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Volume 1, 2011

Mathematical Control & Related Fields

2011 , Volume 1 , Issue 3

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Recent results on the controllability of linear coupled parabolic problems: A survey
Farid Ammar-Khodja , Assia Benabdallah , Manuel González-Burgos and  Luz de Teresa
2011, 1(3): 267-306 doi: 10.3934/mcrf.2011.1.267 +[Abstract](166) +[PDF](671.6KB)
Abstract:
This paper tries to summarize recent results on the controllability of systems of (several) parabolic equations. The emphasis is placed on the extension of the Kalman rank condition (for finite dimensional systems of differential equations) to parabolic systems. This question is itself tied with the proof of global Carleman estimates for systems and leads to a wide field of open problems.
Global Carleman estimate on a network for the wave equation and application to an inverse problem
Lucie Baudouin , Emmanuelle Crépeau and  Julie Valein
2011, 1(3): 307-330 doi: 10.3934/mcrf.2011.1.307 +[Abstract](71) +[PDF](449.3KB)
Abstract:
We are interested in an inverse problem for the wave equation with potential on a star-shaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.
Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks
Serge Nicaise
2011, 1(3): 331-352 doi: 10.3934/mcrf.2011.1.331 +[Abstract](32) +[PDF](468.3KB)
Abstract:
A Mindlin-Timoshenko model with non constant and non smooth coefficients set in a bounded domain of $\mathbb{R}^d, d\geq 1$ with some internal dissipations is proposed. It corresponds to the coupling between the wave equation and the dynamical elastic system. If the dissipation acts on both equations, we show an exponential decay rate. On the contrary if the dissipation is only active on the elasticity equation, a polynomial decay is shown; a similar result is proved in one dimension if the dissipation is only active on the wave equation.
Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain
Dugan Nina , Ademir Fernando Pazoto and  Lionel Rosier
2011, 1(3): 353-389 doi: 10.3934/mcrf.2011.1.353 +[Abstract](66) +[PDF](533.1KB)
Abstract:
The purpose of this work is to study the internal stabilization of a coupled system of two generalized Korteweg-de Vries equations under the effect of a localized damping term. The exponential stability, as well as, the global existence of weak solutions are investigated when the exponent in the nonlinear term ranges over the interval $[1, 4)$. To obtain the decay we use multiplier techniques combined with compactness arguments and reduce the problem to prove a unique continuation property for weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.
Coefficient identification and fault detection in linear elastic systems; one dimensional problems
David L. Russell
2011, 1(3): 391-411 doi: 10.3934/mcrf.2011.1.391 +[Abstract](33) +[PDF](515.7KB)
Abstract:
The determination of parameter distributions, including fault detection, in elastic structures is a subject of great importance in structural engineering and related areas of applied mathematics. In this article we explore, in both continuous and discrete settings, some methods for approximate solution of such identification problems in a one dimensional linear elasticity framework. Methods for related optimization problems based on the matrix trace norm are described. The main objective of the paper is to introduce a method, believed new with this article, for which we suggest the names adjoint null space method or complementary projection method. Computational results for sample problems based on this technique are presented.

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