Calculus of Variations and Geometric Measure Theory

Optimality of observability inequalities for parabolic and hyperbolic systems with potentials

Enrique Zuazua ((Universidad Autónoma de Madrid)

created by alberti on 23 Feb 2006

1 mar 2006


when: Wednesday, March 1st, at 4 pm

where: Sala delle Riunioni, Dipartimento di Matematica

speaker: Enrique Zuazua (Universidad Autónoma de Madrid)

title: "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 be estimated above in terms of the energy localized in a subdomain with an observability constant of the order of $exp(C |a|^{2/3}_\infty)$, $a$ 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 $R^2$ and a non-trivial solution $u$ of $\Delta u=q(x)u$ with the decay property $
\le exp(-
^{4/3})$. 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. 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 order $exp(C |a|^{1/2}_\infty)$. For systems in even space dimensions $n \ge 2$ we prove that the best constant one can expect is of the order of $exp(C |a|^{2/3}_\infty)$ 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. 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.