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Volume 1, 2011

Mathematical Control & Related Fields

2011 , Volume 1 , Issue 2

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Control of a network of magnetic ellipsoidal samples
Shruti Agarwal, Gilles Carbou, Stéphane Labbé and Christophe Prieur
2011, 1(2): 129-147 doi: 10.3934/mcrf.2011.1.129 +[Abstract](235) +[PDF](421.2KB)
Abstract:
In this work, we present a mathematical study of stability and controllability of one-dimensional network of ferromagnetic particles. The control is the magnetic field generated by a dipole whose position and whose amplitude can be selected. The evolution of the magnetic field in the network of particles is described by the Landau-Lifschitz equation. First, we model a network of ellipsoidal shape ferromagnetic particles. Then, we prove the stability of relevant configurations and discuss the controllability by the means of the external magnetic field induced by the magnetic dipole. Finally some numerical results illustrate the stability and the controllability results.
Global Carleman inequalities for Stokes and penalized Stokes equations
Mehdi Badra
2011, 1(2): 149-175 doi: 10.3934/mcrf.2011.1.149 +[Abstract](205) +[PDF](329.8KB)
Abstract:
In this note we use the result of [22] to prove a global Carleman inequality related to the null controllability of penalized Stokes kind systems. The constants of the obtained Carleman inequality are uniform in terms of the penalization parameter $\varepsilon$. It then provides a null control with a uniformly (in $\varepsilon$) bounded $L^2$ norm. With a limiting argument we also deduce a new Carleman inequality for Stokes type system. Thus, we apply theses results to obtain the null controllability of Oseen and Navier-Stokes system in the penalized and in the non penalized cases.
Observability of heat processes by transmutation without geometric restrictions
Sylvain Ervedoza and Enrique Zuazua
2011, 1(2): 177-187 doi: 10.3934/mcrf.2011.1.177 +[Abstract](263) +[PDF](348.7KB)
Abstract:
The goal of this note is to explain how transmutation techniques (originally introduced in [14] in the context of the control of the heat equation, inspired on the classical Kannai transform, and recently revisited in [4] and adapted to deal with observability problems) can be applied to derive observability results for the heat equation without any geometric restriction on the subset in which the control is being applied, from a good understanding of the wave equation. Our arguments are based on the recent results in [15] on the frequency depending observability inequalities for waves without geometric restrictions, an iteration argument recently developed in [13] and the new representation formulas in [4] allowing to make a link between heat and wave trajectories.
Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions
Scott W. Hansen and Oleg Yu Imanuvilov
2011, 1(2): 189-230 doi: 10.3934/mcrf.2011.1.189 +[Abstract](254) +[PDF](670.0KB)
Abstract:
Exact controllability of a multilayer plate system with free boundary conditions are obtained by the method of Carleman estimates. The multilayer plate system is a natural multilayer generalization of a three-layer "sandwich plate'' system due to Rao and Nakra. In the multilayer version, $m$ shear deformable layers alternate with $m+1$ layers modeled under Kirchoff plate assumptions. The resulting system involves $m+1$ Lamé systems coupled with a scalar Kirchhoff plate equation. The controls are taken to be distributed in a neighborhood of the boundary. This paper is the sequel to [2] in which only clamped and hinged boundary conditions are considered.
Strict Lyapunov functions for semilinear parabolic partial differential equations
Frédéric Mazenc and Christophe Prieur
2011, 1(2): 231-250 doi: 10.3934/mcrf.2011.1.231 +[Abstract](351) +[PDF](929.8KB)
Abstract:
For families of partial differential equations (PDEs) with particular boundary conditions, strict Lyapunov functions are constructed. The PDEs under consideration are parabolic and, in addition to the diffusion term, may contain a nonlinear source term plus a convection term. The boundary conditions may be either the classical Dirichlet conditions, or the Neumann boundary conditions or a periodic one. The constructions rely on the knowledge of weak Lyapunov functions for the nonlinear source term. The strict Lyapunov functions are used to prove asymptotic stability in the framework of an appropriate topology. Moreover, when an uncertainty is considered, our construction of a strict Lyapunov function makes it possible to establish some robustness properties of Input-to-State Stability (ISS) type.
Decay of solutions of the wave equation with localized nonlinear damping and trapped rays
Kim Dang Phung
2011, 1(2): 251-265 doi: 10.3934/mcrf.2011.1.251 +[Abstract](191) +[PDF](375.8KB)
Abstract:
We prove some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain in which trapped rays may occur. The approach is based on a comparison with the linear damped wave equation and an interpolation argument. Our result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data.

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