• S Kesavan

Articles written in Proceedings – Mathematical Sciences

• Application of Newton’s method to a homogenization problem

The homogenization of a family (Pε) of uniformly elliptic semilinear partial differential equations of second order is studied. The main result is that any non-singular solutionu of the homogenized problem (P) is the limit of non-singular solutions of (Pε). The method consists of specifying a functionwε starting from which the Newton iterates converge to a solutionuε ofPε. These solutionsuε converge to the given solutionu of (P).

• Homogenization of periodic optimal control problems via multi-scale convergence

The aim of this paper is to provide an alternate treatment of the homogenization of an optimal control problem in the framework of two-scale (multi-scale) convergence in the periodic case. The main advantage of this method is that we are able to show the convergence of cost functionals directly without going through the adjoint equation. We use a corrector result for the solution of the state equation to achieve this.

• On the limit matrix obtained in the homogenization of an optimal control problem

A new formulation for the limit matrix occurring in the cost functional of an optimal control problem on homogenization is obtained. It is used to obtain an upper bound for this matrix (in the sense of positive definite matrices).

• Low-Cost Control Problems on Perforated and Non-Perforated Domains

We study the homogenization of a class of optimal control problems whose state equations are given by second order elliptic boundary value problems with oscillating coefficients posed on perforated and non-perforated domains. We attempt to describe the limit problem when the cost of the control is also of the same order as that describing the oscillations of the coefficients. We study the situations where the control and the state are both defined over the entire domain or when both are defined on the boundary.

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