1,768 research outputs found
Controllability of the heat equation with an inverse-square potential localized on the boundary
This article is devoted to analyze control properties for the heat equation
with singular potential arising at the boundary of a smooth domain
\Omega\subset \rr^N, . This problem was firstly studied by
Vancostenoble and Zuazua [20] and then generalized by Ervedoza [10]in the
context of interior singularity. Roughly speaking, these results showed that
for any value of parameters , the corresponding
parabolic system can be controlled to zero with the control distributed in any
open subset of the domain. The critical value stands for the best
constant in the Hardy inequality with interior singularity. When considering
the case of boundary singularity a better critical Hardy constant is obtained,
namely . In this article we extend the previous results in
[18],[8], to the case of boundary singularity. More precisely, we show that for
any , we can lead the system to zero state using a distributed
control in any open subset. We emphasize that our results cannot be obtained
straightforwardly from the previous works [20], [10].Comment: 32 page
Null controllability of one-dimensional parabolic equations by the flatness approach
We consider linear one-dimensional parabolic equations with space dependent
coefficients that are only measurable and that may be degenerate or
singular.Considering generalized Robin-Neumann boundary conditions at both
extremities, we prove the null controllability with one boundary control by
following the flatness approach, which providesexplicitly the control and the
associated trajectory as series. Both the control and the trajectory have a
Gevrey regularity in time related to the class of the coefficient in
front of .The approach applies in particular to the (possibly degenerate
or singular) heat equation with a(x)\textgreater{}0
for a.e. and , or to the heat equation with
inverse square potential with
Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function
This article is devoted to the analysis of control properties for a heat equation with a singular potential μ/δ2, defined on a bounded C2 domain Ω⊂RN, where δ is the distance to the boundary function. More precisely, we show that for any μ≤1/4 the system is exactly null controllable using a distributed control located in any open subset of Ω, while for μ>1/4 there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate
An obstruction to small time local null controllability for a viscous Burgers' equation
In this work, we are interested in the small time local null controllability
for the viscous Burgers' equation on the line
segment , with null boundary conditions. The second-hand side is a
scalar control playing a role similar to that of a pressure. In this setting,
the classical Lie bracket necessary condition introduced by
Sussmann fails to conclude. However, using a quadratic expansion of our system,
we exhibit a second order obstruction to small time local null controllability.
This obstruction holds although the information propagation speed is infinite
for the Burgers equation. Our obstruction involves the weak norm of
the control . The proof requires the careful derivation of an integral
kernel operator and the estimation of residues by means of weakly singular
integral operator estimates
On the controllability of Partial Differential Equations involving non-local terms and singular potentials
In this thesis, we investigate controllability and observability properties of Partial Differential Equations describing various phenomena appearing in several fields of the applied sciences such as elasticity theory, ecology, anomalous transport and diffusion, material science, porous media flow and quantum mechanics. In particular, we focus on evolution Partial Differential Equations with non-local and singular terms.
Concerning non-local problems, we analyse the interior controllability of a Schr\"odinger and a wave-type equation in which the Laplace operator is replaced by the fractional Laplacian . Under appropriate assumptions on the order of the fractional Laplace operator involved, we prove the exact null controllability of both equations, employing a control supported in a neighbourhood of the boundary of a bounded domain . More precisely, we show that both the Schrodinger and the wave equation are null-controllable, for and for respectively. Furthermore, these exponents are sharp and controllability fails for (resp. ) for the Schrödinger (resp. wave) equation. Our proof is based on multiplier techniques and the very classical Hilbert Uniqueness Method.
For models involving singular terms, we firstly address the boundary controllability problem for a one-dimensional heat equation with the singular inverse-square potential , whose singularity is localised at one extreme of the space interval in which the PDE is defined. For all , we obtain the null controllability of the equation, acting with a control located at , which is both a boundary point and the pole of the potential. This result follows from analogous ones presented in \cite{gueye2014exact} for parabolic equations with variable degenerate coefficients.
Finally, we study the interior controllability of a heat equation with the singular inverse-square potential , involving the distance to the boundary of a bounded and domain , . For all (the critical Hardy constant associated to the potential ), we obtain the null controllability employing a control supported in an open subset . Moreover, we show that the upper bound is sharp. Our proof relies on a new Carleman estimate, obtained employing a weight properly designed for compensating the singularities of the potential
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