Optimal design problems for a non-linear cost in the gradient: numerical results

Abstract

The aim of this article is the numerical study of a control problem for a linear elliptic partial differential equation. The control variable is the matrix diffusion and the functional depends non-linearly on the gradient of the state function. We consider the relaxed formulation of this problem. One of the main difficulties is that the functional which appears in this relaxed problem is not explicitly known. We show that in the discrete approximation, we can replace this functional by an upper or lower one.

Keywords:

• control in the coefficients
• elliptic PDE
• composite optimal design
• numerical analysis

AMS Subject Classifications: :

• 49M25
• 49J20

Acknowledgements

This work has been partially supported by projects MTM2005-04914 of the ‘Ministerio de Educación y Ciencia’ space of Spain and FQM-309 of the ‘Junta de Andalucía’.