lateral
boundary measurement. New roles of the plane progressive wave solutions or
their complex versions for the backward heat equation are given.Comment: 23page
Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing
We study controllability issues for the 2D Euler and Navier-
Stokes (NS) systems under periodic boundary conditions. These systems
describe motion of homogeneous ideal or viscous incompressible fluid on
a two-dimensional torus T^2. We assume the system to be controlled by
a degenerate forcing applied to fixed number of modes.
In our previous work [3, 5, 4] we studied global controllability by
means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing
viscosity (\nu > 0). Methods of dfferential geometric/Lie algebraic
control theory have been used for that study. In [3] criteria for
global controllability of nite-dimensional Galerkin approximations of
2D and 3D NS systems have been established. It is almost immediate
to see that these criteria are also valid for the Galerkin approximations
of the Euler systems. In [5, 4] we established a much more intricate suf-
cient criteria for global controllability in finite-dimensional observed
component and for L2-approximate controllability for 2D NS system.
The justication of these criteria was based on a Lyapunov-Schmidt
reduction to a finite-dimensional system. Possibility of such a reduction
rested upon the dissipativity of NS system, and hence the previous
approach can not be adapted for Euler system.
In the present contribution we improve and extend the controllability
results in several aspects: 1) we obtain a stronger sufficient condition for
controllability of 2D NS system in an observed component and for L2-
approximate controllability; 2) we prove that these criteria are valid for
the case of ideal incompressible uid (\nu = 0); 3) we study solid controllability
in projection on any finite-dimensional subspace and establish a
sufficient criterion for such controllability
Fullerene Black: Relationship between Catalytic Activity in n-alkanes Dehydrocyclization and Reactivity in Oxidation, Bromination and Hydrogenolysis
The reactivity of fullerene black in oxidation (by air oxygen or ions MnO4–or Cr2O7 2– in solution), bromination (by Br2 or (C4H9)4NBr3) and hydrogenolysis (without hydrogenation catalyst) are studied. The dehydrocyclization of n-alkanes over fullerene black is realized via the monofunctional mechanism, i.e. the dehydrogenation and cyclization stages proceed on the same catalytic center. The addition of alumina to the catalyst transforms dehydrocyclization mechanism to bifunctional one, when fullerene black acts as dehydrogenation agent. Reactivity studies and ESR spectroscopy data for initial and annealed fullerene black show the presence in fullerene black structure of both non-conjugated multiple and dangling bonds. Nonconjugated bonds determine catalytic activity and reactivity of fullerene black. They are localized in amorphous part of fullerene black. Technological aspects of fullerene black as alkanes dehydrocyclization catalyst are discussed
A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Random Parabolic PDE with Constrained Control
A stochastic Galerkin approximation scheme is proposed for an optimal control problem governed by a parabolic PDE with random perturbation in its coefficients. The objective functional is to minimize the expectation of a cost functional, and the deterministic control is of the obstacle constrained type. We obtain the necessary and sufficient optimality conditions and establish a scheme to approximate the optimality system through the discretization with respect to both the spatial space and the probability space by Galerkin method and with respect to time by the backward Euler scheme. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results
Global Carleman inequalities for parabolic systems and applications to controllability
This paper has been conceived as an overview on the controllability properties of some relevant (linear and nonlinear) parabolic systems. Specifically, we deal with the null controllability and the exact controllability to the trajectories. We try to explain the role played by the observability
inequalities in this context and the need of global Carleman estimates. We also recall the main ideas used to overcome the difficulties motivated by nonlinearities. First, we considered the classical heat equation with Dirichlet conditions and distributed controls. Then we analyze recent extensions to
other linear and semilinear parabolic systems and/or boundary controls. Finally, we review the controllability properties for the Stokes and Navier–Stokes equations that are known to date. In this context, we have paid special attention to obtaining the necessary Carleman estimates. Some open questions are mentioned throughout the paper. We hope that this unified presentation will be useful for those researchers interested in the field.Ministerio de Educación y Cienci
Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions
The objective of the paper is to investigate the approximate controllability
property of a linear stochastic control system with values in a separable real
Hilbert space. In a first step we prove the existence and uniqueness for the
solution of the dual linear backward stochastic differential equation. This
equation has the particularity that in addition to an unbounded operator acting
on the Y-component of the solution there is still another one acting on the
Z-component. With the help of this dual equation we then deduce the duality
between approximate controllability and observability. Finally, under the
assumption that the unbounded operator acting on the state process of the
forward equation is an infinitesimal generator of an exponentially stable
semigroup, we show that the generalized Hautus test provides a necessary
condition for the approximate controllability. The paper generalizes former
results by Buckdahn, Quincampoix and Tessitore (2006) and Goreac (2007) from
the finite dimensional to the infinite dimensional case.Comment: 31 pages, submitted to AM
Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates
In this paper we will review the main results concerning the issue of
stability for the determination unknown boundary portion of a thermic
conducting body from Cauchy data for parabolic equations. We give detailed and
selfcontained proofs. We prove that such problems are severely ill-posed in the
sense that under a priori regularity assumptions on the unknown boundaries, up
to any finite order of differentiability, the continuous dependence of unknown
boundary from the measured data is, at best, of logarithmic type
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