[CvGmt News] Fwd: Seminario di Analisi: Enrique Zuazua

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Tue Feb 21 12:09:55 CET 2006

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    Date: Tue, 21 Feb 2006 11:59:26 +0100 (CET)
    From: Alberto Abbondandolo <abbondandolo at dm.unipi.it>
Reply-To: Alberto Abbondandolo <abbondandolo at dm.unipi.it>
 Subject: Seminario di Analisi
      To: Seminario di Analisi <abbondandolo at dm.unipi.it>,
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Quando: Mercoledi' 1 Marzo, ore 16

Dove: Sala delle Riunioni, Dipartimento di Matematica

Speaker:  Enrique Zuazua (Universita' autonoma di Madrid)

Titolo: "Optimality of observability inequalities for parabolic
and hyperbolic systems with potentials"

Abstract:  "In this talk we discuss the optimality of the observability
inequality for parabolic systems with potentials  This inequality
asserts, roughly, that for small time, the total energy of solutions can 
estimated above in terms of the energy localized in a subdomain
with an observability constant of the order of
$\exp\left(C\parallel a
\parallel^{2/3}_\infty\right)$, $a$ being the potential involved in
the system.  The problem of observability is relevant both in control and
inverse problems.

The optimality is a consequence of a construction due to V. Z.
Meshkov of a complex-valued bounded potential $q=q(x)$ in
$\mathbb{R}^2$ and a non-trivial solution $u$ of $\Delta u=q(x)u$ with
the decay property $\mid u(x)\mid\leq\exp\left(-|x|^{4/3}\right)$. Meshkov's
construction may be generalized to any even dimension. We
give an extension to it to odd dimensions, which gives a sharp decay rate up to
some logarithmic factor and yields a weaker optimality result in odd
space-dimensions. \par
  We address the same problem for the wave equation. In
this case it is well known that, in space-dimension $n=1$,
observability holds with a sharp constant of the order of
$\exp\left(C\parallel a\parallel^{1/2}_\infty\right)$. For systems in
even space dimensions $n\geq2$ we prove that the best constant one
can expect is of the order of $\exp\Big(C\parallel
a\parallel^{2/3}_\infty\Big)$ for any $T>0$ and any observation
domain. Based on Carleman inequalities, we show that the positive
counterpart is also true when $T$ is large enough and the
observation is made in a neighborhood of the boundary.
As in the context of
the heat equation, the optimality of this estimate is open for
scalar equations.\par
We address similar questions, for both equations, with potentials involving the
first order term. We also discuss issues related with the impact of the  growth
rates of  the nonlinearities on the controllability of semilinear
equations. Some other open problems are raised. The content of this lecture is
based on joint work with Th. Duyckaerts and Xu Zhang."

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