IFCPAR AR (ENGLISH) for CD - CEFIPRA

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CEFIPRA Centre Franco-Indien pour la Promotion de la Recherche Avancée Duration: Three years and six months (February, 2008 to July, 2011) Objectives i) Giving a precise mathematical framework ii) Exploring the controllability question (this means to investigate the possibility to steer the system to any final state by choosing an appropriate control) iii) Study the related question of stabilizability iv) Analyse stabilizability of feedback control, search for local feedback stabilization results in the cases of complete and partial observation of the state v) Explore related optimal control problems vi) Numerical analysis and computation vii) Understand the limiting behaviour of the control problems, for example, under homogenization process Accomplishments i) Different models for fluid-solid interaction have been studied in appropriate mathematical framework for controllability questions : a) Coupled system with Stokes equation for the fluid in 2 dimensional domain and an ordinary differential equation (o.d.e) for the structure, modeling the deformations of an elastic body b) Helmholtz equation to model the vibrations of a coupled fluid-solid system ii) The study of meta-materials which are electromagnetic materials having negative permittivities and/or permeabilities has been initiated by setting up a mathematical model and is ready for further study of control and homogenization iii) A numerical implementation of feedback control for the important problem of fluid control modeled by Navier- Stokes equation has been intiated and is to be investigated further in the coming year iv) The practical problem of Data assimilation has been tried numerically for the Burgers' equation model using optimal control techniques. Further applications with models used in atmospheric sciences is now possible v) Compressible Navier-Stokes system has been taken up for the study of controllability and stabilizability and optimal control Research papers published: Nine Project 3701-1 CONTROL OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS Research Activities 2010-11 Pure and Applied Mathematics Prof. Mythily Ramaswamy T.I.F.R Centre for Applicable Mathematics Bangalore Prof. Jean-Pierre Raymond Institut de Mathématiques Université Paul Sabatier Toulouse 31
32 Pure and Applied Mathematics Prof. Rekha P. Kulkarni Department of Mathematics, Indian Institute of Technology Bombay Mumbai Professor Mario Ahues Laboratoire de Mathématiques de l’Université de Saint-Etienne (LaMUSE EA 3989) Université de Lyon Saint Etienne Project 4101-1 NUMERICAL TREATMENT OF INTEGRAL OPERATORS WITH NON-SMOOTH KERNELS IFCPAR Indo-French Centre for the Promotion of Advanced Research Duration: Three years (September, 2009 to August, 2012) Objectives The focus of this project is numerical solution of single variable and multi-variable Fredholm integral equations of the second kind and of the associated eigenvalue problems. The collaborators would like to extend the available results for integral operators with smooth kernels to the case of integral operators with non-smooth kernels of the following two types: (i) continuous kernels which are non-differentiable across the diagonal, (ii) weakly singular kernels having algebraic/logarithmic singularities. The main idea for treating the continuous kernels is a careful choice of the subintervals in the composite numerical quadrature so as to restore the order of convergence in spite of the nonsmoothness of the kernel across the diagonal. For weakly singular kernels, a comparative study of singularity subtraction and product integration techniques is envisaged. Also, a two-grid method based on a new approximating operator introduced and studied by one of the investigators will be proposed. Iterative refinement schemes will be considered for the eigenvalue problems. For multivariable integral equations, the collaborators would like to study collocation and other projection methods as well as global approximation methods. It is intended to develop software packages for the algorithms developed during the project. Accomplishments i) Study of rates of convergence and asymptotic series expansion in the case of a smooth kernel i) Study of appropriate modification in a quadrature rule so as to take into consideration the lack of smoothness along the diagonal of the kernel of the type of Green's function ii) Asymptotic expansions for solutions of second kind Fredholm integral equations with kernel of the type of Green's function iii) Convolution-truncation for weakly singular kernels iv) Comparative analysis "discretization followed by linearization" against "linearization followed by discretization" Papers presented in conferences: Five Papers presented in conferences: 4
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